We relate fillability of two link exteriors,
and the question when two links admit homeomorphic
surface systems to (a refinement of) Milnor’s triple
linking numbers. This extends a theorem of Davis-Roth
to include also links with non-vanishing linking numbers.
This is joint work with C. Davis, P. Orson, and M. Powell.

Title: Optimal Large Deviation Theory for analytic quasi-periodic Schr\"odinger cocycle and H\"older regularity of the Lyapunov exponent

Date: 03/01/2018

Time: 3:00 PM - 4:00 PM

Place: C517 Wells Hall

Abstract:
We consider 1-d discrete quasi-periodic Schr\"odinger equations and the associated Schr\"odinger cocycles. Suppose the potential is real analytic function with bounded extension, assume positive Lyapunov exponents. We prove some refined Large Deviation Theory (LDT) for any irrational frequency in an exponential regime with respect to the Lyapunov exponent. The large deviation estimates imply some optimal H\"older continuity results of the Lyapunov exponents and the integrated density of states. For small Lyapunov exponent regime, we show that the local H\"older exponent is independent of energy E for Liouville frequency. In the large coupling regime, we show that the local H\"older exponent is independent of the coupling constant. Previously, such coupling independency is only known in the case where the potential is a trigonometric polynomial with (Strong) Diophantine frequency.

Title: Upper cluster algebras and choice of ground ring

Date: 03/01/2018

Time: 3:10 PM - 4:00 PM

Place: C329 Wells Hall

Cluster algebra structures often appear naturally in coordinate rings of algebraic varieties. In many cases the coordinate ring ends up being isomorphic to the corresponding upper cluster algebra. The choice of ground ring the cluster algebra is generated over determines if the cluster algebra consists of regular functions on the algebraic variety. We will discuss how to the choice of ground effects wether or not the cluster algebra coincides with its upper cluster algebra.

Title: Algorithms for mean curvature motion of networks

Date: 03/01/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Motion by mean curvature for networks of surfaces arises in a variety of
applications, such as the dynamics of foam and the evolution of
microstructure in polycrystalline materials. It is steepest descent
(gradient flow) for an energy: the sum of the areas of the surfaces
constituting the network.
During the evolution, surfaces may collide and junctions (where three or
more surfaces meet) may merge and split off in myriad ways as the
network coarsens in the process of decreasing its energy. The first idea
that comes to mind for simulating this evolution -- parametrizing the
surfaces and explicitly specifying rules for cutting and pasting when
collisions occur -- gets hopelessly complicated. Instead, one looks for
algorithms that generate the correct motion, including all the necessary
topological changes, indirectly but automatically via just a couple of
simple operations.
An almost miraculously elegant such algorithm, known as threshold
dynamics, was proposed by Merriman, Bence, and Osher in 1992. Extending
this algorithm, while preserving its simplicity, to more general
energies where each surface in the network is measured by a different,
possibly anisotropic, notion of area requires new mathematical
understanding of the original version, which then elucidates a
systematic path to new algorithms.

Title: Schubert Calculus and Cohomology of Grassmannians

Date: 03/12/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

In this first lecture, we shall begin by constructing a CW-structure for the complex Grassmannian. We will then begin our exploitation of this CW-structure with the goal of answering questions in enumerative geometry. These questions include “How many lines lie on the intersection of two quadric hypersurfaces in P^4?” and “Given four smooth curves in P^3, how many lines will intersect all four curves?”

Title: Real applications of non-real numbers: Ramanujan graphs (First Phillips Lecture)

Date: 03/12/2018

Time: 5:30 PM - 6:30 PM

Place:

The real numbers form a completion of the field of rational numbers. We will describe the fields of p-adic numbers which are different completions of the rationals. Once they are defined, one can study analysis and geometry over them. While being very abstract, the main motivation for studying them came from number theory. Developments in the last 2-3 decades shows various applications to the real world: communication networks, etc. This is done via expander graphs and Ramanujna grpahs which are "Riemann surfaces over these p-adic fields". All notions will be explained.

Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes (Second Phillips Lecture)

Date: 03/13/2018

Time: 4:00 PM - 5:00 PM

Place: 115 International Center

Expander graphs in general, and Ramanujan graphs, in particular, have played a major role in combinatorics and computer science in the last 4 decades and more recently also in pure math. Approximately 10 years ago, a theory of Ramanujan complexes was developed by Li, Lubotzky-Samuels-Vishne and others. In recent years a high dimensional theory of expanders is emerging. The notions of geometric and topological expanders were defined by Gromov in 2010 who proved that the complete d dimensional simplicial complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d greater than 1. Ramanujan complexes were shown to be geometric expanders by Fox-Gromov-Lafforgue-Naor-Pach in 2013, but it was left open if they are also topological expanders. By developing new isoperimetric methods for “locally minimal small” F_2-co-chains, it was shown recently by Kaufman- Kazdhan- Lubotzky for small dimensions and Evra-Kaufman for all dimensions that the d-skeletons of (d+1)-dimensional Ramanujan complexes provide bounded degree topological expanders. This answers Gromov’s original problem, but still leaves open whether the Ramanujan complexes themselves are topological expanders. We will describe these developments and the general area of high dimensional expanders and some of its open problems.

