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.
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.
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.
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.
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.
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.
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.
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'.
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 Néron models, autoduality, and recent work of R. Hartshorne and C. Polini.
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.
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.
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.
Thomas Lam : First steps in affine Schubert calculus
- Algebraic Geometry ( 156 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 ( 151 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.
Zhiwei Yun : Rigid local systems coming from automorphic forms
- Algebraic Geometry ( 151 Views )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.
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
Nicolas Addington : Cubic fourfolds and K3 surfaces
- Algebraic Geometry ( 140 Views )Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics -- conjecturally, the ones which are rational -- have K3s associated to them geometrically. Hassett has studied the cubics with K3s associated to them at the level of Hodge theory, and Kuznetsov has studied the cubics with K3s associated to them at the level of derived categories. These two notions of having an associated K3 should coincide. We prove they coincide generically. That is, Hassett's cubics form a countable union of irreducible Noether-Lefschetz divisors in moduli space, and Kuznetsov's cubics are a dense subset of these, forming a non-empty, Zariski open subset in each divisor.
John Calabrese : Gabriels theorem and points
- Algebraic Geometry ( 133 Views )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.
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.
Jie Wang : The primitive cohomology of the theta divisor of an abelian fivefold
- Algebraic Geometry ( 127 Views )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.
Christine Berkesch Zamaere : Torus actions and holonomic D-modules
- Algebraic Geometry ( 100 Views )Just as algebraic varieties with group actions admit quotients, we provide a quotient construction for D-modules with torus actions that is with several important properties in algebraic analysis. As an application, we apply tools from toric geometry to obtain new information about hypergeometric systems of PDEs studied by Gauss, Appell, and Lauricella, among others. In particular, we determine when such "Horn systems" are regular holonomic. This is joint work with Laura Felicia Matusevich and Uli Walther.
Speaker unknown : On the converse theorem in the theory of Borcherds products
- Algebraic Geometry ( 32 Views )R. Borcherds constructed a lifting from elliptic modular forms of weight $1-n/2$ to meromorphic modular forms on the orthogonal group $O(2,n)$. The lifted modular forms can be written as infinite products analogous to the Dedekind $\eta$-function (``Borcherds products''). Moreover, their divisors are always linear combinations of Heegner divisors; these are algebraic divisors that come from embedded quotients of $O(2,n-1)$. We address the natural question, whether every meromorphic modular form on $O(2,n)$, whose divisor is a linear combination of Heegner divisors, is a Borcherds product? We discuss some recent results that answer this question in the affirmative in a large class of cases.
Jianqiang Zhao : Arithmetic and Geometry of Multiple Polylogarithms
- Algebraic Geometry ( 32 Views )In this talk I will describe a proof of a conjecture made by Beilinson et al concerning the motivic complex in the weight three case. Then I will explain a new way to define the analytic continuation of the multiple polylogarithms which provides an appproach to defining the good variations of mixed Hodge-Tate structures associated to the multiple polylogarithms explicitly. As an application I will define the single-valued real analytic version of these multi-valued functions and which should be connected to the special values of multiple Dedekind zeta functions over general number fields.