Ohio State University Algebraic Geometry Seminar 

  Year 2010-2011

Time/Location: Tuesdays 4:30pm/CH240 (unless otherwise noted)

Schedule of talks:


 

TIME  SPEAKER TITLE HOST
September 28 
Tue, 4:30pm 
Scott Lab E0241 
 Anvar Mavlyutov 
(Oklahoma State University) 
Complete intersection toric ideals with applications to deformations of toric and Calabi-Yau varieties Clemens
October 2--3  Aprodu, Arapura,
Katz, Nori 
OSU/UM/UIC weekend algebraic geometry workshop at UIC
October 5 
Tue, 4:30pm 
No Seminar 
() 
N/A
October 12 
Tue, 4:30pm 
No Seminar 
() 
N/A
October 19 
Tue, 4:30pm 
Chen-Yu Chi 
(Harvard University) 
Pluricanonical spaces and their canonical distance structures
(see also Chi's talk in Differential Geometry seminar)
Tseng
Wu
October 26 
Tue, 4:30pm 
Prabhakar Rao 
(University of Missouri - St. Louis) 
ACM vector bundles on hypersurfaces Joshua
November 2 
Tue, 4:30pm 
No Seminar 
() 
N/A
November 9 
Tue, 4:30pm 
Michael Shapiro 
(Michigan State University) 
Quivers of finite mutation type Chmutov
November 16 
Tue, 4:30pm 
Tony Pantev 
(University of Pennsylvania) 
On the derived structure of the moduli of A-branes Clemens
November 23 
Tue, 4:30pm 
Dawei Chen 
(University of Illinois at Chicago) 
Geometry of Teichmueller Curves Tseng
November 30 
Tue, 4:30pm 
Scott Lab E0241 
Chenyang Xu 
(MIT) 
Boundedness of algebraic varieties Tseng
January 4 
Tue, 4:30pm 
No Seminar 
() 
N/A
January 10 
Mon, 4:30pm
 
Jarod Alper 
(Columbia University) 
Recruitment Talk Department
January 11 
Tue, 4:30pm 
Jarod Alper 
(Columbia University) 
Constructing moduli spaces of objects with infinite automorphisms Department
January 13 
Thu, 4:30pm
 
Emanuele Macri 
(Universität Bonn) 
Recruitment Talk Department
January 14 
Fri, 4:30pm
 
Ana-Maria Castravet 
(University of Arizona) 
Recruitment Talk Department
January 18 
Tue, 4:30pm 
No Seminar 
() 
N/A
January 19 
Wed, 4:30pm
 
Dawei Chen 
(University of Illinois at Chicago) 
Recruitment Talk Department
January 25 
Tue, 4:30pm 
Florian Block 
(University of Michigan) 
Universal Polynomials for Severi Degrees of Toric Surfaces Kennedy
February 1 
Tue, 4:30pm 
Jie Wang 
(OSU) 
Geometry of general curves via degenerations and deformations N/A
February 8 
Tue, 4:30pm 
Susan Cooper 
(University of Nebraska - Lincoln) 
Making Fat Points Simple Loper
February 15 
Tue, 4:30pm 
Joseph Marsano 
(University of Chicago) 
F-theory GUTs: Particle Physics from Algebraic Geometry Clemens
Tseng
February 22 
Tue, 4:30pm 
Alex Castro 
(University of Toronto) 
Curve Singularities and Semple Towers Kennedy
March 1 
Tue, 4:30pm 
No Seminar 
() 
N/A
March 5--6  Abramovich, Getzler, Graham, Liu,
Minicozzi, Rong, Schoen, Tolland 
Great Lakes Geometry Conference 2011 in OSU
March 8 
Tue, 4:30pm 
No Seminar 
() 
N/A
March 10 
Thu, 4:30pm
 
János Kollár 
(Princeton University) 
Varieties with quasi projective universal cover Clemens
March 29 
Tue, 4:30pm 
No Seminar 
 ()
N/A
April 2--3  Borisov, Gross, Hu,
Keum, Lunts, Markman   
OSU/UM/UIC weekend algebraic geometry workshop at U Michigan (in honor of the retirement of I. Dolgachev)
April 05 
Tue, 4:30pm 
Tony Pantev  
(University of Pennsylvania) 
Geometric Langlands Duality and its classical limit Clemens
April 08 
Fri, 4:30pm
 
