Donaldson polynomials are powerful invariants associated to smooth four-manifolds. The introduction by Floer of Instanton homology groups, associated to some 3-manifolds, allowed to define analogs of such polynomials for (some) four-manifolds with boundary, that have a structure similar with a TQFT.
Wehrheim and Woodward developed a framework called "Floer field theory" which, according to the Atiyah-Floer conjecture, should permit to recover Donaldson invariants from a 2-functor from the 2-category Cob_{2+1+1} to a 2-category Symp they defined, which is an enrichment of Weinstein's symplectic category.
I will describe a framework that should permit to extend such a 2-functor to lower dimensions. This framework should permit to define new invariants in Manolescu and Woodward's symplectic instanton homology (sutured theory, equivariant version). This is work in progress.

Title: Geodesic flow in non-positive curvature: An inspiration for new techniques in ergodic theory

Date: 02/01/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

We discuss some recent progress in the smooth ergodic theory of geodesic flows. This talk will be suitable for a general mathematical audience, and will start with an intuitive overview of the classic results developed by luminaries such as Anosov, Bowen and Ruelle in the well understood setting of surfaces with variable negative curvature. Efforts to understand the much more difficult case of non-positive curvature were initiated by Pesin in the 1970’s. However, despite substantial successes, the picture has remained far from complete. There has been a great deal of recent progress in this area, which has required, and motivated, the development of new machinery in the abstract theory. I will give an overview of some recent developments, including:
1) General machinery developed by Vaughn Climenhaga and myself, which gives “non-uniform" dynamical criteria for uniqueness of equilibrium measures;
2) Joint work with Keith Burns, Vaughn Climenhaga and Todd Fisher, where we apply this machinery to geodesic flow on non-positive curvature manifolds;
3) If time permits, I will also mention related joint work with Jean-Francois Lafont and Dave Constantine, where we develop the theory of equilibrium measures for geodesic flow on locally CAT(-1) spaces; these are geodesic metric spaces which generalize negative curvature Riemannian manifolds by having the “thin triangle” property.

Speaker: Anton M. Zeitlin, Louisiana State University

Title: Quantum Integrable Systems and Enumerative Geometry

Date: 02/05/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The correspondence between integrable systems and enumerative geometry
started roughly 25 years ago in the works of Givental and his collaborators,
studying quantum cohomology and quantum K-theory. Around 10 years ago,
physicists Nekrasov and Shatashvili proposed an unexpected relation between
quantum K-theory and quantum integrable systems based on quantum groups
within their studies of 3-dimensional gauge theories. Their bold proposal
led to a lot of interesting developments in mathematics, bringing a new life
to older ideas of Givental, and enriching it with flavors of geometric
representation theory via the results of Braverman, Maulik, Nakajima, Okounkov
and many others. In this talk I will focus on recent breakthroughs in the
subject, leading to the proper mathematical understanding of Nekrasov-Shatashvili
original papers as well as some other subsequent conjectures made by physicists.
Our main illustration of such a relation is an interplay between equivariant quantum K-theory of the cotangent bundles to Grassmanians and the Heisenberg XXZ spin chain. We will also
discuss relation of equivariant quantum K-theory of flag varieties and
many-body integrable systems of Ruijsenaars-Schneider and Toda.

Title: Rectifiability and Minkowski bounds for the singular sets of multiple-valued harmonic spinors

Date: 02/06/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

We prove that the singular set of a multiple-valued harmonic spinor on a 4-manifold is 2-rectifiable and has finite Minkowski content. This result improves a regularity result of Taubes in 2014. It implies more precise descriptions for the limit behavior of non-convergent sequences of solutions to many important gauge-theoretic equations, such as the Kapustin-Witten equations, the Vafa-Witten equations, and the Seiberg-Witten equations with multiple spinors.

In this talk I will give a constructive proof to " Let k be a knot in S1 ×S2 freely homotopic to S1 ×pt then S1 × pt bounds an invertible concordance and k splits (S1 × pt) × [0, 1]."

Title: Dynamical System Seminar Almost sure invariance principle for hyperbolic systems with singularities.

Date: 02/08/2018

Time: 3:00 PM - 4:00 PM

Place: C517 Wells Hall

Speaker: Jianyu Chen, University of Massachusetts Amherst
Title: Almost sure invariance principle for hyperbolic systems with singularities.
Abstract: We investigate a wide class of two-dimensional hyperbolic systems with singularities, and prove the almost sure invariance principle (ASIP) for the random process generated by sequences of dynamically H\"older observables. The observables could be unbounded, and the process may be non-stationary
and need not have linearly growing variances.
Our results apply to Sinai dispersing billiards and their conservative perturbations, as well as the induced systems of Bunimovich billiards. The random processes are not restricted to the ergodic sum, but applicable to entropy fluctuation, shrinking target problems, etc.

