Title and Abstract

Gwyn Bellamy

Title: Towards classification of symplectic resolutions

Abstract: In this talk I will describe progress on a program, joint with Schedler, to classify those symplectic quotient singularities that admit symplectic resolutions I will explain how one can use the representation theory of symplectic reflection algebras in order to do this. I will also explain how one can use these algebras, combined with general theory developed by Namikawa, to compute the ample and movable cones of the minimal models of these quotient singularities. As a consequence, one can explicitly count the number of minimal models.

Huayi Chen

Title: Convex bodies of divisors: an approach of function field arithmetic

Abstract: The works of Okounkov, Lazasfeld-Mustata and Kaveh-Khovanskii permit to associate to each big Cartier divisor on a projective variety a convex body whose measure identifies with the volume of the Cartier divisor. However, their constructions require the existence of a regular rational point in the projective variety, which implies that the variety is geometrically integral. In this talk, I will explain a new construction of convex bodies associated to big Cartier divisors on a general projective variety, by using Arakelov geometry in the function field setting.

Hideaki Ikoma

Title: Adelic Cartier divisors with base conditions and the Bonnesen–Diskant–type inequalities

Abstract: We introduce positivity notions for pairs of adelic R- Cartier divisors and base conditions, and study fundamental properties of the arithmetic volumes associated to such pairs. As a main result, we show that the Gâteaux derivatives of the arithmetic volume function at big pairs along the directions of adelic R-Cartier divisors are given by suitable arithmetic positive intersection numbers.

Young-Hoon Kiem

Title: Critical virtual manifolds and Donaldson-Thomas invariants

Abstract: A critical virtual manifold is an analytic space obtained by gluing critical loci of holomorphic functions on finite dimensional complex manifolds. I will talk about a joint work with Jun Li (arXiv:1212.6444) in which we introduced the notion of a critical virtual manifold and proved the following.

1. Moduli spaces of stable sheaves on Calabi-Yau 3-folds are critical virtual manifolds.

2. If a critical virtual manifold is orientable, the local perverse sheaves of vanishing cycles glue to a globally defined perverse sheaf which underlies a polarizable mixed Hodge module.

3. When a moduli space of stable sheaves admits a universal family, the critical virtual manifold structure is orientable.

4. The hypercohomology of the perverse sheaf on a moduli space of stable sheaves on a CalabiYau 3-fold provides us with a mathematical theory of Gopakumar-Vafa invariant.

5. Critical virtual manifolds are Joyce’s d-critical loci (arXiv:1304.4508) and vice versa.

Taro Kimura

Title: From Seiberg-Witten curve to quiver W-algebras

Abstract: Seiberg-Witten curve is an algebraic curve which characterizes the vacuum of supersymmetric gauge theory. We show that quantization of Seiberg-Witten curve for Γ-quiver gauge theory provides a generating current of W(Γ)-algebra in the free field realization. We also show that the gauge theory partition function is given as a correlator of the corresponding W(Γ)-algebra, which is equivalent to the AGT relation under the gauge/quiver (base/fibre; spectral) duality. This talk is based on a collaboration with V. Pestun [arxiv:1512.08533] [arxiv:1608.04651]. See also an overview article [arxiv:1612.07590].

Conan Leung

Title: Categorical Plucker formula and Homological Projective Duality

Abstract:

Vincent Maillot

Title: Arakelov Geometry : Old and New

Abstract: I will report on my 2010 proof (joint work with D. Roessler) of a weak version of a conjecture of Fang, Lu and Yoshikawa on the birational invariance of BCOV torsion. I will then describe new methods for computing holomorphic torsion in some interesting modular situations.

Keiji Oguiso

Title: Primitive birational automorphisms of projective manifolds

Abstract:After explaining background materials, I would like to show the following result:

in any dimension n> 2, there are smooth projective rational manifolds and Calabi-Yau manifolds of primitive birational automorphisms of first dynamical degree >1, and if dimension N is even or 3, then we also find smooth projective rational manifolds and Calabi-Yau manifold with primitive biregular automorphisms of positive entropy.

