Talk_id  Date  Speaker  Title 
4643

Wednesday 1/11 4:10 PM

Xuancheng Fernando Shao, Oxford

Understanding sieve via additive combinatorics
 Understanding sieve via additive combinatorics
 01/11/2017
 4:10 PM  5:00 PM
 C304 WH
 Xuancheng Fernando Shao, Oxford
Many of the most interesting problems in number theory can be phrased under the general framework of sieve problems. For example, the ancient sieve of Eratosthenes is an algorithm to produce primes up to a given threshold. Sieve problems are in general very difficult, and a class of clever techniques have been discovered in the last 100 years to yield stronger and stronger results. In this talk I will discuss the significance of understanding general sieve problems, and present a novel approach to study them via additive combinatorics. This is joint work with Kaisa Matomaki.

4640

Friday 1/13 4:10 PM

Sanchayan Sen, McGill University

Random discrete structures: Phase transitions, scaling limits, and universality
 Random discrete structures: Phase transitions, scaling limits, and universality
 01/13/2017
 4:10 PM  5:00 PM
 C304 WH
 Sanchayan Sen, McGill University
The aim of this talk is to give an overview of some recent results in two interconnected areas:
a) Random graphs and complex networks: The last decade of the 20th century saw significant growth in the availability of empirical data on networks, and their relevance in our daily lives. This stimulated activity in a multitude of fields to formulate and study models of network formation and dynamic processes on networks to understand realworld systems.
One major conjecture in probabilistic combinatorics, formulated by statistical physicists using nonrigorous arguments and enormous simulations in the early 2000s, is as follows: for a wide array of random graph models on n vertices and degree exponent \tau>3, typical distance both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n^{\frac{\tau\wedge 4 3}{\tau\wedge 4 1}}. In other words, the degree exponent determines the universality class the random graph belongs to. The mathematical machinery available at the time was insufficient for providing a rigorous justification of this conjecture.
More generally, recent research has provided strong evidence to believe that several objects, including
(i) components under critical percolation,
(ii) the vacant set left by a random walk, and
(iii) the minimal spanning tree,
constructed on a wide class of random discrete structures converge, when viewed as metric measure spaces, to some random fractals in the GromovHausdorffProkhorov sense, and these limiting objects are universal under some general assumptions. We report on recent progress in proving these conjectures.
b) Stochastic geometry: In contrast, less precise results are known in the case of spatial systems. We discuss a recent result concerning the length of spatial minimal spanning trees that answers a question raised by Kesten and Lee in the 90's, the proof of which relies on a variation of Stein's method and a quantification of the classical BurtonKeane argument in percolation theory.
Based on joint work with Louigi AddarioBerry, Shankar Bhamidi, Nicolas Broutin, Sourav Chatterjee, Remco van der Hofstad, and Xuan Wang.

4641

Tuesday 1/17 4:10 PM

Thomas Walpuski, MIT

Gauge Theory in Higher Dimensions
 Gauge Theory in Higher Dimensions
 01/17/2017
 4:10 PM  5:00 PM
 C304 WH
 Thomas Walpuski, MIT
Gauge theory is a subject that has emerged from theoretical physics. It has deep links with many areas of mathematics (including partial differential equations, representation theory, algebraic geometry, differential geometry and topology). Most of the mathematical work on gauge theory has focused on low dimensions, where one can exploit the antiselfdual YangMills equation and the analytic difficulties are still quite tractable.
In this talk I will discuss three concrete questions that arise in gauge theory in higher dimensions. First, I will discuss gauge theory on Kähler manifolds, with a particular focus on singular Hermitian YangMills connections. Afterwards, I will move on to the more exotic topic of gauge theory on G2manifolds. I will discuss a method to construct solutions of the YangMills equation on a class of G2manifolds called twisted connected sums. Finally, I will talk about the prospects of defining enumerate gauge theoretical invariants for G2manifolds and the difficulties arising from codimension four bubbling.

