Chad Schoen : Threefolds with trivial canonical sheaf in positive characteristic
- Algebraic Geometry ( 153 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.
John Swallow : Galois module structure of Galois cohomology
- Algebraic Geometry ( 163 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.
Farbod Shokrieh : Divisors on graphs, connected flags, and syzygies
- Algebraic Geometry ( 126 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.
Max Lieblich : K3 surfaces in positive characteristic
- Algebraic Geometry ( 199 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.
Paul Johnson : Topology and combinatorics of Hilbert schemes of points on orbifolds
- Algebraic Geometry ( 118 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.
Remy van Dobben de Bruyn : A variety that cannot be dominated by one that lifts.
- Algebraic Geometry ( 164 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.
Paul Aspinwall : D-Branes and Triangulated Categories of Matrix Factorizations
- Algebraic Geometry ( 163 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.
Yifeng Liu : Relative trace formulas and restriction problems for unitary groups
- Algebraic Geometry ( 194 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.
Romyar Sharifi : A modular interpretation of a pairing on cyclotomic units
- Algebraic Geometry ( 162 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.
Eric Cances : Perturbation of nonlinear self-adjoint operators - Theory and applications
- Algebraic Geometry ( 131 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.
Zhiwei Yun : Rigid local systems coming from automorphic forms
- Algebraic Geometry ( 136 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.
Emanuele Macri : MMP for moduli spaces of sheaves on K3 surfaces and Cone Conjectures
- Algebraic Geometry ( 133 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.
Christine Berkesch Zamaere : Torus actions and holonomic D-modules
- Algebraic Geometry ( 88 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.
David Geraghty : Modularity lifting beyond the numerical coincidence of Taylor and Wiles
- Algebraic Geometry ( 90 Views )Modularity lifting theorems were introduced by Taylor and Wiles and formed a key part of the proof of Fermat's Last Theorem. Their method has been generalized successfully by a number authors but always with the restriction that the Galois representations in question have regular weight. Moreover, the sought after automorphic representation must come from a group that admits Shimura varieties. I will describe a method to overcome these restrictions, conditional on certain conjectures which themselves can be established in a number of cases. This is joint with Frank Calegari.
Benjamin Bakker : o-minimal GAGA and applications to Hodge theory
- Algebraic Geometry ( 168 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.
Luca Schaffler : The KSBA compactification of the moduli space of D(1,6)-polarized Enriques surfaces.
- Algebraic Geometry ( 113 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.
Prakash Belkale : Topology of hyperplane arrangements and tensor product invariants
- Algebraic Geometry ( 151 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.
Richard Schoen : An optimal eigenvalue problem and minimal surfaces in the ball
- Algebraic Geometry ( 88 Views )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.
Richard Rimanyi : Thom polynomials
- Algebraic Geometry ( 132 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.
Izzet Coskun : Brill-Noether Theorems for moduli spaces of sheaves on surfaces
- Algebraic Geometry ( 172 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.
Dick Hain : What is an algebraic group?
- Algebraic Geometry ( 112 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, GLn, ...) and then introduce linear algebraic groups and affine algebraic groups. There will be lots of examples, which will help explain why they are important.
Christopher O'Neill : Shifting numerical monoids
- Algebraic Geometry ( 97 Views )A numerical monoid is a subset of the nonnegative integers that is closed under addition. Given a numerical monoid S, consider the shifted monoid S_n obtained by adding n to each minimal generator of S. In this talk, we examine minimal relations between the generators of S_n when n is sufficiently large, culminating in a description that is periodic in the shift parameter n. We also explore several consequences, some old and some new, in the realm of factorization theory. No background in numerical monoids or factorization theory is assumed for this talk.
Pete Clark : (Postponed to a later date) Algebraic Curves Violating the Hasse Principle
- Algebraic Geometry ( 118 Views )The celebrated "Hasse Principle" holds for plane conics over a number field, but generally not for algebraic curves of positive genus. Isolated examples of curves violating the Hasse Principle go back to Lind, Reichardt and Selmer in the 1940s and 1950s. Many more examples have been found since, and it now seems likely that the Hasse principle should, in some suitable sense, most often be false. However it is challenging to make, let alone prove, a precise statement to this effect. In talk I will discuss certain "anti-Hasse principles", some which are conjectural and others (more modest) which are known to hold. In particular I will address the problem of constructing curves of any given genus g >= 1 over any global field which violate the Hasse