Speaker: Anna Mazzucato, Pennsylvania State University

Title: Optimal mixing and irregular transport by incompressible flows

Date: 02/09/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

I will discuss transport of passive scalars by incompressible flows (such as a die in a fluid) and measures of optimal mixing and stirring under physical constraint on the flow. In particular, I will present recent results concerning examples of flows that achieve the optimal theoretical rate in the case of flows with a prescribed bound on certain Sobolev norms of the associated velocity, such as under an energy or an enstrophy budget. These examples are related to examples of (instantaneous) loss of Sobolev regularity for solutions to linear transport equation with non-Lipschitz velocity.

Title: Spectral Optimization and Free Boundary Problems

Date: 02/12/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

A classic subject in analysis is the relationship between the spectrum of the Laplacian on a domain and that domain's geometry. One approach to understanding this relationship is to study domains which extremize some function of their spectrum under geometric constraints. I will explain how to attack these problems using tools from the calculus of variations to find solutions. A key difficulty with this method is showing that the optimizers (which are a priori very weak) are actually smooth domains, and I address this issue in some recent work with Fanghua Lin. Our results are based on relating spectral optimization problems to certain vector-valued free boundary problems of Bernoulli type.

We're hoping to share our experiences from different projects related to supporting student learning outside of the classroom to generate ideas about how the MLC can shift to facilitate more productive learning for our students.

We will begin by outlining the construction of a cluster algebra associated to any surface with boundary (and marked points). Then we will discuss a formula, due to Schiffler, which explicitly gives an arbitrary cluster variable as a Laurent monomial in the initial variables, using the perfect matchings of an associated graph, called a "snake graph".

This talk will be a brief introduction to the Turaev-Viro Invariant. The Turaev-Viro Invariant is a 3-manifold invariant defined on a triangulation of a manifold. Using skein-theoretic methods, I will demonstrate a proof of its invariance with a technique known as chain-mail. This technique illustrates a close relationship between the Turaev-Viro Invariant and the surgery-presentation invariants originally defined by Reshetikhin and Turaev.

Title: Quantization of conductance in gapped interacting systems

Date: 02/15/2018

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

I will present two closely connected results. The first is the linear response theory in gapped interacting systems, and a proof of the associated Kubo formula. The second is a short proof of the quantization of the Hall conductance for gapped interacting quantum lattice systems on the two-dimensional torus.

Von Neumann algebras are certain *-subalgebras of bounded operators acting on a Hilbert space. They are generally thought of as non-commutative measure spaces and offer connections to many fields of mathematics (e.g. group theory, low-dimensional topology, logic, ergodic theory, and random matrix theory to name a few). In some instances an analogy with probability spaces is more appropriate, and indeed this is precisely what informs the field of free probability, wherein one uses non-commutative analogs of probabilistic notions to study the structure of von Neumann algebras. One particular example of this is free transport. In probability theory, transport refers to a measurable map between probability spaces that pushes one measure onto the other. Following work of Brenier in 1991, transportation theory has known great success. Free transport, the non-commutative analog that was introduced by Guionnet and Shlyakhtenko in 2014, offers methods for proving isomorphisms between von Neumann algebras. In this talk, I will discuss these ideas as well my work, which used free transport to prove isomorphisms between certain so-called "non-tracial" von Neumann algebras.

Title: Auction Dynamics for Semi-Supervised Data Classification

Date: 02/16/2018

Time: 4:10 PM - 5:00 PM

Place: B117 Wells Hall

We reinterpret the semi-supervised data classification problem using an auction dynamics framework (inspired by real life auctions) in which elements of the data set make bids to the class of their choice. This leads to a novel forward and reverse auction method for data classification that readily incorporates volume/class-size constraints into an accurate and efficient algorithm requiring remarkably little training/labeled data. We prove that the algorithm is unconditionally stable, and state its average and worst case time complexity.

Title: Determining the Regularity of Formal Languages

Date: 02/20/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

The concept of a regular language is very useful for computers parsing data and generally in theoretical computer science. We will define a formal language, what makes a formal language regular, and methods to decide whether a language is regular.

