This year, the UChicago Algebraic Geometry seminar will be jointly organized by the Departments of Mathematics and Statistics. A departure from previous years is that in addition to topics in mainstream algebraic geometry, we will also occasionally feature topics that are of interest to applied mathematicians, computer scientists, and statisticians. Algebraic geometry has reached a level of maturity that many concrete aspects of the subject have now found important applications in science and engineering. We welcome all those who are interested to join us every alternate Tuesdays in Eckhart 202.
Note: The Algebraic Geometry Seminar will meet in Eckhart 202 in Spring 2013.
Alternate Tuesdays, 4:30–6:00PM, Eckhart Hall, Room 202, unless noted otherwise
|Apr 09, 2013||Jason Morton (Pennsylvania State University)|
|Apr 23, 2013||Lawrence Ein (University of Illinois, Chicago)|
|May 07, 2013||Stephen Miller (Rutgers University)|
|May 17, 2013||Việt Trung Ngô (Hanoi Institute of Mathematics)|
|May 21, 2013||Dawei Chen (Boston College)|
|Jan 15, 2013||Ketan Mulmuley (University of Chicago)|
|Jan 29, 2013||Harm Derksen (University of Michigan, Ann Arbor)|
|Feb 13, 2013||Matthew Morrow (University of Chicago)|
|Feb 26, 2013||J.M. Landsberg (Texas A&M University)|
|Mar 12, 2013||Mihnea Popa (University of Illinois, Chicago)|
|Oct 09, 2012||Shmuel Friedland (University of Illinois, Chicago)|
|Oct 23, 2012||Luke Oeding (University of California, Berkeley)|
|Oct 30, 2012||Rob de Jeu (Vrije Universiteit Amsterdam)|
|Nov 06, 2012||Chung-Pang Mok (McMaster University)|
|Nov 20, 2012||Burt Totaro (University of Cambridge/University of California, Los Angeles)|
|Dec 04, 2012||Greg Blekherman (Georgia Institute of Technology)|
Shmuel Friedland, Department of Mathematics, Statistics and Computer Science, University of Illionis, Chicago
Eigenvalues and singular values of tensors: The eigenvalues (eigenvectors) and the singular values (singular vectors) of tensors can be defined naturally for real tensors as solutions of corresponding extremal problems of maximizing-minimizing certain polynomial and multilinear forms and finding best low rank approximation of tensors. To find the number of eigenvectors and singular vectors of tensors one needs to pass to complex tensors and use some basic tools of algebraic geometry: degree theory and top Chern numbers of corresponding vector bundles. To establish uniqueness of best rank one approximation, in particular for partially symmetric tensors, one needs to use some soft analysis and and some specific techniques. Some open problems will be presented. This talk will mostly based on joint work in progress with G. Ottaviani from U. Florence, and the recent preprint of the speaker http://arxiv.org/abs/1110.5689.
Luke Oeding, Department of Mathematics, University of California, Berkeley
Hyperdeterminants of polynomials: Hyperdeterminants were brought into a modern light by Gelfand, Kapranov, and Zelevinsky in the 1990's. Inspired by their work, I will answer the question of what happens when you apply a hyperdeterminant to a polynomial (interpreted as a symmetric tensor).The hyperdeterminant of a polynomial factors into several irreducible factors with multiplicities. I identify these factors along with their degrees and their multiplicities, which both have a nice combinatorial interpretation. The analogous decomposition for the mu-discriminant of polynomial is also found. The methods I use to solve this algebraic problem come from geometry of dual varieties, Segre-Veronese varieties, and Chow varieties; as well as representation theory of products of general linear groups.
Rob de Jeu, Department of Mathematics, Vrije Universiteit Amsterdam
The syntomic regulator for K4 of curves: Let C be a curve defined over a discrete valuation field of characteristic zero where the residue field has positive characteristic. Assuming that C has good reduction over the residue field, we compute the syntomic regulator on a certain part of K4(3)of the function field of C. The result can be expressed in terms of p-adic polylogarithms and Coleman integration, or by using a trilinear map ("triple index") on certain functions.
