University of Chicago Algebraic Geometry Seminar

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SYNOPSIS

The UChicago Algebraic Geometry seminar will be jointly organized by the Departments of Mathematics and Statistics. In addition to topics in mainstream algebraic geometry, we will also occasionally feature topics that are of interest to applied mathematicians, computer scientists, physicists, 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 Wednesdays in Eckhart 312, from 4:30–6:00 PM.

SCHEDULE

Wednesdays, 4:30–6:00 PM, Eckhart Hall, Room 312, unless noted otherwise

ABSTRACTS

Oct 09, 2012 Tue, 4:30–6:00PM, Eckhart 312

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.

Oct 23, 2012 Tue, 4:30–6:00PM, Eckhart 312

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).

Oct 30, 2012 Tue, 4:30–6:00PM, Eckhart 312

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.

Nov 06, 2012 Tue, 4:30–6:00PM, Eckhart 312

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.

Nov 20, 2012 Tue, 4:30–6:00PM, Eckhart 312

Burt 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.

Dec 04, 2012 Tue, 4:30–6:00PM, Eckhart 312

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.

Jan 15, 2013 Tue, 4:30–6:00PM, Eckhart 312

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.

Jan 29, 2013 Tue, 4:30–6:00PM, Eckhart 312

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.

Feb 12, 2013 Wed, 4:00–5:30PM, Eckhart 308 (note special day/time/venue)

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.

Feb 26, 2013 Tue, 4:30–6:00PM, Eckhart 312

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.

Mar 12, 2013 Tue, 4:30–6:00PM, Eckhart 312

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.

Apr 09, 2013 Tue, 4:30–6:00PM, Eckhart 202

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.

Apr 23, 2013 Tue, 4:30–6:00PM, Eckhart 202

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.

May 07, 2013 Tue, 4:30–6:00PM, Eckhart 202

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).

May 17, 2013 Fri, 4:00–5:30PM, Eckhart 206 (note special day/time/venue)

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.

May 21, 2013 Tue, 4:30–6:00PM, Eckhart 202

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).

Oct 02, 2013 Wed, 4:30–6:00PM, Eckhart 312

Frank Sottile, Department of Mathematics, Texas A&M University

Symmetric output feedback control and isotropic Schubert calculus: One area of application of algebraic geometry has been in the theory of the control of linear systems. In a very precise way, a system of linear differential equations corresponds to a rational curve on a Grassmannian. Many fundamental questions about the output feedback control of such systems have been answered by appealing to the geometry of Grassmann manifolds. This includes work of Hermann, Martin, Brockett, and Byrnes. Helmke, Rosenthal, and Wang initiated the extension of this to linear systems with structure corresponding to symmetric matrices, showing that for static feedback it is the geometry of the Lagrangian Grassmannian which is relevant. In my talk, I will explain this relation between geometry and systems theory, and give an extension of the work of Helmke, et al. to linear systems with skew-symmetric structure. For static feedback, it is the geometry of spinor varieties which is relevant, and for dynamic feedback it is quantum cohomology and orbifold quantum cohomology of Lagrangian and orthogonal Grassmannians. This is joint work with Chris Hillar.

Oct 23, 2013 Wed, 4:30–6:00PM, Eckhart 312

Mohan Ramachandran, Department of Mathematics, State University of New York, Buffalo

On Holomorphic convexity of reductive coverings of compact Kahler surfaces: I will talk about the following result which is joint work with Terrence Napier. If a reductive covering of a compact Kahler surface does not have two ends then it is holomorphically convex.

Nov 06, 2013 Wed, 4:30–6:00PM, Eckhart 312

Maurice Rojas, Department of Mathematics, Texas A&M University

From complexity to geometry over local fields: The Shub-Smale tau-Conjecture is a hitherto unproven statement (on integer roots of polynomials in one variable) whose truth would resolve two variants of the P vs. NP Problem. We give a simpler statement, potentially easier to prove, whose truth implies the hardness of the permanent. Along the way, we discuss new upper bounds on the number of p-adic valuations of roots of certain sparse polynomial systems, culminating in a purely tropical geometric statement that implies the hardness of the permanent. Our framework also yields new complexity lower bounds for the permanent, even if only weaker versions of our conjectures are proved. Some of the results presented are joint work with Pascal Koiran and Natacha Portier. We assume no background in complexity theory.

