Travel Diary of Makoto Sakurai

Earth Top and Spin Rotation

2012-01-01T23:59:00.002+09:00

Greetings.

Today I prepared some academic documents to submit, so that I did not have time to take a sufficient rest.

Indeed, we had another earthquake (fortunately far southern to Japanese islands and Tokyo) with Magnitude M=7.0. As the Energy Magnitude is (in a conventional definition) defined as a logarithmic scale of effective Joule energy, we had 1/1000 times Energy if compared to that of the Tohoku earthquake (M=9.0).

The image above "The Earth Top" (taken by me) is what I am preparing for the last two lectures of dynamics and electro-magnetics at the year 2012 (Winter semester of 2011) at Tokyo. We will compute the inertia momentum "I", diagonalization by diag(I1, I2, I3) for non-symmetric gyro top, and the introduction of the coordinate transformation.

Galileo's episode (made up by historians?) is quoted as "Yes, that's right" (Answer to "Isn't the earth moving?") This saying has several variations, such as "Yet the earth does move." However, at the era of Galileo Galilei, it was still possible to study mechanics as a mathematical hypothesis like the heliocentric model by Nicolaus Copernicus in the 15C's and 16C's. Galileo's accused point [1564-1642] was to study his relativity principle (coordinate changes between two inertia frames) as a guiding principle of real-world problem.

I wish a happy new year and good documents for (special) relativity.

Newton's Festival at University of Tokyo

2011-12-21T22:38:00.009+09:00

Dear friends;

My homepage is temporarily closed for some reasons (, but I preserve its files and information). For almost one year, I worked for my mathematics lecture [brush-up and undergraduate mathematics at the summer semester] as well as my physics lecture [brush-up and introduction to dynamics / analytical mechanics (including some classical integrable systems of tops with inertia momentum tensor), and electro-magnetic special relativity at winter semester].

Here is some mini-blogs (I'm sorry it is written in Japanese) for this year; I have a Facebook account for more English posts (or sometimes in French, German, or Japanese [from Twitter or Android cell phone]).

http://twilog.org/makotosakuraijp/

My old diary described my trip to a university of Japan, which had an "apple tree of Newton". Today was the "Newton Festival 2011" of Department of Physics, the University of Tokyo. However, as I explained above, I could not leave the lecture until 18:15, so that I could not attend the party at Sanjo-Kaikan (山上会館) in the Hongo Campus of University of Tokyo.

As all of you must know, Japan is under the revival from the Tohoku earthquake at 11-March-2011, 14:46, and the nuclear crisis afterwards. (I explained some of the radioactivity information and radioactivity protection to my colleagues and students.) Therefore, I could not travel much for conferences or workshops far from Tokyo, so that it was very difficult to keep my research activity.

I wish you would have happy holidays and a "better" new year!

Best wishes,

Makoto Sakurai

Japanese Rising Sun 2011

2011-01-01T07:53:00.003+09:00

The sun is rising. I had observed before the dawn to the sunrise of New Year's Day 2011. The photos are listed anti-chronologically.

Have happy holidays and a good new year.

Makoto Sakurai

Japan Physical Society 2010 Fall in Kyushu

2010-09-15T19:01:00.000+09:00

The Japan Math-Phys Society became separated after the Nobel winning paper by Yukawa at about 75 years ago. So my presentation to JPS (Japan Physical Society) is the first time since I did at the year 2006; which was a joint meeting with USA at Hawaii. (See the past blog.)

In this week, I visited the Kyushu-Kogyo University by staying only 2(+1) days in a hotel near Kokura (小倉:bullet train station). The travel was nice, because I took a very-early-morning flight from Haneda to Kita-Kyushu; my flight was the retirement of the aircraft (type MD87) for 25 years, whose schedule for departure was 5 minute delayed, but the arrival was just on time. (The speed of flight was 820 km/hour: the second figure is the Kohu-basin 甲府盆地) Then the third figure is the mono-rail and bullet-train stain Kokura seen from the station square.

Then I took a shuttle bus, early-checked-in at the hotel for suit case, and checked the way to the university presentation room. It (Sep-12) was Sunday and we just have accelerator experimentalists and I heard a public lecture at the auditorium.

The second day was by the lattice gauge theory, quantum integrable systems, AdS/CFT, and quantum mechanics. I heard some physics discussions by senior staffs, and I had become very exhausted.

The final day was my presentation day at the string theory session, for the illumination of my past 2-3 years of development of "pure math" to see how the S-duality conjecture is seen from an algebraic-geometer with some physics background. The slide is available at:
Moduli space of chiral conformal field theories and compactification to Kahler manifolds

The final figure is the panel guiding the participants to the presentation halls.

Virtual visit at KITP California, QFT and Langlands

2010-08-11T07:49:00.000+09:00

Langlands-Type Dualities in Quantum Field Theory, KITP, California, USA, Aug 9, 2010 - Aug 27, 2010 is delivering the videos (and lecture notes) in several formats at the wikispace. This conference is more on physi-cians' mathematics on low dimensional topology (or sometimes with extra dimensions or relative dimension of embeddings).

Unfortunately, I do not have time nor budgets to travel this month, but I hope to learn (and contribute to in some sense) something fruitful in this occasion. My old colleagues and friends attend the meetings, so I will do what I can do by myself.

Jet sky and Differential Graded Lie Algebras

2010-08-05T13:25:00.000+09:00

Today I have a fine sky at Komaba, Tokyo and I am going to visit the string topology of Voronov at Kashiwa campus of University of Tokyo. The video is the jet sky over the ground of Komaba campus, where we have an "open campus".

Rainbow Connection

2010-06-30T22:24:00.000+09:00

Today's post is a déjà-vu of Strings 2005 conference in Toronto.

See the original: 3rd day of Strings 2005

Firework in Tokyo

2010-06-03T11:05:00.000+09:00

Today I observed a firework in this season. Tokyo is about to welcome the summer season.

Level structures and moduli space of chiral conformal field theories

2010-03-27T18:20:00.001+09:00

This week, I delivered a talk on holonomic $\mathcal{D}$-modules and their level structure. In order to explain the moduli space of chiral (holomorphic) conformal field theory, I explained some categorical aspects of topological string theory as well as the DG-scheme of Fontaine-Kapranov theory of integration over formal loop space on (hyper-)kahler manifolds with mild singularity.

As is always, the slides are avaible on the official website:

Level structures and moduli space of chiral conformal field theories.

In summary, I explained the "super-"conformal field theories as well as the "supersymmetric" gauge theory, in the language of Differential Graded Category as well as the stack singularity for the Chan-Paton factor, as a generalization of quiver gauge theory.

Eseen 2010 5th day, Algebraic Geometry and Arithmetic

2010-02-24T16:35:00.001+09:00

2010-Feb-20th was the final day of the official day of Essen 2010, whose schedule was adapted to the carnival that I mentioned previously.

Thelene (CNRS & Paris) and Lazarsfield were the speakers of this day. Thelene talked about the universal invariant and algebraic cycles of higher codimension. The main result was

Z^{2i} (X) = H^{2i}_{Hodge} (X, ¥mathbb{Z}) / H^{2i}_{alg} (X, ¥mathbb{Z})

(where i = 1: Lefshectz, d-1, or 2). He considered Bloch-Ogus theorem and Betti cohomology.

Lazarsfield talked about the positivity (Eckart Viehweg and Esnault) of cycles on abelian varieties. I. Review of positivity for divisors of X = smooth projective variety / ¥mathbb{C}.

-- 1960's (Kodaira and Kleiman) for numerical theory of positivity.

II. Higher codimension:

-- product structure on the nef class (after the letter of Grothendieck to Mumford in the collection Vol. II.), especially on whether nef class is in the pseudo-effective divisors.

III. Positivity of Abelian varieties of real (k,k)-forms on V, where we set B = V / ¥Lambda for abelian variety. He stated the notion of strong / weak positivity of (k,k)- differential form.

After the conference, some of the participants are guided to the coal mine called Zeche Zollverein, which was about 20 minutes far from the Berliner Platz station by "tram".

I could have talked to famous professors as well as young fellows in this week. Therefore I appreciated this occasion of conference on the memory of Prof. Eckard Viehweg.

Eseen 2010 4th day, Algebraic Geometry and Arithmetic

2010-02-24T16:35:00.000+09:00

The 4th day (February 19) morning was by Takuro Mochizuki, Venjakob, Fantechi. Mochizuki was on the recent development on the irregular singularity after Sabbah. The motivation came from \mathbb{Q}-holonomic D-module \mathcal{M} / X. Regular singularity case has Riemann-Hilbert correspondence for f_{\ast}, f_!, f^{\ast}, f^!, \otimes, RHom, \psi_g (nearby cycle), \phi_g (vanishing cycle). [Saito-Terasoma: \mathcal{M} regular singular meromorphic bundle, Beilinson-Bloch-Deligne-Esnault: regular, dim X = 1]. Then, rather than naive hope for "pre-Betti structure", he worked on an extra condition of "canonical Betti strucutre" for functoriality of "Stokes structure" (Stokes filration rather than Deligne filtration) after gluing of \mathbb{C}-perverse sheaves.

Venjakov was on the non-commutative Iwasawa main conjecture for elliptic curves with complex multiplication (CM). The definition of Iwasawa was on p-adic zeta function \zeta_p up to unit, and L_p for elliptic curve. The CM case was E \ \mathbb{Q} with \mathcal{O}_K = End (E).

Fantechi (SISSA) talked about "Twists or logs as you please", Degeneration form for Gromov-Witten invariants. She defined the Gromov-Witten invariants, which were numerical invariants of smooth projective variety over \mathbb{C}. Rather than computing symplectic invariants of Li-Ruan (and relative version by Ionel-Parker), here came the algebraic geometry of Jun Li (2001). After the talks, I went to the student restaurant and a cafe, and explained to participants about the recent understanding of mirror symmetry conjecture on Gromov-Witten potential and generating function of correlation function of cohomological field theory (and Frobenius structure on the Gauss-Manin integrable connection).

Afternoon talk was only by Prof. Hida on analytic number theory of 'big' Hecke algebra. As is often the case with automorphic forms, he introduced the space of cusp forms S_{k+1} with weight k+1. He explained something important for definition of analytic L-function by Greenberg's definition written in his books.

Eseen 2010 3rd day, Algebraic Geometry and Arithmetic

2010-02-20T16:27:00.001+09:00

3rd day morning (Feb.18) was by Kramer, Geisser, and Kim. Kramer's talk was on the Arakelov geometry (a la Gillet et Soule', Faltings, Bismut) by Quillen metric and arithmetic Riemann-Roch theorem by Burger, Kuehn, utilizing heat kernel regulator. The main goal was on the log singular-metric \mu on modular curves and he assumed the Peterson metric induced by \mu on the relative dualizing sheaf. The spectral zeta function worked for the regularization, which seems to be the anomaly formula of string theory. However, I do not know about how supersymmetry can work to regularize the trace and the determinant bundle.

Geisser was on Suslin homology (which is a Torsion of simplicial complexes with coefficient A: abelian group, assuming the resolution of singularity) and his recent preprint on Parshin's conjecture. Let k be a perfect field, and X / k separated of finite type. We divided the cohomology paring into torsion-free part and torsion part. Then the Suslin-Voevodsky theorem asserts the equivalence of Suslin and etale cohomology with coefficient \mathbb{Z}/m, where k is algebraically closed, char k doesn't divide m. Then he worked on base finite fields. Under the use of Weil group generated by Frobenius, the Conjecture P_0 (by Parshin's conjecture) is the vanishing of non-zero Suslin homology of rational coefficient for smooth proper scheme. Tate conjecture and Beilinson conjecture deduce Parshin's conjecture, and if further assume Kimura-O'Sullivan finite dimensnionality, conjecture P_0 holds. Then he finally asserted the equivalence of Conj. P_0 and generalized Kato conjecutre, in which for smooth X, we have the abelianization of the fundamental group.

Minhhyang Kim was on the Diophantine geometry and Galois theory. 1. Abelian case (elliptic curves, Abelian motives: Fontaine-Mazure & Beilinson & Bloch-Kato), 2. Non-abelian Albanase map (Pro-finite version, the section conjecture of Grothendieck, motivic version).

After the morning talk, we took a group photo which is available on the web. We went to the Mensa restaurant and ate a frankfur sausage with vegetable.

Afternoon was by Beilinson and Huybrechts. Beilinson was on the Ziv Ran space's contractiblity, which is used in the arc space (formal loop space) formalism of factorization algebra. For X: topological space, R(X) := the set of finite irreducible subsets of X. The proposition was that if X is any connected cell complex (CW complex), then R(X) is contractible. Then the story was on the category $\mathcal{S}$ of finite non-empty sets and surjectivity for the index set I and Ran space R(X). The claim was that homotopy colimit of M_{X^I} in \mathcal{S} is homotopic to Rat(X,Y) (presheaf of rational map from X to Y). Then, for X a curve, Maps (X \ {x_i}, Y) is an ind-affine ind-scheme, and conctractible if Y = Z \subset \neq \mathbb{A}^d.

