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.

Elliptic curves are fundamental and well-studied objects in arithmetic geometry. However, much is still not known about many basic properties, such as the number of rational points on a "random" elliptic curve. We will discuss some conjectures and theorems about this "arithmetic statistics" problem, and then show how they can be applied to answer a related question about the number of integral points on elliptic curves over Q. In particular, we show that the second moment (and the average) for the number of integral points on elliptic curves over Q is bounded (joint work with Levent Alpoge)

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.)

Abstract: In the sixties, Serre constructed a smooth projective variety in characteristic p that cannot be lifted to characteristic 0. If a variety does not lift, a natural question is whether some variety related to it does. We construct a smooth projective variety that cannot be rationally dominated by a smooth projective variety that lifts.

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.

        I will report on joint work M.A. Barja (UPC, Barcelona) and L. Stoppino (Universita' dell'Insubria, Como - Italy).
        Given a generically finite map a:X--> A, where X is a smooth projective variety and A is an abelian variety, and given a line bundle L on X, we study the linear system |L+P|, where P is a general element of Pic^0(A). We prove that up to taking base change with a suitable multiplication map A-->A, the map given by |L+P| is independent of P and induces a factorization of the map a. When L is the canonical bundle of X, this factorization is a new geometrical object intrinsically attached to the variety X.
        The factorization theorem also allows us to improve the known Clifford-Severi and Castelnuovo type numerical inequalities for line bundles on X, under certain assumptions on the map a:X-->A. A key tool in these proofs is the introduction of a real function, the continuous rank function, that also allows us to simplify considerably the proof of the general Clifford-Severi inequality.

We discuss compact complex manifolds which ``look like'' complex projective spaces or complex tori. Hirzebruch and Kodaira proved in 1957 that when n is odd, any compact Kähler manifold X which is homeomorphic to Pⁿ is isomorphic to Pⁿ. This holds for all n by Aubin and Yau's proofs of the Calabi conjecture. One may conjecture that it should be sufficient to assume that the integral cohomology rings H^*(X,Z) and H^*(Pⁿ,Z) are isomorphic.

Catanese observed that complex tori are characterized among compact Kähler manifolds X by the fact that their integral cohomology rings are exterior algebras on H¹(X,Z) and asked whether this remains true under the weaker assumption that the rational cohomology ring is an exterior algebra on H¹(X,Q). (We call the corresponding compact Kähler manifolds ``rational cohomology tori".) We give a negative answer to Catanese's question by producing explicit examples. We also prove some structure theorems for rational cohomology tori. This is work in collaboration with Z. Jiang, M. Lahoz, and W. F. Sawin.

In the first part of this talk, we consider, in the context of an arbitrary hyperplane arrangement, the map between compactly supported cohomology to the usual cohomology of a local system. A formula (i.e., an explicit algebraic de Rham representative) for a generalized version of this map is obtained. These results are applied in the second part to invariant theory: Schechtman and Varchenko connect invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements: To determine the image of invariants in cohomology. In suitable cases (e.g., corresponding to positive integral levels), the space of invariants is shown to acquire a mixed Hodge structure over a cyclotomic field. This is joint work with P. Brosnan and S. Mukhopadhyay.

The cohomology rings of flag varieties have long been known to exhibit positivity properties. One such property is that the structure constants of the Schubert basis with respect to the cup product are non-negative. Brion (2002) and Anderson-Griffeth-Miller (2011) have shown that positivity extends to K-theory and T-equivariant K-theory, respectively. In this talk I will discuss recent work (joint with Shrawan Kumar) which generalizes these results to the case of Kac-Moody groups.

One of the main tools available for proving certain cases of the Hodge conjecture for abelian varieties is to compute the Hodge groups of the weight-1 Hodge structures associated to these abelian varieties. Thus Hodge groups of abelian varieties have been extensively investigated. In this talk, we discuss generalizing these results about Hodge groups to arbitrary-weight Hodge structures with Hodge numbers (n,0,…,0,n), particularly when n is prime or twice a prime. These generalizations yield some new results about Hodge classes of 2p-dimensional abelian varieties.

A numerical monoid is a subset of the nonnegative integers that is closed under addition. Given a numerical monoid S, consider the shifted monoid S_n obtained by adding n to each minimal generator of S. In this talk, we examine minimal relations between the generators of S_n when n is sufficiently large, culminating in a description that is periodic in the shift parameter n. We also explore several consequences, some old and some new, in the realm of factorization theory. No background in numerical monoids or factorization theory is assumed for this talk.

In this talk we describe the moduli compactification by stable pairs (also known as KSBA compactification) of a 4-dimensional family of Enriques surfaces, which arise as the $\mathbb{Z}_2^2$-covers of the blow up of $\mathbb{P}^2$ at three general points branched along a configuration of three pairs of lines. The chosen divisor is an appropriate multiple of the ramification locus. Using the theory of stable toric pairs we are able to study the degenerations parametrized by the boundary and its stratification. We relate this compactification to the Baily-Borel compactification of the same family of Enriques surfaces. Part of the boundary of this stable pairs compactification has a toroidal behavior, another part is isomorphic to the Baily-Borel compactification, and what remains is a mixture of these two.

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.

Hyperkahler (HK) manifolds appear in many fields of mathematics, such as differential geometry, mathematical physics, representation theory, and algebraic geometry. Compact HK manifolds are one of the building blocks for algebraic varieties with trivial first Chern class and their role in algebraic geometry has grown immensely over the last 20 year. In this talk I will give an overview of the theory of compact HK manifolds and then focus on some of my work, including a recent joint work with R. Laza and C. Voisin.

