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.

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.

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.

In certain situations global topology may force singularities. For example, the topology of the Klein bottle forces self-intersections when mapped into 3-space. Any map of the projective plane must have at least cusp singularities when mapped into the plane. The topology of a manifold may force any differential form on it to degenerate at certian points. In a family of vector bundles over a complex curve some must degenerate to a non-stable bundle (in the GIT sense), depending on the topology of the family. In a family of vector bundle maps---arranged according to a directed graph (quiver)---some may be forced to degenerate. In families of linear spaces some have special incidence with some other fixed ones (Schubert calculus). These degenerations are governed by a unified notion in equivariant cohomology, the Thom polynomial of "singularities". In the lecture I will review Thom polynomials, computational strategies (interpolation, localization, Grobner basis), show examples and applications.

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.

We consider the spectrum of the Dirichlet-Neumann map. This is the spectrum of the operator which sends a function on the boundary of a domain to the normal derivative of its harmonic extension. Along with the Dirichlet and Neumann spectrum, this problem has been much studied. We show how the problem of finding domains with fixed boundary area and largest first eigenvalue is connected to the study of minimal surfaces in the ball which meet the boundary orthogonally (free boundary solutions). We describe some conjectures on optimal surfaces and some progress toward their resolution. This is joint work with Ailana Fraser.

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.

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.

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.

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.

In 1990, Candelas, de la Ossa, Green, and Parkes used the then-new technique of mirror symmetry to predict the number of rational curves of each fixed degree on a quintic threefold. The techniques used in the prediction were subsequently understood in Hodge-theoretic terms: the predictions are encoded in a power-series expansion of a quantity which describes the variation of Hodge structures, and in particular this power-series expansion is calculated from the periods of the holomorphic three-form on the quintic, which satisfy the Picard-- Fuchs differential equation. In 2006, Johannes Walcher made an analogous prediction for the number of holomorphic disks on the complexification of a real quintic threefold whose boundaries lie on the real quintic, in each fixed relative homology class. (The predictions were subsequently verified by Pandharipande, Solomon, and Walcher.) This talk will report on recent joint work of Walcher and the speaker which gives the Hodge- theoretic context for Walcher's predictions. The crucial physical quantity "domain wall tension" is interpreted as a Poincar\'e normal function, that is, a holomorphic section of the bundle of Griffiths intermediate Jacobians. And the periods are generalized to period integrals of the holomorphic three-form over appropriate 3-chains (not necessarily closed), which leads to a generalization of the Picard--Fuchs equations.

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.

For the general linear group, the Bruhat-Tits building can be realized explicitly in terms of periodic lattice chains in the standard representation. Further, each parahoric group scheme can be described as an automorphism group of a particular chain. I will explain a Tannakian formalism which establishes analogous descriptions for arbitrary connected reductive groups over complete discretely valued fields. This complements previously known results for classical groups, and fits in with Mumford's Geometric Invariant Theory, where spherical buildings are similarly described. This is joint work with Kevin Wilson.

It has been 30 years since Koll\’ar and Shepherd-Barron published their groundbreaking paper describing a compactification of Gieseker’s moduli space of surfaces of general type. As with all compactifications, the work raised natural questions. What is the structure of these moduli spaces and the boundary in particular? What sorts of singularities might we expect to obtain? What types of surfaces give rise to divisors in the moduli space, and are these divisors smooth? We discuss general results bounding types of Wahl singularities, and use them to address these questions in the context of Horikawa-type surfaces.

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.

The (small) quantum cohomology ring of a homogeneous space is a deformation of the classical cohomology ring, which uses the three point, genus zero Gromov-Witten invariants as its structure constants. I will present structure theorems for the quantum cohomology of isotropic Grassmannians, including a quantum Pieri rule for multiplication with the special Schubert classes, and a presentation of the quantum ring over the integers with the special Schubert classes as the generators. These results are new even for the ordinary cohomology of isotropic Grassmannians, and are proved directly from the definition of Gromov-Witten invariants by applying classical Schubert calculus to the kernel and span of a curve. This is joint work with A. Kresch and H. Tamvakis.

        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.

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.

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.

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.