## Wei Ho : Integral points on elliptic curves

- Algebraic Geometry ( 349 Views )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)

## HaoHua Deng : Mumford-Tate Groups and the Hodge locus of period maps

- Algebraic Geometry ( 276 Views )Mumford-Tate groups together with their associated Mumford-Tate domains, as their definitions, tell rich information about Hodge classes. While Abelian varieties with complex multiplication serve as (relatively simple) examples, the study on Mumford-Tate groups in general cases could be much more complicated. In this expository talk I will briefly summarize the literature in the view of algebraic geometry and representation theory. The relation between Mumford-Tate groups and the Hodge-generic properties of period maps will be emphasized. I will also talk about some recent applications, including part of the latest results on the distribution of Hodge locus worked out by Baldi-Klingler-Ullmo. The talk is supposed to be accessible for graduate students in algebraic geometry or related fields.

## Sarah J Frei : Moduli spaces of sheaves on K3 surfaces and Galois representations

- Algebraic Geometry ( 263 Views )Moduli spaces of sheaves on K3 surfaces have been well-studied when defined over the complex numbers, because they are one of the known families of hyperkaehler varieties. However, many of their arithmetic properties when defined over an arbitrary field are still unknown. In this talk, I will tell you about a new result in this direction: two such moduli spaces of the same dimension, when defined over a finite field, have the same number of points defined over every finite field extension of the base field, which is surprising when the moduli spaces are not birational. The way to get at this result is to study the cohomology groups of the moduli spaces as Galois representations. Over an arbitrary field, we find that all of the cohomology groups are isomorphic as Galois representations.

## Nils Bruin : Prym varieties of genus four curves

- Algebraic Geometry ( 257 Views )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.

## Alex Perry : Derived categories of cubic fourfolds and their geometric applications

- Algebraic Geometry ( 225 Views )A fundamental problem in algebraic geometry is to determine whether a given algebraic variety is birational to projective space. This is most prominently open for cubic fourfolds, i.e. hypersurfaces defined by a cubic polynomial in a five-dimensional projective space. A decade ago, Kuznetsov suggested an approach to this problem using the derived category of coherent sheaves. I will explain recent applications of this perspective to fundamental questions in hyperkahler geometry and Hodge theory, which in turn shed light on the original question about cubic fourfolds.

## Chenglong Yu : Moduli of symmetric cubic fourfolds and nodal sextic curves

- Algebraic Geometry ( 222 Views )Period map is a powerful tool to study geometric objects related to K3 surfaces and cubic 4-folds. In this talk, we focus on moduli of cubic 4-folds and sextic curves with specified symmetries and singularities. We identify the geometric (GIT) compactifications with the Hodge theoretic (Looijenga, mostly Baily-Borel) compactifications of locally symmetric varieties. As a corollary, the algebra of GIT invariants is identified with the algebra of automorphic forms on the corresponding period domains. One of the key inputs is the functorial property of semi-toric compactifications of locally symmetric varieties. Our work generalizes results of Matsumoto-Sasaki-Yoshida, Allcock-Carlson-Toledo, Looijenga-Swierstra and Laza-Pearlstein-Zhang. This is joint work with Zhiwei Zheng.

## Max Lieblich : K3 surfaces in positive characteristic

- Algebraic Geometry ( 216 Views )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.

## 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.

## Greg Pearlstein : Boundary components of Mumford-Tate domains

- Algebraic Geometry ( 212 Views )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.

## 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\ar and Shepherd-Barron published their groundbreaking paper describing a compactification of Giesekers 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.

## Chad Schoen : A family of surfaces constructed from genus 2 curves

- Algebraic Geometry ( 198 Views )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.

## 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.

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

- Algebraic Geometry ( 185 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.

## Benjamin Bakker : o-minimal GAGA and applications to Hodge theory

- Algebraic Geometry ( 184 Views )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.

## John Swallow : Galois module structure of Galois cohomology

- Algebraic Geometry ( 183 Views )NOTE SEMINAR TIME: NOON!! Abstract: Let p be a prime number, F a field containing a primitive pth root of unity, and E/F a cyclic extension of degree p, with Galois group G. Let G_E be the absolute Galois group of E. The cohomology groups H^i(E,Fp)=Hî(G_E,Fp) possess a natural structure as FpG-modules and decompose into direct sums of indecomposables. In the 1960s Boreviè and Faddeev gave decompositions of E^*/E^*p -- the case i=1 -- for local fields. We describe the case i=1 for arbitrary fields, and then, using the Bloch-Kato Conjecture, we also determine the case i>1. No small surprise arises from the fact that there exist indecomposable FpG-modules which never appear in these module decompositions. We give several consequences of these results, notably a generalization of the Schreier formula for G_E, connections with Demu¹kin groups, and new families of pro-p-groups that cannot be realized as absolute Galois groups. These results have been obtained in collaboration with D. Benson, J. Labute, N. Lemire, and J. Mináè.

## 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.

## Anders Buch : Quantum cohomology of isotropic Grassmannians

- Algebraic Geometry ( 179 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.

## Ben Howard : Twisted Gross-Zagier theorems and central derivatives in Hida families

- Algebraic Geometry ( 177 Views )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.

## Remy van Dobben de Bruyn : A variety that cannot be dominated by one that lifts.

- Algebraic Geometry ( 176 Views )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.

## 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.

## Paolo Stellari : Derived Torelli Theorem and Orientation

- Algebraic Geometry ( 167 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'.

## Olivier Debarre : Fake projective spaces and fake tori

- Algebraic Geometry ( 167 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 ( 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.

## Patricia Hersh : Topology and combinatorics of regular CW complexes

- Algebraic Geometry ( 164 Views )Anders Björner characterized which finite, graded partially ordered sets (posets) are closure posets of finite, regular CW complexes, and he also observed that a finite, regular CW complex is homeomorphic to the order complex of its closure poset. One might therefore hope to use combinatorics to determine topological structure of stratified spaces by studying their closure posets; however, it is possible for two different CW complexes with very different topological structure to have the same closure poset if one of them is not regular. I will talk about a new criterion for determining whether a finite CW complex is regular (with respect to a choice of characteristic functions); this will involve a mixture of combinatorics and topology. Along the way, I will review the notions from topology and combinatorics we will need. Finally I will discuss an application: the proof of a conjecture of Fomin and Shapiro, a special case of which says that the Schubert cell decomposition of the totally nonnegative part of the space of upper triangular matrices with 1's on the diagonal is a regular CW complex homeomorphic to a ball.

## Melanie Matchett Wood : Motivic Discriminants

- Algebraic Geometry ( 163 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.

## Humberto Diaz : The Rost nilpotence principle

- Algebraic Geometry ( 159 Views )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.