• Speaker: Matthew Ballard, University of South Carolina
• Title: Exceptional collections: what they are and where to find them (special colloquium)
• Date: 01/07/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

Analogous to orthonormal bases in linear algebra, exceptional collections in triangulated categories are the most atomic means of decomposition. In this talk, we will introduce exceptional collections drawing heavily on examples from noncommutative algebra, algebraic geometry and symplectic geometry. We will then address the question of where (and how) to find them.

• Speaker: Pavlo Pylyavskyy, University of Minnesota
• Title: Zamolodchikov periodicity and integrability (special colloquium)
• Date: 01/11/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

T-systems are certain discrete dynamical systems associated with quivers. They appear in several different contexts: quantum affine algebras and Yangians, commuting transfer matrices of vertex models, character theory of quantum groups, analytic Bethe ansatz, Wronskian-Casoratian duality in ODE, gauge/string theories, etc. Periodicity of certain T-systems was the main conjecture in the area until it was proven by Keller in 2013 using cluster categories. In this work we completely classify periodic T-systems, which turn out to consist of 5 infinite families and 4 exceptional cases, only one of the infinite families being known previously. We then proceed to classify T-systems that exhibit two forms of integrability: linearization and zero algebraic entropy. All three classifications rely on reduction of the problem to study of commuting Cartan matrices, either of finite or affine types. The finite type classification was obtained by Stembridge in his study of Kazhdan-Lusztig theory for dihedral groups, the other two classifications are new. This is joint work with Pavel Galashin.

• Speaker: Goncalo Oliveira, Universidade Federal Fluminense, Rio de Janeiro
• Title: Gauge theory on Aloff-Wallach spaces
• Date: 01/16/2019
• Time: 1:40 PM - 3:00 PM
• Place: C517 Wells Hall

I will describe joint work with Gavin Ball constructing and classifying G2-instantons on Aloff-Wallach spaces, which are the most interesting known examples of compact "nearly-parallel" G2-manifolds.

• Speaker: Daping Weng, MSU
• Title: More on Scattering Diagram and Theta Functions
• Date: 01/17/2019
• Time: 3:00 PM - 4:00 PM
• Place: C117 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: Wencai Liu, University of California, Irvine
• Title: Universal arithmetical hierarchy of eigenfunctions for supercritical almost Mathieu operators (special colloquium)
• Date: 01/21/2019
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

The Harper's model is a tight-binding description of Bloch electrons on $\mathbb{Z}^2$ under a constant transverse magnetic field. In 1964, Mark Azbel predicted that both spectra and eigenfunctions of this model have self-similar hierarchical structure driven by the continued fraction expansion of the irrational magnetic flux. In 1976, the hierarchical structure of spectra was discovered numerically by Douglas Hofstadter, and was later observed in various experiments. The mathematical study of Harper's model led to the development of spectral theory of the almost Mathieu operator, with the solution of the Ten Martini Problem partially confirming the fractal structure of the spectrum. In this talk we will present necessary background and discuss the main ideas behind our confirmation (joint with S. Jitomirskaya) of Azbel's second prediction of the structure of the eigenfunctions. More precisely, we show that the eigenfunctions of the almost Mathieu operators in the localization regime, feature self-similarity governed by the continued fraction expansion of the frequency. These results also lead to the proof of sharp arithmetic transitions between pure point and singular continuous spectra, both in the frequency and the phase, as conjectured in 1994.

• Speaker: Rebecca Winarski, University of Michigan
• Title: Solving the Twisted Rabbit Problem using trees
• Date: 01/24/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

The twisted rabbit problem is a celebrated problem in complex dynamics. Work of Thurston proves that up to equivalence, there are exactly three branched coverings of the sphere to itself satisfying certain conditions. When one of these branched coverings is modified by a mapping class, a map equivalent to one of the three coverings results. Which one? After remaining open for 25 years, this problem was solved by Bartholdi-Nekyrashevych using iterated monodromy groups. In joint work with Belk, Lanier, and Margalit, we present an alternate solution using topology and geometric group theory that allows us to solve a more general problem.

• Speaker: Zhe Zhang, MSU
• Title: Harmonic map flow in dimension two, II
• Date: 01/28/2019
• Time: 1:40 PM - 3:00 PM
• Place: C517 Wells Hall

Bubble tree analysis and energy identity.

• Speaker: Tom Parker, MSU
• Title: Convergence modulo diffeomorphisms of maps from Riemann surfaces
• Date: 02/06/2019
• Time: 1:40 PM - 3:00 PM
• Place: C517 Wells Hall

For the theory of both harmonic and holomorphic maps, one needs a notion of convergence maps modulo diffeomorphisms of the domain. I will describe an approach (developed with Woongbae Park) that uses Kuranishi families in place of the standard approach based on bubble tree convergence.

