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).