Tony Pantev  
(University of Pennsylvania) 
Geometric Langlands and non-abelian Hodge theory Clemens
April 12 
Tue, 4:30pm 
No seminar 
() 
N/A
April 14 
Thu, 4:30pm
 
Eric Friedlander 
(University of Southern California) 
Elementary modular representation theory (Colloquium) Colloquium
April 15 
Fri, 4:30pm
 
Eric Friedlander 
(University of Southern California) 
Classical conjectures on algebraic cycles Colloquium
Joshua
April 19 
Tue, 4:30pm 
Jordan Ellenberg 
(University of Wisconsin-Madison) 
Stable cohomology of Hurwitz spaces and arithmetic applications Tseng
April 26 
Tue, 4:30pm 
CH 312 
Brian Lehmann 
(University of Michigan) 
Algebraic bounds on analytic multiplier ideals Clemens
May 3 
Tue, 4:30pm 
No seminar 
() 
N/A
May 10 
Tue, 4:30pm 
Aravind Asok 
(University of Southern California) 
Rational points vs zero cycles of degree one in stable A^1-homotopy Joshua
May 17 
Tue, 4:30pm 
CH 312 
Kyungyong Lee 
(University of Connecticut) 
Cluster Algebras and Quiver Grassmannians Kennedy
May 24 
Tue, 4:30pm 
No seminar 
() 
N/A
May 31 
Tue, 4:30pm 
No seminar 
() 
N/A

Abstracts

(Mavlyutov): In the 90's, Klaus Altmann constructed deformations of affine toric varieties as complete intersections in another toric variety using Minkowski sums of polyhedra. We will show how Altmann's construction can be extended using a categorical quotient presentation of a subvariety of a toric variety. In our construction the toric variety is embedded into a higher dimensional toric variety where the image is given by a complete intersection toric ideal in Cox homogeneous coordinates. In the case of Fano toric varieties, we show how their deformations induce deformations Calabi-Yau hypersurfaces.

(Chi): Given a Riemenn surface $X$ there is a canonical norm on its space of holomorphic quadratic differentials $H^0(X, 2K_X)$. If $X\to Y$ is an isomorphism between Riemann surfaces the induced map $H^0(Y, 2K_Y)\to H^0(X, 2K_X)$ will actually be a linear isometry with respect to the canonical norms. In 1971, Royden showed that the converse is true if $X$ and $Y$ both have genus $g\geq 2$. In this talk we will discuss the proof of his theorem and talk about a higher dimensional generalization of this result in birational geometry.

(Rao): An Arithmetically Cohen Macaulay vector bundle on a hypersurface X of projective space is a bundle E for which $H^i(X,E(a))=0$ for any integer a and any i between 1 and dim(X)-1. Indecomposable ACM bundles of rank >1 exist on any smooth hypersurface. We will discuss constructions of such ACM bundles. Recent theorems indicate that ACM bundles of small rank should not exist on hypersurfaces of large dimension. We discuss work of Kleppe, Chiantini-Madonna and Kumar-Rao-Ravindra in this direction. In another direction, we discuss how ACM bundles produce subvarieties of X that are not cut out properly on X by a subvariety of projective space.

(Shapiro): Cluster algebras were introduced by Fomin and Zelevinsky to create an algebraic framework for total positivity and canonical bases in semisimple algebraic groups. Generators of cluster algebra are collected into clusters that mutate one into another. Mutations of clusters of skew-symmetric cluster algebra induce mutations of quivers. Any skew-symmetric cluster algebra defines a class of mutation equivalent quivers. We classify all finite classes of mutation equivalent quivers. More exactly, we proved conjecture by Derksen and Owen that a connected quiver with at least three vertices whose mutation class is finite either corresponds to a triangulated bordered surface or is of one of eleven exceptional types. As application we obtained complete classification of skew-symmetric cluster algebras by the growth rate (finite, polynomial, or exponential).