Title: Groups' approximation, stability and high dimensional expanders (Third Phillips Lecture)

Date: 03/14/2018

Time: 10:00 AM - 11:00 AM

Place: C304 Wells Hall

Several well-known open questions (such as: are all groups sofic or hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)?
In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2)norm.
The strategy is via the notion of “stability”: some higher dimensional cohomology vanishing phenomena is proven to imply stability and using higher dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated.
All notions will be explained. Joint work with M. De Chiffre, L. Glebsky and A. Thom.

Title: Hiring Cycle and Expectations for Faculty at Liberal Arts Schools

Date: 03/14/2018

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

This seminar is aimed at postdocs and graduate students who will be on the job market soon, but all are welcome.
The seminar will start with a short (20 minute) presentation on Chris Marx about the hiring cycle and expectations for faculty at liberal arts schools. Afterwards, there will be a discussion/question and answer period.

Title: Correction term, diagonalization theorem and the sliceness of 2-bridge knots

Date: 03/14/2018

Time: 4:10 PM - 5:00 PM

Place: C204A Wells Hall

About a decade ago, Lisca classified which 2-bridge knots are smoothly slice using an obstruction derived from Donaldson's diagonaliztion theorem. It is known that the diagonalization theorem can be proved using the Heegaard Floer correction term. Moreover, this correction term can also be used to construct a slicing obstruction for knots. In this expository talk, I will explain Josh Greene's proof that these two slicing obstructions actually coincide for 2-bridge knots.

Title: Dependence of the density of states on the probability distribution for discrete random Schrödinger operators

Date: 03/15/2018

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

We prove the Hölder-continuity of the density of states measure (DOSm) and the integrated density of states (IDS) with respect to the probability distribution for discrete random Schrödinger operators with a finite-range potential. In particular, our result implies that the DOSm and the IDS for smooth approximations of the Bernoulli distribution converge to the corresponding quantities for the Bernoulli-Anderson model. Other applications of the techniques are given to the dependency of the DOSm and IDS on the disorder, and the continuity of the Lyapunov exponent in the weak-disorder regime. The talk is based on recent joint work with Peter Hislop (Univ. of Kentucky).

Title: Connected Heegaard Floer homology and homology cobordism

Date: 03/15/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

We study applications of Heegaard Floer homology to homology cobordism. In particular, to a homology sphere Y, we define a module HF_conn(Y), called the connected Heegaard Floer homology of Y, and show that this module is invariant under homology cobordism and isomorphic to a summand of HF_red(Y). The definition of this invariant relies on involutive Heegaard Floer homology. We use this to define a new filtration on the homology cobordism group, and to give a reproof of Furuta's theorem. This is joint work with Jen Hom and Tye Lidman.

Title: Unstable entropy and pressure for partially hyperbolic systems

Date: 03/15/2018

Time: 3:00 PM - 4:00 PM

Place: C517 Wells Hall

We study ergodic properties caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy, topological entropy and pressures, and prove the corresponding variational principles. For unstable metric entropy we obtain affineness, upper semi-continuity and a version of Shannon-McMillan-Breiman theorem. We also obtain existence of Gibbs u-states, differentiability properties of unstable pressure, such as tangent functionals, Gateaux differentiability and Frechet differentiability.

Title: Schubert Calculus and Cohomology of Grassmannains

Date: 03/19/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

In this second talk, we will present formulas for the multiplicative structure of the Grassmannian’s cohomology. Using these formulas, we shall answer the problems raised last time and more. These problems will include “Fix four lines in P^3. How many lines will intersect all four of these lines?” and “Fix four curves in P^3. How many lines will intersect all four of these curves?”

These talks are aimed at first and second year students. Faculty will give an overview of problems that a student could work on. At the 3/20 seminar, we will have 1) Guowei Wei, Is it time for a great chemistry between mathematics and biology?; 2) Vladimir Peller, Contemporary problems of operator theory; and 3) Mark Iwen, Computational Nonlinear Approximation in Signal Processing and Inverse Problems.