Huybrechts was on the stability conditions on derived categoies of (polarized) K3 surfaces, after Douglas and Bridgeland's pi-stability condition, which is a modern reformulation of Harder-Narasimhan property and t-structure on derived category of coherent sheaves with heart. I asked him about the quintic 3-fold, but we do not know about the compact Calabi-Yau 3-folds, except for the local Calabi-Yau 3-fold by Nekrasov-Okounkov, algorithms of Landau-Ginzburg model for quiver variety with path algebra, (and recent work of Walcher for quintic 3-fold and its real Langrangian submanifold).

Eseen 2010 2nd day, Algebraic Geometry and Arithmetic

2010-02-18T23:23:00.000+09:00

Yesterday was the banquet day. But, before reporting the party, I will make an academic staff.

Edinxhoven was on the polynomial time algorithm to compute 2-dimensional Galois representations. To be more precise, if we fix the weight k, we obtain a (deterministic) computation of the Hecke operator T_p within a polynomial time in log p. If we further assume the generalized Riemann hypothesis, it is also in poly. time in the weight k. Jannsen was on excellent schemes of dimension 2 and their strong resolution. This was after the recent progress of posotive characteristic case by Prof. Hironaka et.al. Jannsen mentioned the history of canonical, non-embedding, and functorial resolutions of sigularities. Laza was on the minimal model program (Hatching, de Jong-Oort), ADE hypersurface singularity, and GIT quotient.

Afternoon was by Nicaise and Abramovich. Nicaise talked about Neron model's existence, Grothendieck ring, motivic integration for monodromy of tame ramifications utilizing Chai's base change conductor. Abramovich talked about "Varieties with a twist" rather than "Geometry & moduli of stacks". He mentioned one of the books of Viehweg (Q.P. model of var. of general type) and a new ongoing book by Kollar. He talked about some family (moduli) of stable varieties. He finally mentioned the recent paper by Hacon-Mckernan on the log minimal model program.

Eseen 2010 1st day, Algebraic Geometry and Arithmetic

2010-02-17T16:02:00.000+09:00

Yesterday was the 1st day of the conference on algebraic geometry and arithmetic in Essen, Germany. The internet connection (WiFi) was not established until now, so that I read the e-mail for the first time in Germany at the conference room.

Before the lecture, I had to look for how to reach the yellow building. While I was thinking about a public street (and the tram station) around the campus, it was found out that the lecture room is not at the math department, bur rather at the center of Duisburg-Essen University.

The morning talks were delivered by Vistoli and Chenever. Before that the organizer delivered a speech on the passing away of Prof. Viehweg. Vistoli was on the "essential dimension" of a curve and the genericity theorem. After the definition of Merkuriev, the theorem of Brousnan-Retchtustein-Vistov holds for any ground filed with characteristic 0 (genus 0: 2, genus 1: +infty, genus 2: 5, genus >= 3, 3g - 3). After the theorem on smooth connected Deligne-Mumford stack of finite type over a field, the genericity theorem (Beosnan-Reichstein-Vistoli) holds. After thinking about the Artin stack and stable curves with n points, one conjectured that the automorphism group G should be "extremely reductive". Then we switched the case for d & n (especially for whether 3 divides d).

After the first talk, I could see Prof. Beilinson and talked about my research projects and his recent works with Gaitsogry. I reported my current updates at the topological methods (Jacob Lurie et.al.) in mathematical physics at Komaba compus and IPMU of University of Tokyo.

Chenever was on the analytic number theoretical methods to the modularity problem of complex multiplication. Mazur's results for p > 13 (p not 631) for Ramanujan delta function were stated, and the theorem of Gowea-Mazur was that the quasi-modular points are Zariski dense. After the theorem on the first cohomology on generic situation, its kernel is isomorphic to 2nd cohomology.

The lunch was at the shopping mall, and I took a Thai-Reise Spaghetti, which was very hot and I feel refreshed. After the lunch, we went to a cafe and we saw a carnival with bright contume.

The afternoon session was by Ngo and Brosnan. Ngo talked about the Langlands program. It started from some classical results about Godement, Jacquet, Authur on the L-funciton. The endoscopy and analytic continuation were analyzed on the pole. After the Beilinson-Drinfeld Grassmanian (n=1 case and geometric Langlands), he showed the intersection cohomology of perverse sheaf (Hecke) on the Grassmanian. For the adelization of the trace formula, he utilized the Lie group and its Haar measure, Tamagawa measure on the coset. Then the Poisson resummation formula enables us to make the summation absolutely convergent.

New Year 2010 in Tokyo

2010-01-01T13:02:00.000+09:00

Greetings.

There are several environmental changes in Komaba campus, the University of Tokyo. For example, the entrance of mathematical science building now has a flower garden as is seen in the picture.

Although no colleague I have seen for the first time today, I feel like a freshman of this research institute.

Now that I am writing a long paper, I think I cannot attend a lot of conferences for this month. I also have to prepare for a travel to Europe of next month.

All the best,

Makoto Sakurai

Desplasability and Lagrangean submanifolds

2009-11-27T19:03:00.001+09:00

IPMU is on the verge of winter. Today's seminar by Tamarkin was on Fukaya category and Hamiltonian symplectomorphisms.

Kashiwa has the atmosphere of Christmas.

Visual F# 1.9.7 available

2009-10-22T03:24:00.001+09:00

Don Syme annouced the new version of F# compiler. MSDN subscription enables us to use Visual Studio 2010 Beta 2, but Visual Studio Shell 2008 is ready for today's new F#.

enjoy.

Chiral categories at non-critical levels and generalized localization

2009-09-30T21:52:00.001+09:00

Last week, I went to Shin-Osaka, in order to deliver an oral presentation at the JMS (Japan Mathematical Society). The presentation hall was at a building of Science Department of Osaka University. To reach this campus, I had to transfer at Senri-Chuo, from subway to mono-rail.

As my current specialization is on the algebro-geometry ("algebroid-analysis"), the generalization of Atiyah-Bott fixed point theorem is a long-standing problem to be stated algebraically.

I presented my understanding of the superstring theory within this decade, which is somewhat involved with series of axioms of quantum field theories. Thus I digested my original preprint into a "causality problem" (physicists' localization theorem of $D$-modules). On the contrary, I did not explain the Wightman axiom or Osterwalder-Schrader axioms. It was because I was more on algebraic analysis of holonomic $D$-modules of hyperfunctions of several variables.

OK. Let me summarize. My presentation is available at

Chiral categories at non-critical levels and generalized localization

Like this, I somehow focused on the contradictory problem between quantum conformal field theories and general relativity. I tried to eliminate such contradiction by utilizing space-time supersymmetry (special holonomy), and proposed a compact complex surface, with both positive Ricci scalar curvature and cancellation of gerbe cohomology coming from one part of the gravitational anomalies.

Mobile Phone Posting

2009-09-30T21:50:00.000+09:00

Two weeks ago, I prepared my presentation by a 1.5 hour seminar at my home campus. That day was my birthday so that I have become matured.

Quantization and Hodge structures

2009-09-17T14:42:00.000+09:00

Today was the final day of the celebration of 60th birthday of T.Oda, in Komaba campus, on automorphic forms of several variables. Then I took a short trip (1 hour and a half) to the Kashiwa campus of University of Tokyo. The milestone above is a monument near the entrance of this campus.

We are going to have a seminar on quantization and semi-infinite Hodge theory.

Note also that tomorrow (September 18th) is my 30th birthday, and I will welcome this occasion by a preparation seminar for the JMS (Japan Mathematical Society) workshop at Osaka University.

Total Eclipse

2009-07-22T11:16:00.001+09:00

Recent developments of chiral categories

2009-07-21T16:32:00.000+09:00

2 weeks ago, I presented a 1 hour talk at the mathematics department of Tsukuba Univerisity. The slide is available at:

Recent developments of chiral categories (pdf)

In this talk, I generalized T-duality (of Hodge numbers) into Hyperkaehler / Landau-Ginzburg orbifolds (stacks) correspondence without coordinates (Leray-like theorem of "gerbe cohomology") utilizing BOTH the notion of gluing 2-categories of chiral differential operartors AND a little on the generalized (virtual) localization theorem by quiver gauge theories of chiral multiplets.

IPMU: Focus Week on New Invariants and Wall Crossing

2009-05-23T17:20:00.002+09:00

In this week, I was staying near the Kashiwa campus of the University of Tokyo, which is a little far (> 1 hour) from my regular office at the Komaba campus. The purpose of my visit was to look for the recent developments of my old work on topological string theory (open / closed Gromov-Witten) and topological field theory (generalization of Donaldson type invariants of gauge theories). Although my recent research is more on the mathematics side (because I am now a postdoc of mathematics), I hope there will be a good communication with my old friends.

Well, we were afraid of two things; one is that of new influenza (type A; H1N1) panic from pandemic. Thanks to the efforts of the organizers and staffs, the panic could be avoided, (surprisingly, they prepared medical kits (such as disinfection liquid and surgical masks) as well as a thermography camera). The other is the possible miscommunication between proper mathematicians and sincere physicists. The misunderstandings due to the culture of either physics or mathematics were also overcame by the active questions from the participants from far abroad.

OK, let me summarize some of the academic topics of the conference. The 1st day talks were delivered by Murayama (opening address), Szendroi (refinement of virtual Poincare polynomials by the virtual localization (fixed point) theorem for DT (Donaldson-Thomas) sheaf invariants), Jim Bryan (orbifold & crepant resolution conjecture), Toda (strong rationality conjecture on DT with some examples), and Krefl (work with Walcher, orientifold by "O-plane" by involution on the worldsheet).

The 2nd day was mostly the 2-hour talks by Nakajima, Neitzke, and Verlide. (It was because we were awaiting for the banquet at the evening). Nakajima was on the renewed viewpoint on the old and new on the $t$-structure by Beilinson-Bernstein-Deligne (famous French paper on perverse sheaf) and $Z$: central charge of $\mathcal{C}$: heart (Harder-Narasimhan).
Neitzke was on the 1) review on the Seiberg-Witten data and Kontsevich-Soibelman wall-crossing formula, 2) construction on hyperkahler manifolds (of the moduli space of above-mentioned supersymmetric gauge theory; naive first approximation and quantum corrections by instantons), and 3) examples of moduli space of rank 2 ramified stable Higgs bundles over curve (generalization of Hitchin system). Verlinde was more on vertex algebras (of Borcherds) for $\mathcal{N} = 4$ dyons (electric & magnetic charges) by conformal field theory embedded to K3 x T^2. At the double point ($a = 0$ of a * y^2 + b * y + c = 0), they call a "wall" for Weyl reflection, and Weyl "chamber".

At the 3rd day, everyone became tired. Soibelman was on the joint work with Maxim Kontsevich (, and works in progress). After addressing the tools for construction of motivic integration (motivic functions of Denef-Loeser, their stack version by Joyce, and ind-constructible familes). He also mentioned some Hall algebras, which was reminiscent of the 5-hour talk by Kontserich at IPMU. Mikhalkin was on the tropical geometry over non-field. I am not sure whether its "manifold" is a variety or a scheme of locally ringed space. Cheng was a contributed talk on the Borcherds-Kac-Moody algebra for orbifolds and moduli for degenerated metric inner form, and hyperbolic geometry of Poincare sphere with an infinite set-sum of generically (except the Leech lattice) finte number of chambers. Some recent works on wall-crossing dyons were with Lotte Hollands. Nagao was on a big conjecture on the analogy for "open" non-commutative Donaldson-Thomas invariants by elaborating the work of Szendroi.

The 4th day was Maulik, Denef, Ohkawa, and Yamazaki. Maulik was on the generalization of the work of Bryan-Leung for the higher genus curve of poralized K3 surfaces. Denef was on the "local Calabi-Yau" from moduli space of Seiberg-Witten (\mathcal{N} = 2) gauge theory and its prepotential. Ohkawa was a computation of difference of Betti number under the change of "theta-stability" by flips (wall-crossing). Yamazaki was on the collaboration with Ooguri, on the path-algebra of quiver diagrams for (co-) amoebas.

