If X is a Grassmannian (or an arbitrary homogeneous space) the 3-point, genus 0, Gromov-Witten invariants count rational curves of degree d satisfying certain incidence conditions - if this number is expected to be finite. If the number is infinite, Givental and Lee defined the K-theoretic Gromov-Witten invariants, which compute the sheaf Euler characteristic of the space of rational curves in question, embedded in Kontsevich's moduli space of stable maps. The resulting quantum cohomology theory - the quantum K-theory algebra - encodes the associativity relations satisfied by the K-theoretic Gromov-Witten invariants. In joint work with Anders Buch, we shown that the (equivariant) K-theoretic Gromov-Witten invariants for Grassmannians are equal to structure constants of the ordinary (equivariant) K-theory of certain two-step flag manifolds. We therefore extended - and also reproved - the "quantum=classical" phenomenon earlier discovered by Buch-Kresch-Tamvakis in the case of the usual Gromov-Witten invariants. Further, we obtained a Pieri and a Giambelli rule, which yield an effective algorithm to multiply any two classes in the quantum K algebra.

The usefulness of the various Riemann-Roch formulas as computational tools is well documented in literature. Grothendieck-Riemann-Roch is a commutative diagram relating pullback in K-theory to the pullback of associated Chow invariants for locally complete intersection (l.c.i.) morphisms. We extend this notion to quasi-smooth morphisms between derived schemes, this is the "derived" analog of l.c.i. morphisms and it encompasses relative perfect obstruction theories. We will concentrate on the naturality of the construction from the standpoint of pure intersection theory and how it interacts with the virtual Gysin homomorphism defined by Behrend-Fantechi. Time permitting we will discuss the relationship with existing formulas, i.e., Ciocan-Fonanine, Kapranov, Fantechi, and Goettsche.

Hodge structures on cohomology groups are fundamental invariants of algebraic varieties; they are parametrized by quotients $D/\Gamma$ of period domains by arithmetic groups. Except for a few very special cases, such quotients are never algebraic varieties, and this leads to many difficulties in the general theory. We explain how to partially remedy this situation by equipping $D/\Gamma$ with an o-minimal structure in which any period map is definable. The algebraicity of Hodge loci is an immediate consequence via a theorem of Peterzil--Starchenko. We further prove a general GAGA type theorem in the definable category, and deduce some finer algebraization results. This is joint work with Y. Brunebarbe, B. Klingler, and J.Tsimerman.

We use a problem in arithmetic dynamics as motivation to introduce new computational methods in algebraic number theory, as well as new techniques for studying quadratic points on algebraic curves.

We study smooth, projective varieties with trivial canonical sheaf. Properties of such varieties over the complex numbers will be recalled, especially in dimension 3 in the case that the first cohomology group is zero. We construct examples in positive characteristic which have quite different properties. This leads us to explore the notion of supersingularity and to pose some open questions.

Anders Björner characterized which finite, graded partially ordered sets (posets) are closure posets of finite, regular CW complexes, and he also observed that a finite, regular CW complex is homeomorphic to the order complex of its closure poset. One might therefore hope to use combinatorics to determine topological structure of stratified spaces by studying their closure posets; however, it is possible for two different CW complexes with very different topological structure to have the same closure poset if one of them is not regular. I will talk about a new criterion for determining whether a finite CW complex is regular (with respect to a choice of characteristic functions); this will involve a mixture of combinatorics and topology. Along the way, I will review the notions from topology and combinatorics we will need. Finally I will discuss an application: the proof of a conjecture of Fomin and Shapiro, a special case of which says that the Schubert cell decomposition of the totally nonnegative part of the space of upper triangular matrices with 1's on the diagonal is a regular CW complex homeomorphic to a ball.

We encode arbitrary finite impartial combinatorial games in terms of lattice points in rational convex polyhedra. Encodings provided by these lattice games can be made particularly efficient for octal games, which we generalize to squarefree games. These encompass all heap games in a natural setting where the Sprague-Grundy theorem for normal play manifests itself geometrically. We provide polynomial-time algorithms for computing strategies for lattice games provided that they have a certain algebraic structure, called an affine stratification.

We consider the problem of classifying all positive-definite integer-valued quadratic forms that represent all positive odd integers. Kaplansky considered this problem for ternary forms, giving a list of 23 candidates, and proving that 19 of those represent all positive odds. (Jagy later dealt with a 20th candidate.) Assuming that the remaining three forms represent all positive odds, we prove that an arbitrary, positive-definite quadratic form represents all positive odds if and only if it represents the odd numbers from 1 up to 451. This result is analogous to Bhargava and Hanke's celebrated 290-theorem. In addition, we prove that these three remaining ternaries represent all positive odd integers, assuming the Generalized Riemann Hypothesis. This result is made possible by a new analytic method for bounding the cusp constants of integer-valued quaternary quadratic forms $Q$ with fundamental discriminant. This method is based on the analytic properties of Rankin-Selberg $L$-functions, and we use it to prove that if $Q$ is a quaternary form with fundamental discriminant, the largest locally represented integer $n$ for which $Q(\vec{x}) = n$ has no integer solutions is $O(D^{2 + \epsilon})$.

R. Borcherds constructed a lifting from elliptic modular forms of weight $1-n/2$ to meromorphic modular forms on the orthogonal group $O(2,n)$. The lifted modular forms can be written as infinite products analogous to the Dedekind $\eta$-function (``Borcherds products''). Moreover, their divisors are always linear combinations of Heegner divisors; these are algebraic divisors that come from embedded quotients of $O(2,n-1)$. We address the natural question, whether every meromorphic modular form on $O(2,n)$, whose divisor is a linear combination of Heegner divisors, is a Borcherds product? We discuss some recent results that answer this question in the affirmative in a large class of cases.