Nov 20, 2013 Wed, 4:30–6:00PM, Eckhart 312

Hal Schenck, Department of Mathematics, University of Illinois, Urbana-Champaign

Geometry of Wachspress surfaces: Let P_d be a convex polygon with d vertices. The associated Wachspress surface W_d is a fundamental object in approximation theory, defined as the image of the rational map w_d from P² to P^{d - 1}, determined by the Wachspress barycentric coordinates for P^d. We show w_d is a regular map on a blowup X_d of P², and if d > 4 is given by a very ample divisor on X_d, so has a smooth image W_d. We determine generators for the ideal of W_d, and prove that in graded lex order, the initial ideal of I(W_d) is given by a Stanley–Reisner ideal. As a consequence, we show that the associated surface is arithmetically Cohen–Macaulay, of Castelnuovo–Mumford regularity two, and determine all the graded betti numbers of I(W_d).

Jan 08, 2014 Wed, 4:30–6:00PM, Eckhart 312

Benjamin Bakker, Courant Institute of Mathematical Sciences, New York University

The geometry of the Frey–Mazur conjecture: A crucial step in the proof of Fermat's last theorem was Frey's insight that a nontrivial solution would yield an elliptic curve with modular p-torsion but which was itself not modular. The connection between an elliptic curve and its p-torsion is very deep: a conjecture of Frey and Mazur, stating that the p-torsion group scheme actually determines the elliptic curve up to isogeny (at least when p > 13), implies an asymptotic generalization of Fermat's last theorem. We study a geometric analog of this conjecture, and show that over function fields the map from isogeny classes of elliptic curves to their p-torsion group scheme is one-to-one. Our proof involves understanding curves on a certain Shimura surface, and fundamentally uses the interaction between its hyperbolic and algebraic properties. This is joint work with Jacob Tsimerman.

Feb 05, 2014 Wed, 4:30–6:00PM, Eckhart 312

Prakash Belkale, Department of Mathematics, University of North Carolina, Chapel Hill

Quantum cohomology and conformal block divisors: Recent work of Fakhruddin has refocussed attention on conformal block divisors on moduli spaces of marked curves, in particular to the birational geometry of moduli spaces of genus zero curves with marked points: Conformal blocks give a interesting family of numerically effective divisors, and hence relate to well known conjectures on nef cones of moduli spaces of curves. I will describe joint work with Angela Gibney and Swarnava Mukhopadhyay where we study the higher level theory of these divisors: in particular producing vanishing theorems, new symmetries and non- vanishing properties of these divisors (one of our tools is the relation to quantum cohomology of Grassmannians). These properties are then applied to the study of moduli spaces.

Apr 02, 2014 Wed, 4:30–6:00PM, Ryerson 251 (joint meeting with Theory Seminar)

Jin-Yi Cai, Department of Computer Science, University of Wisconsin, Madison

Siegel's theorem, edge coloring, and a holant dichotomy: What do Siegel's theorem on finiteness of integer solutions have to do with complexity theory? In this talk we discuss a new complexity dichotomy theorem for counting problems. Such a dichotomy is a classification of a class of problems into exactly two kinds: those that are polynomial time computable, and those that are #P-hard, and thus intractable. (For logicians, a complexity dichotomy theorem is a kind of restricted anti-Friedberg–Muchnick Theorem.) An example problem in this dichotomy is the problem of counting the number of valid edge colorings of a graph. We will show that an effective version of Siegel's theorem and some Galois theory are key ingredients in the proof of this dichotomy. Along the way we will also meet the Tutte polynomial, medial graphs, Eulerian orientations, Puiseux series, and a certain lattice condition on the (logarithms of) the roots of polynomials with integer coefficients. Joint work with Heng Guo and Tyson Williams.

Apr 09, 2014 Wed, 4:30–6:00PM, Eckhart 312

Ron Donagi, Department of Mathematics, University of Pennsylvania

Moduli of super Riemann surfaces: We study various aspects of supergeometry, including obstruction, Atiyah, and super-Atiyah classes. This is applied to the geometry of the moduli space of super Riemann surfaces. We prove that for genus greater than or equal to 5, this moduli space is not projected (and in particular is not split): it cannot be holomorphically projected to its underlying reduced manifold. Physically, this means that certain approaches to superstring perturbation theory that are very powerful in low orders have no close analog in higher orders. Mathematically, it means that the moduli space of super Riemann surfaces cannot be constructed in an elementary way starting with the moduli space of ordinary Riemann surfaces. It has a life of its own. Joint work with Edward Witten.