The 5th day was by Fukaya, Hanany, Dimofte, and Konishi. Fukaya was not on Donaldson-Thomas, but on the open Gromov-Witten invariants / Floer homology of Lagrangian submanifold (A-brane with the Maslov (topological) index condition) of symplectic manifold $X = (X, \omega)$. He considered the convergence condition of quantum cohomology utilizing the Novikov ring and Cho-Oh, and the famous preprint by Fukaya-Oh-Ohno-Ohta. The first 1 hour was on the results on toric (Fano) cases, and the latter 1 hour was on the conjecture on Calabi-Yau cases. It was something like the bubbles of Donaldson theory for the Uhlenbeck compactification of Chern-Simons theory. Hanany was on meson / baryon counting of chiral operators, by illuminating algebraic surfaces / orbifolds with at most 2 Kahler moduli, with the help of quiver diagrams, subset of crystals, and toric diagrams. Dimofte was on the refined / motivic wall-crossing, where it was not certain to consider the rigidness or existence of motives (dilogarithm after Beilinson-Deligne). Yukiko Konishi was on the decreasing / increasing filtration structure of mixed Hodge structure for Kaehler variety with singularities ("local Calabi-Yau", namely the total space of canonical bundle of smooth nef surface, e.g. $\mathbb{P}^2$), and its application to the symplectic "open" manifolds for the Yukawa coupling.

Some of the participants will stay in Tokyo for a few more days, and I believe that all appreciated this occasion of meeting.

Re-announcement of quantum geometric Langlands conjecture

2009-03-28T12:16:00.000+09:00

Today, I presented a refinement of my previous presentation at the Strings 2005 conference, Toronto.

My work of this presentation was on the quantization / localization of the chiral (Hecke) - factorization (Galois) equivalence of right /left $\mathcal{D}$-modules, which is a modern version of the quantization of mirror symmetry.

I also tried to explain the recent development of the open string theory coming from the Chan-Paton factor of the gauge theory attached to the orbifold singularity points.

The details of my presentation slides are avaible at my homepage.

http://www.ms.u-tokyo.ac.jp/~makoto/

Deformed chiral algebras and Kac-Moody algebras

By the way, Tokyo has the cherry blossoms at this weekend, as is shown at the right figure.

Algebraic Analysis and Deformation Quantization

2009-03-08T21:11:00.000+09:00

There is a workshop announcement on the "algebraic analytical geometry" at the following website;

http://www.u-bourgogne.fr/monge/g.dito/aadq/index.php?page=home

Take care.

Birth and Origin of Quantum Geometric Langlands Program

2009-03-05T00:06:00.000+09:00

To the best of my knowledge, the conjecutres on quantum geometric Langlands and Mirror symmetry for D-modules (not in the form of quantum D-module of Givental, but in the form of Beilinson-Drinfeld chiral algebras) were first proposed by me (Makoto Sakurai) at the Strings 2005 conference in Toronto (Fields Institute), and my slides of presentations on the web were delivered to Dennis Gaitsgory next month at the Seattle 2005 summer instiute held by AMS. Ivan Mirkovic and David Ben-Zvi know my presentations, becuase I e-mailed to them at the end of the Seattle 2005 conference.

"Duality between open Gromov-Witten invariants and Beilinson-Drinfeld chiral algebras" (July 2005, Fields institute, Strings 2005)(available ppt, pdf)

Of course, I did not write the quantum groups formalism, and I did not mention about the irrational quantum deformation parameter formalism, which was originally claimed at the Kashiwara conference by Dennis Gaitsgory. Rather, I am now working on supersymmetric Poisson sigma model, as a stratification (globalization) of quantum groups (non-commutative schemes), which is a generalization of the quantum flag varieties (affine Kac-Moody algebras as the fiber) and this has the Euler obstruction classes as summation (Riemann-Roch theorem) of the 1st and 2nd cohomology class of the gerbe cohomology of chiral differential operators.

Of course, the quantum deformation by 1 parameter or 2 parameters are not yet sufficiently done in my work, although I am working on the quantum cohomology and elliptic genus (algebraic hom of cobordism due to Thom and character (supertrace of conformal blocks) of super conformal field theory, elaborating Eguchi-Sugawara-Taormina's representation of non-rational conformal field theory). I will just talk about the relation between the algebraic cobordism ("open-closed duality of cylinder amplitude") and the double loop group ("double Kac Moody" algebras) at the end of this month at the Mathematical Society of Japan.

Also, Braverman said that Dennis should mention about the work by Feigin-Stoyanovsky (while at RIMS = arXiv:math/0610974 ?) where I could not find out exacly what was going on.

Since I do not want to argue about who was the first (like Newton and Leibnitz on calculus...), but we must be more careful on which point of our works owes the idea to who.

Kyoto & Clay on strings & gauge theory

2009-03-03T23:41:00.001+09:00

At the morning session, Bezrukavnikov was on noncommutative resolution of singularities. After the lecture, I asked him whether we can construct noncommutative projective schemes after Artin, Stafford, and ven den Bergh, in higher dimensional cases.

We went to a Japanese noodle restaurant for lunch, where we ate Japanese "healthy" noodles with vegetable or fish. In Japan, we have snows everywhere (in Kyoto it was first snowflakes, but soon became stronger). In Boston and New York, it was also snowing.

At the afternoon session (14:30-), Nakajima's quiver gauge theory started from the historic Donaldson theory of instantons where we have two compactifications of translations and bubbles. We dealt with Uhlenbeck's compactification. He explained that the order to take affinization and Langlands dual group (co-weight, co-root) is important, which was different from that of Gaitsgory. We will work on infinite dimensional Grassmannian, but we have a good approximation by finite dimensional geometry; so, don't be afraid. We worked on the intersection cohomology, usual tensor product (not fusion product), and level-rank duality by Igor Frenkel (due to instanton number and the order of the cyclic group acting on the space (orbifold singularities)), which will be talked in more detail.

Gaitsgory's (about one third of) talk was overlapped with Nakajima's introducion to geometric Satake correspondence. He introduced

1) something like a Tannakian category (rigid tensor category, which can be identified with a representation of algebraic group)

and

2) the notion of "compactly generated" category

and

3) many limits... (direct, inverse 2-limit, ...)

and

4) non-degenerated Killing bilinear form

and

5) deformation 1-parameter $\kappa$ for the deformation of quantum groups

To make a good algebraic theory, he made $\kappa$ an irrational parameter at the Kashiwara conference, but there are some progresses after that. He claimed the last theorem / conjecture that the Whittaker sheaf (or rather, Whittaker category) $Whit_{\kappa} (G)$ is equivalent to $KL_{{\kappa}^{-1}}$ (${}^L G$). (not in Riemann-Hilbert correspondence)

In the last note, he used some representation of Heisenberg algebra and semi-infinite cohomology depending on "$n (\kappa)$", which should be a nilradical part (to construct the W-algebras) and I asked that point, because the audience was very silent after the lecture. He said "Yes, it is a nilpotent, Laurent." I am awaiting for the global quantum geometric Langlnads correspondence of tomorrow's talk. (I know local quantum geometric Langlands and affine Kac-Moody algebras, but I am not satisfied with the localization functor.)

See you tomorrow!

Clay public lectures for quantization, Langlands, and Riemann hypothesis

2009-03-03T00:15:00.001+09:00

Today was a public lecture day, so the talks were not aimed at specialists.

The first speaker was Roman (MIT) who suggested, at the positive characteristic cases, the quantum nature becomes easier to study. But, for the time reason, he did not explain its relation to the recent developements of Bridgeland's stability conditions in detail.

The next speaker was Dennis Gaitsgory (Harvard), whom I first met at the Settle'05 summer institute on algebraic geometry. I could have talked to him at the reception time.

The final speaker (James Carlson) was from Clay institute who claims the Riemann hypothesis for the non-trivial zero point of Riemann zeta function. An audience asked what if somebody would find a counter example? Then the speaker said there are some conditions for the $1,000,000 award, but as was in the Hodge conjecture, which was negatively stated by Voisin, the problem statement will be improved. So maybe if someone who might find a zero point of the zeta function outside the critical line (and of course the trivial zero-point in the real axis), they can obtain the money.

The reception was at the Holiday-Inn Kyoto. The toast ("Kanpai") speech was done by Prof. Kashiwara. We could have good conversations with participants from far abroad, although there were many Korean student groups who were invited to the conference. In the middle of the dinner time, we took some group photos in front of the golden wall ("Byobu").

Clay Symposium with Kyoto Univerisity

2009-03-02T01:21:00.000+09:00

From tomorrow, we will have a series of lectures by Bezrukavnikov (MIT), Gaitsgory (Harvard), and Nakajima (Kyoto). The main theme is on the quantum algebras and "quantum" geometric Langlands program, (and a lot of number theory...).

http://www.claymath.org/programs/claylecturesmath/2009CLM/

For the moment, I am preparing (namely, reading the original papers of the speakers and myself, and some recent research books.) for the discussion in my hotel, where the network connection is very unstable. So I am using au WIN cable (cellular phone connection) to write this blog. I will ask my friends to let me use better internet environment from tomorrow.

The picture today is the city center of Kyoto-Fu (local government).

I hope we will enjoy this workshop.

Global COE symposium held at Tokyo

2009-02-09T21:09:00.001+09:00

The new research project at the University of Tokyo (Graduate School of Mathematical Sciences) is now launching.

This project is entitled as "The research and training center for new development in mathematics", 拠点リーダー(Leader)：川又雄二郎(Yujiro KAWAMATA）.

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/index_e.html

Its international opening symposium was held by some honourable people in algebra, geometry, and analysis from the abroad: Mathematics from Today to Tomorrow.

We are going to have a lot of workshops in Tokyo (Komaba campus); for example, a spring school on the representation theory.

Ramification theory in Arithmetic Geometry

2009-01-18T21:54:00.000+09:00

In this week, I went to a training camp held by Seidai Yasuda (RIMS), at the space called "Fruit Flower Park" in Kobe (western region of Japan).

The abstract and schedules are available on the web:

http://www.kurims.kyoto-u.ac.jp/~yasuda/seminar.html

, although there was a slight change in the contents of speakers.

The scenary was so nice that we had plenty pounds of snowflakes twice within the week from Monday to Friday.

We seeked for the analogy between algebraic analysis of $D$-modules with irregular singularities with char = 0 or p > 0 (prime, positive level) AND arithmetic geometry / algebraic number theory in the sence of ramification of l-adic cohomology with crystalline structures in level 0 (Takeshi Saito, not necessary Lagrangian, but also co-isotropic submanifolds as the Characteristic cycles , or I suppose some calibrated geometry).

We also had some conversations interdisciplinarily at the residence from the morning to the midnight.

The plan of next year's conference is not scheduled yet, but RIMS (Kyoto Univerisity) has some plans to organize workshops like this.

The Rising Sun

2009-01-03T02:59:00.001+09:00

Happy new year from Japan!

2008-12-31T20:21:00.001+09:00

In Japan, where the time is defined as the eastest (+9:00 hour), we are now about to finish the year 2008 and welcome the year 2009. The above photo is a Japanese style decoration of happy new year.

The ending of the year 2008

2008-12-30T23:22:00.001+09:00

In Japan, it is about to finish the end of 30th of December, 2008.

I went to the Komaba campus, and then to the Shibuya station, in order to work through the academic works as well as some shopping of digital devices.

The photos are what the Komaba campus of the University of Tokyo looks like at the end of the year of European calender.

Langlands correspondence for loop groups

2008-12-28T18:21:00.000+09:00

Today I went to IPMU to discuss the abstract idea of geometric Langlands program from geometric conformal field theory.

The above picture is what the train station provides for the WiFi wireless LAN at the station or inside the train.

Have a good holiday!

Makoto Sakurai

Edward Frenkel (not Igor Frenkel) and my conjecture

2008-12-27T20:33:00.000+09:00

We will have an informal seminar on the books of Edward Frenkel at tomorrow and the day after tomorrow.

I will try to comment on the definition, the meaning, and the significance of the Beilinson-Drinfeld chiral algebra theory for (twisted) loop groups and its foundation and applications to geometric representation theory and even mathematical physics of quantum cosmology beyond the standard model with or without Higgs boson(s) (or even Higgsinos?).

My originarity is on the conjecutre of quantum (q-analogue of Poisson bracket) version of geometric Langlands program, which was proposed by me (Makoto Sakurai) at the Strings 2005 conference, Toronto.

Xmas Festival and Newtonian birthday

2008-12-25T23:59:00.001+09:00

I went to Shibuya to first eat and then drink.

So, what did we learn about the quantum universe of gravity?

Do we believe in God or in human?

Of course, I am talking about some moonshiny topics.

Xmas Cake

2008-12-24T12:10:00.000+09:00

Moon sky at the Christmas Eve

2008-12-24T07:09:00.000+09:00

IPMU miniworkshop: A New Recursion from Random Matrices and Topological String Theory

2008-12-11T19:52:00.000+09:00

Today we have an opportunity to meet our friends working on the algebro-geometry, the symplectic geometry, and some topological string theory:

http://www.ipmu.jp/seminars/20081211-MiniWorkshop.html

Here Bertrand Eynard (Saclay) a little informally introduced some historical aspects of matrix models of Kontsevich-Witten and Mulase was more on the algebraic geometry of the characteristic classes of Hodge bundles on the Deligne-Mumford compactification of stable maps.

