• Speaker: Eugenia Malinnikova, Norwegian University of Science and Technology
• Title: Quantitative unique continuation for elliptic PDEs and application (special colloquium)
• Date: 01/09/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

If a solution to a uniformly elliptic second order PDE with smooth coefficients vanishes on an open subset of a domain then it is zero on the whole domain. This is a classical result known as weak unique continuation. We will discuss stronger versions, including some recent quantitative results and outline applications to the study of eigenfunctions of Laplace-Beltrami operator on compact manifolds.

• Speaker: Alex Blumenthal, University of Maryland
• Title: Chaotic regimes for random dynamical systems (special colloquium)
• Date: 01/14/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

It is anticipated that chaotic regimes (characterized by, e.g., sensitivity with respect to initial conditions and loss of memory) arise in a wide variety of dynamical systems, including those arising from the study of ensembles of gas particles and fluid mechanics. However, in most cases the problem of rigorously verifying asymptotic chaotic regimes is notoriously difficult. For volume-preserving systems (e.g., incompressible fluid flow or Hamiltonian systems), these issues are exemplified by coexistence phenomena: even in quite simple models which should be chaotic, e.g. the Chirikov standard map, completely opposite dynamical regimes (elliptic islands vs. hyperbolic sets) can be tangled together in phase space in a convoluted way. Recent developments have indicated, however, that verifying chaos is tractable for systems subjected to a small amount of noise— from the perspective of modeling, this is not so unnatural, as the real world is inherently noisy. In this talk, I will discuss two recent results: (1) a large positive Lyapunov exponent for (extremely small) random perturbations of the Chirikov standard map, and (2) a positive Lyapunov exponent for the Lagrangian flow corresponding to various incompressible stochastic fluids models, including stochastic 2D Navier-Stokes and 3D hyperviscous Navier-Stokes on the periodic box. The work in this talk is joint with Jacob Bedrossian, Samuel Punshon-Smith, Jinxin Xue and Lai-Sang Young.

• Speaker: Chris Kottke, New College Florida
• Title: Compactification of monopole moduli spaces
• Date: 01/17/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

I will discuss joint work with Michael Singer and Karsten Fritzsch on compactifications of the moduli spaces $M_k$ of $\mathrm{SU}(2)$ magnetic monopoles on $\mathbf{R}^3$ . Via a geometric gluing procedure, we construct manifolds with corners compactifying the $M_k$ , the boundaries of which represent monopoles of charge $k$ decomposing into widely separated ‘monopole clusters' of lower charge. The hyperkahler metric on $M_k$ has a complete asymptotic expansion, the leading terms of which generalize the asymptotic metric discovered by Bielawski, Gibbons and Manton in the case that the monopoles are all widely separated. From the structure of the compactification, we are able to make partial progress toward proving Sen's conjecture for $L^2$ cohomology of the moduli spaces.

• Speaker: Zhe Zhang, MSU
• Title: Harmonic map flow in dimension two, I
• Date: 01/21/2019
• Time: 1:40 PM - 3:00 PM
• Place: C517 Wells Hall

Bubbling analysis due to Struwe.

• Speaker: Stavros Garoufalidis, Georgia Institute of Technology
• Title: A brief history of quantum topology (special colloquium)
• Date: 01/23/2019
• Time: 10:20 AM - 11:10 AM
• Place: C304 Wells Hall

Quantum topology originated from Vaughan Jones's discovery of the Jones polynomial of a knot in 1985. I will explain the area and its interaction with mathematical physics, algebra, analysis, number theory and combinatorics.

• Speaker: Daping Weng, MSU
• Title: More on Scattering Diagram and Theta Functions
• Date: 01/24/2019
• Time: 3:00 PM - 4:00 PM
• Place: C204A Wells Hall

I will continue the discussion on scattering diagram and theta functions and relate them to the classical cluster theories. I will sketch Gross-Hacking-Keel-Kontsevich’s proofs of positive Laurent phenomenon, sign coherence, and a weak version of the cluster duality conjecture.

• Speaker: Nick Ovenhouse, MSU
• Title: Total Positivity
• Date: 01/28/2019
• Time: 3:00 PM - 4:00 PM
• Place: C517 Wells Hall

A matrix is "totally positive" if all of its minors are positive. We will discuss combinatorial models of total positivity using weighted graphs, as well as some criteria and characterizations for checking total positivity. If there is time, we will also discuss total positivity in the Grassmannian manifold, along with combinatorial models.

