Speaker: Keerthi Madapusi Pera, University of Chicago

Title: Periods, L-functions and abelian varieties

Date: 02/01/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Periods are a special class of complex numbers, arising as integrals of differential forms on algebraic varieties. L-functions 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 L-functions, 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 Andre-Oort conjecture for Siegel modular varieties.

Speaker: Leonid Chekhov, Steklov Mathematical Institute

Title: Quantum cluster algebras from geometry

Date: 02/02/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

We identify the Teichmuller space $T_{g,s,n}$ of (decorated) Riemann
surfaces $\Sigma_{g,s,n}$ of genus $g$, with $s>0$ holes and $n>0$
bordered cusps located on boundaries of holes uniformized by Poincare with
the character variety of $SL(2,R)$-monodromy problem. The effective
combinatorial description uses the fat graph technique; observables are
geodesic functions of closed curves and $\lambda$-lengths of paths
starting and terminating at bordered cusps decorated by horocycles. Such
geometry stems from special 'chewing gum' moves corresponding to colliding
holes (or sides of the same hole) in a Riemann surface with holes. We
derive Poisson and quantum structures on sets of observables relating them
to quantum cluster algebras of Berenstein and Zelevinsky. A seed of the
corresponding quantum cluster algebra corresponds to the partition of
$\Sigma_{g,s,n}$ into ideal triangles, $\lambda$-lengths of their sides
are cluster variables constituting a seed of the algebra; their number
$6g-6+3s+2n$ (and, correspondingly, the seed dimension) coincides with the
dimension of $SL(2,R)$-character variety given by
$[SL(2,R)]^{2g+s+n-2}/\prod_{i=1}^n B_i$,
where $B_i$ are Borel subgroups associated with bordered cusps. I also discuss the
very recent results enabling constructing monodromy matrices of SL(2)-connections out of
the corresponding cluster variables.
The talk is based on the joint papers with with M.Mazzocco and V.Roubtsov

Title: Cutting plane theorems for Integer Optimization and computer-assisted proofs

Date: 02/03/2017

Time: 10:00 AM - 11:00 AM

Place: 1502 Engineering Building

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 state-of-the-art solvers for integer optimization problems use cutting-plane
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 hand-written, and were dominated by
tedious and error-prone 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 next-generation cutting planes that answers the
needs prompted by ever-larger applications and models.

Speaker: Wenqiang Feng, The University of Tennessee, Knoxville

Title: Linearly Preconditioned Nonlinear Solvers for Phase Field Equations

Date: 02/06/2017

Time: 10:00 AM - 11:00 AM

Place: C100 Wells Hall

Many unconditionally energy stable schemes for the physical models will lead to a highly nonlinear elliptic PDE systems which arise from time discretization of parabolic equations. I will discuss two efficient and practical preconditioned solvers- Preconditioned Steepest Descent (PSD) solver and Preconditioned Nonlinear Conjugate Gradient (PNCG) solver - for the nonlinear elliptic PDE systems. The main idea of the preconditioned solvers is to use a linearized version of the nonlinear operator as a pre-conditioner, or in other words, as a metric for choosing the search direction. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. Numerical simulations for some important physical application problems - including Cahn-Hillilard equation, epitaxial thin film equation with slope selection, square phase field crystal equation and Functionalized Cahn-Hilliard equation- are carried out to verify the efficiency of the solvers.

This is a continuation of the previous talks. We will be working through the paper of Karp and Williams, to understand the Amplituhedron as the image of a matrix map from the totally positive Grassmannian. This talk will work through the basic definitions of the amplituhedron, and then walk through some examples when k=1. It will follow closely to section 1 and section 3 of the Karp and Williams paper.

This is an expository talk and no background will be assumed. Given two integral vectors R = (r_1,...,r_m) and S = (s_1,...,s_n) we wish to know whether there exists an m x n matrix A whose ith row has sum r_i and whose jth column has sum s_j for all i, j. Such matrices have applications via the Transportation Problem. We will discuss the fundamental results in this area, including the Gale-Ryser Theorem.

Title: Spectral dimension and quantum dynamical bound for quasiperiodic Schrodinger operators.