Apr 23, 2014 Wed, 4:30–6:00PM, Eckhart 312

Saugata Basu, Department of Mathematics, Purdue University

Some quantitative results in real algebraic geometry: In this talk I will discuss two recent results on bounding the topological complexity of real semi-algebraic sets. The first result (joint with Sal Barone) aims at proving an analogue of the Bezout inequality for varieties defined over real closed fields. This result is motivated partly by the requirements of the new "polynomial partitioning" method recently introduced into discrete geometry by Guth and Katz. The second result (joint with Cordian Riener) is a polynomial bound (for fixed degrees) on the S_n-equivariant Betti numbers of symmetric semi-algebraic sets. The underlying leitmotif of the talk is to contrast the real and complex cases, and to point out important ways in which real algebraic geometry differs from complex algebraic geometry in certain respects. The talk is based on the results contained in the papers arXiv:1303.1577 [math.AG] and arXiv:1312.6582 [math.AG].

May 21, 2014 Wed, 4:30–6:00PM, Eckhart 312

David Morrison, Department of Mathematics, University of California, Santa Barbara

Hodge theory and Gromov–Witten invariants: The original mirror symmetry predictions of Gromov–Witten invariants of Calabi–Yau threefolds relied heavily on the behavior of a degenerating variation of Hodge structure near the boundary of Calabi–Yau moduli space. This led to a definition in the early 1990's of the "A-variation of Hodge structure": a degenerating variation of Hodge structure directly constructed from the Gromov–Witten invariants themselves.

Recently, there have been advances in the physical study of the "A-model" (the physical theory leading to Gromov–Witten invariants), which have revealed that one aspect of the original definition of A-VHS needs clarification and modification. The modification involves the Gamma class, a characteristic class closely related to the Gamma function.

We will explain this modification, and discuss some interesting examples. If time permits, we will also describe the new physics calculation which directly leads to Gromov–Witten invariants without invoking a mirror.

May 28, 2014 Wed, 4:30–6:00PM, Eckhart 312

Kapil Paranjape, Indian Institute of Science Education and Research, Mohali

Special Calabi–Yau varieties and modular forms: In this talk, we will examine joint work of the speaker with Dinakar Ramakrishnan where a relation between modular forms and certain special types of Calabi–Yau varieties is explored. We will begin by spelling out the proposed relation and some of the consequences of this. Finally, we will discuss one example in some detail.

Jun 04, 2014 Wed, 4:30–6:00PM, Eckhart 203 (note special venue)

Domingo Toledo, Department of Mathematics, University of Utah

Fundamental groups of compact Kähler manifolds: Survey of results and open problems: There has been great interest in the study of the topology of complex algebraic vatieties at least since the foundational work of Lefschetz and Hodge in the 1920's and 30's. The fundamental group was emphasized by Serre in the 1950's and 60's. The basic problem is to know if there is a group-theoretic characterization of the fundamental groups of a given class of algebraic varieties, say of smooth projective varieties. This seems presently out of reach. In this talk I will attempt to summarize the present state of knowledge: A number of restrictions are known, usually obtained by some kind of Hodge theory and apply to compact Kähler manifolds. A number of interesting examples are also known. I will also discuss a number of open problems.

Dec 03, 2014 Wed, 4:30–6:00PM, Eckhart 312

Andreea Nicoara, University of Pennsylvania

Direct proof of termination of the Kohn algorithm in the real-analytic case: In 1979 J.J. Kohn gave an indirect argument using the Diederich-Fornaess theorem that his algorithm terminates on a pseudoconvex real-analytic domain of finite D'Angelo type. I will give a direct argument for the same assertion by constructing subelliptic multipliers that give a subelliptic estimate at each boundary point in terms of Catlin's boundary system at that point. I will also show what else is needed (two ingredients) in order to turn this argument into one that yields an effective lower bound for the subelliptic gain in terms of the dimension, D'Angelo type, and order of the forms for any pseudoconvex real-analytic domain of finite D'Angelo type.

Jan 21, 2015 Wed, 4:30–6:00PM, Eckhart 312

Vasudevan Srinivas, School of Mathematics, Tata Institute of Fundamental Research

Étale motivic cohomology and algebraic cycles: This talk will report on joint work with A. Rosenschon. There are examples showing that the torsion and co-torsion of Chow groups are complicated, in general, except in the "classical" cases (divisors and 0-cycles, and torsion in codimension 2). Instead, we may (following Lichtenbaum) consider the etale Chow groups, which coincide with the usual ones if we use rational coefficients; we show that they have better torsion and cotorsion if we work over the complex numbers. In contrast, they can have infinite torsion in some arithmetic situations (the usual Chow groups are conjectured to be finitely generated).

