Algebraic geometry and number theory seminar schedule
Department of Mathematics, Johns Hopkins University
Spring  2011

All talks are from 4:30 to 5:30 unless noted.

Room: 308
Title: Compactifying the space of relative stable maps using logarithmic structures.
Speaker: Qile Chen, Brown University
Abstract: For the purpose of computing Gromov-Witten invariants, the compactification of the space of relative stable maps with respect to a smooth divisor were introduced and studied using expanded degeneration during the past decade. Starting from the backgrounds and basic definitions, I will introduce a new way of compactification using logarithmic structures in the sense of Kato-Fontaine-Illusie. In particular, this covers many interesting cases, such as the target variety with a simple normal crossings divisor, or a simple normal crossings degeneration of a variety with simple normal crossings singularities. This is in part joint work with Dan Abramovich. Room: 308
Title: Ulrich Bundles on del Pezzo Surfaces.
Speaker: Yusuf Mustopa, The University of Michigan at Ann Arbor
Abstract: Ulrich bundles occur naturally in a variety of algebraic and algebro-geometric topics, including determinantal and Pfaffian descriptions of hypersurfaces, the computation of resultants, and the representation theory of generalized Clifford algebras. In this talk I will discuss the connection between the existence of rank r Ulrich bundles on a degree-d del Pezzo surface X and the geometry of curves of degree dr on X. This is joint work with Emre Coskun and Rajesh Kulkarni Room:
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 Room: 302
Title: Iwasawa Theory for Supersingular Modular Forms.
Speaker: Sarah Zerbes, University of Exeter
Abstract:The cyclotomic Iwasawa theory for modular forms which are ordinary at a prime p is very well understood. If the modular form is supersingular at p, then some curious phenomena occur which made it difficult to develop a theory analogous to the ordinary case. In my talk, I will discuss joint work with Antonio Lei and David Loeffler, where we give a joint framework for ordinary and supersingular Iwasawa theory and extend the known results in the supersingular case.

 Room: 308
Title: The Monodromy Conjecture
Speaker: Nero Budur, University of Notre Dame
Abstract: The Monodromy Conjecture relates the number of solutions of a polynomial modulo prime powers to singularity invariants. It has directly influenced the development of a number of techniques in algebraic geometry, such as motivic integration over jet schemes. However, the list of cases for which this conjecture is known is very small. We will present the state of the art concerning this conjecture. Room: 308
Title:  The Congruence Subgroup Kernel and the reductive Borel-Serre compactification
Speaker:  Leslie Saper, Duke University
Abstract:  Let G be a reductive algebraic group defined over a number field.  The congruence subgroup kernel quantifies to what extent is  every arithmetic subgroup of G a congruence subgroup.  It has been studied extensively.  On the other hand, the reductive Borel-Serre compactification of an arithmetic quotient of the symmetric space associated to G reflects the geometry of this quotient at infinity.  Its cohomology has been extensively studied in view of applications to automorphic forms.  After describing these two seemingly disparate objects I will show how the congruence subgroup kernel can be related to the fundamental group of the reductive Borel-Serre compactification. There is a generalization to S-arithmetic subgroups.  This is joint work with Lizhen Ji, V. Kumar Murty, and John Scherk. Room: 308
Title: Higher dimensional moduli and related problems
Speaker: Zsolt Patakfalvi, University of Washington
Abstract: The moduli space of stable schemes, or equivalently of semi-log canonical models, is the higher dimensional generalization of the widely investigated space of stable curves. From another point of view, it is the compact moduli space classifying varieties of general type up to birational equivalence. I will list a few results concerning the global geometry of this space and discuss in detail some questions about the base change behavior of relative canonical sheaves, motivated by the geometry and construction of the moduli space of stable schemes.  N/A
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 Room: 308
Title: Conformal blocks and rational normal curves.
Speaker: Noah Giansiracusa, Brown University
Abstract: I'll discuss a result that the Chow quotient parametrizing configurations of n points in P^d which generically lie on a rational normal curve is isomorphic to M_{0,n}, generalizing the well-known d=1 result of Kapranov. The corresponding GIT quotients, for symmetric linearization, are related to certain line bundles coming from the genus zero WZW model in conformal field theory. A representation-theoretic symmetry is manifest as the classical Gale transform in this setting. Room: 308
Title: Lifting Tropical Curves and Linear Systems on Graphs.
Speaker: Eric Katz, the University of Texas at Austin
Abstract: Tropicalization is a procedure for associating a polyhedral complex to a subvariety of an algebraic torus.  We explain the method of tropicalization and study the question of which graphs arise from tropicalizing algebraic curves. By applying Baker's technique of specialization of linear systems from curves to graphs, we are able to give a necessary condition for a balanced weighted graph to be the tropicalization of a curve. Our condition is phrased in terms of the harmonic theory of graphs, reproduces the known necessary conditions, and also gives new conditions.  Moreover, our method gives a combinatorial way of thinking about the deformation theory of algebraic varieties.

Room: 308
Title: Hyperelliptic jacobians and prymians: endomorphisms and Hodge classes.
Speaker: Yuri Zarhin, Penn State University
Abstract: Using Galois theory, we compute the endomorphism rings of jacobians and prymians of ``generic" hyperelliptic curves. In addition, we describe Hodge classes on self-products of these abelian varieties.

Room: 302
Title: Abelian varieties and the Kervaire invariant problem.
Speaker: Jack Morava,  Johns Hopkins University
Abstract: The Kervaire invariant problem has stood for nearly fifty years as the last open question in the classification of differentiable structures on spheres. It was recently solved by Hill, Hopkins, and Ravenel

    http://arxiv.org/abs/0908.3724
                                                                               
using a host of new techniques, some of which involve the Shimura moduli stack of Abelian varieties of type U(1,3) at the prime two, cf
                                                                               
    http://arxiv.org/abs/0910.0617.
                                                                               
[None of this is my work; this is intended as an expository talk.]

Room: 308
Title: Finite subgroups of Cremona groups
Speakers: Ivan Cheltsov, University of Edinburgh (UK) and Higher School of Economics (Russia)
                         Constantin Shramov, Steklov Institute of Mathematics (Russia) and  Higher School of Economics (Russia)
Abstract:    Part I (I.Cheltsov) The Cremona group of rank N is the group of birational selfmaps of the projective space of dimension N. We give a survey of old and new results on finite subgroups in Cremona groups of small rank.
                           Part II. (C.Shramov)We show how to apply Nadel-Shokurov vanishing  and  Kawamata subadjunction to study conjugacy classes of finite subgroups in the Cremona groups. In particular, we give a partial answer to a question of Serre on normalizers of finite simple subgroups in the Cremona of rank 3.


Room: 308
Title: Curvature and Rationality of Complex Manifolds
Speaker: Bun Wong, UC Riverside
Abstract: We will explore the rational connectivity of a projective variety from the viewpoint of curvature. One can relate the rational dimension with the intermediate Ricci curvature. In particular, there is a connection between the scalar curvature (and holomorphic sectional curvature) with the uniruledness of a projective manifold. The other case is a known result of Kollar/Miyaoka/Mori on the rational connectedness of a complex manifold with positive definite Chern class. This  is a joint work with Gordon Heier at Houston University.