Date: 02/09/2017

Time: 11:00 AM - 11:50 AM

Place: C304 Wells Hall

We introduce a notion of beta-almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded beta-almost periodic potentials. Applications include a sharp arithmetic criterion of full spectral dimensionality for analytic quasiperiodic Schrodinger operators in the positive Lyapunov exponent regime and arithmetic criteria for families with zero Lyapunov exponents, with applications to Sturmian potentials and the critical almost Mathieu operator.

Speaker: Christine Lee, University of Texas, Austin

Title: Jones slopes and Murasugi sums of links

Date: 02/09/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

A Jones surface for a knot in the three-sphere is an essential surface whose boundary slopes, Euler characteristic, and number of sheets correspond to quantities defined from the asymptotics of the degrees of colored Jones polynomial. The Strong Slope Conjecture by Garoufalidis and Kalfagianni-Tran predicts that there are Jones surfaces for every knot.
A link diagram D is said to be a Murasugi sum of two links D' and D' if a state graph of D has a cut vertex, which separates the graph into two state graphs of D' and D', respectively. We may obtain a state surface in the complement of the link K represented by D by gluing the state surface for D and the state surface for D' along the disk filling the circle represented by the cut vertex in the state graph. The resulting surface is called the Murasugi sum of the two state surfaces.
We consider near-adequate links which are Murasugi sums of certain non-adequate link diagrams with an adequate link diagram along their all-A state graphs with an additional graphical constraint. For a near-adequate knot, the Murasugi sum of the corresponding state surface is a Jones surface by the work of Ozawa. We discuss how this proves the Strong Slope Conjecture for this class of knots and raises interesting questions about constraints on the possible Murasugi sum-decompositions of a link diagram.

Title: Spatial and stochastic dynamics in development and regeneration

Date: 02/09/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

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.

Title: Polytopes and the Problem with Pick's Theorem

Date: 02/10/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Polytopes are a generalization of convex polygons into higher dimensions. In two dimensions, we have Pick's Theorem: a simple way of relating the area of a polygon with the number of ordered pairs it contains where each coordinate is an integer. So, why not try to do the same thing with polytopes in general? Problems quickly arise, even in three dimensions. In this talk, I will describe what these problems are and how we can overcome them. The answers aren't obvious, but we will see just how much it pays off in the end.

Speaker: Joshua Mike, The University of Tennessee, Knoxville

Title: Statistics in Topological Data Analysis

Date: 02/10/2017

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

In this talk, we will explore two applications: The first considers simplicial cohomology as a tool to investigate and eliminate inequity in kidney paired donation (KPD). The KPD pool is modeled as a graph wherein cocycles represent fair organ exchanges. Helmholtz decomposition is used to split donation utilities into gradient and harmonic portions. The gradient portion yields a preference score for cocycle allocation. The harmonic portion is isomorphic to the 1-cohomology and is used to guide a new algorithmic search for exchange cocycles. We examine correlation between a patient’s chance to obtain a kidney and their score under various allocation methods and conclude by showing that traditional methods are biased, while our new algorithm is not.
The second considers the persistent homology of a smoothed noisy dynamic. The machinery of persistent homology yields topological structure for discrete data within a metric space. Homology in dynamical systems can capture important features such as periodicity, multistability, and chaos. We consider a hidden Markov dynamic and compare particle filter to optimal smoothing a posteriori. We conclude with a stability theorem for the convergence of the persistent homology of the particle filtered path to that of the optimal smoothed path.

We will continue reading Preiss's paper [Geometry of measures in R^n: Distribution, rectifiability, and densities]. We will first review some necessary notations and properties of uniformly distributed measure discussed last semester and then study whether a measure is flat or curved at infinity.

Title: Matrices with given row and column sums, II

Date: 02/14/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In this continuation of the first talk, I will discuss new results by Brualdi and myself where we impose various symmetry conditions on the desired matrix A via the action of the dihedral group of the square.