Feb 4, 2015 Wed, 4:30–6:00PM, Eckhart 312

Dawei Chen, Boston College

Boundary behavior of strata of holomorphic one-forms: Consider the stratum of holomorphic one-forms on Riemann surfaces that have a fixed number of zeros and multiplicities. They define flat structures that can realize the underlying surfaces as plane polygons of similar type. In this talk, I will report some recent results on the degeneration of holomorphic one-forms in a stratum when the underlying Riemann surfaces become nodal, with a focus on the interplay between algebraic geometry and flat geometry.

Feb 11, 2015 Wed, 4:30–6:00PM, Eckhart 312

Dan Halpern-Leistner, Institute for Advanced Study

Reductive moduli problems, stratifications, and applications: Many moduli problems in algebraic geometry are "too big" to possibly be parameterized by a quasiprojective scheme. Nevertheless one can find a stratification of the moduli problem for which the large open stratum has a good moduli space, and the remaining strata have nice modular interpretations as well. I will introduce a framework for generalizing and analyzing stratifications of this kind arising in geometric invariant theory and in moduli problems for objects in derived categories of coherent sheaves, and I will discuss some applications of these stratifications to understanding the geometry of these moduli problems. This framework leads to the notion of a "reductive moduli problem" (which generalizes the notion of a reductive group) -- these are the moduli problems for which the results of geometric invariant theory generalize in a nice way.

Mar 04, 2015 Wed, 4:30–6:00PM, Eckhart 312

Jerzy Weyman, University of Connecticut

Semi-invariants of quivers, cluster algebras and the hive model: The saturation theorem for Littlewood-Richardson coefficients was a fashionable subject about a decade ago. There are two completely different proofs of the theorem: the original one by Knutson-Tao based on their hive model, and a proof based on quiver representations given by Harm Derksen and myself. So far there was no link between these two proofs.

Recently Jiarui Fei discovered a remarkable cluster algebra structure on the ring SI(T_n,n,n,β(n)) of semi-invariants of a triple flag quiver, whose weight spaces have dimensions that are Littlewood-Richardson coefficients.

In proving his result he uses both the hive model and the quiver representations. It turns out that the link between the two approaches is the quiver with potential underlying the cluster algebra structure. The combinatorics of g-vectors for this quiver with potential turns out to be identical to the hive model.

In my talk I will explain the notions involved and basic ideas behind Jiarui Fei's proof.

Mar 11, 2015 Wed, 4:30–6:00PM, Eckhart 312

Andrew Obus, University of Virginia

The local lifting problem, the Oort Conjecture, and its generalizations. Let k be an algebraically closed field of characteristic p. The local lifting problem asks if the action of a finite group G by k-automorphisms on k[[t]] can be lifted to an action of G on R[[t]], where R is some characteristic zero DVR with residue field k. This is motivated by the problem of lifting a Galois branched cover of smooth projective algebraic curves from characteristic p to characteristic zero.

The Oort conjecture (now a theorem of Obus-Wewers and Pop) states that cyclic actions can always be lifted (for some R). We will discuss a generalization of this conjecture to the case of metacyclic actions, as well as recent progress by the speaker on this problem. A fundamental technique is the use of Kato's generalization of the Swan conductor.

Mar 18, 2015 Wed, 4:30–6:00PM, Eckhart 312

Jochen Heinloth, University of Essen

An introduction to the P=W conjecture and related conjectures of Hausel.The intersection form on moduli of Higgs bundles vanishes - a conjecture of Hausel.

Hausel made a series of conjectures on the global geometry of moduli spaces of Higgs bundles. One of these conjectures turns out to be closely related to a boundary case of the support theorem. In this talk I'd like to explain the conjecture and how one can prove it geometrically.

Apr 01, 2015 Wed, 4:30–6:00PM, Eckhart 312

Alena Pirutka, École Polytechnique

Stable rationality and quartic threefolds.

Let X be a smooth complex algebraic variety. Recall that X is rational if it is birational to a projective space, X is stably rational if a product of X with some projective space becomes rational and X is unirational if it is rationally dominated by a projective space. A classical question is to distinguish the properties of rationality and unirationality. In 1970s three examples of unirational but not rational varieties were discovered : cubic threefolds (Clemens and Griffiths), some quartic threefolds (Iskovskikh and Manin) and some conic bundles (Artin and Mumford). The example of Artin and Mumford is not stably rational, but it was not known if this property holds for other examples. In this talk we will discuss the case of quartic threefolds and show that many of them are not stably rational. This is a work in common with J.-L. Colliot-Thélène. The methods we use are based on the properties of the diagonal decomposition in the Chow groups, the universal properties of the Chow group of zero cycles as well as some specialization techniques.

Apr 02, 2015 Wed, 4:30–6:00PM, Eckhart 206 (note special venue).