We also heard the talk of Akaho on the immersion version of Floer theory of Fukaya category of $A_{\infty}$-algebra structure, which look alike the intersecting branes of recent topological string theory (of A-model by FOOO (2008)).

Virtualization Forum 2008

2008-11-18T17:57:00.000+09:00

After the morning seminar of vertex operator algebras in Komaba, I went to the Virtualization Forum 2008 held by VMWare.

http://info.vmware.com/content/VirtualizationForum_JP?src=WWW_08Q2VMW_VFORUM_JP&ossrc=WWW_08Q2VMW_VFORUM_JP&elq=WWW_08Q2VMW_VFORUM_JP

I heard the talks of HP, NEC, and Hitachi. They distributed some handouts of the presentations as well as some "commercial bag" of VMWare. At the coffee break, I went to the panel session, to see a blade server of NEC as well as the desktop virtualization by DELL or HP. They did not explain how they were involved with the Xen virtualization or which one was the smartest virtualization machine.

There was also the annoucement of Xen Summit 2009 North America. http://www.xen.org/community/xensummit.html

(See, also 2008 Tokyo

http://xen.org/xensummit_japan.html)

IPMU Seminar: Conjectures on mixed motives

2008-11-05T18:35:00.000+09:00

Sergey Gorchinsky had a 2-hour seminar from 10:30 - 12:30. It started from the Chow groups of smooth irreducible variety $X$ over a field $k$. This was a story on the algebraic cycles modulo rational equivalence on $Z^p (X)$. Then it was on the conjecture by Bloch in algebraic K-theory; if $h^{(0,2)} > 0$, then the kernel of the Albanese map $T (X) = 0$. In the 70's, it is known for $X$ NOT of general type (with $h^{0,2}=0$). He also explained that (by Beilinson?) for $X$-smooth projective surface over $\mathbb{C}$, suppose $\alpha \in CH^2 (X \times X)$ the Hodge realization $h (\alpha) = 0$. Then $\alpha_{\ast}|_{T (X)} = 0$. The remark was on the preservation of the filtration.

After a break, $X$ will be more generally an arbitrary variety over a field $k$. The corresponding categories are the de Rham, e'tale, and Betti (mixed motives like onions (by Bloch)). He then worked on the mixed motive version $\mathcal{MM}_k$ as a right tensor abelian $\mathbb{Q}$-linear category from $Vect_k$. Of course we assumed the vanishing conjecture of Beilinson-Soule' for the existence of the category of mixed motives. Well, he then functored $Vect_k$ to a triangulated category $DM_k$, which has a non-degenerated $t$-structure s.t. its heart is the image of the inclusion of $\mathcal{MM}_k$ to $DM_k$. The story came to the Chow motive $CHM_k$ to pure motive $\mathcal{PM}_k$. And there was a "believable" conjecture of Beilinson. Then Sergei showed some examples of smooth projective $X$ of Murre. Finally he explained the theorem of he and Guletskii of when $X$ is a smooth projective 3-folds over $\mathbb{C}$, then there exists a split (or decomposition?) of $M(X)$.

After the talk, I asked him whether there exists a relation between Kontsevich's motivic integration and the conjectural category of mixed (Tate) motives after Bloch-Kriz. He informed me of the paper by Lunts, which showed that there is not a homemorphism from a motivic measure category $M_k$ to the $K_0$ group of smooth scheme $X$. I am not sure about the case of "refined" motivic integration, which has the value not on the Grothendieck group, but rather a trianglated category.

Informal Seminar with Maxim Kontsevich in Keio

2008-10-15T22:30:00.000+09:00

I went to Keio University, Yagami Campus, to hear the informal seminar with Maxim Kontsevich from 13:30 to about 17:40. There were three talks: Toric degenerations, A_\infty category, and pseudo-groups. Maxim made some comments on the talks of speakers.

Then we took a dinner of Japanese sake bar (Izakaya), so that we brightened the conversation on some culture (not always on mathematics).

Today was the end of Kontsevich's visit in Tokyo, and he is going to Kyoto University.

Kontsevich's 5 hour seminar at the IPMU on wall-crossing

2008-10-14T22:33:00.000+09:00

Maxim Kontsevoch explained his recent joint work "wall-crossing" with Y.Soibleman at the IPMU (University of Tokyo, Kashiwa campus) from 13:30 o'clock to after 19:00, including a half hour break and the dinner at one of the cafeterias (also he had a half-hour interview with Kyoji Saito for some journal).

His first half of the talk was

1. Physics background \mathcal{N} = 2, d = 4

super Yang-Mills

supergravity (black hole)

superstring

2. Mathematics background

Stability condition in 3-dim Calabi-Yau categories

Donaldson-Thomas invariants (and their generalization)

Hyperkaehler metrics

? Donaldson invariant, and Borcherds Kac-Moody algebras

3. Stability data (Z, (a_{\gamma})_{\gamma \in \Gamma} (infinite set)), where \Gamma \cong \mathbb{Z}^d is a lattice, and \mathcal{G} = \oplus_{\gamma \in \Gamma} \mathcal{G}_{\gamma} / \mathbb{Q}

Then he defined quadratic form Q on \Gamma_{\mathbb{R}} := \Gamma \otimes \mathbb{R}. Thereafter he fixed Q, and he reformulated the stability data (Z, (A_V)), where V is a strict sector ("chamber" of Donaldson theory) in \mathbb{R}^2 \cong \mathbb{C}.

He claimed that, under some assumptions, even if move Z which a little bit preserves "the" condition, A_{\Triangle} should not change. (He was working on a topology structure.)

Now he defined a continuous path in stab (\mathcal{G}) with a parameter in [0,1] for a continuous family Z_t in order to study 2 examples. The 1st example was simpler than the 2nd example: \mathcal{G} = \mathcal{gl}_n and \Gamma = \mathbb{Z}^{n-1}. In this case the walls look like a geometric vector of 2-dim Euclidean space.

4. After the tea break (15:30-), M. Kontsevich examined the 2nd example: \Gamma skew symmetric, with < , >: \Gamma \otimes \Gamma \to \mathbb{Z}, and \Gamma = \otimes_{\gamma \in \Gamma} \mathbb{Q} e_{\gamma} (e is a basis of the Fourier modes). Then he introduced a conjecture on the Donaldson-Thomas invariant \Omega (\gamma) \in \mathbb{Q} (\gamma \neq 0).

Here came the non-commutative geometry and motives; X will be a 3-dim compact Calabi-Yau equipped with a holomorphic (nowhere vanishing (3,0)-form) \Omega^{3,0}; rather than just a K3 surface. Assume that the Calabi-Yau 3-folds are endowed with a Kaehler form [\omega^{1,1}] \in H^2 (X, \mathbb{Z}). We therefore have a \Gamma = H_3 (X, \mathbb{Z}), < , > := intersection Z(\Gamma) := \int_{\gamma} \Omega^{3, 0} \in \mathbb{C}, and \Omega (\gamma) := "number" of special submanifolds of X (whose definition for the "calibrated submanifold" L \in X is such that \omega^{1,1}|_L = 0 and vol (L) = |Z (\gamma)|, but I was not certain about how to define a computable definition of this "number".) Well, we will deform Z <=> change complex str on \Omega^{3,0} by tensoring $\mathbb{C}$ with "B-field" or complexified Kaehler form.

Hereafter he explained the categorical aspects of A_{\infty}-categories / field k (similar to the Fukaya category). The object is once defined as a set of object(s) \mathcal{E}, \mathcal{F}, and the morphism is \mathbb{Z}-graded Hom^\ast (\mathcal{E}, \mathcal{F}) such that \Sum_i dim Hom^i (\mathcal{E}, \mathcal{F}) < \infty, and for all n \ge 1; \Sigma_0, ..., \Sigma_n; m_n of the following;

Hom(\mathcal{E}_0, \mathcal{E}_1) \otimes ... \otimes Hom(\mathcal{E}_{n-1}, \mathcal{E}_n) \to \Hom(\mathcal{E}, \mathbb{E}_n) of degree Z:

\Sum \pm m_i (\alpha_1 \otimes \alpha, m_1 (\alpha...) ... \alpha_n) = 0,

which is something like a unital DGA of homotopy algebras of some "mutation" like enhanced triangulated category with multiplucation structure. Then he changed the object of the category from "set" to a constructible set for any triangulated category. He compared semi-stable objects a_{\gamma} with Lie algebra A (V) of the left and the right. This comparison was done in a following way; Stab(\mathcal{C}) \to \Gamma^{\dual} \otimes \mathbb{C} is locally homeomorphism. And 2 special cases:

1. the base field k = \mathbb{F}_q

2. Hall algebras H (\mathbb{C}) (after To\"en's formula which was more complicated) = \oplus_{\gamma \in \Gamma} \H (\mathcal{C}) \gamma

The equivalence from associative algebras to Lie algebras was;

\mathcal{A} (V) := \Sum_{\mathcal{E} \in \mathcal{A} (V) / isom.} \frac{[\mathcal{E}]}{# \Aut (\mathcal{E})}. This Hall algebra in the char k = 0 will be, for example, a 3-dim Calabi-Yau. In this Calabi-Yau case, we will deal with ind-constructible A_{\infty} algebras / k. The target space is a scheme of finite type, and the object is a constructible maps. The Hall algebras will be "motivic" in the sense of motivic integration in terms of symbol $\mathbb{L}$. He used the motivic integration by means of \mathcal{U} := Ext^1 (\mathcal{E}, \mathcal{E}), with a use of formal (non-convergent) power series on \mathcal{U}. He introduced the "(super)potential" (Morse function) W (\alpha) (\alpha \in \mathbb{C} infinitely close to 0) to study the Milnor fiber (also Denef-Loeser) and vanishing cycles. The talk came to the computation of the "weight" of \mathcal{E} (with a 5-term relation for quantum torus).

Prof. Kontsevich on BPS counting and wall-crossing

2008-10-09T14:48:00.000+09:00

Maxim Kontsevich (IHES) was on "BPS counting as hyperkaehler metric".

He started from axioms, its background physics and mathematics such as Donaldson-Thomas, quiver cluster algebras, collapsing hyperkaehler geometry, geometric Langlands and classical integrable systems (Hitchin system and Seiberg-Witten theory).

Today was the end of the joint meeting of Japan and France.

Prof. Kazuya Kato and Voisin at the IHES 50th anniversary and the Mathematical Society of Japan joint workshop,"Perspectives in mathematical sciences"

2008-10-08T23:36:00.000+09:00

We are holding a joint meeting of Universities of Tokyo, Keio, and IHES (France) on the mathematical science in Japan.

Today I heard the talks of Kato and Voisin. Kato was on the moduli space and complex / p-adic mixed Hodge structure analogies. Voisin was on the geometric structure on the cohomology algebras of compact Kaehler manifolds. (Kontsevich will talk tomorrow morning at Keio, Mita Campus.)

The talk of Tohru Eguchi (and conversation between Prof. Kontsevich and Prof. Eguchi) was cancelled because of the Nobel prize in physics. Note that Eguchi was a student of Emer. Prof. Yoichiro Nambu in Chicago University more than 30 years ago. As I used to be a Ph.D. student of Eguchi, I am one of the grandchildren of Nambu.

My work on chiral algebra depends on the Higgs mechanism, which is a kind of symmetry breaking of gauge theory, is somehow a generalization of Nambu-Goldstone bosons (massless particles due to broken "global" symmetry, such as pi mesons inside atoms) into the "local" gauge symmetry (acquiring masses) of "the" standard model of elementary particle physics.

Although the Higgs boson(s) of the standard model is (or "are") now in search of LHC (Large Hadron Collider) in Europe (Geneve, Swiss), the symmetry mechanism in the quantum / statistical field theory is everywhere in theoretical physics (including condensed matter physics) or even in modern mathematics including algebro-geometry. (This standard model is still not established experimentally, because we do not know about the Higgs boson masses or their supersymmetric version (e.g. MSSM), or string theory versions.)

Nobel Laureate Yoichiro Nambu

2008-10-07T21:16:00.000+09:00

Dear all;

Prof. Nambu will receive the Nobel prize in physics 2008. Nambu was the supervisor of my previous supervisor Tohru Eguchi.

http://nobelprize.org/nobel_prizes/physics/laureates/2008/

We must celebrate this occasion.

5th Takagi Lectures

2008-10-05T20:20:00.000+09:00

This weekend we held the 5th Takagi Lectures at the University of Tokyo, Komaba Campus.

