• Title:  Low-temperature localization of directed polymers
• Date: 09/07/2017
• Time: 11:00 AM - 12:00 PM
• Place: C304 Wells Hall
• Speaker: Erik Bates, Stanford University

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: Postdoc Lightning Talks
• Date: 09/07/2017
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: MSU Postdocs

No abstract available.

• Title: The Burnside Problem and PI-Theory
• Date: 09/11/2017
• Time: 3:00 PM - 3:50 PM
• Place: C304 Wells Hall
• Speaker: Charlotte Ure, MSU

No abstract available.

• Title: Reading seminar on Topological Insulators
• Date: 09/12/2017
• Time: 11:00 AM - 12:00 PM
• Place: C304 Wells Hall
• Speaker: 

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: Stability of superselection sectors in infinite quantum spin systems
• Date: 09/14/2017
• Time: 11:00 AM - 12:00 PM
• Place: C304 Wells Hall
• Speaker: Matthew Cha, MSU

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: Second order Lyapunov exponent for the hyperbolic Anderson model
• Date: 09/14/2017
• Time: 4:10 PM - 5:00 PM
• Place: C405 Wells Hall
• Speaker: Raluca Balan, University of Ottawa

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
• Speaker: Jason Starr, Stony Brook University

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".

• 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
• Speaker: Katherine Raoux, MSU

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.

• Title: Roots of descent polynomials
• Date: 09/19/2017
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Bruce Sagan, Michigan State University

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.

• Title: What is a cluster algebra?
• Date: 09/21/2017
• Time: 10:00 AM - 10:50 AM
• Place: C304 Wells Hall
• Speaker: Eric Bucher, MSU

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.

• Title: The deformed Hermitian-Yang-Mills equation
• Date: 09/21/2017
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Speaker: Adam Jacob, UC Davis

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: Introduction to ergodic Schrodinger operators I
• Date: 09/22/2017
• Time: 4:00 PM - 5:00 PM
• Place: C117 Wells Hall
• Speaker: Rodrigo Bezerra Matos, MSU

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.

• Title: Band surgeries between knots
• Date: 09/25/2017
• Time: 4:10 PM - 5:30 PM
• Place: C304 Wells Hall
• Speaker: Tye Lidman, NCSU

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.

• Title: Introduction to the arithmetic of modular forms.
• Date: 09/26/2017
• Time: 3:00 PM - 3:50 PM
• Place: C304 Wells Hall
• Speaker: John Bergdall, MSU

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: Positive representations of quantum groups
• Date: 09/28/2017
• Time: 10:00 AM - 10:50 AM
• Place: C304 Wells Hall
• Speaker: Alexander Shapiro, University of Toronto

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.

• Title: Turaev Viro invariants and Gromov norm
• Date: 09/28/2017
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Speaker: Renaud Detcherry, MSU

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.

• Title: An Introduction to Intersection Theory
• Date: 10/02/2017
• Time: 3:00 PM - 3:50 PM
• Place: C304 Wells Hall
• Speaker: Thomas Plante

No abstract available.

• Title: BPS monopoles with non-maximal symmetry breaking and the Nahm transform
• Date: 10/03/2017
• Time: 11:00 AM - 12:00 PM
• Place: C304 Wells Hall
• Speaker: Akos Nagy, Fields Institute/University of Waterloo

The notion of “broken symmetry” is central in gauge theories. For BPS monopoles, symmetry breaking can be defined in terms of the eigenvalues of the Higgs-field at infinity. The symmetry breaking is maximal if the eigenvalues are distinct. Monopoles with maximal symmetry breaking have been studied extensively by both mathematicians and physicists for decades now, but little is known about the general case. In this talk, I will show how to produce monopoles with arbitrary symmetry breaking using the Nahm transform, and I will also outline the construction of its inverse. The inversion heavily uses first order elliptic PDE's on non-compact spaces, more concretely, the theory of Callias-type operators in 3D. This is a joint project with Benoit Charbonneau.

• Title: The Development of Shocks in Compressible Fluids
• Date: 10/04/2017
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Demetrios Christodoulou, ETH Zurich

The lecture shall trace the history of the theoretical study of the formation and evolution of shocks in compressible fluids, starting with the fundamental work of Riemann, the first work on nonlinear hyperbolic partial differential equations. Riemann considered the case of plane symmetry where the problem reduces to 1 spatial dimension. One milestone in the development of the theory was the work of Sideris who gave the first general proof of the finite time breakdown of smooth solutions in 3 spatial dimensions. Another milestone was the work of Majda who first addressed the problem of the local in time continuation of a shock front as a nonlinear free boundary problem for a nonlinear hyperbolic system of partial differential equations. I shall then discuss my own work, which uses differential geometric methods and resolves the resulting singularities giving a complete description in terms of smooth functions. My first work studies the maximal smooth development of given smooth initial data, the boundary of the domain of this development, and the behavior of the solution at this boundary. The boundary contains certain singular hypersurfaces which originate from certain singular surfaces. The singular surfaces do occur in nature, but not the singular hypersurfaces. My second work studies the physical evolution beyond the singular surfaces by solving a nonlinear free boundary problem with singular initial conditions associated to each of the singular surfaces. From each singular surface a shock hypersurface issues which appears as the corresponding free boundary.