Yuji Odaka

Title: On Compactifying Moduli of Kahler-Einstein manifolds

Abstract: Many of classically studied varieties, such as curves, Calabi-Yau varieties ("in most general" sense), canonical models, many Fano manifolds are known to admit ``canonical" Kahler metrics, the so-called Kahler-Einstein metrics. We will start with discussing a part of unifying framework of moduli of such varieties, in particular towards algebra-geometric compactifications, and briefly introduce our recent developments. (If time permits, I will briefly mention relations with non-archimedean geometry and Arakelov geometry as well.) Latter half will be, in turn, to exhibit some ``weird looking" compactifications of the same moduli spaces - what I called ``tropical geometric compactifications". It is in connection with tropical geometry, inspired by the developments of the Strominger-Yau-Zaslow Mirror symmetry. In particular, I will explain about the explicit structures of such compactifications for M_g (curves case), A_g (abelian varieties case), and K3 surfaces case. The latter half is a joint work with Yoshiki Oshima. (Older/original version is available at 1406.7772).

Travis Schedler

Title: Towards classification of symplectic resolutions: quiver and character varieties and Higgs bundle moduli spaces

Abstract: I will explain joint work with Bellamy classifying which quiver varieties admit symplectic resolutions: roughly, that they are those that are products of symmetric powers of du Val singularities, varieties for indivisible dimension vectors, and one special type related to O'Grady's example (for which the resolution is not given by GIT). I will then explain how we apply the same technique to the analogous situation of character varieties (moduli of local systems on compact Riemann surfaces). Finally, I will mention work of my student Tirelli applying this to the analogous moduli spaces of Higgs bundles of degree zero, via Simpson's isosingularity theorem.

Yuji Tachikawa

Title: Instantons on ALE spaces and supersymmetric quantum field theories

Abstract: Supersymmetric quantum field theories closely related to instantons on ALE spaces will be discussed. They predict a certain mathematical property of the moduli spaces of instantons on ALE spaces, seemingly unknown in the mathematical side of the literature.

Yukinobu Toda

Title: Gopakumar-Vafa type invariants for Calabi-Yau 4-folds

Abstract: As an analogy of Gopakumar-Vafa conjecture for CY 3-folds, Klemm-Pandharipande proposed GV type invariants on CY 4-folds using GW theory and conjectured their integrality. In this talk, I propose a sheaf theoretical interpretation to these invariants using Donaldson-Thomas theory on CY 4-folds introduced by Cao-Leung and Borisov-Joyce. This is a joint work with Yalong Cao and Davesh Maulik.

Kenichi Yoshikawa

Title: BCOV invariant for Calabi-Yau orbifolds

Abstract:After the discovery of physicists Bershadsky-Cecotti-Ooguri-Vafa, Fang, Lu and I introduced a holomorphic torsion invariant of Calabi-Yau threefolds,

which we call BCOV invariant. For Calabi-Yau orbifolds of dimension three, it is possible to construct a similar invariant. Moreover, for Calabi-Yau orbifolds

satisfying certain additional condition, we can construct its twisted version. In the talk, I will explain the construction of these BCOV invariants for Calabi-Yau orbifolds and give their variational formulae on the moduli space (holomorphic anomaly equation). In some cases, I compare the BCOV invariants between Calabi-Yau orbifolds and their crepant resolutions.

Yutaka Yoshida

Title: Equivariant A-twisted GLSM and Calabi-Yau 3-folds in Grassmannians

Abstract: Recently, one parameter deformation of A-twisted gauged linear sigma model (GLSM), called equivariant A-twist was introduced by Closset, Cremonesi and Park. In my talk, I will explain applications of equivariant A-twisted GLSM to computations of genus zero instanton numbers of Calabi-Yau 3-folds in Grassmannians. My talk is based on arXiv:1602.02487 [hep-th] with Kazushi Ueda and arXiv:1607.08317 [math.AG] with Bumsig Kim, Jeongseok Oh and Kazushi Ueda.