4654

Monday 1/23 4:10 PM

Sean Li, University of Chicago

Beyond Euclidean rectifiability
 Beyond Euclidean rectifiability
 01/23/2017
 4:10 PM  5:00 PM
 C304 WH
 Sean Li, University of Chicago
Rectifiable spaces are a class of metric measure spaces that are Lipschitz analogues of differentiable manifolds (for example, they admit a parameterization by Lipschitz charts) and arise naturally in many areas of analysis and geometry. Due to the important works of Federer, Mattila, Preiss, and many others, we now have a good understanding of the geometric properties of rectifiability in Euclidean spaces. In this talk, we examine some generalizations of rectifiability to the setting of nonEuclidean spaces and discuss the similarities and differences between rectifiability in the Euclidean setting and these generalizations.

4658

Wednesday 1/25 4:10 PM

Jenya Sapir, UIUC

Geodesics on surfaces
 Geodesics on surfaces
 01/25/2017
 4:10 PM  5:00 PM
 C304 WH
 Jenya Sapir, UIUC
Let S be a hyperbolic surface. We will give a history of counting results for geodesics on S. In particular, we will give estimates that fill the gap between the classical results of Margulis and the more recent results of Mirzakhani. We will then give some applications of these results to the geometry of curves. In the process we highlight how combinatorial properties of curves, such as selfintersection number, influence their geometry.

4659

Friday 1/27 4:10 PM

Linhui Shen, Northwestern

Representations, Combinatorics, and Configurations.
 Representations, Combinatorics, and Configurations.
 01/27/2017
 4:10 PM  5:00 PM
 C304 WH
 Linhui Shen, Northwestern
We briefly recall KnutsonTao’s hive model that calculates the LittlewoodRichardson coefficients. We consider the configuration spaces of decorated flags introduced by Fock and Goncharov. The configuration spaces admit natural functions called potentials introduced by Goncharov and myself. We prove that the tropicalization of configuration spaces with potentials recovers KnutsonTao’s hives. As an application, Hong and I solve the Saturation problem for the Lie algebra so(2n+1). If time permits, I will further explain their deep connections with geometric Satake correspondence, homological mirror symmetry, and DonaldsonThomas theory.

4665

Wednesday 2/1 4:10 PM

Keerthi Madapusi Pera, University of Chicago

Periods, Lfunctions and abelian varieties
 Periods, Lfunctions and abelian varieties
 02/01/2017
 4:10 PM  5:00 PM
 C304 WH
 Keerthi Madapusi Pera, University of Chicago
Periods are a special class of complex numbers, arising as integrals of differential forms on algebraic varieties. Lfunctions are analytic objects that generalize the Riemann zeta function. Both are objects admitting deceptively simple definitions, but carry deep arithmetic information.
In this talk, I'll explain a relationship between periods of abelian varieties with complex multiplication, and certain Artin Lfunctions, originally conjectured by P. Colmez, and sketch a proof of it that arose out of joint work with Andreatta, Goren and Ben Howard. Among other applications, this relationship has led to a proof by J. Tsimerman of the AndreOort conjecture for Siegel modular varieties.

4671

Friday 2/3 10:00 AM

Yuan Zhou, UC Davis

Cutting plane theorems for Integer Optimization and computerassisted proofs
 Cutting plane theorems for Integer Optimization and computerassisted proofs
 02/03/2017
 10:00 AM  11:00 AM
 1502 EB
 Yuan Zhou, UC Davis
Optimization problems with integer variables form a class of mathematical
models that are widely used in Operations Research and Mathematical Analytics.
They provide a great modeling power, but it comes at a high price: Integer
optimization problems are typically very hard to solve, both in theory and practice.
The stateoftheart solvers for integer optimization problems use cuttingplane
algorithms. Inspired by the breakthroughs of the polyhedral method for
combinatorial optimization in the 1980s, generations of researchers have studied the
facet structure of convex hulls to develop strong cutting planes. However, the
proofs of cutting planes theorems were handwritten, and were dominated by
tedious and errorprone case analysis.
We ask how much of this process can be automated: In particular, can we use
algorithms to discover and prove theorems about cutting planes? I will present our
recent work towards this objective. We hope that the success of this project would
lead to a tool for developing the nextgeneration cutting planes that answers the
needs prompted by everlarger applications and models.

