Title: Low-temperature localization of directed polymers

Date: 09/07/2017

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

On the d-dimensional integer lattice, directed polymers can be seen as paths of a random walk in random environment, except that the environment updates at each time step. The result is a statistical mechanical system, whose qualitative behavior is governed by a temperature parameter and the law of the environment. Historically, the phase transitions of this system have been best understood by whether or not the path’s endpoint localizes. While the endpoint is no longer a Markov process as in a random walk, its quenched distribution is. The key difficulty is that the space of measures is too large for one to expect convergence results. By adapting methods recently used by Mukherjee and Varadhan, we develop a compactification theory to resolve the issue. In this talk, we will discuss this intriguing abstraction, as well as new concrete theorems it allows us to prove for directed polymers constructed from SRW or any other walk. (This talk is based on joint work with Sourav Chatterjee.)

Title: Thurston's metric on Teichmueller spaces of flat n-tori

Date: 09/07/2017

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Several interesting metrics have been defined for Teichmueller spaces of hyperbolic surfaces. However, analogous metrics on the Teichmueller space of flat n-tori have not been as well studied. After reviewing some background on Teichmueller theory, we will define an analog of Thurston's metric for these spaces. We find that in dimension n=2, it agrees with the hyperbolic metric. In particular, this gives a new way to realize the hyperbolic plane as the moduli space of marked flat tori. Time permitting, we will describe the corresponding situation in dimension n>2. This work is joint with Lizhen Ji.

Title: τ-invariants for knots in rational homology spheres

Date: 09/11/2017

Time: 4:10 PM - 5:30 PM

Place: C304 Wells Hall

Using the knot filtration on the Heegaard Floer chain complex, Ozsváth and Szabó defined an invariant of knots in the 3-sphere called τ(K). In particular, they showed that τ(K) is a lower bound for the 4-ball genus of K. Generalizing their construction, I will show that for a (not necessarily null-homologous) knot, K, in a rational homology sphere, Y, we can define a collection of τ-invariants, one for each spin-c structure on Y. In addition, these invariants give a lower bound for the genus of a surface with boundary K properly embedded in a negative definite 4-manifold with boundary Y.

In the study of quantum phases, the concept of topological invariant has emerged as a new paradigm beyond that of Landau theory. The relevance of topology for the classification of phases has been known since the discovery of the quantum hall effect. However, recent theoretical and experimental discoveries of new topological insulators has led to a renewed interest. The purpose of this reading group is to explore both recent and classical results for topological insulators including but not limited to (1) bulk-boundary correspondence (2) K-theoretic classification of topological insulators (3) topological invariants in the presence of disorder (4) quantization of Hall conductance in interacting systems.

We will present methods for bounding the modulus of the complex roots of a polynomial. These include the Cauchy bound and the use of Newton polynomials, the latter also being useful in interpolation problems. No background outside of elementary calculus will be assumed. In a subsequent talk, we will use these techniques to make progress on a conjecture about the roots of a polynomial of combinatorial interest.

Title: Stability of superselection sectors in infinite quantum spin systems

Date: 09/14/2017

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

Superselection sectors are equivalence classes of unitarily equivalent representations and can be used to label charges in a quantum system. We consider a family of superselection sectors for infinite quantum spin systems corresponding to almost localized endomorphisms. If the vacuum state is pure and satisfies certain locality conditions, we show how to recover the charge statistics. In particular, the superselection structure is that of a braided tensor category, and further, is stable against deformations by a quasi-local dynamics. We apply our results to prove stability of anyons in Kitaev's quantum double. Braided tensor categories naturally appear as the algebraic theory of anyons in topological phases of matter. Our results provide evidence that the anyonic structure is an invariant of topologically ordered states. This is work is joint with Pieter Naaijkens and Bruno Nachtergaele.

Title: On the braid index and the fractional Dehn twist coefficient

Date: 09/14/2017

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

The braid index of a knot is the least number of strands necessary to represent the knot as a closure of a braid on that many strands. If we view a braid as an element of the mapping class group of the punctured disk, its fractional Dehn twist coefficient (FDTC) is a number that measures the amount of twisting it exerts about the boundary of the disk. In this talk I will demonstrate that n-braids with FDTC larger than n-1 realize the braid index of their closure. The proof uses the concordance homomorphism Upsilon arising from knot Floer homology as a crucial tool. This is joint work with Peter Feller.

We consider an instance of the phase-retrieval problem, where one wishes to recover a signal (viewed as a vector) from the noisy magnitudes of its inner products with locally supported vectors. Such measurements arise, for example, in ptychography, which is an imaging technique used in lense-less X-ray microscopes and in optical microscopes with increased fields of view.
Starting with the setup where the signal is one-dimensional, we present theoretical and numerical results on an approach that has two important properties. First, it allows deterministic measurement constructions (which we give examples of). Second, it uses a robust, fast recovery algorithm that consists of solving a system of linear equations in a lied space, followed by finding an eigenvector (e.g., via an inverse power iteration). We also present extensions to the two-dimensional setting.
This is joint work with M. Iwen, B. Preskit, and A. Viswanathan.

