Abstract. In this talk, I will discuss Toen's theory of affine stacks and their relation to unipotent group schemes. I will also briefly talk about how this is applied in studying certain unipotent homotopy group schemes associated with algebraic varieties studied in joint work with Emanuel Reinecke.

Abstract. Suppose X is a smooth projective variety, E and F are vector bundles on X, and M: E —> F is a map of vector bundles. More concretely, M defines a family of matrices {M_x}, parametrized by the points x of the variety X. For a positive integer k, we can define the kth determinantal variety of M to be the locus of points x in X for which the linear map M_x has rank at most k. Such varieties give some of the simplest examples of subvarieties of X which are not complete intersections. Determinantal varieties are almost always singular, however there are two natural desingularizations, called the PAX and PAXY models, defined using basic concepts from linear algebra. It is natural to ask what the relationship is between these two resolutions: How do their cohomology rings compare? their derived categories? etc… In this talk I will describe a beautiful correspondence, conjectured by physicists, between the number of curves in each of the two resolutions. I will give a sketch of the proof and connect this correspondence to mutations of quiver varieties. This is based on joint work with Nathan Priddis and Yaoxiong Wen.

Abstract. A Landau—Ginzburg model (Y, w) consists of a smooth complex variety Y together with a function w: Y—> C. Curve counting invariants have recently been defined for a large class Landau—Ginzburg models. The theory goes by the name of the gauged linear sigma model, and generalizes, among other things, the Gromov—Witten theory of hypersurfaces. In this talk I will give a gentle introduction to the theory, focusing on examples.

Abstract. This talk is based on a joint work with Charles De Clercq. The main result is a classification theorem for some Chow motives with finite coefficients, which applies, notably, to motives of projective homogeneous varieties under some semi-simple algebraic groups. The result uses a new invariant, the Tate trace of a motive, defined as a pure Tate summand of maximal rank. If time permits, the notion of critical variety will be presented, as an application of our result.

Abstract. A Kazhdan-Lusztig atlas on a stratified variety is a structure that models the stratification of the variety on that of Schubert varieties. We will describe a classification of Kazhdan-Lusztig atlases on toric surfaces. We will touch on topics such as pizzas, nutrition, toppings and braids.

Abstract. The Hodge numbers of a Calabi-Yau threefold X are determined by the two numbers h^{1,1}(X) and h^{1,2}(X) which can be interpreted respectively as the dimensions of the space of Kähler forms and complex deformations respectively. We construct four new examples X_N, where N \in {5,6,8,9}, of rigid Calabi-Yau threefolds (h^{2,1}=0) with Picard number 4 (h^{1,1}=4). These manifolds are of “banana type” and have among the smallest known values for Calabi-Yau Hodge numbers. We (partially) compute the Donaldson-Thomas partition functions of these manifolds and in particular show that the associated genus g Gromov-Witten potential is given by a weight 2g-2 Siegel paramodular form of index N. These manifolds are also modular in the arithmetic sense: there is a weight 4 modular form whose Fourier coefficients are obtained by counting points over finite fields on X_N. We compute this form and observe that it is the unique cusp form of weight 4 and index N. This is joint work with Stephen Pietromonaco.

Abstract. (Based on joint work with Ruofan Jiang) The Hilbert scheme of points on a variety X parametrizes 0-dimensional quotients of the structure sheaf. When X is a planar singular curve, its enumerative invariants are closely related to mathematical physics, knot theory and combinatorics. In this talk, we investigate two analogous moduli spaces, one being a direct generalization of the Hilbert scheme. Our results reveal their surprising relations to Hall polynomials, matrix equations, modular forms, etc.

Abstract. For X a smooth variety or Deligne-Mumford stack, the quantum cohomology ring QH^*(X) is a deformation of the usual cohomology ring H*(X), where the product structure is modified to incorporate quantum corrections. These correction terms are defined using Gromov-Witten invariants. For a GIT quotient V//G, the cohomology ring H*(V//G) also has the structure of a H^*(G)-module. In this work, we use quasimap invariants with light points and a modified version of the WDVV equation to define a quantum deformation of this H*(G)-module structure. Using localization, we explicitly compute this structure for the Hirzebruch surface of type 2. We conjecture that this new quantum module structure is isomorphic to the Batyrev ring when the target is a semipositive smooth projective toric variety.

Abstract. Given a finite group G and a faithful representation G -> GL(V) where V is a finite dimensional k-vector space, the essential dimension of G over k is the minimal dimension dim(Y) of an irreducible faithful G-variety Y with a dominant G-equivariant rational map V -> Y. The essential dimension of a finite group G can be viewed as the minimum number of independent objects needed to define roots of a "generic" polynomial with Galois group G. It is believed that the essential dimension of the symmetric group Sn is n-3 for n>=5, at least in characteristic 0. We show that the essential dimension of Sn can be n-4 in prime characteristic.

Abstract. A minimal free resolution of an ideal produces a sequence of integers which are called “Betti numbers". It's usually computational expansive to calculate the Betti numbers of a monomial ideal(a ideal generated by monomials). In this talk, I will introduce a constructive method to add generators to a monomial ideal $I$ while preserving most Betti numbers of $I$. I will start with explaining how one can label the generators of the monomial ideal to the vertices of a simplicial complex and converts it to a free resolution of $S/I$ under certain condition. Then, by adding a generator (monomial $m$) to the monomial ideal (adding a vertex to the labeled simplicial complex), the new labeled simplicial complex can convert to a free resolution of $S/(I+(m))$ under certain condition. Next, I will conclude how the Betti numbers of $I$ and $I+(m)$ relate. Finally, with this method, and by a sequence of operations, we can create infinitely many monomial ideals with arbitrarily many generators which have similar Betti numbers as $I$. And we can also create infinitely many monomial ideals with the same number generators as $I$ which have exactly the same Betti numbers as $I$.

Abstract. Introduced in conjectural form 35 years ago and put on a solid foundation only 10 years ago, Arthur packets now play an inportant role in the Langlands program. In this talk I will describe some fundamental properties of local Arthur packets, review a conjectural generalization due to Adams, Barbash and Vogan, made 30 years ago, and explain the current status of this conjecture.