## 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.

## Ravindra Girivaru : Lefschetz type theorems for algebraic cycles and vector bundles.

- Algebraic Geometry ( 214 Views )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.

## 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.

## David Morrison : Normal functions and disk counting

- Algebraic Geometry ( 209 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.

## Julie Rana : Moduli of general type surfaces

- Algebraic Geometry ( 203 Views )It has been 30 years since Koll\Â?ar and Shepherd-Barron published their groundbreaking paper describing a compactification of GiesekerÂ?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.

## Sebastian Casalaina-Martin : Distinguished models of intermediate Jacobians

- Algebraic Geometry ( 198 Views )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.

## 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.

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

- Algebraic Geometry ( 180 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.

## 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.

## Rita Pardini : Linear systems on irregular varieties

- Algebraic Geometry ( 174 Views )
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.

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

- Algebraic Geometry ( 166 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.

## 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 NÃ©ron 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.

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

- Algebraic Geometry ( 149 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.

## 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

## Richard Rimanyi : Thom polynomials

- Algebraic Geometry ( 143 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.

## Jeffrey Giansiracusa : Equations of tropical varieties

- Algebraic Geometry ( 142 Views )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.

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

- Algebraic Geometry ( 142 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.

## Arend Bayer : Stability conditions on the local P2 revisited

- Algebraic Geometry ( 136 Views )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.

## Giulia Sacca : Compact Hyperkahler manifolds in algebraic geometry

- Algebraic Geometry ( 131 Views )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.

## Sam Payne : Boundary complexes and weight filtrations

- Algebraic Geometry ( 130 Views )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.

## Jeremy Rouse : Quadratic forms representing all odd positive integers

- Algebraic Geometry ( 129 Views )We consider the problem of classifying all positive-definite integer-valued quadratic forms that represent all positive odd integers. Kaplansky considered this problem for ternary forms, giving a list of 23 candidates, and proving that 19 of those represent all positive odds. (Jagy later dealt with a 20th candidate.) Assuming that the remaining three forms represent all positive odds, we prove that an arbitrary, positive-definite quadratic form represents all positive odds if and only if it represents the odd numbers from 1 up to 451. This result is analogous to Bhargava and Hanke's celebrated 290-theorem. In addition, we prove that these three remaining ternaries represent all positive odd integers, assuming the Generalized Riemann Hypothesis. This result is made possible by a new analytic method for bounding the cusp constants of integer-valued quaternary quadratic forms $Q$ with fundamental discriminant. This method is based on the analytic properties of Rankin-Selberg $L$-functions, and we use it to prove that if $Q$ is a quaternary form with fundamental discriminant, the largest locally represented integer $n$ for which $Q(\vec{x}) = n$ has no integer solutions is $O(D^{2 + \epsilon})$.

## Franziska Hinkelmann : Analysis of discrete models of biological systems using computer algebra

- Algebraic Geometry ( 126 Views )Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, bounded Petri nets, and agent-based models. Simulation is a common practice for analyzing discrete models, but many systems are far too large to capture all the relevant dynamical features through simulation alone. We convert discrete models into algebraic models and apply tools from computational algebra to analyze their dynamics. The key feature of biological systems that is exploited by our algorithms is their sparsity: while the number of nodes in a biological network may be quite large, each node is affected only by a small number of other nodes. In our experience with models arising in systems biology and random models, this structure leads to fast computations when using algebraic models, and thus efficient analysis. All algorithms and methods are available in our package Analysis of Dynamic Algebraic Models (ADAM), a user friendly web-interface that allows for fast analysis of large models, without requiring understanding of the underlying mathematics or any software installation. ADAM is available as a web tool, so it runs platform independent on all systems.

## Luca Schaffler : The KSBA compactification of the moduli space of D(1,6)-polarized Enriques surfaces.

- Algebraic Geometry ( 125 Views )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.

## Dick Hain : What is an algebraic group?

- Algebraic Geometry ( 120 Views )Algebraic groups are important in algebraic and arithmetic
geometry. This talk will be a general introduction to them. I will
discuss some basic example (elliptic curves, **GL _{n}**, ...) and then
introduce linear algebraic groups and affine algebraic groups. There
will be lots of examples, which will help explain why
they are important.

## Bernd Sturmfels : Real rank-two geometry

- Algebraic Geometry ( 119 Views )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.