At the conference, I talked to several well-known people on several well-known conjectures on mathematics and mathematical physics.

The staffs offered speeches on the occasion of 50th anniversary of IHES, France. They also encouraged younger researchers by inspiration from honored professors from abroad.

Finally, the wine party today was nice, so that we appreciated our conversations.

Presentation at the Mathematical Society of Japan, 2008 Fall

2008-09-26T20:30:00.001+09:00

At the Tokyo Institute for Technology (TITech), a mathematics domestic conference is running. TITech is close to my home university, so that I did not reserve a hotel nor a flight.

My presentation was at the first of morning session of algebra session (Session I) at the following building. [see left]

The slides of this presentation are now uploaded at my official web-site, where there exist the ppt file as well as the pdf file. The slides are consisted of two-folds; one is on the gerbes of chiral differential operators and the other is more on recent developtment of mine on motivic nature of chiral algebras of Beilinson and Drinfeld. My points on quantum geometry were almost abbreviated, because the audience must be mostly interested in the chiral Hecke algberas of algebro-geomteric nature; rather than generalized differential geometry.

At the afternoon, I heard two 1-hour talks of representation of W-algebras (by Arakawa), and of polylogs, Eisenstein classes, and p-adic L-functions (by Bannai). Arakawa's talk was on the non-linear vertex algebras and around, including the history such as Victor Kac, Igor Frenkel, Feigin-Frenkel, abd Miura pairs. This talk also emphasized the role of chiralization and Hamiltonian reduction. Bannai's talk was mostly on the movie like demostration of elementary theory of generalized polylogarithm well based on the classic work of Beilinson and Deligne, and the relation to local systems after the Riemann-Hilbert correspondence (but not on the monodromy problem so much). Bannai explained his personal motivation at his student period from the paper by Kazuya Kato on the absence of fundamental theorem.

By the way, the sound of bell-ring insect at night is getting louder in Japan.

Third day of micro-local analysis

2008-09-18T20:43:00.000+09:00

Today was my birthday.

After the Hilbert scheme and the Hamiltonian reduction of quiver variety and (particular) Frobenius action, Kashiwara's lecture was going on. For rational Cherednik algebra of type A, $A_c^{loc}, and e H_c e$, we were talking about something like a noncommutative coherent ring (finite represented but not always finitely generated).

We ate lunch at the cafeteria (to be the truth, there are 2 cafeterias), Leclerc explained his article (based on a proceeding of a conference).

1. Facts about the preprojective algebra

ADE (not $\hat$ case; we study finite-dimensional one of $\Lambda$'s, where $\Lambda := \mathbb{C} \bar{Q}$ /(Gelfand-Ponomarev equation)), $\Lambda$ is self-adjoint, and simple $\Lambda$-modules case.

2. Relation between mod $\Lambda$ and $\mathbb{C} [N]$

3. Mutation for rigid $\Lambda$-modules

Thm.[Geiss-Schr\"oer] and Thm[GLS].

I walked around and near the campus in the spare time, and there exists a park in front of the gate of the Kashiwa campus. It is said that new office building will be installed to the empty grass field.

Today was the end of the workshop on Micro-local analysis and Symplectic manifolds.

Second day of micro-local analysis

2008-09-17T20:05:00.000+09:00

Today's talks were by Kashiwara, Leclerc, and Nagao. Masaki Kashiwara ("Quantization of symplectic manifolds") was on DGA (Differential Graded Algebra; and its localization algebra), the dualizing complex, Frobenius action, Beilinson-Bernstein theory, and a little bit of (spherical) rational Cherednik algebra of type $A_n$. Bernard Leclerc was on symmetric Kac-Moody algebras $\mathcal{g}$ of type ADE (and its maximal nilpotent subalgebra $\mathcal{n}$; tomorrow will be $A^{(i)}, D^{(i)})$), under the title "Nilpotent varieties and cluster algebras".

See you again.

IPMU (Institute for the Physics and Mathematics of the Universe) workshop on micro-local analysis

2008-09-16T20:06:00.000+09:00

Today we have a workshop, which consists of review / lecture and original works of 3 speakers with about 20 participants. This workshop was mainly on the (complex) symplectic geometry of several (deformation / resolution) quantization procedures.

The atmosphere of the conference was informal, in the sense that the organizers encourage the questoions from the audience, from both technical and conceptual meanings.

By the way, the workshop residence was at the Kashiwa campus of the University of Tokyo. However, strictly speaking, Kashiwa is inside the Chiba prefecture, and is not in Tokyo. It takes about 2 hours from the center of Tokyo; although the "Tsukuba express" -- newly constructed modern railway -- is available.

The weather is getting better since the rainy season and Summer are over. We are awaiting for Autumn.

Cheers.

Back to home

2008-09-10T19:42:00.001+09:00

For the moment, I do not go abroad. I stay at the mathematical science building in Komaba, the University of Tokyo.

Well, I am preparing for workshop presentations, seminars, and a paper submission. I hope I can do a good work.

Summary of Geometric Langlands at Lorentz Center, Leiden

2008-07-21T22:31:00.000+09:00

Today I arrived at the Lorentz Center, Leiden, the Netherlands. This is a workshop entitled as The Geometric Langlangds Program, which has the emphasis more on arithmetic geometry or geometry of conformal field theories, if compared to Miniprogram for Gauge Theory and Langlands Duality at KITP.

During the visit, I could have informal talks with several acquiaintances from both the east and the west. I could especially make some discusssions with "ancestors" of the (quantum) geometric Langlands program of D-modules.

The weather was not stable in the sense that we have both sunshines and showers within each day. At the Oort Building, we have a coffee machine (at right) of common room.

I wish we would have a good occasion in the near future, where we can come together somewhere else and appreciate our community.

Take care.

Arrived at Kyoto again

2007-06-24T18:56:00.001+09:00

I printed out the OHP which will be used for the 1 minute introduction talk before the poster presentation at June 27.

Today I rode on the bullet train from Shin-Yokohama to Kyoto at about 13 o'clock. And it heavily rained in Kyoto. I walked around of Shijo, where my hotel exists, and I went to a Japanese roast meat restaurant for dinner.

3rd day of Joint Meeting of Pacific Region Particle Physics Communities

2006-11-01T17:39:00.000+09:00

I felt really bad today, so that I could not go to the conference room until 20:00 0'clock for the poster session. I didn't eat the breakfast nor the lunch. I just could eat the D bone steak for the dinner and then I regained to go out to the ABC store to buy an umbrella.

There was no wine left when I reached at the poster session. I walked around, but there was no such presentation as the chiral algebras or geometric Langlands programs.

2nd day of Joint Meeting of Pacific Region Particle Physics Communities

2006-10-31T17:24:00.000+09:00

The morning session started from 8:30 and not 8:00 because of the jet rags of Japanese participants. I heard the talk of Ookouchi and the next speaker, but I could not help going back to my hotel to take a strong nap until 14:00 o'clock. So that I didn't eat the lunch before my presentation at 15:30. Silverstein left the position of chair at the point of Akishi Kato, about which I am not sure why.

My presentation was prepared for a little longer than the scheduled time, so that I had 3 pages left when Ooguri (Chair) presented "0 minitue left" card. Therefore I could not explain well about the open Gromov-Witten invariants part. Ooguri questioned about that part and I said "The open Gromov-Witten is still immature, but for the closed Gromov-Witten invariant for toric Fano, it is a theorem.".

I could talk to Ookouchi again and he was discussing with Intriligator (17:30-18:00 talk at the Field Theory session) and Kato.

I turned back to the hotel and I took the D bone steak again with 2 rices and 1 mushed potato. Today was the Haloween so that there were many girls with costumes. The waitresses were also dressed and I did a vote for the best costumer.

1st day of Joint Meeting of Pacific Region Particle Physics Communities

2006-10-30T17:04:00.000+09:00

After the first morning session, I asked Ooguri about whether the presentation time is 10 min (talk) + 5 (questions). He asked Kazama and confirmed my question. I was misunderstanding the time schedule to be 15 min (talk) + extra (questions) so that I was anxious so much, which was pointed out at the reception by him.

The reception was beautiful with musics and delicious foods (some were sold out very quickly).

I could talk to several people today, which I do not write explicitly in this diary. Note that Ooguri's talk was changed for some reasons, which can be seen from the time table of the home page of the conference (google Hawaii 2006 and JPS!). I still needed a strong nap after Ooguri's talk and before Ovrut's talk. It was because of the jet lag but maybe I was nervous about the English presentation tomorrow.

Arrived at the Honolulu City

2006-10-30T16:22:00.001+09:00

I took a 7 hour flight from Narita to Honolulu. When I arrived, it was the morning of Sunday. We took a guide from the JTB staff about half an hour on the gift, the optional tour, and the "Oli Oli coupon" for cruising and lunch. As we still have time before the check-in (12:00), we went to 2 hour cruising until 13:30.

The cruising has several optional activities such as the feeding of fishes, making a human figure from bamboo leaf, and the ukulele with the song of Sukiyaki.

At last, there was a dance with beautiful songs. I really enjoyed it.

After the cursing, we went to a Hong-Kong restraint and I took a fried rice and we left the place because some of the participants were thinking about using a rent-a-car to run all around the Oahu island.

I took the dinner after a strong nap at about 8:00 pm. Fortunately, there was a restaurant called IHOP (International House Of Pancake) and I didn't have to go outside the hotel. I took a D bone steak with rice, vegetables, and mashed potatoes.

3rd day of Mathematical Society of Japan

2006-09-22T00:29:00.000+09:00

Today, the morning session of algebra started from 9 am, so that I waked up at 7 am. I went to the topological string session intermediately included in the geometry session at the morning. Then I heard the talks of Painleve equations and algebraic geometry by Masahiko Saito (Kobe), global structure of quantum D-modules (Wall-crossing and Chen-Ruan orbifold Gromov-Witten) by Iritani, and higher dimensional (Log) Minimal Model Program and Standard Models by Shokurov and Kollar et.al. The existence is proved inductively but the terminality is not.

I was not sure what is the relation between the wall-crossing and the geometric transition for open-closed duality of generating function of Gromov-Witten invariants under flip.

2nd day of Mathematical Society of Japan

2006-09-20T22:23:00.000+09:00

Today I skipped the morning session, and heard the session on the infinite analysis on the quantum invariants and modular forms. I also saw several aquaintances before and after the award ceremony, which was dedicated to the work of inverse scattering problem of 3 particles.

We also see another workshop on the Quantum Mechanics and Chaos.

1st day of Mathematical Society of Japan

2006-09-19T20:34:00.000+09:00

I did my oral presentaion at the morning session. There was a interval before my presentation. I did my trial towards automorphic representationｓ, but I could not reproduce the modular form of 9 point generic point blowup of CP^2.

Arrived at Ten-noji again

2006-09-18T21:29:00.000+09:00

I arrived at Shin-osaka by the bullet train to stay at the Osaka City University again. Fortunately, it was not raining because the typhoon vanished. Today was my birthday, so that I ate a birthday cake and the barbecue at the dinner.

4th day of Affine Hecke Algebras

2006-06-23T04:09:00.000+09:00

Today's schedule has changed; it was because Kapranov didn't come. The morning session was by David Kazhdan and Iain Gordon, instead. Kazhdan was on "Iwahori-Hecke algebras for reductive groups over higher dimensional fields" -- which started from "what is integration? -- it is Grothendieck's K_0 group of differential invariant functionals" then some examples of dependable sets, definable but not differentiable maps, and almost representability. Then he looked at his small memo from his pocket and explained the Iwahori subgroup of SL_2(F = Q_p ((t))), H: its definable functions with bounded support, and its convolution. He defined the rational section of H on M and its relation to representability. Ian Gordon was on the rational Cherednik algebras by OHP. PBW by Cherednik and Etingof-Ginzburg and the Theorem by Ginzburg-Guay-Opdam-Rouquier on the existence of KZ functor which is fully faithful on projections and surjective on objects, utilizing the Ariki-Koike algebra. And also the theorem of Rouquier for the decomposition of chambers, and the theorem
(Gordon-Stafford, Musson, Boyarchenko, Vale) the Gordon-Stafford paper on the Hilbert scheme.