Title: Topological recursion for matrix models and abstract topological recursion (a course of 3 lectures)

Date: 02/15/2017

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

I will begin with the description of a generating function for numbers of
Grothendieck's dessins d'enfant, or Belyi pairs. This generating function
is given by a random matrix model integral, I describe what is a
topological expansion (AKA genus expansion) of such models. The method
allowing finding corrections in all orders of the genus expansion is
Topological Recursion formulated in its present form by Eynard, Orantin
and the speaker in 2005-2006. This method had already found numerous
applications in mathematics and mathematical physics, so I describe the
general construction underlying the topological recursion and present an
(incomplete) list of its applications. Very recently, this method was
developed into an abstract topological recursion by Kontsevich and
Soibelman. In my last lecture I explain their construction and our
interpretation of it (forthcoming paper by J.Andersen, G.Borot, L.Ch., and
N.Orantin).

Speaker: Maria del Rosario Zavala, San Francisco State University

Title: How does racial identity matter in the mathematics classroom?

Date: 02/15/2017

Time: 3:30 PM - 5:00 PM

Place: 252 EH

In this talk I will use contemporary research in mathematics learning as a racialized form of experience to take stock of where the field is in relation to the question of “How does racial identity matter in the mathematics classroom?” and examine where we are headed in research and classroom practice. One aspect of my talk is to complicate the master narrative of achievement motivation, which ascribes high achievement solely to effort, and analyze the racial dimensions of such narratives using examples from research with Latinx high school students.

First, I will give the definition of CAT(0) spaces. Then I will describe the boundary of these spaces and how the geodesic flow behave on these spaces.

Title: Non-equilibrium transitions between metastable patterns in populations of motile bacteria

Date: 02/16/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Active materials can self-organize in many more ways than their
equilibrium counterparts. For example, self-propelled particles whose
velocity decreases with their density can display motility-induced
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
detailed-balance 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 noise-induced
non-equilibrium transitions in a variety of other non-equilibrium
set-ups, and lead to predictions that are verifiable experimentally.

We will continue reading Theorem 3.14 of Preiss's paper [Geometry of measures in R^n: Distribution, rectifiability, and densities]. We will study whether a uniformly distributed measure is flat or curved at infinity. The proof is based on the previous Lemma 3.13 and some basic properties of a symmetric bi-linear form.

Title: On Distance Preserving and Sequentially Distance Preserving Graphs

Date: 02/21/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

A graph H is an isometric subgraph of G if d_H(u,v) = d_G(u,v), for every pair u,v in V(H), where d denotes distance. A graph is distance preserving (dp) if it has an isometric subgraph of every possible order. We consider how to add a vertex to a dp graph so that the result is a dp graph. This condition implies that chordal graphs are dp. A graph is sequentially distance preserving (sdp) if its vertices can be ordered such that deleting the first i vertices results in an isometric subgraph, for all i at least 1. We give an equivalent condition to sequentially distance preserving based upon simplicial orderings. Using this condition, we prove that if a graph does not contain any induced cycles of length 5 or greater, then it is sdp. In closing, we discuss our results, other work and open problems concerning dp graphs.

Speaker: Ramis Movassagh, IBM TJ Watson Division of Mathematical Sciences

Title: Density of States and Gap of Generic Quantum Hamiltonians

Date: 02/23/2017

Time: 11:00 AM - 11:50 AM

Place: C304 Wells Hall

We propose a method, inspired by Free Probability Theory and Random Matrix Theory, that predicts the eigenvalue distribution of quantum many-body systems with generic interactions [1]. At the heart is a 'Slider', which interpolates between two extremes by matching fourth moments. The first extreme treats the non-commuting terms classically and the second treats them 'free'. By 'free' we mean that the eigenvectors are in generic positions. We prove that the interpolation is universal. We then show that free probability theory also captures the density of states of the Anderson model with an arbitrary disorder and with high accuracy [2]. Theory will be illustrated by numerical experiments.
[Joint work with Alan Edelman]
Time permitting we will prove that quantum local Hamiltonians with generic interactions are gapless [3]. In fact, we prove that there is a continuous density of states arbitrary close to the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. We calculate the scaling of the gap with the system's size in the case that the local terms are distributed according to gaussian β−orthogonal random matrix ensemble.
References:
- - - - - - - - -
[1] Phys. Rev. Lett. 107, 097205 (2011)
[2] Phys. Rev. Lett. 109, 036403 (2012)
[3] R. Movassagh 'Generic Local Hamiltonians are Gapless', (2017)
arXiv:1606.09313v2 [quant-ph]