4642

Thursday 2/9 4:10 PM

Qing Nie, UC Irvine

Spatial and stochastic dynamics in development and regeneration
 Spatial and stochastic dynamics in development and regeneration
 02/09/2017
 4:10 PM  5:00 PM
 C304 WH
 Qing Nie, UC Irvine
In developing and renewing tissues, the properties of cells are controlled by secreted molecules from cells, their actions to the downstream regulators and genes, and transitions among different types of cells. The multiscale and stochastic nature of such spatial and dynamic systems presents tremendous challenges in synthesizing experimental observations and their understanding. In this talk, I will present several mathematical modeling frameworks with different complexity for systems ranging from single cells to multistage cell lineages. Questions of our interests include roles of feedbacks in regeneration speed, stem cell niche for tissue spatial organization, and crosstalk between epigenetic and genetic regulations. In addition to comparing our modeling outputs with experimental data, we will emphasize development of various mathematical and computational tools critical to success of using models in analyzing complex biological systems.

4661

Thursday 2/16 4:10 PM

Eric VandenEijnden, NYU

Nonequilibrium transitions between metastable patterns in populations of motile bacteria
 Nonequilibrium transitions between metastable patterns in populations of motile bacteria
 02/16/2017
 4:10 PM  5:00 PM
 C304 WH
 Eric VandenEijnden, NYU
Active materials can selforganize in many more ways than their
equilibrium counterparts. For example, selfpropelled particles whose
velocity decreases with their density can display motilityinduced
phase separation (MIPS), a phenomenon building on a positive feedback
loop in which patterns emerge in locations where the particles slow
down. Here, we investigate the effects of intrinsic fluctuations in
the system's dynamics on MIPS, using a field theoretic description
building on results by Cates and collaborators. We show that these
fluctuations can lead to transitions between metastable patterns. The
pathway and rate of these transitions is analyzed within the realm of
large deviation theory, and they are shown to proceed in a very
different way than one would predict from arguments based on
detailedbalance and microscopic reversibility. Specifically, we show
that these transitions involve fluctuations in diffiusivity of the
bacteria followed by fluctuations in their population, in a specific
sequence. The methods of analysis proposed here, including their
numerical components, can be used to study noiseinduced
nonequilibrium transitions in a variety of other nonequilibrium
setups, and lead to predictions that are verifiable experimentally.

4662

Thursday 2/23 4:10 PM

Ramis Movassagh, MIT

Supercritical Entanglement: counterexample to the area law for quantum matter
 Supercritical Entanglement: counterexample to the area law for quantum matter
 02/23/2017
 4:10 PM  5:00 PM
 C304 WH
 Ramis Movassagh, MIT
In recent years, there has been a surge of activities in proposing exactly solvable quantum spin chains with the surprisingly high amount of entanglement entropies (superlogarithmic violations of the area law). We will introduce entanglement and discuss these models. These models have rich connections with combinatorics, random walks, and universality of Brownian excursions. Lastly, we develop techniques for proving the gap and conclude that these models do not have a relativistic conformal field theory description.

4660

Thursday 3/16 4:10 PM

Federico Ardilla, SFSU

Using geometry and combinatorics to move robots quickly.
 Using geometry and combinatorics to move robots quickly.
 03/16/2017
 4:10 PM  5:00 PM
 C304 WH
 Federico Ardilla, SFSU
How do we move a robot quickly from one position to another? To answer this question, we need to understand the 'space of possibilities” containing all possible positions of the robot. Unfortunately, these spaces are tremendously large and highdimensional, and are very difficult to visualize. Fortunately, geometers and algebraists have encountered and studied these kinds of spaces before. Thanks to the tools they’ve developed, we can build “remote controls” to navigate these complicated spaces, and move (some) robots optimally.
This talk is based on joint work with my students Arlys Asprilla, Tia Baker, Hanner Bastidas, César Ceballos, John Guo, and Rika Yatchak. It will be accessible to undergraduate students, and assume no previous knowledge of the subject.