Afternoon session was done by Ian Grojnowski on "some semi-infinite geometry" which is "work in progress, many details to check... attempt to understand work of Feigin-Frenkel [FF] and Beilinson-Tate linear algebra". The first problem was to define a category of D-modules on semi-infinity flag variety by loop space LG / (LB)^0 [well-defined homotopy type, tangent space: Beilinson-Tate space] and an equivalence of categories D^{\le 0} (D-modules) to D (\hat{g}-enhanced). The main problem was to define the category. Examples started from sl_2 / P^1 case and its Iwahori subalgebra. [FF] defined \hat{g}-modules, which they called Wakimoto modules, which look like they should be D-modules attached to Iwahori orbits. When we wish to "localize LG / (LB)^0" around the middle dimension with finite codimension intersections, it is like to defining by a manifold y gluing net along open embeddings but closed also. Then he defined lci of mappings for the Quillen model category of dg algebra if it is locally finitely presented (lfp) and L_{B/A} has Tor dim \le 1. In the commutative algebra case, it coincides with usual notion. However, by gluing the \Triangle^{op} Presh(Aff)_{\tau}-stacks, we had to deal with some of the Grothendieck topology and the surjectivity seems not enough for the definition of Cech covering for the half-infinite geometry.

3rd day of Affine Hecke Algebras

2006-06-22T02:03:00.000+09:00

I felt very bad at the morning; in addition, the bus neglected me while I was sitting to wait. So that it was just on time 9:00 when I arrived at the auditorium.

Etingof was on the rational Cherednik algebras and Hecke algebras attached to complex orbifolds, which started from the past work of Hochshild cohomology, the definition of rational Cherednik algebras, PBW theorem, Etingof-Ginzburg theorem on H_{1,c} (with formal c), and polynomial representations. Then it turned to the generalization to the case of any irreducible smooth algebraic variety X over complex numbers with a faithful G-action, which is also devided into the case |G|=1 and X affine. Some definitions of reflection hypersurface, Dunkl-Opdam operators, and filtration by order. The main theorem for the Hecke algebras was on the flat deformation when X: complex manifolds has 2nd cohomology 0, which includes the lower degree del Pezzo surfaces. The idea of proof of theorem was by the KZ functor. Next, Rouqier was on the representations of rational Cherednik algebras, which started from the classical classification of finite Hecke, affine Hecke, double affine Hecke (by the order of affinization), -, deg. affine Hecke, trigonometric DAHA, -, -. rational DAHA (by the order of degeneration). Type A was done by Suzuki and also we used the Hilbert scheme of points of C^2 and PBW then a correspondence towards the Cartan subalgebras and nilpotent radical of Borel subalgebras is obtained. The problem is 1) the functor CKZ (Cherednik-Knizknik-Zamolodchikov) / CKZ with regular singularitiy connection, 2) compute local monodromy, 3) CKZs fully faithful on projective objects.

The lunch was a fried rice with cheese and cafe. After that, I turned back to the hotel to take a rest because of the sickness. Then I went to a supermarket and bought food and the meal including sandwiches.

2nd day of Affine Hecke Algebras

2006-06-20T19:18:00.000+09:00

Today I safely took the bus 21 and used the ethernet (Gigabit) as well as the WiFi connection.

The morning talks were by three people, Delorme / Solleveld, and Stokman. Delorme was on the Schwarz algebra of affine Hecke algebra using his collabolation with Opdam. It started from the discrete series, cross relations, interwining operators, and the Fourier transforms for the Schwartz algebra. The main theorem has corollaries of Chandra completeness theorem and an analogue of Langlands disjointness theorem. Solleveld was also on the Schwartz algebra, but from topological K-theoretical language of cyclic homology HP_*, which has the properties of additivity, Morita invariance, diffeotopy invariance, and excision. The 2nd lecture was done by Stokman on the periodic quantum integrable systems with delta-interactions which consists of 1) integrability / "double degeneration" of DAHA (Dunkl operators), 2) spectral theory / Bethe ansatz equations, and 3) spin models / degenerate intertwiners. However, 3) cannot be told because of the time limit.

After the lunch, I could discuss with Mirkovic on my research project and difficulties on the construction of coherent sheaves in the non-abelian Fourier-Mukai.

Afternoon session was on the Yangian and Mickelsson algebras by Nazakov on the composition of the Cherednik functor and Drinfeld functor and its generalization for groups other than gl_n. On the Mickelsson algebras, he talked about propositions of Zhelobenko on cocycles.

After the lectures, I went to the McDonald's and did the laundries.

1st day of Affine Hecke Algebras

2006-06-19T23:38:00.000+09:00

I walked up at 7 am but I was waiting for the bus number 21 (towards CIRM in Luminy) at the opposite terminal so that I was late for the 9 am lecture by Cherednik -- it was on the construction of Fourier transform for Double Affine Hecke Algebras (DAHA) by the analytic continuation for the convergence. Next one was by Victor Ginzburg on the noncommutative geometry and quiver diagrams -- something like construction of the superpotential from the loop space data. (See the right photo of the auditorium.)

After the lunch, Ariki -- to whom I talked a little at the break -- talked about the tensor product of A_n type quantum groups, which demands the crystal basis of Kashiwara and is the opposite direction to the Nakajima's loop Grassmanian / quiver varieties. Last one was the seminar by Di Francesco on the combinatorics in the physics -- more precisely the statistical 6-vertex model of ice by use of the Yang-Baxter equations and unitarity.

I ate lunch at a French restaurant near the Citadine metro station.

Arrived at Marseille

2006-06-19T02:51:00.000+09:00

After the 14+2 hours flights, I arrived at Marseille. It is more radical than Paris.

From the metro terminal "Castellane" (see the statue), it was about 5 minutes walk. I'm staying at the Citadines hotel. I tried the WiFi spot at the hotel, but it was unstable and I could not read my e-mails.

3rd day of JPS March 2006 -- my presentation

2006-03-29T20:30:00.000+09:00

I did my presentation from 15:30 because the former presenter canceled; therefore I could adjust my PC to the projector. It worked in a non-landscape mode when I tried in the double screen at the coffee break, so that it was happy for me to know the trick.

My presentation was a little long with 20 pages because I supplemented some 2 page backgrounds for non-Calabi-Yau homological mirror symmetry and genus 0 modular forms. However, I skipped some of the reviews; and there was no question. As an alternative, I uploaded the powerpoint file to my homepage.

By the way, I managed to walk to the Matsuyama Castle from the Shinonome Shrine as far as the eighth where we had a park, which was closed because it was later than 9:00 pm. However, I could see the castle illuminated and with cherry blossoms.

I hope I can post my paper within few weeks.

2nd day of JPS March 2006

2006-03-28T18:52:00.000+09:00

Some people went to the spa or the Matsuyama Castle, but I came back to the Hotel to prepare for tomorrow's presentation. In stead, I went to an Italian restaurant, Pizzeria Mar-de Napoli and chose a course of salad (antipasto), carbonara, coke and dessert.

Click the candle, and you will see the flickering without breeze.

1st day of JPS March 2006

2006-03-28T16:02:00.002+09:00

We went to the Ehime University for the meeting hall, where we could see the red cross hospital. Such kinds of institution unevenly distribute around here.

Today was the longest day starting from 9 am to 18 pm, which included two special lectures of 50 minutes each. Therefore we got tired after the talks, and took a view of the tree of Confucius and the apple tree of Newton (see the picture) around the gate of the university.

Then we walked to the same beef restaurant as I had gone last time, as a recursion. During the dinner, I went to an electronics shop to buy the ink for my portable printer but it worked very unstably with stacks when I tried to make the OHP sheets for emergency.

From Tokyo, the Haneda airport, to the Ehime-Matsuyama University

2006-03-26T19:53:00.000+09:00

This year has 1 week earlier cherry blossoms in Tokyo -- it will be flawn down before the first lunch seminar of April. Look at the left picture, which I took at the Yasuda Auditorium by a digital camera of cellular phone -- whose precision has become much higher than several years ago.

After an election of Yokohama city at the morning, I took trains to the Haneda airport. I was surprised that the airport had changed greatly with a beautiful shopping mall.

It was about 5 pm when I arrived at the hotel -- because I got lost after leaving the shuttle bus. I again losed my way when I was looking for the restaurant -- in the way, I saw the beautiful setting sun.

I'm printing out specific pages of the review of Edward Frenkel to prepare for the presentation on the geometric Langlnads - quantum chiral de Rham complex in the JPS (Japan Physical Society).

Black hole entropy, special geometry, and dynamical systems

2005-10-08T18:09:00.000+09:00

I ended my presentation at the JPS on Septermber 14 and then I heard the talk of Y. Yasui (Osaka City Univ.) on the relation between Kerr black holes (black holes with several charges) and Sasaki-Einstein real 5-fold. I asked whether we can explicitly write "many" metrics for the compact G_2 holonomy manifolds of Joyce and Kovalev after the development of Bryant-Salamon and he said it is like asking whether we can write metrics of K3 surfaces. (which I heard that Eguchi and Yau once tried.) Although string theorists are ordinary interested in non-compact G_2 holonomy manifolds, works like Atiyah-Witten of cone over CP^3 after Kovalev should be an example that we should take care of. One month later, Choissi-Fino(math.DG/0510087) seem to construct an exactly G_2 holonomy metric possibly related to the Hitchin flow equation.

OK, I will write somehow on the story after I went back to the University of Tokyo. Masaki Shigemori did two seminars (formal and informal) on "Massless black holes and black rings as effective geometries of the D1-D5 system" (hep-th/0508110) and black rings. In that place, we heard something about the Poincare recurrence. He said that the unitarity (preservation of probability) of blackhole systems is apparently broken at the long period, but it will be restored in the longer period. However, Eguchi said we should think about infinite-degree systems because it is a (quantum) field theory.

After the lecture, I reviewed some 19th century physics of statistics from modern ergodic theory point of view, something like Wiener recurrence theorem and infinite Markov process by books of Bunimovich and Aaronson. And then I looked up some papers on the relation between the Banach space theory and entropies, which include "Dynamical entropy in Banach spaces" by David Kerr and Hanfeng Li (Invent. math. (2005)) or "Schreier sets in Ramsey theory" by V. Farmaki AND S. Negrepontis (math.CO/0510102). I hope something like non-separable Banach space theory and the theory of Gowers will find some realistic applications.

1st day of JPS 2005 Autumn

2005-09-13T14:51:00.000+09:00

Today I got up early and went to Umeda to buy the Pixus portable printer (Canon) and transparency sheets. Then I went to the JPS conference to hear the talks on the string field theory.

I moved to the Kyoto University by the express Haruka and heard the talk on the covering of algebraic curves and arithmetic geometry. Then I talked a little with my senior member, who is now in the RIMS.

After that, I went back to Osaka and buy a laser pointer mouse at Namba. I cought a slight cough.

Arrived at Osaka -- JPS Autumn conference

2005-09-12T15:36:00.000+09:00

I got up about 7 am and went to the election of the Lower House. Then I took the bullet train ("Hikari") from Shin-Yokohama to Shin-Osaka within about 2 hours and 40 minutes. I was a little tired and take a 1 hour nup before the supper of Spaghatti.

I was surprised that the result of election was obvious at 8.00 pm, namely, just after the deadline of election. Before going to bed, I confirmed tomorrow's schedule of mine to go to Kyoto (RIMS) after the JPS (Japan Physical Society). Then I checked my presentation file, and it turned out that there are some mistakes on the transparency and I have to reprint them by my portable printer tomorrow.

5th day of Seattle'05 week 3

2005-08-13T14:59:00.000+09:00

Gaitsgory's talk was entitles as "(Overview of the work of Beilinson and Drinfeld)". His talk was on the D-schemes, hotizontal sections, critical levels, formal punctured disc, Hecke eigensheaves, and Beilinson-Bernstein. I asked Gaitsgory about the equivalence of jet schemes and n-jets of motivic integration. Loeser was on cell decompositions theorem of Denef-Pas, Tarski' theorem, theorem of Cluckers-Loeser, or such kind of model theoretical stuffs. Miles Reid talked during 12:00 to 12:20 about the history from 31 years ago in Arkaida (a name of village) near Vancouver. His story included Deligne, Griffiths, Dixon and football on TV. And then Deligne-Mumford, computational algebra, MacPherson, moduli of curves, 3-folds of Mori, and Kawamata. Something like music school was stated, and he said they had only tiny black boards and dispatched to buy a bigger black board.

I selected Mirkovic, Bloch, and Kaledin. Milkovic was on unramified global geometric conjecture (Beilinson and Drinfeld), loop Grassmannians (Drinfeld, Ginzburg, Lustzig, Vilonen), affine flag vaeirty of G and dual{G} (Bezrukavnikov,Arkhipov,Ginzburg), application to Lie algebras for p>0, exotic coherent sheaves on the tangent bundles and flag variety, and Springer fibers (Bezrukavnikov, Milkovic, and Rumynin). Bloch was on the motives for graphs, which was similar to the Feynmann diagrams of "Schwinger trick". The rests were on Kontsevich conjecture & motives associated ot graph hypersurface, and disproof by Belkale-Brosaa of Kontsevich conjecture (Hopf algebras for renormalization and for mixed Tate motives). Kaledin was again on the deformation of structure sheaf for flag varieties. During the afternoon session, I could talk to Mirkovic and he said he knew some of the physics works. I show the poster presentation and my drafts of research paper to Mirkovic and Gaitsgory.