• Speaker: Abhishek Mallick
• Title: Computations in equivariant Floer homology
• Date: 02/06/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

Following Hendricks-Lipshitz -Sarkar we discuss the construction of equivariant Floer homology and a few cases where it can be computed.

• Speaker: Adam Sikora, SUNY at Buffalo
• Title:  New Approach to Quantum Teichmuller Theory
• Date: 02/07/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

The Jones polynomial invariant of links in R^3 extends to links in thickened surfaces, leading to the notion of the skein algebra of a surface, which is a version of Chekhov-Fock quantum Teichmuller space. The algebraic structure of skein algebras is quite rich and mysterious. We will approach it using the theory of measured foliations and pseudo-Anosov diffeomorphisms of surfaces

• Speaker: Alek Vainshtein, University of Haifa
• Title: Exotic cluster structures on SL_n
• Date: 02/07/2019
• Time: 3:00 PM - 4:00 PM
• Place: C204A Wells Hall

Back in 2005, Berenstein, Fomin and Zelevinsky discovered a cluster structure in the ring of regular functions on a double Bruhat cell in a semisimple Lie group, in particular, SL_n. This structure can be easily extended to the whole group. The compatible Poisson bracket is given by the standard r-matrix Poisson-Lie structure on SL_n. The latter is a particular case of Poisson-Lie structures corresponding to quasi-triangular Lie bialgebras. Such structures where classified in 1982 by Belavin and Drinfeld. In 2012, we have conjectured that each Poisson-Lie structure on SL_n gives rise to a cluster structure, and gave several examples of exotic cluster structures corresponding to Poisson-Lie structures distinct from the standard one. In my talk I will tell about the progress in the proof of this conjecture and its modifications. Joint with M.Gekhtman and M.Shapiro.

• Speaker: Cori Fata-Hartley and Cheryl Sisk
• Title: Designation B Process
• Date: 02/11/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

Dr. Fata-Hartley and Dr. Sisk will describe the Designation B requirements and application process. They will answer questions about eligibility, timeline, procedures, etc.

• Speaker: Keshav Sutrave
• Title: Harmonic map heat flow
• Date: 02/13/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

The harmonic map flow equation and Eells-Sampson theorem.

• Speaker: Alexander Shapiro, The University of Edinburgh
• Title: Positive Peter-Weyl theorem
• Date: 02/14/2019
• Time: 3:00 PM - 4:00 PM
• Place: C204A Wells Hall

The classical Peter-Weyl theorem asserts that the regular representation of a compact Lie group $G$ on the space of square-integrable functions $L^2(G)$ decomposes as the direct sum of all irreducible unitary representations of $G$. In the talk, I will use positive representations of cluster varieties, to obtain a "non-compact" quantum analogue of the Peter-Weyl theorem. This is joint work with Ivan Ip and Gus Schrader.

• Speaker: Willie Wong, MSU
• Title: In the beginning ... : Big Bang and Bianchi IX
• Date: 02/15/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

Big Bang Theory is now an established part of cosmology. A common mental picture of the big bang singularity is that of the Friedman model, where, looking backwards in time, the universe collapses uniformly to one point leaving the shape unchanged. In 1970, however, the Russian physicists Belinskii, Khalatnikov, and Lifshitz showed that the Friedman-inspired picture is far from common. In fact they proposed an alternative picture where near the beginning of time, the generic universe rapidly and chaotically oscillates between different "shapes" as it shrinks down to a single point. I aim to explain some of the things we now know about this BKL conjecture, including an interesting connection to the continued fraction expansion of real numbers and the golden ratio.