• Speaker: Mihai Tohaneanu, University of Kentucky
• Title: Local energy estimates on black hole backgrounds
• Date: 02/06/2019
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall

Local energy estimates are a robust way to measure decay of solutions to linear wave equations. I will discuss several such results on black hole backgrounds, such as Schwarzschild, Kerr, and suitable perturbations converging at various rates, and briefly discuss applications to nonlinear problems. The most challenging geometric feature one needs to deal with is the presence of trapped null geodesics, whose presence yield unavoidable losses in the estimates. This is joint work with Lindblad, Marzuola, Metcalfe, and Tataru.

• Speaker: Elizabeth Munch, MSU
• Title: The Interleaving Distance for a Category with a Flow
• Date: 02/07/2019
• Time: 3:00 PM - 3:50 PM
• Place: C304 Wells Hall

All data has noise, and rigorously understanding how your analysis fares in the face of that noise requires a notion of a metric. The idea of the interleaving distance arose in the context of generalizing metrics for persistence modules from the field of topological data analysis (TDA). Essentially, the idea is that two objects in a category should be distance 0 if there is an isomorphism between them; the distance between two objects should be "almost" 0 if there is "almost" an isomorphism between them. Placed in the right context, we can measure what we mean by an "almost'' isomorphism and use this to define a distance. Building on the work of Chazal et al.; and Bubenick, Scott, and de Silva, we will discuss the generalization of the notion of the interleaving distance to a so-called "category with a flow". We will show that this generalization provides metrics for many different categories of interest in TDA and beyond, including Reeb graphs, merge trees, phylogenetic trees, and mapper graphs. This work is the result of collaborations with Anastasios Stefanou, Vin de Silva, and Amit Patel.

• Speaker: Xiaodong Wang, MSU
• Title: Harmonic maps and superrigidity (after Mok-Siu-Yeung)
• Date: 02/11/2019
• Time: 1:40 PM - 3:00 PM
• Place: C517 Wells Hall

No abstract available.

• Speaker: Albert Ai, UC Berkeley
• Title: Low Regularity Solutions for Gravity Water Waves
• Date: 02/13/2019
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall

We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness, below that which is attainable by energy estimates alone. This program was initiated for gravity water waves by Alazard-Burq-Zuily, by proving Strichartz estimates with loss. We discuss how these Strichartz estimates, and thus the low regularity threshold, can be sharpened by applying an integration along the Hamilton flow combined with local smoothing estimates.

• Speaker: Melissa Zhang, Boston College
• Title: Localization in Khovanov homology
• Date: 02/14/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

When a topological object admits a group action, we expect that our invariants reflect this symmetry in their structure. This talk will tour the expression of link symmetries in three generations of related invariants: the Jones polynomial; its categorification, Khovanov homology; and the youngest invariant in the family, the Khovanov stable homotopy type, introduced by Lipshitz and Sarkar. I will describe how to use Lawson-Lipshitz-Sarkar's Burnside functor construction of the Lipshitz-Sarkar Khovanov homotopy type to produce localization theorems and Smith-type inequalities for the Khovanov homology of periodic links. This joint work with Matthew Stoffregen.

• Speaker: Nicolas Garcia Trillos, University of Wisconsin-Madison
• Title: Large sample asymptotics of spectra of Laplacians and semilinear elliptic PDEs on random geometric graphs
• Date: 02/15/2019
• Time: 4:10 PM - 5:00 PM
• Place: 1502 Engineering Building

Given a data set $\mathcal{X}=\{x_1, \dots, x_n\}$ and a weighted graph structure $\Gamma= (\mathcal{X},W)$ on $\mathcal{X}$, graph based methods for learning use analytical notions like graph Laplacians, graph cuts, and Sobolev semi-norms to formulate optimization problems whose solutions serve as sensible approaches to machine learning tasks. When the data set consists of samples from a distribution supported on a manifold (or at least approximately so), and the weights depend inversely on the distance between the points, a natural question to study concerns the behavior of those optimization problems as the number of samples goes to infinity. In this talk I will focus on optimization problems closely connected to clustering and supervised regression that involve the graph Laplacian. For clustering, the spectrum of the graph Laplacian is the fundamental object used in the popular spectral clustering algorithm. For regression, the solution to a semilinear elliptic PDE on the graph provides the minimizer of an energy balancing regularization and data fidelity, a sensible object to use in non-parametric regression. Using tools from optimal transport, calculus of variations, and analysis of PDEs, I will discuss a series of results establishing the asymptotic consistency (with rates of convergence) of many of these analytical objects, as well as provide some perspectives on future research directions.

• 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.