Today was the end of my long trip to Canada (Toronto) and America (Seattle).

4th day of Seattle'05 week 3

2005-08-11T20:31:00.000+09:00

Gaitsgory was on the central extension of loop algebras, ind-scheme (D-modules for the affine Grassmann), criticality, and Beilinson-Bernstein for the flag variety. Loeser was on motivic Milnor fiber, starting from the theorem of Denef-Loeser, monodromy conjecture, Steenbrink's mixed Hodge structure, and motivic zeta function (and a little on Voevodsky?). At the last of morning session, one organizer talked about some history of the Summer Institute in Algebraic Geometry, which started at 1954. It was after the WW2 that Kunihiko Kodaira was invited to the IAS by Spencer (Kodaira-Spencer theory and Spencer-Zariski) and he mantioned something like Seminar Bourbaki, Thom's paper, the lecture by Hironaka on resolution of singularity (just sketch of proofs?), and Grothendieck, something about going to the beach.

The afternoon was Olsson, Vistoli, and Kaledin. Olsson was on the nonabelian p-adic Hodge theroy, which was similar to the recent work \pi_1 (\mathbb{P}^1 \ {0,1,\infty}, b) of Deligne-Terasoma from e'tale, de Rham to crystalline. It also started from Hain (mixed Hodge structure) and relative Malcev completion. The organization of talk was 1. abelian p-adic HT (Theorem of Fortaine Messng, Faltings, Tsuji, Niziol and definition of crystalline), 2. p-adic HT for \pi_1 (Tannaka duality), 3.Computing Lie algebras, 4.Idea of proofs (Toeu's theory: Higher Tannaka duality). Vistoli was on tame Artin (with Abramovich) stacks and he said he thought nobody would come for the title of Artin stacks. It was a generalization of "tame"-ness to Artin stacks, which is defined as the order of the action group to the Deligne-Mumford stack is prime to the char k, where k is an algebraically closed field. He then talked about the linearly reductive finite group schemes, relation of tameness to properness (Theorem of Abramonich-Corti-Vistoli), and the final remark on a problem: "Is there a modular description of the closure?"Kaledin was on Fedosov quantization in X: smooth over k: positive characteristic p, especially for p>2. The details were on the Poisson brackets (Poisson manifolds and Frobenius structure), central quantization, Frobenius-constant quotient, (restricted) Poisson structure, canonical filgration, and extension of Poisson structure to Frobenius quantization, and the Azumaya algebras.

At the BBQ, I heard that there is an international conference: "Algebraic geometry and beyond" in Kyoto in the middle of December. Then I went to the drop-in center to ask Arinkin about the paper of Drinfeld (2003).

3rd day of Seattle'05 week 3

2005-08-10T20:22:00.000+09:00

Today I heard only the talk of Gaitsgory and skipped the rest (Loeser and Conrad). His talk was on 1. the classical local Langlands correspondence, 2. geometric Langlands, and 3. abelian categories over stacks. First one was on the class field theory of Takagi and Weyl group (rather than Galois group). Second one was on the geometric representation of loop group (probably the affine D-module). Third was something on the base change. I could asked to Gaitsgory about the Beilinson-Drinfeld and local / global geometric Langlands personally.

2nd day of Seattle'05 week 3

2005-08-09T17:44:00.000+09:00

Griffiths' talk was based on the recent book of AMS 157 with Green. It has a background of Spencer Bloch on Chow groups, Duisequx series, Mumford (infinite dimensionality), and Bloch-Suslin. Conrad wore a T-shirt with a brief proof of Fermat-Wiles-Taylor last theorem proved 10 years ago when I was a high school student. His talk was on an elementary explanation of modularity, p-adic Galois representations, and Hecke rings.

Bondal, Hain, and Kaledin were what I chose today. Bondal was on the derived category of toric varieties with a background of homological mirror symmetry with superpotential for Landau-Ginzburg. I asked him whether we need the DERIVED Fukaya category, but he said the Fukaya category is already a triangulated category. He seemed not to use symplectic sides, so that it was not a serious problem anyway. He also mentioned that the SCFT before topological twists does not have a categorical formulation yet. The method was essentially exceptional collections for the toric actions. Hain was on the elliptic cohomology theory: stable homotopy theory, topological modular forms, which was done by Landweber and Hopkins et.al. Kaledin was on the first lecture of his series of three talks. It is essentially closed string noncommutative algebraic geometry. It started from 3-dimensional McKay correspondence by Miles Reid (1997), the conjecture solved by Bridgeland-King-Reid 1999 and its generalization by Bezrukavnikiov to any dimension, any resolution, and symplective group G contained in Sp(V), and then Bridgeland-Van den Bergh. p-adic version was also mentioned with a question from the audience on the recent work of Kontsevich on p-adic D-modules and Jacobian conjecture.

From 7:30, we had an extended talk by Miles Reid on K3s and Fano 3-folds, which was postponed last week. He reviewed the Mori category of Q-factorial terminal singularity, the Hilbert series (Syzyny), the orbifold Riemann-Roch theorem, and the Gorenstein rings. Half of his talk was done by transparency (OHP) and available at his website, where we can try his computer programme of Fano data base with Type I projection / unprojection and Tom / Jerry.

1st day of Seattle'05 week 3

2005-08-09T06:40:00.000+09:00

I heard the talks of Griffiths and Loeser for the morning session. Griffiths was on the Hodge cycles, generalized Hodge conjectures, Bloch-Beilinson conjecture, and , after assuming GHC and BB, he thought of some problems arising from, for example, the codimension 2 case. He finished his talk 15 minutes before the schedule. Loeser was an introduction to the definition and history of motivic integration from the birational viewpoint of Denef-Loeser (1987) and Batyrev (1995). I could talk to him a little after the presentation.

For the afternoon session, I chose Arinkin, Hoboush, and Nadler. Arinkin was on the quantum Liouville theory from the (polarized) deformation quantization and some problems of quantum Hitchin system of affine curve (fibers not connected, not smooth, no Lagrangian sections). Hoboush was on representation theroy for flag varieties including the Kazhdan-Lusztig conjecture. Nadler was on the perverse sheaves and introduction of the notion of tilting and some application to G=GL_2 and X=P^1 constructing the tiltings via Morse theory.

At night, I summarized the article of Drinfeld (2003) at the terrace to understand his notion of closed string heterotic CFT.

Weekend holiday

2005-08-06T16:13:00.000+09:00

On Saturday, I took a strong nup until 4:00 pm to be refleshed. I went to the Ave to take two pieces of Pizzas with Coke and then to the Starbacks to write an e-mail. Sunday was also a good holiday without going out and just reviewing the materials last week.

5th day of Seattle'05 week 2

2005-08-05T16:13:00.000+09:00

Morning session was performed by V.V.Shokurov and C.Voison. Sokorov's slides were too small to read so that I can not follow all. His talk consists of 1.Flips and flops, 2.Functional algebras, 3.Reductions (char=0) following the book of Ambro, Corti, Fujino, Mckernan and Takagi (Oxford). The first was on the existence and uniqueness of (log) flips and MMP for 4-folds. The second was on the canonical embedding as well as mobile and characteristic system, and discrepancies / FGA conjectures. The third was on flliping algebras. Voison was on the examples of Kaehler manifolds which cannot be polarized utilizing torus / Kummer surfaces. Her final remark was on Tsunoda-Campana's question.

Afternoon sesson, I went to hear the talks of Yum-Tong Siu, Lev Borisov, and Mikhail Kapranov. The talk of Siu was on the techniques towards the conjecture of finite generation of canonical rings; from the viewpoint of complex analysis of several variables such as Skoda's estimate, irreducible Lelong sets, finiteness of Lelong numbers, Shokorov's theorem, Pemailly's observation, and Fujita's conjecture. Borisov started from very elementary level of cones in connection with Hartshorne, Griffith-Harris, and string theory (Batyrev: normal curve). The original work was on non-normal toric varieties and Eisenbud-Goto conjecture. Kapranov was on the definition of formal loop space in Zariski topology (locally compact ind-schemes) and its application to chiral de Rham complex (Kapranov-Vasserot) and small quantum cohomolgoy (Arkhipov-Kapranov) with a few comments on Beilinson-Drinfeld chiral / factorization algebras and J-function of Iritani. I could talk to him personally a little after his talk on possible application towards non-toric target space other than flag manifolds, which I cannot wite here.

4th day of Seattle'05 week 2 -- Temporary update

2005-08-04T16:12:00.000+09:00

Baseball game to the Safeco Field

2005-07-31T16:10:00.000+09:00

I went with 4 people to the Sefeco Field at the right edge of downtown. To be more precise, we meet together at 12:30 in front of the left gate. Only I have reserved the seat so that, when they bought the ticket, it was already 13:05, that is, the time the game starts.

The game itself was interesting because we could see many hits and scores with a fine play by Ichiro. However, he had no hits today although he did 2 homeruns yesterday. The last batting of Ichiro at the bottom of 9th was especially exciting. The audience stands and shouted "Ichiro-Ichiro". If he had a timely hit, it was "Sayonara" (come-from-behind), but he couldn't.

We went to the international areas to take the dinner of soup and they let us come back to the doom by their car.

Weekend trip to the downtown, Temporary update

2005-07-30T16:09:00.000+09:00

Albert Biestadt (American 1830-1902), Puget Sound on the Pacific Coast, 1870.

5th day of Seattle'05 week 1

2005-07-29T16:08:00.001+09:00

Pandharipande was on the old topic of GW/DT correspondence including relative GW / DT. Hori was extensive including topological open string theory (with short comment on the hetetoric string), homological mirror symmetry with dg category language / Fukaya cateory (FOOO), RR charge, Uhlenbeck compactification of bubbles, and Elementary T-duality / SYZ.

Afternoon sessions, I took Arcara, Y.Ruan, and Jun Li. Arcara was on the moduli space in the derived category of a K3 surface including a review of (semi-)stable bundles, Harder-Narasimhan filtration, Bridgeland's sense of alpha-stability and central charges, tilting, and Mukai flops (Bertram). Y.Ruan changed his title from "Landau-Ginzburg Gromov-Witten theory" to "Twisted K-theory and its product". It is the so-called Chen-Ruan orbifold cohomology or twisted K-theory of Freed-Hopkins-Teleman. The 1st main focus was how we can modify the native definition of exteritor product in order that they have the associativity incorpolaring the obstruction class. The 2nd was on the transgression map, multiplicative property, and Pontrajgin product. Jun Li was on "Dimension zero Donaldson-Thomas invariants"; a trial on defining Gromov-Witten invariants in terms of stable sheaves on X = 3-fold target spaces. However, his talk included the case of X=C smooth curve, X=smooth alg. surface (with some speculation). When it came to the story on the 3-fold case, he talked about the Gopakumar-Vafa and Donaldson-Thomas invariants with proof of MNOP [Maulik-Nekrasov-Okounkov-Pandharipande] conjecture utilizing some standard facts of Hilbert scheme of points. How do you derive the Hilbert scheme of points from the Ziv Ran space of Beilinson-Drinfeld chiral algebras? After his talk, I asked him several questions personally.

After the lecture, we took lunch at the terrace of MacMahon with some friends. I caught a cold and so that I could not go, but some people left the Hall at the evening or at the next morning to the Olympic National Park.

4th day of Seattle'05 week 1

2005-07-28T16:27:00.000+09:00

Okounkov was on the equivariant localization in Donaldson-Thomas theory. It was something like melting crystals. The closing was something like this: What about non-toric case? Not a rhetorical question.

Huybrechts wes on "twisted"/ polarized K3 surfaces, in the same sense of Yoshioka. It is a generalization of the global Torelli theorem of Mukai and Orlov. Katzarkov was on generalized Hodge structures (GHS) & homological mirror symmetry, using the white board. The question was "Are there examples of 4 dimensional simply connected homeomorphic symplectic manifolds X1, X2 such that GW(X1) = GW (X2) but not X1 \cong X2". He talked some definitions and exmaples of genux two curves and del Pezzo surfaces. The final theorem was "GHS forms a Tannakian category". Teleman was on the quantum Hitchin systems (and some topological field theories). I was first surprised that the number of audience was 18 before the lecture starts, although it became about 35. The talk started from some history like Atiyah-Bott, but reached Beilinson-Drinfeld chiral algebras (D-modules), chiral de Rham complex, Verlinde formula for WZW model (flag variety or affine D-module), the equivariant GW theory. "Moral of understanding so far" was "K-theory better than cohomology", "non-abelian localization", Teleman-Woodward theorem, and Newstead conjecture. I asked whether he knew about the recent work of Drinfeld (2003) personally.