Bruno Klingler, Jussieu

The hyperbolic Ax-Lindemann-Weierstrass conjecture.

The hyperbolic Ax-Lindemann-Weierstrass conjecture is a functional algebraic independence statement for the uniformizing map of an arithmetic variety. In this talk I will describe the conjecture, its role and its proof (joint work with E. Ullmo and A. Yafaev).

Apr 08, 2015 Wed, 4:30–6:00PM, Eckhart 312.

Tatsunari Watanabe, Duke

Rational Points of Generic Curves in Positive Characteristics

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. Using the theory of weighted completion, we prove the analogous result in positive characteristics. The theory of weighted completion was developed by Richard Hain and Makoto Matsumoto. It is a variant of relative completion due to Deligne and can be used to "linearize" a profinite group such as the arithmetic mapping class groups.

Hain used it to show that the section conjecture holds for the generic curve of type (g,0) in characteristic zero for g>2. Using comparison theorems, we can also prove the analogous result in positive characteristics.

Apr 22, 2015 Wed, 4:30–6:00PM, Eckhart 312.

Chris Hall, IAS

Sequences of curves with growing gonality

Given a smooth irreducible complex curve C, there are several isomorphism invariants one can attach to C. One invariant is the genus of C, that is, the number of handles in the corresponding Riemann surface. A subtler invariant is the gonality of C, that is, the minimal degree of a dominant map from C to P¹. A lower bound for either invariant has diophantine consequences when bounds depends on how C is presented. In this talk we will consider a sequence of finite unramified covers of C and give spectral criteria for the gonality of the curves in the sequence to tend to infinity.

May 06, 2015 Wed, 4:30–6:00PM, Eckhart 312.

Jesse Kass, University of South Carolina

How to make Poincaré Duality into a regular morphism

Poincaré Duality of a smooth complex curve — the duality isomorphism that describes how cycles intersect — can be realized by a holomorphic map between complex manifolds called the Abel map. Starting with the definition of the Abel map, I review this result and then explain how it extends to singular curves. In doing so, I describe the compactified Jacobian of a curve with ordinary n-fold singularities and, if time permits, discuss some connections with Dima Arinkin's work on autoduality. This work is joint with Kirsten Wickelgren.

May 13, 2015 Wed, 4:30–6:00PM, Eckhart 312.

Adam Topaz, University of California, Berkeley

On mod-l birational anabelian geometry

In the early 90's, Bogomolov introduced a program whose ultimate goal is to reconstruct function fields of dimension > 1 over algebraically closed fields from their pro-l 2-step nilpotent Galois groups. Although it is far from being resolved in full generality, this program has since been carried through for function fields over the algebraic closure of a prime field. Unfortunately, when passing to the mod-l 2-step nilpotent Galois group, one can no longer use the fundamental theorem of projective geometry, which plays a crucial role in the pro-l situation. After an introduction to Bogomolov's program, in this talk I will describe some progress in the mod-l context which overcomes this difficulty.

May 27, 2015 Wed, 5:00–6:00PM, Eckhart 312.

Tyler Kelly, University of Cambridge

Equivalences of Calabi-Yau hypersurfaces in Toric Varieties

Given Calabi-Yau hypersurfaces in a fixed toric variety, there are various constructions to find its mirror. Sometimes they are isomorphic, but sometimes they are not. Mirror symmetry predicts they still should be equivalent in some sense. In this talk, we will show that these (stacky) mirrors are birational and derived equivalent. If we have time, we will describe applications to more general contexts, depending on audience interest, about either lattice polarisations of families of K3 surfaces in toric varieties or extensions to Calabi-Yau complete intersections in toric varieties.

June 03, 2015 Wed, 4:30–6:00PM, Eckhart 312.

Paolo Rossi, CNRS, Institut de Mathématiques de Bourgogne

Double ramification cycles and integrable systems

In a series of papers with A. Buryak and B. Dubrovin we are studying the algebraic structure behind the intersection theory of the double ramification cycle, a cycle inside the moduli space of stable marked curves, heuristically representing (a compactification of) the locus of those Riemann surfaces whose marked points support a principal divisor. It turns out that natural generating functions of intersection numbers produce infinite dimensional integrable Hamiltonian systems (this part was inspired by Eliashberg's symplectic field theory) and even previously unknown quantization of such field theories. Buryak has conjectured that these systems are equivalent to the Dubrovin-Zhang systems involved in Witten's conjecture and we have gathered quite some evidence of this fact, proving it in a number of special cases. This is a typical and quite new application of moduli space geometry to mathematical physics and I will outline its main ideas and results.

October 20, 2015 Tue, 5:30–7:00PM, Eckhart 312.