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.

Tropical geometry is a combinatorial shadow of algebraic geometry over a nonarchimedean field that encodes information about things like intersections and enumerative invariants. Usually one defines tropical varieties as certain polyhedral subsets of R^n satisfying a balancing condition. I'll show how these arise as the solution sets to certain systems of polynomial equations over the tropical semiring T = (R union -infinity, max, +) related to matroids. This yields a notion of tropical Hilbert polynomials, and in this framework there is a universal tropicalization that is closely related to the Berkovich analytification and the moduli space of valuations.

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.

An old theorem of Gabriel says that a variety X can be reconstructed by the category Coh(X) of coherent sheaves on it. This result has seen a few generalizations over the years. I will present a different and more geometric proof, with new generalizations. The idea being that X can be recovered as a moduli space of "points" in Coh(X). This is joint work with Michael Groechenig.

This talk will report on joint work with A. Rosenschon. There are examples showing that the torsion and co-torsion of Chow groups are complicated, in general, except in the ``classical'' cases (divisors and 0-cycles, and torsion in codimension 2). Instead, we may (following Lichtenbaum) consider the etale Chow groups, which coincide with the usual ones if we use rational coefficients; we show that they have better torsion and cotorsion if we work over the complex numbers. In contrast, they can have infinite torsion in some arithmetic situations (the usual Chow groups are conjectured to be finitely generated).

I will explain why it is expected that the Chow ring of hyperKaehler varieties has a similar structure as the Chow ring of abelian varieties. Examples of hyperKaehler varieties are given by K3 surfaces, and Hilbert schemes of length-n subschemes on K3 surfaces and their deformations. In fact I will introduce the notion of ``multiplicative Chow-Kuenneth decomposition'' and provide examples of varieties that can be endowed with such a decomposition. In the case of curves, or regular surfaces, this notion is intimately linked to the vanishing of a so-called "modified diagonal cycle". For example, a very general curve of genus >2 does not have vanishing modified diagonal cycle, but a result of Ben Gross and Chad Schoen establishes the vanishing of a modified diagonal cycle for hyperelliptic curves.

Chow groups give functors from algebraic varieties to abelian groups which are related to (co)homology. However Chow groups frequently contain more information than (co)homology. The construction of Chow groups is easy. Their computation is often difficult. This talk has two aims. First of all it will serve as an introduction to Chow groups which should be accessible to those who have taken a one semester course in Riemann surfaces, two semesters of algebraic topology, and have a passing acquaintance with affine and projective algebraic varieties. (One month in an algebraic geometry course may suffice for the latter.) Given that the next two talks in the algebraic geometry seminar will discuss various aspects of Chow groups, this talk may function as a warm up. The second aim is to introduce Bloch's conjecture on the Chow group of zero dimensional algebraic cycles on a non-singular projective surface. Throughout the talk one may assume that the base field is the complex numbers.

It follows from results in Teichmüller Theory that generic curves of type (g,n) in characteristic zero have only n rational points that come from the tautological points. Richard Hain gave an algebraic proof of the theorem. Extending his algebraic method to positive characteristics, we prove the analogous result for generic curves in positive characteristics. The primary tool used is the theory of weighted completion, which was developed by Richard Hain and Makoto Matsumoto. It linearises a profinite group such as arithmetic mapping class groups. In our case, the weighted completion connects topology and algebraic geometry in positive characteristics.

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.

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

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.

Given a smooth irreducible complex curve $C$, there are several isomorphism invariants one can attach to $C$. One invariant is the genus of $C$, that is, the number of handles in the corresponding Riemann surface. A subtler invariant is the gonality of $C$, that is, the minimal degree of a dominant map from $C$ of $\mathbb{P}^1$. A lower bound for either invariant has diophantine consequences when $C$ can be defined over a number field, but the ability to give non-trivial lower bounds depends on how $C$ is presented. In this talk we will consider a sequence $C_1,C_2,\ldots$ of finite unramified covers of $C$ and give spectral criteria for the gonality of $C_n$ to tend to infinity.

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.

Mumford-Tate groups arise as the natural symmetry groups of Hodge structures and their variations. I describe recent work with Matt Kerr on computing the Mumford-Tate group of the Kato-Usui boundary components of a degeneration of Hodge structure.

Let X be a connected and geometrically reduced variety over a field k, with a fixed rational point x_0 in X(k). Nori defined a profinite group scheme N(X,x_0), usually called Nori's fundamental group scheme, with the property that homomorphisms N(X,x_0) to a fixed finite group scheme G correspond to G-torsors P--> X with a fixed rational point in the inverse image of x_0 in P. If k is algebraically closed of characteristic 0 this coincides with Grothendieck's fundamental group, but is in general very different. Nori's main theorem is that if X is complete, the category of finite-dimensional representations of N(X,x_0) is equivalent to an abelian subcategory of the category of vector bundles on X, the category of essentially finite bundles. In my talk I will recall the basics of the theory of group schemes and torsors, and give a detailed description of Nori's results. Then I will explain my work in collaboration with Niels Borne, from the University of Lille, in which we extend them by removing the dependence on the base point, substituting Nori's fundamental group with a gerbe (in characteristic 0 this had already been done by Deligne), and give a simpler definition of essentially finite bundle, and a more direct and general proof of the correspondence between representations and essentially finite bundles.

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.