Abstract: The Grothendieck ring of varieties is the target of a rich invariant associated to any algebraic variety which witnesses the interplay between geometric, topological and arithmetic properties of the variety. The motivic Hilbert zeta function is the generating series for classes in this ring associated to a certain compactification of the unordered configuration space, the Hilbert scheme of points, of a variety. In this talk I will discuss the behavior of the motivic Hilbert zeta function of a reduced curve with arbitrary singularities. For planar singularities, there is a large body of work detailing beautiful connections with enumerative geometry, representation theory and topology. I will discuss some possible extensions of this picture to non-planar curves.

Abstract: Gelfand-Cetlin systems are completely integrable systems on partial flag varieties constructed by Guillemin and Sternberg. The completely integrable systems are useful to produce Lagrangian submanifolds in flag varieties which are essential ingredients for studying SYZ mirror symmetry and Floer theory. In this talk, I will talk about the peculiarities of the systems such as isotropic property and smoothness of fibers and explain how to describe Gelfand-Cetlin polytopes and Lagrangian fibers in terms of combinatorics on ladder diagrams.

Abstract: (joint with I. Biswas and C. Garay López)

According to Plucker's formula, the total inflection of a linear series (L,V) on a complex algebraic curve C is fixed by numerical data, namely the degree of L and the dimension of V. Equipping C and (L,V) with compatible real structures, it is more interesting to ask about the total real inflection of (L,V). The topology of the real inflectionary locus depends in a nontrivial way on the topology of the real locus of C. We study this dependency when C is hyperelliptic and (L,V) is a complete series. We first use a nonarchimedean degeneration to relate the (real) inflection of complete series to the (real) inflection of incomplete series on elliptic curves; we then analyze the real loci of Wronskians along an elliptic curve, and formulate some conjectural quantitative estimates.

Abstract: The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. If a curve is embedded in projective space, it is natural to ask whether the gonality is related to the embedding. In my talk, I will discuss work with James Hotchkiss. Our main result is that, under mild degree hypotheses, the gonality of a complete intersection curve in projective space is computed by projection from a codimension 2 linear space, and any minimal degree branched covering of P^1 arises in this way.

Abstract: I will discuss recent work on constructing small quantum groups–also known as Frobenius-Lusztig kernels–at even roots of unity. In particular, for any simple Lie algebra g and even order q, we would like to associate a corresponding finite-dimensional, factorizable, ribbon (i.e. log-modular) quasi-Hopf algebra. The main issue here is that, for g=sl2 at any even root of unity, for example, naive construction of such quantum groups produce finite tensor categories which admit no braidings. Our investigation is motivated by conjectural relations between non-rational vertex operator algebras and such log-modular quantum groups, which I will also discuss.

Abstract: Introduced by Konno, hyperpolygon spaces are examples of Nakajima quiver varieties. The simplest of these is a noncompact complex surface admitting the structure of a gravitational instanton, and therefore fits nicely into the Kronheimer-Nakajima classification of complete ALE hyperkaehler 4-manifolds, which is a geometric realization of the McKay correspondence for finite subgroups of SU⁡(2). For more general hyperpolygon spaces, we can speculate on how this classification might be extended by studying the geometry of hyperpolygons at "infinity". This is ongoing work with Hartmut Weiss.

Abstract: The syzygies of an algebraic variety are the relations amongst its equations in projective space. They often encode subtle geometric properties of this variety, such as the presence of special secant spaces. In my talk, I would like to present this relation, focusing on the case of smooth curves and surfaces, and explain a natural connection to the theory of tautological bundles on the Hilbert scheme of points.