When studying knots, it is common to look at their complement to find invariants of the knot. One way to do this is to put a geometric structure on the complement, and look at common geometric invariants, such as volume. This talk is an introduction to hyperbolic knots, knots whose complement admits a hyperbolic structure. This will include a couple of diagramatic conditions to detect hyperbolicity, as well as using the structure to calculate bounds on the volume of the complement.

Speaker: Huyi Hu, MSU
Title: Infimum of the Metric Entropy of Anosov Systems
Abstract: We show that any Anosov diffeomorphism can be deformed continuously within the space of all Anosov diffeomorphisms in a way that the metric entropy with respect to the SRB measure can be arbitrarily close to $0$. That is, there is a path $\{f_t: t\in (0,1]\}$ such that $f_1=f$, and for each $t$, $f_t$ is an Anosov diffeomorphism, and $\lim_{t\to 0} h_{\mu_t}(f_t)=0$, where $\mu_t$ is an SRB measure of $f_t$ and $h_{\mu_t}(f_t)$ denote the metric entropy.
Similar results can be obtained within the space of volume preserving Anosov diffeomorphisms.

Speaker: Mohammad Jahangoshahi, University of Chicago

Title: Smoothness of the partition function for multiple Schramm-Loewner evolutions in simply connected domains

Date: 02/22/2018

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

We consider the measure on multiple chordal Schramm-Loewner evolution curves. We establish a derivative estimate using properties of the Poisson kernel and use it to give a direct proof that the partition function is C^2 if \kappa<4.

Title: Fast high-order CAD-independent Nystrom methods for frequency-domain electromagnetics

Date: 02/23/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Over the past three decades, there has been a myriad of advances in fast algorithms, singular quadrature, and integral equation theory relevant to the computational solution of partial diﬀerential equations, namely Maxwell’s equations, which govern the propagation of electromagnetic radiation. These advances have culminated in the ability to perform large-scale computations, but high-order accurate applications to solving integral equations has mostly been restricted to trivial geometries deﬁned by analytic formulas or large analytically deﬁned patches. These geometric descriptions are very limiting, given the advances that have been made in three-dimensional modeling software and fabrication. In this talk, I will describe recent advances in the numerical discretization of boundary integral equations along surfaces in three dimensions, new techniques for computing the resulting singular integrals, and the coupling of these techniques to fast algorithms, such as the fast multipole method.

Title: Mathematical Knowledge for Teaching Chemistry

Date: 02/26/2018

Time: 12:00 PM - 1:00 PM

Place: 252 EH

Progress toward STEM degree depends not only on completing required mathematics courses but also being able to successfully use mathematics to support learning in science courses. Introductory college chemistry courses are often the first place where inadequate preparation in mathematics impedes students’ learning in science. In this talk, Drs. Posey and Bieda will share their efforts to strategically incorporate mathematics support for students in Introductory Chemistry. Our findings suggest important implications for developing students’ conceptual understanding in mathematics courses. We will also share what we have learned about forging and sustaining an interdisciplinary research project.

Title: Taking the long way home: Orbits of plane partitions

Date: 02/27/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Plane partitions are piles of cubes stacked in the corner of a room. P. Cameron and D. Fon-der-Flaass (1995) studied a simple action on such piles, whose dynamics are nonetheless quite mysterious. In particular, repeating this action will always eventually return the original pile, but sometimes the voyage is much longer than expected. Motivated by some deep problems in algebraic geometry, H. Thomas and A. Yong (2009) introduced a suite of combinatorial algorithms on certain grids of numbers. In particular, there is a beautiful K-theoretic promotion operator, which again has some mysteriously large orbits, despite its simple combinatorial definition. We'll see how these two mysteries are in fact the same mystery, and use this relation to explain special cases of both actions. (Based on joint work with Kevin Dilks and Jessica Striker)

Speaker: Jennifer Langer-Osuna, Stanford University

Title: Fostering productive and inclusive collaborative mathematics classrooms

Date: 02/28/2018

Time: 1:45 PM - 3:15 PM

Place: 252 EH

Student-led group work is an increasingly common activity in K-12 mathematics classrooms. Students are expected to debate ideas, justify conjectures, and come to consensus on reasonable approaches to solving problems. Yet several studies have shown that some students become unduly influential, while others' contributions are routinely marginalized. This talk pursues the question, how can collaborative mathematics classrooms foster both equity and productivity? To do so, this talk begins with an exploration of the role of authority relations during collaborative math activity, followed by new design research, in partnership with local schools, based on the results of earlier, exploratory work. The talk closes by contextualizing these projects in a broader body of work focused on examining classrooms designed to equitably engage students from diverse backgrounds in intellectually productive mathematical activity.