Chung-Pang Mok, Department of Mathematics, McMaster University
Introduction to endoscopic classification of automorphic representations on classical groups: The recent work of Arthur on endoscopic classification of automorphic representations on classical groups is a landmark result in the Langlands' program. In this talk we will try to indicate the nature of the classification and the tools that are used in the proof.
Totaro, Department of
Mathematics, University of California, Los Angeles, and Department of Pure
Mathematics and Mathematical Statistics,
University of Cambridge
The integral Hodge conjecture for 3-folds: The Hodge conjecture predicts which rational cohomology classes on a smooth complex projective variety can be represented by linear combinations of complex subvarieties. In other words, it is about the difference between topology and algebraic geometry. The integral Hodge conjecture, the analogous conjecture for integral homology classes, is false in general. We discuss negative results and some new positive results on the integral Hodge conjecture for 3-folds.
Greg Blekherman, Department of Mathematics, Georgia Institute of Mathematics
Real symmetric tensor decomposition: Symmetric tensor decomposition (also known as the Waring problem for forms) asks for a minimal decomposition of a symmetric tensor in terms of rank 1 tensors. Equivalently the Waring problem for forms asks for a minimal decomposition of a form of degree d as a linear combination d-th powers of linear forms. These problems are usually studied over complex numbers, while it is of definite interest to only consider real decompositions for real tensors (or equivalently real forms). I will explain several ways in which the situation is different for real tensors. For instance, a generic form with complex coefficients has a well-defined unique rank, which is given by the Alexander-Hirschowitz theorem. This is no longer the case over real numbers and there can be several "typical" ranks, while no generic rank exists. I will show how classical tools, such as the Apolarity Lemma can be used to study the typical ranks of real tensors.
Ketan Mulmuley, Department of Computer Science, University of Chicago
The GCT chasm: We show that the problem of derandomizing Noether's Normalization Lemma (NNL) that lies at the heart of the wild problem of classifying tuples of matrices can be brought down from EXPSPACE, where it was earlier, to PSPACE unconditionally, to PH assuming the Generalized Riemann Hypothesis (GRH), and even further to P assuming the black-box derandomization hypothesis for symbolic trace (or equivalently determinant) identity testing. Furthermore, we show that the problem of derandomizing Noether's Normalization Lemma for any explicit variety can be brought down from EXPSPACE, where it is currently, to P assuming a strengthened form of the black-box derandomization hypothesis for polynomial identity testing (PIT). These and related results reveal that the fundamental problems of Geometry (classification) and Complexity Theory (lower bounds and derandomization) share a common root difficulty, namely, the problem of overcoming the formidable EXPSPACE vs. P gap in the complexity of NNL for explicit varieties. We call this gap the GCT chasm. On the positive side, we show that NNL for the ring of invariants for any finite dimensional rational representation of the special linear group of fixed dimension can be brought down from EXPSPACE to quasi-P unconditionally.
Harm Derksen, Department of Mathematics, University of Michigan
Ranks and nuclear norms of tensors: The rank of a matrix generalizes to higher order tensors. There are many applications of the rank of a tensor in applied and pure mathematics. For example, the rank of a certain tensor related to matrix multiplication is closely related to the complexity of matrix multiplication. An important tool in applied math is low rank matrix completion. Matrix completion is the problem of finding missing entries in a low rank matrix. I will explain how the low rank matrix completion problem can be reduced to finding the rank of a certain tensor. Unfortunately, it is often difficult to determine the rank of higher order tensors. A common simplification is convex relaxation: instead of the rank of a tensor, we may consider its nuclear norm. For many tensors, for which we do not know the rank, we can determine the nuclear norm. Examples are: the matrix multiplication tensor, the determinant, permanent, and the multiplication tensor in group algebras. We also will generalize the notion of the Singular Value Decomposition (at least for some tensors) and find the singular values of some tensors of interest.