• Title: Reading Seminar on Topological Insulators
• Date: 10/05/2017
• Time: 11:00 AM - 12:00 PM
• Place: C304 Wells Hall
• Speaker: Matthew Cha

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: Nonlinear stability of sources
• Date: 10/05/2017
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Björn Sandstede, Brown University

Defects are interfaces that mediate between two wave trains with possibly different wave numbers. Of particular interest in applications are sources for which the group velocities of the wave trains to either side of the defect point away from the interface. While sources are ubiquitous in experiments and can be found easily in numerical simulations of appropriate models, their stability analysis still presents many challenges. One difficulty is that sources are not travelling waves but are time-periodic in an appropriate moving coordinate frame. A second difficulty is that perturbations are transported towards infinity, which makes it difficult to apply various commonly used approaches. In this talk, I will discuss nonlinear stability results for sources in general reaction-diffusion system and outline a proof that utilizes pointwise estimates.

• Title: A geometric approach for the heteroclinic connection problem for multi-well gradient systems
• Date: 10/06/2017
• Time: 4:10 PM - 5:00 PM
• Place: C100 Wells Hall
• Speaker: Andres Zuniga , Indiana University

Linked Abstract

• Title:  Counting associatives and Seiberg-Witten equations (2)
• Date: 10/09/2017
• Time: 4:10 PM - 5:30 PM
• Place: C304 Wells Hall
• Speaker: Thomas Walpuski, MSU

There is a natural functional on the space of orientation 3-dimensional submanifolds in a G2-manifold. Its critical points are associative submanifolds, a special class of volume-minimizing submanifolds which obey an elliptic deformation theory. Given this, it is a natural question whether one can count associative submanifolds in order to construct an enumerative invariant for G2–manifolds. I will explain several geometric scenarios, which prohibit a naive count of such submanifolds cannot possible be invariant. I will then go on to discuss how (generalized) Seiberg-Witten equations might help cure these problems.

• Title: Mathematics Education Colloquium
• Date: 10/10/2017
• Time: 1:30 PM - 3:00 PM
• Place: 252 EH
• Speaker: 

Title: Students’ graphing activities: Re-presentations of what? Abstract: Students’ representational activities are key to their mathematical development. Specifically, students’ representational activities in constructing displayed graphs can afford them the figurative material necessary to engage in and abstract mental operations. In this talk, I draw on Piagetian ideas to frame the sophistication of students’ ways of thinking for graphing. Namely, I illustrate distinctions between those ways of thinking dominated by sensorimotor experience and those ways of thinking dominated by the coordination of mental actions. Against the backdrop of these distinctions, I argue that we, as educators and researchers, need to broaden students’ representational experiences. Instructionally, doing so can afford students increased opportunities to construct productive and generative ways of thinking for mathematical ideas and concepts. In terms of research, broadening students’ representational experiences enables researchers to form more viable and detailed working hypotheses of students’ ways of thinking for graphing and related topics.

• Title: Schur Symmetric Functions and Their Analogues, Part 1
• Date: 10/10/2017
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Duff Baker-Jarvis, MSU

I will continue to discuss some properties of the Schur symmetric functions and then talk about some algebras related to the algebra of symmetric functions, such as noncommutative symmetric functions. In these related algebras I will discuss some analogues of the Schur symmetric functions and their properties.

• Title: Stretching and Rotation Sets of Quasiconformal Maps
• Date: 10/11/2017
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall
• Speaker: Tyler Bongers, MSU

Quasiconformal maps in the plane are orientation preserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of bounded eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible and powerful generalization of conformal mappings. In this work, we study the singularities of these maps, in particular the sizes of the sets where a quasiconformal map can exhibit given stretching and rotation behavior. We improve results by Astala-Iwaniec-Prause-Saksman and Hitruhin to give examples of stretching and rotation sets with non-sigma-finite measure at the appropriate Hausdorff dimension.

• Title: Large scale geometry of asymptotically flat 3-manifolds
• Date: 10/12/2017
• Time: 11:00 AM - 12:00 PM
• Place: C304 Wells Hall
• Speaker: Otis Chodosh

Abstract: I will discuss recent work concerning the isoperimetric structure of asymptotically flat 3-manifolds and its relationship to the ADM and Hawking masses. This is joint work with M. Eichmair, Y. Shi, and H. Yu.

• Title: Galois groups in Enumerative Geometry and Applications
• Date: 10/12/2017
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Frank Sottile, Texas A&M University

In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem. I will describe this background and discuss some work in a long-term project to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry.

• Title: Schur Symmetric Functions and Their Analogues, Part 2
• Date: 10/17/2017
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Duff Baker-Jarvis, MSU

I will continue to discuss some properties of the Schur symmetric functions and then talk about some algebras related to the algebra of symmetric functions, such as noncommutative symmetric functions. In these related algebras I will discuss some analogues of the Schur symmetric functions and their properties.