Title: Abundant quasifuchsian surfaces in cusped hyperbolic 3-manifolds

Date: 02/23/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

I will discuss a proof that every finite volume hyperbolic 3-manifold M contains an abundant collection of immersed, $\pi_1$-injective surfaces. These surfaces are abundant in the sense that their lifts to the universal cover separate any pair of disjoint geodesic planes. The proof relies in a major way on the corresponding theorem of Kahn and Markovic for closed 3-manifolds. As a corollary, we recover Wise's theorem that the fundamental group of M is acts properly and cocompactly on a cube complex. This is joint work with Daryl Cooper.

Title: Supercritical Entanglement: counter-example to the area law for quantum matter

Date: 02/23/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

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 (super-logarithmic 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.

Title: AIMS: Access, Agency, and Allies in Mathematical Systems

Date: 02/27/2017

Time: 12:00 PM - 1:00 PM

Place: 133G Erickson Hall

In this presentation, we will share information about the AIMS project, on which faculty and doctoral students from eight universities are studying how mathematics teacher educators, mathematics teachers, and students work together to support the fair distribution of opportunities to learn. We will highlight our theoretical focus on access, agency, and allies and the ways in which this has translated into professional development design and will share the story of how two teachers’ adaptations of a PD task supported them in connecting to authentic student experiences and supporting students’ opportunities to learn rigorous mathematics.

Title: On the motion of a slightly compressible liquid

Date: 02/27/2017

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

I would like to go over some recent results on the compressible Euler equations with free boundary. We first provide a new a priori energy estimates which are uniform in the sound speed, which leads to the convergence to the solutions of the incompressible Euler equations. This is a joint work with Hans Lindblad. On the other hand, the energy estimates can be generalized to the compressible water wave problem, i.e., the domain that occupied by the fluid is assumed to be unbounded. We are also able to prove weighted energy estimates for a compressible water wave. Our method requires the detailed analysis of the geometry of the moving boundary.

Rachael, Andy, and Tsveta will be leading a discussion about RCPD accommodations. The goal of the discussion is to gather questions regarding common RCPD accommodations and bring those to the RCPD staff. We hope this will result in developing a guide for instructors and GTAs about how to help students with various RCPD accommodations. For example:
• What are some suggestions for providing 50% extended time on a 30-minute quiz if you have another class 20-minutes later?
• What are good ways to provide reduced distraction seating for exams?
• What are some good ways to work with students who have unexpected absences due to their condition who might miss a quiz or exam that you plan to hand back before they return to school?
We invite all of you to share your questions with us so we can bring those to the RCPD to develop a partnership with them to better serve our students.

Speaker: Hugh Thomas, University of Quebec at Montreal

Title: Describing the amplituhedron

Date: 02/28/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Amplituhedra were introduced by Arkani-Hamed and Trnka as part of a program to provide an alternative to the classical Feynman diagram approach to scattering amplitudes, via a surprising link with Grassmannian geometry. I will attempt to provide some physics context, and then give the original definition of the amplituhedron (as a certain image of a totally non-negative Grassmannian) and a new proposed definition. This talk is based on joint work with Nima Arkani-Hamed and Jaroslav Trnka.

Title: On Distance Preserving and Sequentially Distance Preserving Graphs

Date: 02/28/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

A graph H is an isometric subgraph of G if d_H(u,v) = d_G(u,v), for every pair u,v in V(H), where d denotes distance. A graph is distance preserving (dp) if it has an isometric subgraph of every possible order. We consider how to add a vertex to a dp graph so that the result is a dp graph. This condition implies that chordal graphs are dp. A graph is sequentially distance preserving (sdp) if its vertices can be ordered such that deleting the first i vertices results in an isometric subgraph, for all i at least 1. We give an equivalent condition to sequentially distance preserving based upon simplicial orderings. Using this condition, we prove that if a graph does not contain any induced cycles of length 5 or greater, then it is sdp. In closing, we discuss our results, other work and open problems concerning dp graphs.