Abstract: We will consider a class of Calabi-Yau varieties given by cyclic branched covers of a fixed semi Fano manifold. The first prototype example goes back to Euler, Gauss and Legendre, who considered 2-fold covers of P^1 branched over 4 points. Two-fold covers of P^2 branched over 6 lines have been studied more recently by many authors, including Matsumoto, Sasaki, Yoshida and others, mainly from the viewpoint of their moduli spaces and their comparisons. I will outline a higher dimensional generalization from the viewpoint of mirror symmetry. We will introduce a new compactification of the moduli space cyclic covers, using the idea of ‘abelian gauge fixing’ and ‘fractional complete intersections’. This produces a moduli problem that is amenable to tools in toric geometry, particularly those that we have developed jointly in the mid-90’s with S. Hosono and S.-T. Yau in our study of toric Calabi-Yau complete intersections. In dimension 2, this construction gives rise to new and interesting identities of modular forms and mirror maps associated to certain K3 surfaces. The lecture is based on on-going joint work with S. Hosono, T.-J. Lee, H. Takagi, S.-T. Yau, and D. Zhang.

Abstract: We present an operadic approach to vertex algebra cohomology. A general construction associates a cohomology complex to a linear operad. In the case of the chiral operads we get a vertex algebra cohomology complex. Likewise, the associated graded of the chiral operad leads to a classical operad, which produces a Poisson vertex algebra cohomology complex. The latter is closely related to the variational Poisson cohomology.

Abstract: I will describe a special class of odd-dimensional sphere bundles over symplectic manifolds where the topological invariants of these sphere bundles are related to the symplectic invariants of the base manifolds. Such a relation gives a novel viewpoint to search for and understand symplectic invariants. This talk is based on a joint work with Hiro Tanaka and Jiawei Zhou.

Abstract: In the present talk, I will define a family of infinite-dimensional Lie algebras associated with a "continuum" analog of Kac--Moody Lie algebras. They depend on a topological version of the notion of the quiver. These Lie algebras have some peculiar properties: for example, they do not have simple roots and in the description of them in terms of generators and relations, only quadratic (!) Serre type relations appear. I will discuss also their quantizations, which go under the name of "continuum quantum groups". In particular, in the second part of the talk, I will focus on the case when the "topological quiver" is a circle: in this case, the continuum quantum group can be realized by means of the theory of classical Hall algebras. If time permits, I will discuss the representation theory of the continuum quantum group of the circle (in particular, the construction of the Fock space). This is based on joint works with Andrea Appel and Olivier Schiffmann.

Abstract: Let G be a simply connected complex linear algebraic group, and X be a smooth projective G-variety. In this talk, we will construct an action of the equivariant homology of the affine Grassmannian Gr_G on the G-equivariant quantum cohomology of X. The construction uses shift operators in quantum cohomologies. When G=SL⁡(2) and X is the projective line, this action is compatible with the Peterson isomorphism. Joint work with Alexander Braverman.

Abstract: In 1997, Kontsevich gave a universal solution to the "deformation quantization" problem in mathematical physics: starting from any Poisson manifold (the classical phase space), it produces a noncommutative algebra of quantum observables by deforming the ordinary multiplication of functions. His formula is an example of a Feynman expansion, involving an infinite sum over graphs, weighted by volume integrals on the moduli space of marked holomorphic disks. The precise values of these integrals are currently unknown. I will describe recent joint work with Banks and Panzer, in which we develop a theory of integration on these moduli spaces via suitable sheaves of polylogarithms, and use it to prove that Kontsevich's integrals evaluate to integer-linear combinations of special transcendental constants called multiple zeta values, yielding the first algorithm for their calculation.

Abstract: A generalized Kummer variety of dimension 2n is the fiber of the Albanese map from the Hilbert scheme of n+1 points on an abelian surface to the surface. We compute the monodromy group of a generalized Kummer variety via equivalences of derived categories of abelian surfaces. As an application we prove the Hodge conjecture for the generic abelian fourfold of Weil type with complex multiplication by an arbitrary imaginary quadratic number field K, but with trivial discriminant invariant. The latter result is inspired by a recent observation of O'Grady that the third intermediate Jacobians of smooth projective varieties of generalized Kummer deformation type form complete families of abelian fourfolds of Weil type.