(Pantev): I will explain how essential information about the structure of symplectic manifolds is captured by algebraic data, and specifically by the non-commutative mixed Hodge structure on the cohomology of the Fukaya category. I will also explain how the instanton-corrected Chern-Simons theory fits in the framework of normal functions in non-commutative Hodge theory and will give applications to explicit descriptions of quantum Lagrangian branes. This is a joint work with L. Katzarkov and M. Kontsevich.

(Chen): Teichmueller curves are central objects in dynamics and geometry. They provide fertile connections between polygon billiards, flat surfaces and moduli spaces. A special class of Teichmueller curves arise from a branched cover construction. I will use them as examples to illustrate the beautiful interplay between combinatorics, dynamics, number theory and algebraic geometry. This talk will be accessible to a general audience.

(Xu): (joint with Christopher Hacon and James McKernan) We will discuss our recent investigation on certain boundedness behavior of volumes of canonical classes. It has a number of applications, including results on bounding automorphisms, bounding Fano varieties and general type varieties, and limit behavior of log canonical thresholds.

(Alper): Moduli problems parameterizing objects with infinite automorphisms (e.g., semi-stable vector bundles) often do not admit coarse moduli varieties. However, they may admit moduli varieties identifying certain non-isomorphic objects. The goal of this talk is to introduce a framework to construct and study such moduli spaces. The crucial ingredient is the notion of a good moduli space, which associates to a moduli stack a variety which parameterizes objects up to a certain equivalence relation. I will present a general theorem providing sufficient conditions for an algebraic stack to admit a good moduli space. This theory will be applied to construct various projective modular compactifications of $M_g$ which arise as log canonical models of $\bar{M}_g$.

(Block): The degree of the Severi variety of plane curves of degree d with delta nodes is eventually polynomial in d (Fomin-Mikhalkin, 2009). In this talk, I will discuss a generalization of this result for a family of (in general non-smooth) toric surfaces. We will show that the degrees of the respective Severi varieties are eventually polynomial in the multidegree and "as a function of the surface." Our strategy is combinatorial and motivated by tropical geometry. This is joint work with Federico Ardila.

(Wang): A central problem in curve theory is to describe algebraic curves in a given projective space with fixed genus and degree. One wants to know the extrinsic geometry of the curve, i.e information on the equations defining the curve. koszul cohomology groups in some sense carry 'everything one wants to know' about the extrinsic geometry of curves in projective space: the number of equations of each degree needed to define the curve, the relations between the equations, etc. In this talk, I will present a new method using deformation theory to study koszul cohomology of general curves. Using this method, I will describe a way to determine number of defining equations of a general curve in some special degree range (but for any genus).

(Cooper): Hilbert functions of ideals defining reduced 0-dimensional schemes encode non-trivial geometric and algebraic data about the scheme, and hence play central roles in numerous interesting problems. These sequences of numbers have a well-known characterization. However, we are left perplexed when trying to characterize Hilbert functions of symbolic powers of such ideals (which define non-reduced schemes called fat point schemes). In this talk we'll compare the two situations and provide some insight into the Hilbert function of a fat point scheme.

(Marsano): String compactifications described by F-theory represent a framework where most important ingredients of particle physics are 'geometrized'. We will review the basic structure of F-theory and the techniques of geometric engineering that are used to build particle physics models in this setting. To address several important issues, we must make use of dualities that connect F-theory compactifications on Calabi-Yau 4-folds to Higgs bundles and to Heterotic compactifications on Calabi-Yau 3-folds with E_8 x E_8 vector bundles. We will review these dualities and describe a recent formalism for constructing fluxes that connects them.

(Castro): Around 1954 J.G. Semple proposed the contraction of a two-level tower of circle bundles encoding information about differential elements of second order (2-jets of curves) on the projective plane. This now classic algebraic geometric construction made a reappearance 40 years later in the works of Demailly, Kennedy, Colley, Lejeune-Jalabert among others.

A second reincarnation of Semple's idea occurred in the works of Montgomery and Zhitomirskii in 1998 when they were studying the classification problem of a certain class of non-holonomic distributions called Goursat. Much to their surprise the Semple tower was a universal classifying space for Goursat distributions.