Sergey Galkin, Moscow

Measurements of varieties

I will discuss a few ways to define various rings of varieties and rings of categories, and how to introduce algebraic operations on these rings. These rings can be thought as decategorifications of the respective categories, and some of them possess natural algebraic operations which can be recategorified. It is possible to prove identities in these rings, which might be powerful enough to answer geometric questions. This talk is based on joint work with Evgeny Shinder.

November 10, 2015 Tue, 5:30–7:00PM, Eckhart 312.

Alexandru Buium, University of New Mexico

Connections and curvature on Spec Z

An arithmetic analogue of differential geometry can be developed in which functions are replaced by integer numbers and partial derivatives are replaced by Fermat quotient operators. Chern and Levi-Civita connections are shown to exist in this context. The Christoffel symbols have, as analogues, "higher dimensional Legendre symbols". Curvatures of these connections can be introduced and computed via "analytic continuation between primes". As a result the spectrum of the integers appears, in this setting, as an "infinite dimensional manifold" that is "naturally curved".

February 17, 2016 Wed, 4:30–6:00PM, Eckhart 312.

John Lesieutre, University of Illinois at Chicago

Dynamical Mordell--Lang and automorphisms of higher-dimensional varieties

The dynamical Mordell--Lang conjecture states that if $f$ is an endomorphism of a complex variety $X$, with $p$ a point of $X$ and $V$ a subvariety, then the set of $n$ for which $f^n(p)$ lands in $V$ is a union of a finite set and finitely many arithmetic progressions. When $f$ is \'etale, this is a result of Bell--Ghioca--Tucker. I'll discuss an extension of this result to the setting in which $p$ and $V$ are non-reduced closed subschemes of $X$, and show how this statement can be applied to study dynamically interesting (e.g. positive entropy) automorphisms of complex varieties in higher dimensions. This is joint work with Daniel Litt.

February 24, 2016 Wed, 4:30–6:00PM, Eckhart 312.

Martin Helmer, University of California, Berkeley.

Algorithms to Compute Characteristic Classes of Subschemes of Certain Toric Varieties

Let X be a complete smooth toric variety where all Cartier divisors in Pic(X) are nef and let V be a subscheme of X. We give a new expression for the Segre class of the subscheme V, s(V,X), in terms of the projective degrees of a rational map associated to V. We also give a concrete and computable expression for these projective degrees. These results are applied to develop effective algorithms for the computation of the Chern-Schwartz-MacPherson class, Segre class and the Euler characteristic of V. The algorithms will, in particular, be applicable to any subscheme of a product of projective spaces. In the case of smooth subschemes V this will also allow us to compute the total Chern class of V. The algorithms may be implemented symbolically using Groebner basis or numerically using homotopy continuation via a package such as Bertini or PHCPack. The algorithms have been implemented in Macaulay2 and an M2 package is available. The algorithms described perform favourably on a wide selection of examples in comparison to other known algorithms. Theoretical running time bounds for several of the algorithms are also given. In the talk we will focus first on describing the algorithm in the special case where X is a projective space of dimension n to allow for a cleaner and more concrete exposition of concepts; the more general case will be seen to follow from this discussion.

April 13, 2016 Wed, 4:30–6:00PM, Eckhart 312.

Ivan Losev, Northeastern University

Deformations of symplectic singularities and the orbit method

Symplectic singularities were introduced by Beauville in 2000. These are especially nice singular Poisson algebraic varieties that include symplectic quotient singularities and the normalizations of orbit closures in semisimple Lie algebras. Poisson deformations of conical symplectic singularities were studied by Namikawa in 2009 who proved that they are classified by points of a vector space. Recently I have proved that quantizations of conical symplectic singularities are still classified by the points of the same vector spaces. I will explain these results and then apply them to establish a version of Kirillov's orbit method for semisimple Lie algebras.

April 20, 2016 Wed, 4:30–6:00PM, Eckhart 312.

Joe Kileel , University of California, Berkeley

The Chow form of the essential variety in computer vision

In computer vision, 3D reconstruction is a fundamental task: starting from photographs of a world scene, taken by cameras with unknown positions and orientations, how can we best create a 3D model of that world scene? In this talk, we will introduce and answer a basic mathematical problem when the number of cameras is two, left open by Googler Sameer Agarwal and his co-authors. The answer is a determinantal formula for the Chow form of the configuration space of two calibrated cameras, which is a special five-dimensional variety in P8. The formula is in the spirit of classical Bezoutian formulas for resultants, but we need secant varieties, representations of GL4 and Ulrich sheaves to derive it. At the end, I will report on numerical experiments that illustrate the robustness of this result, and I will indicate where it might fit inside 3D reconstruction algorithms. Joint with Gunnar Floystad and Giorgio Ottaviani.

