## Bhargav Bhatt : Interpolating p-adic cohomology theories

- Algebraic Geometry ( 318 Views )Integration of differential forms against cycles on a complex manifold helps relate de Rham cohomology to singular cohomology, which forms the beginning of Hodge theory. The analogous story for p-adic manifolds, which is the subject of p-adic Hodge theory, is richer due to a wider variety of available cohomology theories (de Rham, etale, crystalline, and more) and torsion phenomena. In this talk, I will give a bird's eye view of this picture, guided by the recently discovered notion of prismatic cohomology that provides some cohesion to the story. (Based on joint work with Morrow and Scholze as well as work in progress with Scholze.)

## Yiannis Sakellaridis : Moment map and orbital integrals

- Algebraic Geometry ( 284 Views )In the Langlands program, it is essential to understand spaces of Schwartz measures on quotient stacks like the (twisted) adjoint quotient of a reductive group. The generalization of this problem to spherical varieties calls for an understanding of the double quotient H\G/H, where H is a spherical subgroup of G. This has been studied by Richardson for symmetric spaces. In this talk, I will present a new approach, for spherical varieties "of rank one", based on Friedrich Knop's theory of the moment map and the invariant collective motion.

## Yifeng Liu : Relative trace formulas and restriction problems for unitary groups

- Algebraic Geometry ( 212 Views )In this talk, I will introduce some new relative trace formulas toward the global Gan-Gross-Prasad conjecture for unitary groups, which generalize the trace formulas of Jacquet-Rallis and Flicker. In particular, I will state the corresponding conjecture of relative fundamental lemmas. A relation between the well-studied Jacquet-Rallis case the equal-rank case will also be discussed.

## Jeff Achter : Divisibility of the number of points on Jacobians

- Algebraic Geometry ( 202 Views )Given an elliptic curve over a finite field, one might reasonably ask for the chance that it has a rational point of order $\ell$. More generally, what is the chance that a curve drawn from a family over a finite field has a point of order $\ell$ on its Jacobian? The answer is encoded in an $\ell$-adic representation associated to the family in question. In this talk, I'll answer this question for hyper- or trielliptic curves, and give some results concerning an arbitrary family of curves. ** Keeping in mind what you said about the audience, I'll focus on the geometric and topological ideas.

## Izzet Coskun : Brill-Noether Theorems for moduli spaces of sheaves on surfaces

- Algebraic Geometry ( 188 Views )In this talk, I will discuss the problem of computing the cohomology of the general sheaf in a moduli space of sheaves on a surface. I will concentrate on the case of rational and K3 surfaces. The case of rational surfaces is joint work with Jack Huizenga and the case of K3 surfaces is joint work with Howard Nuer and Kota Yoshioka.

## Jimmy Dillies : On some K3 automorphisms

- Algebraic Geometry ( 183 Views )In order to construct a viable model of string theory, one seeks to build Calabi Yau threefolds with prescribed conditions. Borcea and Voisin were able to built a family of Calabi-Yau threefolds using elliptic curves and K3 surfaces admitting non symplectic involutions. We will display how the construction can be generalized by studying higher order non symplectic automorphisms on K3 surfaces

## Romyar Sharifi : A modular interpretation of a pairing on cyclotomic units

- Algebraic Geometry ( 177 Views )Class groups of cyclotomic fields have long been of central interest in number theory. We consider elements of these class groups that arise as values of a cup product pairing on cyclotomic units. These pairing values yield information on a wealth of algebraic objects, but any analytic interpretation of them was heretofore unknown. We will describe how, conjecturally, modular representations can be used to relate the pairing values to p-adic L-values of cusp forms.

## Paul Aspinwall : D-Branes and Triangulated Categories of Matrix Factorizations

- Algebraic Geometry ( 177 Views )Orlov has recently proven a remarkable equivalence between the derived category of coherent sheaves on a Calabi-Yau variety and a particular category of matrix factorizations. I review this work and explain why it's so interesting to string theorists.

## Wenjing Liao : Spectral estimation on a continuum

- Algebraic Geometry ( 166 Views )The problem of spectral estimation, namely recovering the frequency contents of a signal arises in various applications, including array imaging and remote sensing. In these fields, the spectrum of natural signals is composed of a few atoms on the continuum of a bounded domain. After the emergence of compressive sensing, spectral estimation was widely explored with an emphasis on sparse measurements. However, with a few exceptions, the spectrum considered in the compressive sensing community is assumed to be located on a DFT grid, which results in a large gridding error.

In this talk, I will present the MUltiple SIgnal Classification (MUSIC) algorithm and some modified greedy algorithms, and show how the problem of gridding error can be resolved by these methods. Our work focuses on a stability analysis as well as numerical studies on the performance of these algorithms. Moreover, the MUSIC algorithm features its super-resolution effect, i.e., the capability of resolving closely spaced frequencies. We will provide some numerical experiments and theoretical justifications to show that the resolution length of MUSIC follows a power law with respect to the minimum separation of frequencies.

## Jesse Kass : What is the limit of a line bundle on a nonnormal variety

- Algebraic Geometry ( 164 Views )On a nonnormal variety, the limit of a family of line bundles is not always a line bundle. What is the limit? I will present an answer to this question and give some applications. If time permits, I will discuss connections with Néron models, autoduality, and recent work of R. Hartshorne and C. Polini.

## Parker Lowrey : Virtual Grothendieck-Riemann-Roch via derived schemes

- Algebraic Geometry ( 163 Views )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.

## Chad Schoen : Threefolds with trivial canonical sheaf in positive characteristic

- Algebraic Geometry ( 163 Views )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.