The fundamental group of a topological space is usually defined in terms of homotopy classes of based loops. The group structure is given by composition of loops. If X is a complex algebraic variety, one has an underlying topological space, and hence a fundamental group. Hain showed that the nilpotent completion of the group ring of this topological fundamental group carries a mixed Hodge structure. Grothendieck defined a fundamental group for schemes defined over any field. Applying this to a complex algebraic variety, one obtains the profinite completion of the topological fundamental group. This group comes with a natural action of the absolute Galois group of the field of definition. The above indicates that varieties over fields of characteristic zero have two notions of fundamental group, armed with either a Galois action or a mixed Hodge structure. This is similar to the situation with homology and cohomology groups, where one has both an etale and Betti version carrying Galois actions and Hodge structures. An important guiding principle is that both of these versions of homology and cohomology should come from an underlying ``motivic'' theory. This is a homology and cohomology theory for algebraic varieties over a field k taking values the abelian tensor category of mixed motives over k, denoted M_k. There should be functors from M_k to both the category of Galois modules and mixed Hodge structures. When applied to the motivic homology of a variety X, these functors should yield the etale homology or Betti homology of X. In this way, motives glue these two different theories together more strongly than just the comparison isomorphisms. Nori has recently given a definition of the category of mixed motives. In this talk, we will show how this category relates to the fundamental group. In fact, more generally there is a motivic version of paths between two different points x and y of X which is important for applications. We also show that the multiplication and comultiplication maps are motivic, and compare with Hain's theory.

Many arithmetic properties of hyperbolic curves become apparent from embeddings into abelian varieties, in particular their Jacobians. For special curves, particularly those that arise as unramified double covers of another curve (of genus g), the Jacobian variety itself is decomposable. This leads to Prym varieties. These are principally polarized abelian varieties of dimension g-1. Having an explicit description of these varieties is an essential ingredient in many computational methods. We discuss an explicit construction for g equal to 4. This is joint work with Emre Can Sertoz.

Inspired by Kontsevich's homological mirror symmetry conjecture, I will show how to construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth quasi-projective Calabi-Yau variety. Examples will be presented. I will also explain the 'local' character of the picture.

In a seminal Inventiones 1976 paper, Hirzebruch and Zagier produced a set of cycles on certain Hilbert modular surfaces whose intersection numbers are the Fourier coefficients of elliptic modular forms with nebentypus. Their result can be viewed as a geometric manifestation of the Naganuma lift from elliptic modular forms to Hilbert modular forms. We discuss a general analogue of this result where the real quadratic extension is replaced by an arbitrary quadratic extension of totally real fields. Our result can be viewed as a geometric manifestation of quadratic base change for GL_2 over totally real fields. (joint work with Mark Goresky).

In this talk, I will discuss Chow motives and the Rost nilpotence principle, which played a role in Voevodsky's celebrated proof of the Milnor conjecture. Conjectural in general, this principle is useful in determining when motivic decompositions obey Galois descent. After covering some preliminaries, I will give an overview of a new proof of this principle for surfaces over a perfect field.

In this talk I will describe a proof of a conjecture made by Beilinson et al concerning the motivic complex in the weight three case. Then I will explain a new way to define the analytic continuation of the multiple polylogarithms which provides an appproach to defining the good variations of mixed Hodge-Tate structures associated to the multiple polylogarithms explicitly. As an application I will define the single-valued real analytic version of these multi-valued functions and which should be connected to the special values of multiple Dedekind zeta functions over general number fields.

Monstrous Moonshine asserts that the coefficients of the modular j-function are given in terms of ''dimensions'' of virtual character for the Monster group. There are 194 irreducible representations of the Monster, the largest of the sporadic simple groups, and it has been a longstanding open problem to determine their distribution in Moonshine. Witten and others have demonstrated deep connections between Monstrous Moonshine and quantum physics. The distributions of the Monster representations in Moonshine can be interpreted as the distributions of black hole states in 3 dimensional quantum gravity. In joint work with Ono and Duncan, we obtain exact formulas for these distributions. Moonshine type-phenomena have been observed for other finite simple groups besides the Monster. The Umbral Moonshine conjectures of Cheng, Duncan, and Harvey asserts that the Moonshine extends to 24 isomorphism classes of even unimodular positive-definite rank 24 lattices. Monstrous Moonshine can be regarded as the case of the Leech lattice. In 2013, Gannon proved the case for the Mathieu group M24. We offer a method of proof for the remaining 22 cases.

This talk is about complex analytic geometry, the field of mathematics concerned with complex manifolds and more generally with complex analytic spaces. The "curves" of the title are compact Riemann surfaces and the "surfaces" in the title are compact complex manifolds of dimension 2 over the complex numbers (and hence dimension 4 over the real numbers). The talk will explore the problem of constructing two dimensional complex manifolds by deforming known complex analytic spaces. It will focus on a single example. The talk should be quasi-accessible to anyone who has courses in Riemann surfaces and algebraic topology.

The real rank-two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two. This is joint with Anna Seigal.

Mumford-Tate groups together with their associated Mumford-Tate domains, as their definitions, tell rich information about Hodge classes. While Abelian varieties with complex multiplication serve as (relatively simple) examples, the study on Mumford-Tate groups in general cases could be much more complicated. In this expository talk I will briefly summarize the literature in the view of algebraic geometry and representation theory. The relation between Mumford-Tate groups and the Hodge-generic properties of period maps will be emphasized. I will also talk about some recent applications, including part of the latest results on the distribution of Hodge locus worked out by Baldi-Klingler-Ullmo. The talk is supposed to be accessible for graduate students in algebraic geometry or related fields.