• Speaker: Paul Dawkins, Northern Illinois University
• Title: The use(s) of “is” in Mathematics
• Date: 02/19/2019
• Time: 2:00 PM - 3:30 PM
• Place: 252 EH

This talk presents analysis of some of the ambiguities that arise among statements with the copular verb “is" in the mathematical language of textbooks as compared to day-to-day English language. We identify patterns in the construction and meaning of is statements using randomly selected examples from corpora representing the two linguistic registers. We categorize these examples according to the part of speech of the object word in the grammatical form “[subject] is [object].” In each such grammatical category, we compare the relative frequencies of the subcategories of logical relations conveyed by that construction. Within some categories we observe that the same grammatical structure alternatively conveys different logical relations and that the intended logical relation can only sometimes be inferred from the grammatical cues in the statement itself. This means that one can only interpret the intended logical relation by already knowing the relation among the semantic categories in question. Such ambiguity clearly poses a communicative challenge for teachers and students. We discuss the pedagogical significance of these patterns in mathematical language and consider the relationship between these patterns and mathematical practices.

• Speaker: Ilya Gekhtman , University of Toronto
• Title: Growth rates of invariant random subgroups of hyperbolic groups and rank 1 Lie groups.
• Date: 02/21/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

Abstract: Invariant random subgroups (IRS) are conjugacy invariant probability measures on the space of subgroups of a given group G. They arise naturally as point stabilizers of probability measure preserving actions. The space of invariant random subgroups of SL_{2}R can be regarded as a natural compactification of the moduli space of Riemann surfaces, related to the Deligne-Mumford compactification. Invariant random subgroups can be regarded as a generalization both of normal subgroups and of lattices in topological groups. As such, it is interesting to extend results from the theories of normal subgroups and of lattices to the IRS setting. Jointly with Arie Levit, we prove such a result: the critical exponent (exponential growth rate) of an infinite IRS in an isometry group of a Gromov hyperbolic space (such as a rank 1 Lie group, or a hyperbolic group) is almost surely greater than half the Hausdorff dimension of the boundary. This generalizes an analogous result of Matsuzaki-Yabuki-Jaerisch for normal s As a corollary, we obtain that if $\Gamma$ is a typical subgroup and $X$ a rank 1 symmetric space then $\lambda_{0}(X/\Gamma)<\lambda_{0}(X)$ where $\lambda_0$ is the bottom of the spectrum of the Laplacian. The proof uses ergodic theorems for actions of hyperbolic groups. I will also talk about results about growth rates of normal subgroups of hyperbolic groups that inspired this work.

• Speaker: Round Table Discussion
• Title: Pros and Cons for Evening Exams for Uniform Courses
• Date: 02/25/2019
• Time: 4:10 PM - 5:00 PM
• Place: C109 Wells Hall

No abstract available.

• Speaker: Eva Belmont, Northwestern University
• Title: TBA
• Date: 02/28/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

No abstract available.

• Speaker: Round Table Discussion
• Title: Poll Everywhere and other in-class participation technologies
• Date: 03/11/2019
• Time: 4:10 PM - 5:00 PM
• Place: C109 Wells Hall

No abstract available.

• Speaker: Jonas Lührmann, Johns Hopkins
• Title: Local smoothing estimates for Schrödinger equations on hyperbolic space and applications
• Date: 03/13/2019
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall

We establish frequency-localized local smoothing estimates for Schrödinger equations on hyperbolic space. The proof is based on the positive commutator method and a heat flow based Littlewood-Paley theory. Our results and techniques are motivated by applications to the problem of stability of solitary waves to nonlinear Schrödinger-type equations on hyperbolic space. The talk is based on joint work with Andrew Lawrie, Sung-Jin Oh, and Sohrab Shahshahani.

• Speaker: Peter Magyar, MSU
• Title: TBA
• Date: 03/15/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

No abstract available.

• Speaker: Wilfrid Gangbo, University of California, Los Angeles
• Title: TBA
• Date: 03/21/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

No abstract available.

• Speaker: Dongsoo Lee
• Title: TBA
• Date: 03/27/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

No abstract available.

• Speaker: Brandon Bavier
• Title: TBA
• Date: 04/03/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

No abstract available.

• Speaker: Mona Merling, University of Pennsylvania
• Title: TBA
• Date: 04/08/2019
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall

No abstract available.

• Speaker: Jennifer Hom, Georgia Tech
• Title: TBA
• Date: 04/11/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

No abstract available.

• Speaker: Emmy Murphy, Northwestern University
• Title: TBA
• Date: 04/18/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

No abstract available.

• Speaker: André Neves, University of Chicago
• Title: TBA
• Date: 04/25/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

No abstract available.