4714

Monday 3/20 5:30 PM

Peter Ozsvath, Princeton

An introduction to Heegaard Floer homology
 An introduction to Heegaard Floer homology
 03/20/2017
 5:30 PM  6:30 PM
 KCA 0
 Peter Ozsvath, Princeton
'Knot theory' is the study of closed, embedded curves in
threedimensional space. Classically, knots can be studied via a
various computable polynomial invariants, such as the Alexander
polynomial. In this first talk, I will recall the basics of knot
theory and the Alexander polynomial, and then move on to a more modern
knot invariant, 'knot Floer homology', a knot invariant with more
algebraic structure associated to a knot. I will describe applications
of knot Floer homology to traditional questions in knot theory, and
sketch its definition. This knot invariant was originally defined in
2003 in joint work with Zoltan Szabo, and independently by Jake
Rasmussen. A combinatorial formulation was given in joint work with
Ciprian Manolescu and Sucharit Sarkar in 2006.

4715

Tuesday 3/21 5:20 PM

Peter Ozsvath, Princeton

Bordered techniques in Heegaard Floer homology.
 Bordered techniques in Heegaard Floer homology.
 03/21/2017
 5:20 PM  6:20 PM
 115 IC
 Peter Ozsvath, Princeton
Heegaard Floer homology is a closed threemanifold invariant, defined
in joint work with Zoltan Szabo, using methods from symplectic
geometry (specifically, the theory of pseudoholomorphic disks). The
inspiration for this invariant comes from gauge theory. In joint work
with Robert Lipshitz and Dylan Thurston from 2008, the theory was
extended to an invariant for threemanifolds with boundary,
'bordered
Floer homology'. I will describe Heegaard Floer homology, motivate
its construction, list some of its key properties and applicat

4716

Wednesday 3/22 4:10 PM

Peter Ozsvath, Princeton

Bordered knot invariants
 Bordered knot invariants
 03/22/2017
 4:10 PM  5:00 PM
 C304 WH
 Peter Ozsvath, Princeton
I will describe a bordered construction of knot Floer homology,
defined as a computable, combinatorial knot invariant. Generators
correspond to Kauffman states, and the differentials have an algebraic
interpretation in terms of a certain derived tensor product. I will
also explain how methods from bordered Floer homology prove that this
invariant indeed computes the holomorphically defined knot Floer
homology. This is joint work with Zoltan Szabo.

4668

Thursday 4/6 4:10 PM

Wolfgang Ziller, UPENN

Finsler metrics with constant curvature
 Finsler metrics with constant curvature
 04/06/2017
 4:10 PM  5:00 PM
 C304 WH
 Wolfgang Ziller, UPENN
Finsler metrics are a generalization of Riemannian metrics (a norm in each tangent space) and occur naturally in various areas in physics and mathematics. Unlike for Riemannian metrics, there exists a large interesting class of Finsler metrics with constant (flag) curvature. We discuss joint work with R.Bryant, P. Foulon, S. Ivanov and V. S. Matveev on a characterization of the geodesic flow of such metrics in terms of the length of the shortest periodic orbit.

4663

Thursday 4/13 4:10 PM

Olga Holtz, UC Berkeley

TBA
 TBA
 04/13/2017
 4:10 PM  5:00 PM
 C304 WH
 Olga Holtz, UC Berkeley
No abstract available.

4664

Thursday 4/20 4:10 PM

Federico Rodriguez Hertz, PSU

TBA
 TBA
 04/20/2017
 4:10 PM  5:00 PM
 C304 WH
 Federico Rodriguez Hertz, PSU
No abstract available.