## Jayce Getz : Hilbert modular generating functions with coefficients in intersection homology

- Algebraic Geometry ( 162 Views )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).

## Andrew Critch : Causality and Algebraic Geometry

- Algebraic Geometry ( 157 Views )Abstract: Science, and perhaps all learning, is the problem of inferring causal relationships from observations. It turns out that algebraic geometry can provide powerful intuition and methods applicable to causal inference. The relevant theory of graphical causal models is a major entry point to the budding field of algebraic statistics, where algebraic geometry meets statistical modeling, and this talk will give an introduction to it from the perspective of an algebraic geometer. I'll introduce some conceptual tools and methods that are peculiar to algebraic statistics, and work through an example such causal inference computation using the commutative algebra software Macaulay2. At the end I'll review some of my research on hidden Markov models and varieties, and their close connection to matrix product state models of quantum-entangled qubits.

## Thomas Lam : First steps in affine Schubert calculus

- Algebraic Geometry ( 156 Views )I will explain some attempts to develop a theory of Schubert calculus on the affine Grassmannian. I will begin with the different descriptions of the (co)homology rings due to Bott, Kostant and Kumar, and Ginzburg. Then I will discuss the problems of finding polynomial representatives for Schubert classes and the explicit determination of structure constants in (co)homology.

## Paul Johnson : Topology and combinatorics of Hilbert schemes of points on orbifolds

- Algebraic Geometry ( 151 Views )The Hilbert scheme of n points on C^2 is a smooth manifold of dimension 2n. The topology and geometry of Hilbert schemes have important connections to physics, representation theory, and combinatorics. Hilbert schemes of points on C^2/G, for G a finite group, are also smooth, and their topology is encoded in the combinatorics of partitions. When G is a subgroup of SL_2, the topology and combinatorics of the situation are well understood, but much less is known for general G. After outlining the well-understood situation, I will discuss some conjectures in the general case, and a combinatorial proof that their homology stabilizes.

## Zhiwei Yun : Rigid local systems coming from automorphic forms

- Algebraic Geometry ( 151 Views )We will give a survey of recent progress on constructing local system over punctured projective lines using techniques from automorphic forms and geometric Langlands. Applications include solutions of particular cases of the inverse Galois problem and existence of motives with exceptional Galois groups.

## Jason Polak : Exposing relative endoscopy

- Algebraic Geometry ( 145 Views )For a reductive algebraic group G with Lie algebra g and involution \theta we define relative orbital integrals with respect to G acting on the -1 eigenspace of \theta on g. We prove some fundamental lemmas relating these orbital integrals to relative orbital integrals on smaller groups, providing the first example of a theory of relative endoscopy in our setting

## Eric Cances : Perturbation of nonlinear self-adjoint operators - Theory and applications

- Algebraic Geometry ( 144 Views )The perturbation theory of linear operators has a long history. Introduced by Rayleigh in the 1870's, it was used for the first time in quantum mechanics in an article by Schrödinger published in 1926. The mathematical study of the perturbation theory of self-adjoint operators was initiated by Rellich in 1937, and has been since then the matter of a large number of contributions in the mathematical literature.

Perturbation theory of nonlinear operators plays a key role in quantum physics and chemistry, where it is used in particular to compute the response properties of molecular systems to external electromagnetic fields (polarizability, hyperpolarizability, magnetic susceptibility, NMR shielding tensor, optical rotation, ...) within the framework of mean-field models.

In this talk, I will recall the basics of linear perturbation linear, present some recent theoretical results [1] on nonlinear perturbation theory, and show how this approach can be also used to speed-up numerical simulations [2,3] and compute effective a posteriori error bounds.

[1] E. Cancès and N. Mourad, A mathematical perspective on density functional perturbation theory, Nonlinearity 27 (2014) 1999-2034.

[2] E. Cancès, G. Dusson, Y. Maday, B. Stamm and M. Vohralik, A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations, CRM 352 (2014) 941-946.

[3] E. Cancès, G. Dusson, Y. Maday, B. Stamm and M. Vohralik, A perturbation-method-based post-processing for the planewave discretization of Kohn-Sham models, in preparation.

## Sam Grushevsky : Stable cohomology of compactifications of the moduli spaces of abelian varieties

- Algebraic Geometry ( 136 Views )Cohomology of A_g, the moduli space of principally polarized complex g-dimensional abelian varieties, is the same as the cohomology of Sp(2g,Z). By Borel's result on group homology it turns out that for g>k the cohomology H^k(A_g) is independent of g - it is then called the stable cohomology of A_g. Similarly, the stable cohomology of the moduli space of curves was the subject of Mumford's conjecture, proven by Madsen and Weiss by topological methods. In a joint work with Klaus Hulek and Orsola Tommasi we show that the cohomology of the perfect cone toroidal compactification of A_g stabilizes, and compute some of this stable cohomology using algebro-geometric methods.

## Jie Wang : The primitive cohomology of the theta divisor of an abelian fivefold

- Algebraic Geometry ( 127 Views )The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. In this talk, I will explain how one can use the Prym map to show that this version of the Hodge conjecture is true for the theta divisor of a general abelian fivefold. This is joint work with Izadi and Tam\'as.

## Humberto Diaz : On Chow groups of Varieties

- Algebraic Geometry ( 125 Views )For a complex algebraic variety, the Chow group is a geometric invariant which is easy to construct but often difficult to compute. In this talk, I will describe the construction of the Chow group, give some key examples and discuss some difficult open questions. I will also present a result about the Chow group of 0-cycles of the surface which parametrizes lines on a cubic 3-fold.