April 20, 2016 Wed, 4:30–6:00PM, Eckhart 312.

Botong Wang , University of Wisconsin - Madison

Cohomology jump loci and examples of nonKahler manifolds

Cohomology jump loci are generalizations of usual cohomology groups of a topological space. I will first give an introduction on the theory of cohomology jump loci of projective, quasi-projective and compact Kahler manifolds. Then I will introduce some concrete examples of (real) 6-dimensional symplectic-complex Calabi-Yau manifolds, which satisfies the standard topological criterions of compact Kahler manifolds such as Hodge theory and Hard Lefschetz theorem, but fail the cohomology jump locus property of compact Kahler manifolds.

May 11, 2016 Wed, 4:30–6:00PM, Eckhart 312.

JM Landsberg , Texas A&M

Geometry of algorithms and the punctual Hilbert schemes

Ever since Strassen discovered in 1969 that the usual method for multiplying matrices is not the optimal one, it has been a central open question to determine just how efficiently matrices can be multiplied. Computer scientists make the astounding conjecture that as the size of the matrices gets large, it becomes nearly as fast to multiply matrices as it is to add them. Geometry has played a role in proving lower complexity bounds for matrix multiplication via equations for secant varieties of Segre varieties. In this talk I will discuss a new role for geometry via a "variety of algorithms". These methods are promising for both further lower bounds and constructing new algorithms. I will discuss two ongoing projects, one with Michalek, the other with Ballard, Ikenmeyer and Ryder.

September 27, 2016 Tue, 4:30–6:00PM, Eckhart 312.

Aaron Silberstein , U Chicago

The Disjoint Divisors Theorem

We will discuss when you can conclude that a collection of divisors on an algebraic variety are contained in the fibers of a morphism from that variety to a curve. Joint with F. Bogomolov and A. Pirutka.

October 4, 2016 Tue, 4:30–6:00PM, Eckhart 312.

Aaron Silberstein , U Chicago

Bogomolov's Program of Birational Anabelian Geometry: Recent Results

We give an introduction to Bogomolov's program of birational anabelian geometry.

October 11, 2016 Tue, 4:30–6:00PM, Eckhart 312.

Aaron Silberstein , U Chicago

TBA

TBA

October 25, 2016 Tue, 4:30–6:00PM, Eckhart 312.

Restricted local volumes and deformation theory

In this talk I will introduce the restricted local volume of a line bundle as a powerful tool of deformation theory. As the name suggests it, the restricted local volume is the local counterpart of the usual restricted volume of a line bundle as introduced by Ein, Lazarsfeld, Mustață, Nakamaye and Popa . I will present a result of mine that determines the change of the restricted local volume across flat families for a fairly general class of line bundles. The change turns out to be the degree of certain projective scheme. Then I will discuss applications to deformation theory of singularities generalizing previous work of Gaffney, Kleiman, Teissier and Hironaka among others.

November 22, 2016 Tue, 4:30–6:00PM, Eckhart 312.

Fedor Bogomolov, New York University

Unramified correspondences and torsion of elliptic curves

I will report on the results of an ongoing project which we began some years ago with Yuri Tschinkel and continued with Hang Fu and Jin Qian. We say that a smooth projective curve \(C\) dominates \(C'\) if there is nonramified covering \(\tilde C\) of \(C\) which has a surjection onto \(C'\). Thanks to Bely's theorem we can show that any curve \(C'\) defined over \(\overline{\mathbb{Q}}\) is dominated by one of the curves \(C_n, y^n-1= x^2\). Over \(\overline{\mathbb{F}}_p\) any curve in fact is dominated by \(C_6\) which is in a way also a minimal possible curve with such a property. Conjecturally the same holds over \(\overline{\mathbb{Q}}\) but at the moment we can prove only partial results in this direction. There are not many methods to establish dominance for a particular pair of curves and the one we use is based on the study of torsion points and finite unramified covers of elliptic curves. In fact for any ellitpic curve \(E\) over the complex numbers there is a uniquely defined subset of \(4\) points in \(\mathbb{P}^1\) modulo projective transformations defining the curve.These points correspond naturally to a subgroup of points of order \(2\) in \(E\) and there is a well defined (modulo projective transformation) subset of the images of torsion points from \(E\) in \(\mathbb{P}^1\). The corresponding subsets \(PE\) in \(P^1\) for different elliptic curves \(E, E'\) have finite intersection and in many cases we can show that such intersection consists of one point. On the other hand there are such subsets with intersection at least \(22\). It raises a question about the existence of a universal upper bound for such intersections. This question is somewhat related to a question of Serre in the theory of Galois representations. Any subset \(PE\) defines a bigger subset \(SE\) in \(\mathbb{P}^1 \) by closing it by elliptic division. Namely \(SE\) contains \(PE\) and \(PE'\) for any \(E'\) defined by four points in \(SE\). In the case when \(E\) is defined over \(\overline{\mathbb{Q}}\) we show that \(SE\) is projectively equivalent to \(\mathbb{P}^1(KE)\) where \(KE\) is an infinite extension of \(\mathbb{Q}\) which is not equal to \(\overline{\mathbb{Q}}\) and varies for different elliptic curves over \(\overline{\mathbb{Q}}\).

