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.

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

WINTER 2017 |
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Oct 18, 2017 | Isabel Vogt (MIT) | ||

Oct 25, 2017 | James Pascaleff (UIUC) | ||

Nov 1, 2017 | Jose Israel Rodriguez (University of Chicago) | ||

SPRING 2017 |
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Mar 28, 2017 | Djordje Milicevic (Bryn Mawr College) | ||

Apr 18, 2017 | Lucia Mocz (Princeton University) | ||

May 2, 2017 | Wei Ho (University of Michigan) | ||

May 23, 2017 | Florian Pop (University of Pennsylvania) | ||

FALL 2016 |
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Sep 27, 2016 | Aaron Silberstein (University of Chicago) | ||

Oct 4, 2016 | Aaron Silberstein (University of Chicago) | ||

Oct 11, 2016 | Aaron Silberstein (University of Chicago) | ||

Oct 25, 2016 | Antoni Rangachev (Northeastern University) | ||

Nov 22, 2016 | Fedor Bogomolov (New York University) | ||

Dec 6, 2016 | Zinovy Reichstein (University of British Columbia) | ||

SPRING 2016 |
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Apr 13, 2016 | Ivan Losev (Northeastern University) | ||

Apr 20, 2016 | Joe Kileel (University of California, Berkeley) | ||

Apr 27, 2016 | Botong Wang (University of Wisconsin Madison) | ||

May 11, 2016 | J.M. Landsberg (Texas A&M University) | ||

WINTER 2016 |
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Feb 17, 2016 | John Lesieutre (University of Illinois, Chicago) | ||

Feb 24, 2016 | Martin Helmer (University of California, Berkeley) | ||

FALL 2015 |
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Oct 20, 2015 | Sergey Galkin (Moscow) | ||

Nov 10, 2015 | Alexandru Buium (University of New Mexico) | ||

SPRING 2015 |
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Apr 01, 2015 | Alena Pirutka (École Polytechnique) | ||

Apr 02, 2015 | Bruno Klingler (Jussieu) In room 206. | ||

Apr 08, 2015 | Tatsunari Watanabe (Duke University) | ||

Apr 22, 2015 | Chris Hall (IAS) | ||

May 06, 2015 | Jesse Kass (University of South Carolina) | ||

May 13, 2015 | Adam Topaz (University of California, Berkeley) | ||

May 20, 2015 | Chris Peterson (Colorado State University) | ||

May 27, 2015 | Tyler Kelly (University of Cambridge) from 5-6 pm | ||

June 03, 2015 | Paolo Rossi (CNRS, Institut de Mathématiques de Bourgogne) | ||

WINTER 2015 |
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Jan 21, 2015 | Vasudevan Srinivas (Tata Institute for Fundamental Research) | ||

Feb 04, 2015 | Dawei Chen (Boston College) | ||

Feb 11, 2015 | Dan Halpern-Leistner (Institute for Advanced Study) | ||

Feb 18, 2015 | Jakob Stix (University of Frankfurt) CANCELLED | ||

Mar 04, 2015 | Jerzy Weyman (University of Connecticut) | ||

Mar 11, 2015 | Andrew Obus (University of Virginia) | ||

Mar 18, 2015 | Jochen Heinloth (University of Essen) | ||

FALL 2014 |
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Dec 03, 2014 | Andreea Nicoara (University of Pennsylvania) | ||

(General) hiatus — the organizers are away. | |||

SPRING 2014 |
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Apr 02, 2014 | Jin-Yi Cai (University of Wisconsin, Madison), joint meeting with Theory Seminar in Ryerson 251 | ||

Apr 09, 2014 | Ron Donagi (University of Pennsylvania) | ||

Apr 23, 2014 | Saugata Basu (Purdue University) | ||

May 21, 2014 | David Morrison (University of California, Santa Barbara) | ||

May 28, 2014 | Kapil Paranjape (Indian Institute of Science Education and Research, Mohali) | ||

Jun 04, 2014 | Domingo Toledo (University of Utah) | ||

WINTER 2014 |
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Jan 08, 2014 | Benjamin Bakker (New York University) | ||

Feb 05, 2014 | Prakash Belkale (University of North Carolina, Chapel Hill) | ||

Unusually severe winter. Cancelled talks have all been rescheduled to Spring. | |||

FALL 2013 |
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Oct 02, 2013 | Frank Sottile (Texas A&M University) | ||

Oct 23, 2013 | Mohan Ramachandran (State University of New York, Buffalo) | ||

Nov 06, 2013 | Maurice Rojas (Texas A&M University) | ||

Nov 20, 2013 | Hal Schenck (University of Illinois, Urbana-Champaign) | ||

SPRING 2013 |
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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) | ||

WINTER 2013 |
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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) | ||

FALL 2012 |
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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).

Rob de Jeu, Department of Mathematics, Vrije Universiteit Amsterdam

**The syntomic regulator for K_{4} of curves:**
Let

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.

**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.

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
|*d**A* + *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).

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.

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.

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.

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
**P**^{2} 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
**P**^{2}, 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}).

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.

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.

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.

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.

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]`.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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).

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.

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 **P**^{1}. 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.

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.

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.

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.

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.

Sergey Galkin, Moscow

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.

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".

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.

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.

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.

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.

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.

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.

Aaron Silberstein , U Chicago

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.

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.

Aaron Silberstein , U Chicago

TBA

Antoni Rangachev, Northeastern Univ

** 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.

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 \(\overline{\mathbb{Q}}\).

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 \(n\)).

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.

Lucia Mocz, Princeton University

Abstract TBD

Wei Ho, University of Michigan

Abstract TBD

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.

You may also want to check out:

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`.