Matthew Morrow, Department of Mathematics, University of Chicago
The K-theory of singular varieties: The study of the K-theory of singular varieties has seen enormous progress in recent years, due both to descent techniques developed by C. Weibel, C. Haesemeyer, et al, and to infinitesimal methods using profinite K-theory. Specific applications which I will discuss include singular analogues of Gersten's injectivity conjecture and cycle-theoretic descriptions of K-groups of singular varieties. This work is joint with A. Krishna.
J.M. Landsberg, Department of Mathematics, Texas A&M University
Algebraic geometry and complexity theory: I will discuss how algebraic geometry and representation theory have been used to prove results in theoretical computer science.
Mihnea Popa, Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago
Kodaira dimension and zeros of holomorphic one-forms: I will report on recent work with C. Schnell, in which we prove that every holomorphic one-form on a variety of general type must vanish at some point (together with a suitable generalization to arbitrary Kodaira dimension). The proof makes use of generic vanishing theory for Hodge D-modules on abelian varieties.
Jason Morton, Departments of Mathematics and Statistics, Pennsylvania State University
Geometry and tensor networks: Tensor networks (or more generally, diagrams in monoidal categories with various additional properties) arise constantly in applications, particularly those involving networks used to process information in some way. Aided by the easy interpretation of the graphical language, they have played an important role in computer science, statistics and machine learning, and quantum information and many-body systems. Tools from algebraic geometry, representation theory, and category theory have recently been applied to problems arising from such networks. Basic questions about each type of information-processing system (such as what probability distributions or quantum states can be represented) turn out to lead to interesting problems in algebraic geometry, representation theory, and category theory.
Lawrence Ein, Department of Mathematics, University of Illinois, Chicago
Asymptotic syzygies of algebraic varieties: We'll discuss my joint work with Rob Lazarsfeld and Daniel Erman. We study the asymptotic behaviors of the Betti table of the minimal resolution of the coordinate ring of a smooth projective variety, when it is embedded into the projective space by a the linear system of the form |dA + B| where A is an ample divisor and B is a fixed divisor and d is sufficiently large integer.
Stephen Miller, Department of Mathematics, Rutgers University
Eisenstein series on affine loop groups: Eisenstein series on exceptional Lie groups are used in a number of constructions in number theory and representation theory. These groups have exotic arithmetic configurations, but are limited in number. It is thus tempting to define Eisenstein series on infinite-dimensional Kac-Moody groups. In the simplest such case of affine loop groups, they were constructed by Garland, who showed convergence in a shifted Weyl chamber. We give the full holomorphic continuation of Garland's cuspidal Eisenstein series to the entire complex plane. We also give the first convergence results for general Kac-Moody groups. I plan to describe these results as well as to indicate possible applications to the Langlands-Shahidi method and some recent work in string theory concerning graviton scattering. (Joint work with Howard Garland and Manish Patnaik).
Việt Trung Ngô, Department of Algebra, Hanoi Institute of Mathematics
Cohen-Macaulayness of monomial ideals: A combinatorial criterion for the Cohen-Macaulayness of monomial ideals will be presented. This criterion helps to explain all previous results on this topics. In particular, there is a striking relationship between the Cohen-Macaulayness of symbolic powers of Stanley-Reisner ideals and matroid complexes.
Dawei Chen, Department of Mathematics, Boston College
Flat surface, moduli of differentials, and Teichmüller dynamics: An abelian differential on a Riemann surface X defines a flat structure, such that X can be realized as a plane polygon. Changing the shape of the polygon induces an SL(2,R)-action on the moduli space of abelian differentials, called the Teichmüller dynamics. A central question is to study the orbit closures of this action and the associated dynamical quantities, like the Lyapunov exponents and the Siegel-Veech constants. In this talk I will focus on the minimal orbit closures, called Teichmüller curves, and introduce tools in algebraic geometry to study them. As an application, we prove a conjecture of Kontsevich-Zorich regarding a special numerical property of Teichmüller curves in low genus (joint work with Martin Möller).
For further information on this event, please email Lek-Heng Lim at lekheng(at)galton.uchicago.edu or Madhav Nori at nori(at)math.uchicago.edu.