Quasiconformal maps in the complex plane are homeomorphisms that satisfy certain geometric distortion inequalities; infinitesimally, they map circles to ellipses with bounded eccentricity. The local distortion properties of these maps give rise to a certain degree of global regularity and Holder continuity. In this talk, we will discuss improved lower bounds for the Holder continuity of these maps; the analysis is based on combining the isoperimetric inequality with a study of the length of quasicircles. Furthermore, the extremizers for Holder continuity can be characterized, and we will also give some applications to solutions to elliptic partial differential equations.

Title: Almost alternating, Turaev genus one, and semi-adequate links

Date: 03/22/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

A link is almost alternating if it is non-alternating and has a diagram such that one crossing change transforms it into an alternating diagram. Turaev genus one links are a certain generalization of non-alternating Montesinos links. A link is semi-adequate if it has a diagram where at least one of the all-A or all-B Kauffman state graphs is loopless. In this talk, we discuss the Jones polynomial and Khovanov homology of links in these three classes, and we discuss open problems about the relationships between the three classes.

These talks are aimed at first and second year students. Faculty will give an overview of problems that a student could work on. At the 3/22 seminar, we will have 1) Rajesh Kulkarni; 2) Matt Hirn, Understanding high dimensional data analysis and machine learning via harmonic analysis; 3) Di Liu

Title: Observability of the visco-elastic wave equation

Date: 03/23/2018

Time: 4:10 PM - 5:00 PM

Place: C100 Wells Hall

In this talk we give a proof of the Neumann boundary observability inequality for the visco-elastic wave equation in an arbitrary space dimension. To do this, we first give a new proof of the boundary observability for the classical wave equation that extends the harmonic analysis perspective of D.L.Russell to higher space dimensions. We then argue by perturbation to show the Riesz sequence property of the corresponding harmonic system for the visco-elastic wave equation.

Title: Avoiding Collisions and Braiding String: Configuration Spaces and the Braid Group

Date: 03/23/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

There are many real-world problems that amount to studying the possible paths of n objects through a space X such that those objects never collide. (Examples include flight traffic patterns around an airport, or automated carts moving around a factory floor.) We think about such problems by studying paths in configuration spaces, spaces consisting of ordered sets of distinct points in X. In one of the simplest cases, these spaces turn out to be closely linked to a seemingly different mathematical object, the n-stranded braid group. We introduce configuration spaces and the braid group and learn about their relationship.

We will give a brief introduction to combinatorial Hopf algebras using a Hopf algebra structure on hypergraphs as the main example. Combinatorial reciprocity results that can be obtain by combining Aguiar-Bergeron-Sottile character theory and antipode formulas will be emphasized. In particular, we will see a generalization of Stanley's theorem on acyclic orientations.

Title: Symplectic Quotients and GIT Quotients : The Kempf-Ness Theorem

Date: 03/28/2018

Time: 4:10 PM - 5:00 PM

Place: C204A Wells Hall

The Kempf-Ness theorem is a fundamental result at the intersection of complex algebraic Geometry and Symplectic Geometry .It states the equivalence of Symplectic and Geometric invariant theory quotients. After brief introduction of each of the quotients we will cover the proof of the theorem.

Title: On minimizers and critical points for anisotropic isoperimetric problems

Date: 03/28/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

Anisotropic surface energies are a natural generalization of the perimeter that arise in models for equilibrium shapes of crystals. We discuss some recent results for anisotropic isoperimetric problems concerning the strong quantitative stability of minimizers, bubbling phenomena for critical points, and a weak Alexandrov theorem for non-smooth anisotropies. Part of this talk is based on joint work with Delgadino, Maggi, and Mihaila.

Speaker: Ramanujan Santharoubane, University of Virginia

Title: Asymptotic of quantum representations of surface groups

Date: 03/29/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

In a previous work with Thomas Koberda we defined actions of surface groups on the vector spaces coming from the Witten-Reshetikhin-Turaev TQFT. For l a given loop in a surface we can define the trace of the associated operator. Actually, this is a sequence of invariant depending on a sequence of roots of unity. For any z on the unit circle, we study the asymptotic of this sequence of invariant when the sequence of roots of unity converges to z. The main theorem says that this asymptotic is determined by the evaluation at z of a Laurent polynomial depending only on l. This polynomial can be viewed as a Jones polynomial for surface groups. The main corollary concerns the so-called AMU conjecture which relates TQFT representations of mapping class groups to the Nielsen-Thurston classification.
This talk represent a joint work with Julien Marché.

Clifford algebras play an important role in the classification of quadratic forms over number fields. Surprisingly, the also play a critical role in studying the isotropy (existence of nontrivial zeros) of quadratic forms over function fields of curves over totally imaginary number fields. We shall explain some open questions concerning isotropy of quadratic forms over function fields of curves over number fields and their connection to Clifford algebras.