Take a point in the k-step Semple tower. This point roughly represents the k-jet of a plane curve. One can extend Semple's original construction and talk about k-jets of curves in n-space. The diffeomorphism group of the n-space naturally acts on the Semple tower and the classification problem consists of understanding the orbits of this action. I will talk about my joint work with Richard Montgomery where we classify the simple low-codimension orbits. To give a partial yet satisfactory answer to the classification problem we need to delve into the theory of singular curves, in the sense of V. Arnol'd. Time allowing I will talk about some open problems.

(Kollár): We prove that the universal cover of a normal, projective variety $X$ is quasi-projective iff a finite, étale cover of $X$ is a fiber bundle over an Abelian variety with simply connected fiber.

(Pantev): This talk will be an overview of the geometric Langlands conjecture and its relation to the geometry of the Hitchin integrable system. I will explain the statement of the conjecture and will discuss its classical limit. I will review some recent results on the classical limit conjecture and will explain the relation to quantization of Fourier-Mukai transforms. I will try to be slow and will be careful to define everything.

(Pantev): I will explain a general framework for understanding the geometric Langlands correspondence via non-abelian Hodge theory. I will show how this framework can be used to produce Hecke eigensheaves explicitly. I will also illustrate the general strategy on the non-trivial example of the projective line with tame ramification at five points. This is a joint work with R. Donagi and C. Simpson.

(Friedlander): Lawson homology theory is a mixture of complex algebraic geometry and algebraic topology. Andrei Suslin has formulated a conjecture comparing Lawson homology and singular homology of projective, smooth complex varieties which is analogous to a recently proved comparison of motivic and \'etale cohomology. Alexander Beilinson has recently shown that a weak version of Suslin's conjecture is equivalent to Grothendieck's Standard Conjectures. This relates to an earlier (stronger) conjecture with Barry Mazur which is equivalent to Grothendieck's Standard Conjectures plus the Generalized Hodge Conjecture. We mention a K-theoretic approach to these conjectures for very special class of varieties.

(Ellenberg): We will discuss a theorem in algebraic geometry (or, better, algebraic topology) with motivation from number theory. Hurwitz spaces are moduli spaces of finite branched covers of P^1 -- for instance, the moduli space of trigonal curves is a Hurwitz space. We will discuss the stabilization of the cohomology of these spaces as the number of branch points (resp. number of strands) grows, with the Galois group of the cover being fixed; this can be thought of as a "Harer theorem" for this family of moduli spaces. It turns out that the function field analogues of many popular conjectures in analytic number theory (due to Cohen-Lenstra, Bhargava, etc.) reduce to topological questions about Hurwitz spaces. We will discuss the arithmetic consequences of the stabilization theorem, and of a geometrically natural conjecture about the stable cohomology classes of Hurwitz spaces. (joint work with Akshay Venkatesh and Craig Westerland).

(Lehmann): A classical theorem of Kodaira states that ample line bundles are characterized by the positivity of their curvature form. More generally, one expects that the geometric "positivity" of a line bundle L can be detected on the metrics carried by L. The key tool relating these two concepts is the multiplier ideal. I will introduce multiplier ideals and explain their importance. I will also discuss joint work with E. Eisenstein giving algebraic bounds on the behavior of analytic multiplier ideals.

(Asok): I will discuss recent work with Christian Haesemeyer showing that existence of a rational point or zero cycle of degree 1 is detected by an appropriate variant of the stable A^1-homotopy category. I will outline the necessary homotopic preliminaries.

(Lee): A cluster algebra, which was introduced by Fomin and Zelevinsky, is a commutative algebra with a family of distinguished generators (the cluster variables) grouped into overlapping subsets (the clusters) which are constructed by mutations. Cluster algebras have found applications and connections in a wide range of subjects including quiver representation theory, Poisson geometry, Teichm\"uller theory, tropical geometry and so forth. A quiver Grassmannian is a projective variety parametrizing subrepresentations of a quiver representation with a given dimension vector. Quiver Grassmannians include the usual Grassmannians and flag varieties as very special cases. After introducing cluster algebras and quiver grassmannians, we show how the Euler characteristics of quiver grassmannians can be explicitly computed by using cluster algebras.


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