December 6, 2016 Tue, 4:30–6:00PM, Eckhart 312.

Zinovy Reichstein, University of British Columbia

The rationality problem for forms of moduli spaces of stable marked curves

Let \(M_{g, n}\) be the moduli space of stable curves of genus \(g\) with \(n\) marked points. It is a classical problem in algebraic geometry to determine which of these spaces are rational over the field of complex numbers. In this talk, based on joint work with Matthieu Florence, I will discuss the rationality problem for twisted forms of \(M_{g, n}\). Twisted forms of \(M_{g, n}\) are of interest because they shed light on the arithmetic geometry of \(M_{g, n}\), and because they are coarse moduli spaces for natural moduli problems in their own right. A classical result of Enriques, Manin, and Swinnerton-Dyer asserts that every form of \(M_{0, 5}\) is rational. (The \(F\)-forms \(M_{0, 5}\) are precisely the del Pezzo surfaces of degree 5 over \(F\).) Matthieu and I have generalized this result to \(M_{0, n}\) for \(n \geq 5\). We also have some positive results for forms of \(M_{g, n}\), where \(g \leq 5\) (for small \(n\)).

March 28, 2017 Tue, 4:30–6:00PM, Eckhart 312.

Djordje Milicevic, Bryn Mawr College

On moments of twisted L-functions

Central values of L-functions encode essential arithmetic information in contexts ranging from distribution of primes to elliptic curves and arithmetic manifolds. In this talk, I will present asymptotic formulas for moments in families of twisted L-functions with all primitive characters modulo q, with a power saving in q, obtained by combining analytic and algebro-geometric techniques. On the one hand, we use the full power of spectral theory of GL(2) automorphic forms to treat a possibly highly unbalanced shifted convolution problem; on the other hand, we combine analytic number theory and input from algebraic geometry (including the Riemann Hypothesis for curves over finite fields and independence of Kloosterman sheaves) to prove estimates on bilinear forms in Kloosterman sums in critical ranges. The emphasis in this talk will be on explaining how these various methods fit together as well as how algebraic geometry naturally enters the analytic problem of asymptotic evaluation of moments. In addition to providing statistical and intrinsic information about the underlying family of automorphic forms, asymptotics of moments are an essential ingredient in analytic approaches to questions of arithmetic importance such as upper bounds, nonvanishing, or extreme values, and I will also survey several of our applications. This is joint work with Blomer, Fouvry, Kowalski, Michel, and Sawin.

April 18, 2017 Tue, 4:30–6:00PM, Eckhart 312.

Lucia Mocz, Princeton University

Title TBD

Abstract TBD

May 2, 2017 Tue, 4:30–6:00PM, Eckhart 312.

Wei Ho, University of Michigan

Title TBD

Abstract TBD

November 1, 2017 Wed, 4:30–6:00PM, Eckhart 312.

Jose Israel Rodriguez, University of Chicago

Numerical computation of Galois groups

Galois groups are an important part of number theory and algebraic geometry. To a parameterized system of polynomial equations one can associate a Galois group whenever the system has k (finitely many) nonsingular solutions generically. This Galois group is a subgroup of the symmetric group on k symbols. Using random monodromy loops it has already been shown how to compute Galois groups that are the full symmetric group. In this talk, we show how to compute Galois groups that are proper subgroups of the full symmetric group. We conclude with an implementation using Bertini.m2, an interface to the numerical algebraic geometry software Bertini through Macaulay2. This is joint work with Jonathan Hauenstein and Frank Sottile.

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CONTACTS

For further information on this seminar, please email Lek-Heng Lim at lekheng(at)galton.uchicago.edu, Madhav Nori at nori(at)math.uchicago.edu, Jose Israel Rodriguez at JoIsRo@UChicago.edu, or Aaron Silberstein at asilbers(at)math.uchicago.edu.