Gaitsgory's talk was entitles as "(Overview of the work of Beilinson and Drinfeld)". His talk was on the D-schemes, horizontal 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 Arcadia (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).

## Saturday, August 13, 2005

## Thursday, August 11, 2005

### 4th day of Seattle'05 week 3

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).

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).

## Wednesday, August 10, 2005

### 3rd day of Seattle'05 week 3

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.

## Tuesday, August 09, 2005

### 2nd day of Seattle'05 week 3

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

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.

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.

## Saturday, August 06, 2005

### Weekend holiday

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.

## Friday, August 05, 2005

### 5th day of Seattle'05 week 2

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 session, 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 write here.

## Thursday, August 04, 2005

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