Title: Second order Lyapunov exponent for the hyperbolic Anderson model

Date: 09/14/2017

Time: 4:10 PM - 5:00 PM

Place: C405 Wells Hall

In this talk, I will present some recent results regarding the asymptotic behavior of the second moment of the solution to the hyperbolic Anderson model in arbitrary spatial dimension d, driven by a Gaussian noise which is white in time. Two cases are considered for the spatial covariance structure of the noise: (i) the Fourier transform of the spectral measure of the noise is a non-negative locally-integrable function; (ii) d=1 and the noise is a fractional Brownian motion in space with index 1/4<H<1/2. These results are derived from a connection between the hyperbolic and parabolic models, and the recent powerful results of Huang, Le and Nualart (2015) for the parabolic model. This talk is based on joint work with Jian Song (University of Hong Kong).

Title: Solving polynomials with (higher) positive curvature

Date: 09/14/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

A smooth solution set of a system of complex polynomials is a manifold that can be studied geometrically. About 15 years ago, two results proved the existence of solutions of the system over a "function field of a complex curve" (Graber-Harris-Starr) and over a finite field (Esnault) provided the associated complex manifolds have positive curvature in a weak sense (rational connectedness). More recently, when
the manifold satisfies a higher version of positive curvature (rational simple connectedness), a similar result was proved over a function field of a complex surface (de Jong-He-Starr). I will explain these results, some applications to algebra (Serre's "Conjecture II", "Period-Index"), and recent extensions, joint with Chenyang Xu, to "function fields over finite fields" and Ax's "PAC fields".

The concept of a groupoid is a very general one, abstracting and generalizing many different concepts and definitions, including groups, group actions, equivalence relations, vector bundles, fundamental groups, and many others. I will define groupoids and give many examples, and then discuss the notion of "groupoid objects" in a category, which is the analogue of the notion of a "group object". Examples include Lie groupoids, symplectic groupoids, and algebraic groupoids.

Title: τ-invariants for knots in rational homology spheres (2)

Date: 09/18/2017

Time: 4:10 PM - 5:30 PM

Place: C304 Wells Hall

Using the knot filtration on the Heegaard Floer chain complex, Ozsváth and Szabó defined an invariant of knots in the 3-sphere called τ(K). In particular, they showed that τ(K) is a lower bound for the 4-ball genus of K. Generalizing their construction, I will show that for a (not necessarily null-homologous) knot, K, in a rational homology sphere, Y, we can define a collection of τ-invariants, one for each spin-c structure on Y. In addition, these invariants give a lower bound for the genus of a surface with boundary K properly embedded in a negative definite 4-manifold with boundary Y.

In the study of quantum phases, the concept of topological invariant has emerged as a new paradigm beyond that of Landau theory. The relevance of topology for the classification of phases has been known since the discovery of the quantum hall effect. However, recent theoretical and experimental discoveries of new topological insulators has led to a renewed interest. The purpose of this reading group is to explore both recent and classical results for topological insulators including but not limited to (1) bulk-boundary correspondence (2) K-theoretic classification of topological insulators (3) topological invariants in the presence of disorder (4) quantization of Hall conductance in interacting systems.

Let S_n be the symmetric group of all permutations p = p_1 ... p_n of the numbers 1, ..., n. The descent set of p is the set of indices i such that p_i > p_{i+1}. Given a set of positive integers I we let d(I;n) be the number of permutations in S_n with descent set I. In 1915 MacMahon proved that d(I;n) is a polynomial in n, but its properties do not seem to have been much studied until now. We apply the method of Newton bases from the previous lecture to make progress on a conjecture about the location of the roots of d(I;n) where n is now a complex number. This is joint work with Alexander Diaz-Lopez, Pamela Harris, Erik Insko, and Mohamed Omar.

This will be the first talk of the fall cluster algebra seminar. We have some new attendees this semester so we will start with the basics, discussing definitions and examples of cluster algebras.

Polaron theory is a model of an electron in a crystal lattice.
It is studied in the framework of nonequilibrium statistic mechanics.
There are two different mathematical models: H. Frohlich proposed a
quantum model in 1937; L. Landau and S. I. Pekar proposed a system of
nonlinear PDEs in 1948. In this talk I will present a proof that these
two models are equivalent to certain orders, and present some other
related works. These are joint works with Rupert Frank.

In this talk I will discuss a complex generalization of the special Lagrangian graph equation of Harvey-Lawson. I will discuss methods for constructing solutions, and relate the solvability of the equation with notions of stability from symplectic and algebraic geometry. This is joint work with T.C. Collins and S.-T. Yau.

Title: Parametrized SLE curves with self-similarity and stationary increments

Date: 09/21/2017

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

We describe an SLE$_\kappa$ curve, $\kappa\in(0,8)$, which is parametrized by $(d:=1+\frac \kappa 8)$-dimensional Minkowski content, and has self-similarity of exponent $1/d$ and stationary increments. We then prove that such SLE$_\kappa$ curve is $\alpha$-H\"older continuous for any $\alpha<1/d$, and Mckean's dimension theorem holds for this curve.

