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)

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.

I will describe some aspects of the geometry of K3 surfaces in positive characteristic, including derived-category replacements for the classical Torelli theorem, supersingular analogues of twistor spaces, and some consequences for the arithmetic of certain elliptic curves over function fields. Some of the work described is joint with Daniel Bragg, and some is joint with Martin Olsson.

The Weak Lefschetz theorem (or the Lefschetz hyperplane theorem) states that for a smooth, projective variety Y and a smooth hyperplane section X in Y, the restriction map of cohomologies H^i(Y) to H^i(X) is an isomorphism for i less than dim{X}, and an injection when i equal to dim{X}. Analogues of this theorem have been conjectured for algebraic cycles. We will talk about some results in this area. We will also talk about such questions for vector bundles.

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.

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.

In this talk I will discuss joint work with J. Achter and C. Vial showing that the image of the Abel--Jacobi map on algebraically trivial cycles descends to the field of definition for smooth projective varieties defined over subfields of the complex numbers. The main focus will be on applications to topics such as: descending cohomology geometrically, a conjecture of Orlov regarding the derived category and Hodge theory, and motivated admissible normal functions.

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.

Abstract: Given a Hida family of modular forms, a conjecture of Greenberg predicts that L-functions of forms in the family should generically vanish to order 0 or 1 at the center of the functional equation. Similarly the Selmer groups of forms in the family should generically be of rank 0 or 1. In this talk I will prove a generalization of the Gross-Zagier theorem, relating Neron-Tate heights of special points on the modular Jacobian J_1(N) to derivatives of L-functions, and explain how this generalization can be used to verify Greenberg's conjecture for any particular Hida family.

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 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 problem of spectral estimation, namely Â? recovering the frequency contents of a signal Â? 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.

We consider the "limiting behavior" of *discriminants* (or their complements), by which we mean informally the closed locus in some parameter space of some type of object where the objects have singularities. We focus on the collection of unordered points on a variety X, and linear systems on X. These are connected --- we use the first to understand the second. We describe their classes in the Grothendieck ring of varieties, as the number of points gets large, or as the line bundle gets very positive. As applications, (i) we show the motivic analogue of Poonen's point-counting result: the motivic probability of a section of L being smooth (as L gets large) is 1 / Z_X( \A^{-\dim X - 1} ) (where Z_X is the motivic zeta function), and (ii) show a priori unexpected structure in configuration spaces of points on a variety, with topological and point-counting consequences. Some low-tech examples: if v is a partition of n \leq 9, and v \neq (1,1,2,2,3), then the v-discriminant in the space of degree n polynomials (those polynomials with those root multiplicities, or worse) can be cut-and-pasted into affine space. (Question: over \C, does the complement have only two nonvanishing cohomology groups? What structure remains when n is larger?) This is joint work with Ravi Vakil.

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.

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.

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.

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.

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.

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.

Associated to every finite graph G there is a canonical ideal which encodes the linear equivalences of divisors on G. We study this ideal and its associated initial ideal. We give an explicit description of their syzygy modules and the Betti numbers in terms of the "connected flags" of G. This resolves open questions posed by Postnikov-Shapiro, Perkinson-Perlmen-Wilmes, and Manjunath-Sturmfels. This is a joint work with Fatemeh Mohammadi.

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.

We will give a description of the space of Bridgeland stability conditions on the derived category of sheaves on P2 sitting inside a compact Calabi-Yau threefold. We will discuss its fractal-like boundary, its relation with the group of auto-equivalences, with mirror symmetry, and with counting invariants for both P2 and the quotient stack [C3/Z_3]. This is joint work with E. Macri.

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.

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.

The boundary complex of an algebraic variety is the dual complex of the boundary divisor in a compactification of a log resolution. I will present recent work showing that the homotopy type of this complex is independent of the choice of resolution and compactification, and give relations between these complexes and Deligne's weight filtration on singular cohomology.

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,Â?,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.

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.

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.

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.

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.