• Title: Component Preserving Mutations and Maximal Green Sequences
• Date: 10/19/2017
• Time: 10:00 AM - 10:50 AM
• Place: C304 Wells Hall
• Speaker: Eric Bucher, MSU

In this talk we will outline the study of maximal green sequences for cluster algebras. We will discuss new results which allow one to define component preserving mutations for quivers, and utilize them to create maximal green sequences by considering maximal green sequences for induced subquivers. This is ongoing work with John Machacek and three undergraduate researchers here at Michigan State University, Ethan Zewde, Evan Runberg, and Abe Yeck

• Title: Complex dynamics and elliptic curves
• Date: 10/19/2017
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Laura DeMarco, Northwestern University

In this talk, I will present some connections between recent research in dynamical systems and the classical theory of elliptic curves and rational points. The main goal is to explain the role of dynamical stability and bifurcations in deducing arithmetic finiteness statements. I will focus on three examples: (1) the theorem of Mordell and Weil from the 1920s, presented from a dynamical point of view; (2) a recent result of Masser and Zannier about torsion points on elliptic curves, and (3) features of the Mandelbrot set.

• Title: Uses of Neurocognitive Measures to Evaluate Cognitive Load During the Process of Proving
• Date: 10/23/2017
• Time: 12:00 PM - 1:00 PM
• Place: 212 North Kedzie
• Speaker: Shiv Karunakaran, MSU

A recent special issue of ZDM (June 2016) made the case for increasing the interdisciplinary collaboration between researchers in the fields of mathematics education and cognitive neuroscience. Specifically, Ansari and Lyons (2016) argued for increasing the ecological validity of the testing situations and specific [neurocognitive] tests used to measure mathematical processing” (pp. 379-380). To this end, Ansari and Lyons (2016) suggest that it would be useful to explore the use of lower-cost and less invasive neuroimaging methods such as Near Infrared Spectroscopy (NIRS). The study reported on in this talk serves as a “proof-of-concept” for the use of Frontal Near-Infrared Spectroscopy (fNIRS) to measure the level of cognitive load of the brain under mathematical processing. The talk will address the pros and cons of using neurocognitive measures, such as the fNIRS, to measure and examine the physiological stresses of the brain under the complex mathematical process of proving. It will also explore future studies designed to better understand the role and influence of neurocognitive structures (such as working memory capacity) in mathematical problem solving.

• Title: Rational and integral homology balls
• Date: 10/23/2017
• Time: 4:10 PM - 5:30 PM
• Place: C304 Wells Hall
• Speaker: Selman Akbulut, MSU

I will explain how to generate infinite families of rational balls from figure eight knot, which bound Rohlin invariant one Brieskorn homology spheres (a joint work with Kyle Larson). I will then discuss the question of when a contractible manifold fails to be a cork; this happens either if there is no exotic involution on its boundary, or it fails to be Stein. I will give examples of Stein non-corks, and non-Stein non-loose corks. I will discuss open questions of whether there are loose-corks that can not be corks, or if there are infinite order corks (this part is a joint work with Danny Ruberman).

• Title: Slopes of modular forms and the ghost conjecture.
• Date: 10/24/2017
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall
• Speaker: John Bergdall, MSU

Abstract: In the previous two talks I discussed modular forms and their generalization p-adic modular forms. The space of p-adic modular forms is a p-adic Banach space equipped with an important operator U (the p-th Hecke operator). In this talk I will explain a conjecture recently formulated by Pollack and myself on the p-adic norms of eigenvalues of the U-operator acting on p-adic modular forms. Surprisingly, our description of these norms is completely elementary. If time permits I will discuss some of the evidence we have for our conjecture and its natural generalizations.

• Title: Arithmetic of Dehn surgery points and Azumaya algebras
• Date: 10/26/2017
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Alan Reid, Rice University

Associated to a finite volume hyperbolic 3-manifold is a number field and a quaternion algebra over that number field. Closed hyperbolic 3-manifolds arising from Dehn surgeries on a hyperbolic knot complement provide a family of number fields and quaternion algebras that can be viewed as "varying" over a certain curve component (the so-called canonical component) of the SL(2,C)-character variety of the knot group. This talk will give examples of different behavior and survey recent work on how the varying behavior can be explained using the language of Azumaya algebras over the canonical curve.

• Title: TBA
• Date: 11/02/2017
• Time: 11:00 AM - 12:00 PM
• Place: C304 Wells Hall
• Speaker: Marius Lemm, IAS

No abstract available.

• Title: TBA
• Date: 11/02/2017
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Mike Hill, UCLA

No abstract available.

• Title: TBA
• Date: 11/10/2017
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Ben Schmidt, Mathematics, MSU

No abstract available.

• Title: TBA
• Date: 11/16/2017
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Speaker: Elizabeth Munch, MSU

No abstract available.

• Title: Thanksgiving---No seminar
• Date: 11/23/2017
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Speaker: 

No abstract available.

• Title: TBA
• Date: 12/01/2017
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Rajesh Kulkarni, Mathematics, MSU

No abstract available.