After the BBQ (6-8pm), I talked to Okada and Iritani for about 1 hour and went to a Chinese restaurant.

3rd day of Seattle'05 week 1

2005-07-28T15:58:00.000+09:00

Today we had only the moning session. Pandharipande talked the obstruction theory, the Hodge bundle, symplectic approaches / creation-annihilation operator representation (Frobenius manifolds), GW-SW for surfaces (Taubes: minimal surfaces), and some conjectures. Seidel was on Karoubi completion, Picard-Lefschetz and the fact that the fuctor between both derived categories induces a subcategory. He also mentioned some reviews like dg-category as A_{\infty} category or new claims including the isotopy invariance of Floer cohomology. Hori illustrated the Landau-Ginzburg (linear sigma model) phase, some N=2 supersymmetry algebras, and mirror symmetry for Fano manifolds (Hori-Vafa) from vortex configuration.

2nd day of Seattle'05 week 1

2005-07-27T14:35:00.000+09:00

I skipped the morning talk because I was a little tired. Seidel's talk should be the definition of Fukaya A^{\infty} category, but I am not sure whether he talked about the Hamiltonian deformation and the completion of Fukaya category by coisotropic manifolds. During the lunch time, I received the parcel from Amazon.com.

Afternoon session was busy; Auroux (Sieg Hall) - Hausel (Elec. Eng.) - Wei-Dong Ruan (Mech. Eng.). Auroux was on the recent work with Katzrkov-Orlov on the homological mirror symmetry for dP_n (n=0,1,\cdtos,9) as a generalization of Seidel's work on CP^2. He first review the HMS (Homological Mirror Symmetry) conjecture and the definition of Fukaya category for Calabi-Yaus. Then the story came to the Fano / Landau-Ginzburg mirror symmetry ulitizing the Lefschetz fibration of disk Lagrangian cycles, which vanished at the critical points with some deformation by B-fields and checks on the areas of vanishing cycles. Hausel was on explicit calculations on the H polynonial for mixed Hodge structures utilizing some non-Abelian Hodge theory (Riemann-Hilbert map) for GL(N) and some finite field harmonic analysis. Wei-Dong Ruan was again on the Homological Mirror, but a little more review-oriented talk. He derived the HMS from SYZ from local picture (cpx=log structure) and global structure(Fourier-Mukai). The new work is on the construction of sheaves as an extended work of Seidel.

At 8 pm, about 12 graduate students came to the lecture room and one of them talked about his recent interest on the "deformation of singularities and variations of GIT quotients". It started from the history of 70s-80s on the low modality (parameters for defining equation): simple singularity of ADE from the viewpoints of Dynkin diagrams including the theorem of Looijenga as well as unimodal singularity. Then he tried to tie to the Hodge theory and more complicated singularities for the GIT construction of parameter space. The conclusion was the dependance of GIT on the line bundle and Thaddeux-Dulgachev-Hu works on the case Picard = Z^2 and the slope of lattice denotes how complicated the singularity is.

1st day of Seattle'05 week 1

2005-07-26T13:44:00.000+09:00

The institute (conference?) started from Pandharipande on GW in low dimensions (0,1,2,3). His talk was on Kontsevich's combinatorial model, the Comb model, the cohomology structure of low genera, and the topological ring. Next was Hori on "Mirror symmetry" starting with an applaud. He reviewed some basic history of Mirror symmetry from the viewpoint of physical strings (non-topological twisted superstrings), but including the Mirror symmetry for Landua-Ginzburg model. His resume on the formal website is not complete, but will include the homological mirror symmetry. The audience's question was whether there is a physical interpretation of Deligne-Mumford compactification and he answered that "To be honest, I do not know it but it works. And other physicists may say another thing." Maybe T.Eguchi once said in the lunch seminar that it was related to the worldsheet instantons, but I am not confident.

For the afternoon session, I heard the talked by Nakajima, Markman, and Donagi. Nakajima was on the works with Yoshioka in K-theoretical interpretation of Donaldson theory (rather than homological theoretic) and rigorous treatment of the work of Nekasov. Markman started from the Mukai paring and Mukai vectors on K3 surfaces, and the work of Yoshioka. We can attach to each orthogonal complementary space of Mukai vector (with some conditions), a monodromy invariant related to a (weighted) Hodge structure / Hodge isometry. The theorem states on the functor Mon: \cal{G}(groupoind) \to Algebras, which consists of two groupoinds of deformation of surface and equivalence of derived categories. Donagi was on the recent work with Dijkgraaf et.al. on the "geometric transition and integrable systems". I was not sure whether he showed the large N duality as a categorical equivalence. Also I was inspired by the statement that the Mukai integrable system degenerates to the Hitchin system.

After a short supper with Okada, Iritani, and Shimizu, we had the reception at the veranda of Hagget Hall at 7-9 pm, where I talked to Iritani, Okada, Lotte, Ben-Zvi, and Donagi. Afterwards I went to the same Indian restaurant as I ate with Bryan et.al last week. It was 11 pm when I came back to the doom and I wrote some abstract mail to the "informal semiar" in some of GSWW members.

Encounter with Japanese participants -- registration of Seattle'05

2005-07-25T07:31:00.000+09:00

After the breakfast and before 9 am, I had to check-in and check-out to another room because the GSWW and the Seattle'05 are organized by different people.

My current room is in the same block as other two Japanese people including Yuji Shimizu. I first talked to him in English but he found out that we both are Japanese. We three left the McMahon during the bed making time, and with another graduate student Okada went to the Student Union, the Mechanical Engineering Buindling (where the registartion will take place in 5-9pm), and the Univeristy Way. In addition to the lunch at the same Thai restrant as Okada and I ate last week, we went to the University Bookstore, the drug store, and the convinient store.
We have a nice scenary of Elliot Bay from the veranda and the "famous" bridge is especially beautiful.

After the registration, where I revised "Makota Sakurai" to "Makoto Sakurai", we met many Japanese teachers including Kawamata and Nakajima. We (9 people) took dinner at the downstairs of McMahon and I saw Iritani there.

Weekend trip to downtown (temporary update)

2005-07-24T09:04:00.000+09:00

Today we (Peter, Okada and I) went to the downtown by bus (line 71-73) because the Seattle Mariners does not have home games until next Monday. The first visit was the bookstore where I bought a map and "Guns, germs, and steel". Then we went to the farmers' market. I again bought a magcup. There was the Elliot Bay (with many seagulls) close to the market slace and so that we went there and take some pictures.

It was about 3.40 and we dicided to go back to the University because we got tired. Here is the picture of Seattle City although we didn't go to the Space Needle nor saw the famous beautiful night spot.

We went strait to the Spike Street through a garden and there was the moving objet in front of the Seattle Museum (linked in the below photo).

5th day of GSWW

2005-07-23T09:04:00.000+09:00

Morning lectures were Geometric Langlands (Ben-Zvi): "Algebraic geometer's harmonic analysis" or "Fourier transformation between Schwartz distributions", Rational points (Hassett), and combinational positivity (Miller).

Afternoon session was a review work on the pseudo-modular form of Bryan-Leung on the blow-ups of CP^2. This is the continuation of the group work of yesterday; division into the each nodal fibration, multiplication counting, and the Cremona relation, and the Russian notation of Young tableau. It was similar to the Hilbert scheme for K3 surface and instanton calculus, but I was not sure about the direct relation between them.

After the reception, we went to a Indian restaurant with Bryan and Miller et.al. I said to Bryan that Ooguri cited Bryan-Pandharipande at the Strings 2005.

4th day of GSWW

2005-07-22T09:03:00.000+09:00

We started from the homework on the large quantum cohomlogy for CP^2, Goettsche-Pandharipande, and the Kontsevich formula. Then we studied 2 intersection generic cubics F_0 and F_1 which define a linear combinationF=s F_0 + t F_1 and this pencil therefore introduce an elliptic fibration. The canonical class for "del Pezzo 9" is -K_X = 3H-E_1-...-E_9 which has 12 nodal curve fibration over the exceptional divisor E_9.

Then we moved on to the classification of the components consisting of \beta = S + bF. This is essentially a study on the universal covering space (or multiplication and orientation) or a nodal curve fibration.

3rd day of GSWW

2005-07-21T15:06:00.000+09:00

Morning lessons were interesting, which consists of Roth (Hilbert schemes: decomposition of Hilbert polynomials), Mantovan (Langlands' and Shimura's abelian varieties) , and Mckernan (Birational classification of surfaces: Mori cone and minimal model programme). The last one was done by Red Hat slides so that I could not take notes sufficiently. At the coffee break in the morning session, we took a photo.

At the workgroup time, we difined small / large quantum cohomolgy as well as Gromov-Witten invariants (virtual fundamental class) although we did not do the virtual localization yet. Actually we did the divisor equation, the point axiom, the splitting axiom, and the associativity (Witten-Dijkgraaf-Verlinde-Verlinde eq.), but we have not yet done the proof of the associativity by potential (partition function), which is a homework in the case X=CP^2.

Those who once did the closed GW for CP^n worked on the small quantum cohomology ring for del Pezzo surfaces, which I suggested to do because it is related to the work of Orlov el.al. / Ueda on the mirror symmety for del Pezzo surfaces. I printed the classic works of Goettsche-Pandharipande and Eguchi-Hori-Xiong during the coffee break. I have to check up some definitions like flatness bacause I learned them long time ago and do not remember well.

2nd day of GSWW

2005-07-20T13:08:00.000+09:00

I had the morning lectures of Garber(stacks), Donagi(hetetoric standard model), and Gibuey(moduli space). I went to the math library to confirm something about champs algebraique (Laumon) and Langlands program(Asterisque).

At the workgroup, we had some explicit homological calculations with moduli space of stable curves. I went to a Japanese-Korean restrant for dinner, but it was not so delicious. There was a music concert outside another hall when I return back to the McMahon.

1st day of GSWW

2005-07-19T14:48:00.000+09:00

I waked up at 6 am and took the breakfast using my pre-paid card. My work group will be group C (Bryan) for one week, which will talk about closed / open Gromov-Witten invariants in the A-model side in the Savery Hall.

However, before that, we had three elementary lectures by Allocok (Chevalley groups), Jim Bryan (algebro-geometry and physics), and Mustata (motivic integration of Kontsevich, Denef-Loeser) in the same building. I could not use my wireless LAN card at the lecture room because it demands the student ID. The lecture of Bryan was almost what I studied until last summer. And there were some handouts.

I went the general library and the communication center for photo copy cards and printing papers. I also went to the math library, but it was closed (opens during 9 - 17 pm).

Afterwards, I went to the University Way (the Ave) again for supper and I spent about 2 hours for reviewing today's lectures. I look forward to seeing the library for some literature.

Reception of GSWW at McMahon Hall

2005-07-18T08:20:00.000+09:00

I left Toronto early around 5 pm -- the weather was terrible and I was saturate with the rain. I took the shuttle bus in stead of TTC because one of my friends recommended me to do so.

After two short flights, I arrived at Seattle. The security was not tight. In deed, there was no checks at the buggage claim. Rather, there was a internal train from the flight gate to the outside.

We have a short (30-40 minutes) drive from Seattle to the University of Washington. I will visit in the McMahon Hall for the next one month. The scenary is beautiful and the air is clean -- more "transparent" than Toronto.

Then we had a reception from 6 to 8 pm. We have non-alcoholic drinks with some cheese. I talked to Bryan and Lotte and went with a Japanese participant to a Thai restaurant. I didn't know it but my room mate came late at night.

6th day of Strings 2005

2005-07-16T22:32:00.000+09:00

Shoke Sen: "Entropy is the legendgre transform of Lagrangian density". Yau wrote on the transparency during the presentation.

After the presentation, we went to the CN (Canadian National) tower: 553 m tall, until 4 pm and then to the Starbucks for tea. It was with other three students in the U.S. Click the photo above for the movie of elevator.

5th day of Strings 2005

2005-07-16T10:12:00.000+09:00

Today was on matrix models, which included the talk of Takayanagi. The formal website of Strings 2005 is preparing to upload the audio and slide files.

At noon, there was an entertainment by several participants of strings 2005. It was like street music -- singing "Strings 2005", and somebody started to throw coins to the guitar case. It was amusing. Then we went with A.Sen et.al. to a Chinese restraurant for lunch, but it was 1 pm when we started to eat, so that we had to hurry to come back to the auditorium.

After 4 pm, I had to clean up my poster at the hall. I sent my poster to Japan at the postoffice in the student office. After that , I went to the mathematics library to confirm something about the central extension of loop groups in the Hitchin systems. However, it was already closed. After that, I went with my laboratory members to "little Italy" for dinner, whose name was Amsterdam (why?).