Ergodic Schrodinger operators and dynamical systems

Speaker: Rodrigo Bezerra Matos, MSU

Title: Introduction to ergodic Schrodinger operators I

Date: 09/22/2017

Time: 4:00 PM - 5:00 PM

Place: C117 Wells Hall

This is the first introductory talk of the reading seminar. The main goals are to show that spectra of ergodic operators are almost surely invariant, and to introduce several important objects such as the integrated density of states.

Attaching a half-twisted band to a knot produces a non-orientable cobordism to a new knot. The knots which admit such band moves to the unknots are quite simple - they are all cabled knots. We characterize when there exists such a band move between other families of knots. This is joint work with Allison Moore.

In the study of quantum phases, the concept of topological invariant has emerged as a new paradigm beyond that of Landau theory. The relevance of topology for the classification of phases has been known since the discovery of the quantum hall effect. However, recent theoretical and experimental discoveries of new topological insulators has led to a renewed interest. The purpose of this reading group is to explore both recent and classical results for topological insulators including but not limited to (1) bulk-boundary correspondence (2) K-theoretic classification of topological insulators (3) topological invariants in the presence of disorder (4) quantization of Hall conductance in interacting systems.

Title: Introduction to the arithmetic of modular forms.

Date: 09/26/2017

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

This is part of a 3 lecture series. The ultimate purpose of this lecture series is to explain a recent conjecture made jointly with Robert Pollack. The conjecture itself is about coarse p-adic invariants of modular forms called slopes, which are nothing other than the norms of the eigenvalues of a certain operator. By way of motivation, I will start by first discussing modular forms with a bias towards the arithmetic context of the Langlands program. The second talk will be reserved for exposing what one might call p-adic methods for modular forms. These have been around since the 70’s and 80’s and they are central to research on “p-adic Langlands” over the past 15 years. Finally, I will aim to state precisely the conjecture (called the ghost conjecture) Pollack and I have made and explain its numerical and theoretical evidence.

Title: A Combinatorial Description of Schur Functions

Date: 09/26/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

A symmetric function is a formal power series in countably many variables that is fixed under any permutation of the indices of the variables. The set of symmetric functions forms a vector space (actually, a graded algebra), and so it is natural to look for convenient bases of the space. In this talk, we will describe four "easy" bases and one more subtle, but much more important, basis, called the basis of Schur functions. We will give the combinatorial definition of Schur functions and highlight some of its uses in various branches of mathematics.

Positive representations are certain bimodules for a quantum group and its modular dual. In 2001, Ponsot and Teschner constructed these representations for U_q(\mathfrak{sl}_2) and proved that they form a continuous braided monoidal category, where the word "continuous" means that a tensor product of two representations decomposes into a direct integral rather than a direct sum. Ten years later, their construction was generalized to all other types by Frenkel and Ip. Although the corresponding categories were braided more or less by construction, it remained a conjecture that they are monoidal. Following a joint work with Gus Schrader, I will discuss the proof of this conjecture for U_q(\mathfrak{sl}_n). The proof is based on our previous result where the quantum group is realized as a quantum cluster \mathcal X-variety. If time permits, I will outline a relation between this story and the modular functor conjecture in higher Teichmüller theory along with several other applications.

Speaker: Günter Stolz, University of Alabama Birmingham

Title: Localization in the droplet spectrum of the random XXZ spin chain

Date: 09/28/2017

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

The XXZ quantum spin chain in random exterior field is one of the models where numerics indicate the existence of a many-body localization transition. We will discuss recent joint work with Alexander Elgart and Abel Klein, which provides rigorous results on the localization side of the expected transition. We show several of the accepted manifestations of MBL at the bottom of the spectrum for the random XXZ chain in the Ising phase. In this regime spins form quasi-particles in the form of droplets (of, say, down-spins in a sea of up-spins), which become fully localized under the addition of a random field.

According to Chen and Yang's volume conjecture, the asymptotics of the Turaev-Viro invariants of a 3-manifold predicts its hyperbolic volume. We show a compatibility between Turaev-Viro invariants and JSJ-decomposition and get an ineqality relating Turaev-Viro invariants and Gromov norm.

Speaker: Günter Stolz, University of Alabama Birmingham

Title: What is many-body localization?

Date: 09/28/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The phenomenon of many-body localization (MBL), as opposed to one-particle or Anderson localization, has recently received strong attention in the physics and quantum information literature. We will discuss the difference between these two concepts and then propose disordered quantum spin systems as a suitable model to study MBL. Among the possible manifestations of MBL we will mention the absence of many-body (or information) transport as well as area laws for the quantum entanglement of eigenstates. Examples where these properties can be proven include the random XY chain and, more recently, the droplet regime of the random XXZ chain. But many problems remain open.