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

- Algebraic Geometry ( 208 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.

## David Morrison : Normal functions and disk counting

- Algebraic Geometry ( 205 Views )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.

## Leonardo Mihalcea : Quantum-K theory of the Grassmannians

- Algebraic Geometry ( 183 Views )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.

## Jimmy Dillies : On some K3 automorphisms

- Algebraic Geometry ( 181 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

## Thomas Haines : A Tannakian approach to Bruhat-Tits buildings and parahoric group schemes

- Algebraic Geometry ( 178 Views )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.

## Anders Buch : Quantum cohomology of isotropic Grassmannians

- Algebraic Geometry ( 177 Views )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.

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

- Algebraic Geometry ( 172 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.

## Olivier Debarre : Fake projective spaces and fake tori

- Algebraic Geometry ( 165 Views )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 **P**^{n} is isomorphic to
**P**^{n}. 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*^{*}(**P**^{n},**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 *H*^{1}(*X*,**Z**) and asked whether this remains true under the
weaker assumption that the rational cohomology ring is an exterior algebra
on *H*^{1}(*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.

## Prakash Belkale : Topology of hyperplane arrangements and tensor product invariants

- Algebraic Geometry ( 163 Views )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.

## Paolo Stellari : Derived Torelli Theorem and Orientation

- Algebraic Geometry ( 163 Views )We will consider the problem of describing the group of autoequivalences of the derived categories of smooth K3 surfaces. After recalling the (Twisted) Derived Torelli Theorem, we will focus on its conjectural refinement, involving the preservation of the orientation of some 4-dimensional space in the total cohomology lattice. The conjecture will be proved in the generic (non-projective) case and we will discuss a few results which will possibly lead to the proof of the conjecture for smooth projective K3 surfaces. This is a joint work with D. Huybrechts and E. Macri'.

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

- Algebraic Geometry ( 162 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.

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

- Algebraic Geometry ( 161 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.

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

- Algebraic Geometry ( 161 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.

## Wenjing Liao : Spectral estimation on a continuum

- Algebraic Geometry ( 161 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.

## Melanie Matchett Wood : Motivic Discriminants

- Algebraic Geometry ( 160 Views )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.

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

- Algebraic Geometry ( 160 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 ( 156 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 ( 154 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.

## Tatsunari Watanabe : Weighted completion and Generic curves in positive characteristics

- Algebraic Geometry ( 150 Views )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.

## Will Donovan : Noncommutative deformations and the birational geometry of 3-folds

- Algebraic Geometry ( 148 Views )I will speak about recent work with Michael Wemyss (arXiv:1309.0698), applying noncommutative deformation theory to study the birational geometry of 3-folds. I will give a brief introduction to noncommutative deformations, and explain how every flippable or floppable rational curve in a 3-fold has a naturally associated algebra of noncommutative deformations. This construction yields new information about the (commutative) geometry of the 3-fold, and provides a new tool to differentiate between flops. As a further application, we show how the noncommutative deformation algebra controls the homological properties of a floppable curve, relating a Fourier-Mukai flop-flop functor and a spherical twist about the universal family over the noncommutative deformation algebra. I will also explain work in progress applying this approach to other geometric situations, and to higher dimensions.

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

- Algebraic Geometry ( 148 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.

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

- Algebraic Geometry ( 143 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.

## Richard Rimanyi : Thom polynomials

- Algebraic Geometry ( 141 Views )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.

## Emanuele Macri : MMP for moduli spaces of sheaves on K3 surfaces and Cone Conjectures

- Algebraic Geometry ( 140 Views )We report on joint work with A. Bayer on how one can use wall-crossing techniques to study the birational geometry of a moduli space M of Gieseker-stable sheaves on a K3 surface X. In particular: (--) We will give a "modular interpretation" for all minimal models of M. (--) We will describe the nef cone, the movable cone, and the effective cone of M in terms of the algebraic Mukai lattice of X. (--) We will establish the so called Tyurin/Bogomolov/Hassett-Tschinkel/Huybrechts/Sawon Conjecture on the existence of Lagrangian fibrations on M.

## Farbod Shokrieh : Divisors on graphs, connected flags, and syzygies

- Algebraic Geometry ( 139 Views )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.