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.

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.

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

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.

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.

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

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.

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: 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: 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.

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.

Title: Prime Torsion of the Brauer Group of an Elliptic Curve

Date: 02/18/2019

Time: 3:00 PM - 4:00 PM

Place: C517 Wells Hall

The Brauer group is an invariant, that can detect arithmetic properties of the underlying variety. In this talk, I will define the Brauer group of a variety, describe it's connection to rational points, and give an algorithm to calculate generators and relations of the q-torsion of the Brauer group of and elliptic curve., where q is prime. This talk will be accessible to all.

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.

Title: A construction of the Deligne-Mumford orbifold

Date: 02/20/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Isomorphism classes $M_{g,n}$ of stable nodal Riemann surfaces of arithmetic genus g with n marked points. A marked nodal Riemann surface is stable if and only if its isomorphism group is finite. A natural construction based on the existence of universal unfoldings endows the Deligne-Mumford moduli space with an orbifold structure. Here we use the methods of differential geometry rather than algebraic geometry.

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.

Title: Berry-Esseen Bounds in the Breuer-Major Central Limit Theorem

Date: 02/21/2019

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

The Breuer-Major theorem provides sufficient conditions in order that a normalized sum of non-linear functionals of Gaussian random fields exhibits Gaussian fluctuation. Such a result has far-reaching applications in statistical inference of Gaussian models. In this talk, I will be focusing on the rate of convergence in the total variation distance of the Breuer-Major theorem. To this end, we apply Malliavin calculus (stochastic calculus of variation) techniques, Stein’s method for normal approximations, and Gebelein’s inequality for functionals of correlated Gaussian fields. Based on joint work with I. Nourdin and G. Peccati.

Title: QSym and the Shuffle Compatibility of Permutation Statistics

Date: 02/25/2019

Time: 3:00 PM - 4:00 PM

Place: C517 Wells Hall

The fundamental basis of the Hopf algebra of quasisymmetric functions can be thought of in terms of shuffling permutations, however we do not distinguish between permutations that have the same descent set. We can thus think of the algebra structure of QSym as having a basis indexed by equivalence classes of permutations. This descent set, Des, is a simple example of a permutation statistic that exhibits a property called being shuffle compatible. We will show that permutation statistics that are shuffle compatible give rise to “shuffle algebras” that are quotients of QSym and then discuss some bijective proofs that certain statistics are shuffle compatible.

Title: Analysis of measures converging to a Dirac-delta measure in Riemann surfaces

Date: 02/27/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

This is a 2-dim case of general energy concentration phenomenon. If we have sequence of harmonic maps with bounded energy defined on a Riemann surface, Uhlenbeck compactness theorem says its subsequence converges away from at most finite points, called bubble points. At the bubble point energy concentrates, so we may blow up the point to capture energy distribution on the bubble. But energy may concentrate again on the bubble, so careful touch is needed to finish this blow up process. In this talk I will introduce a way to choose two marked points with desired properties. The typical example is of harmonic map case, but it may be applied to other energy concentrating cases.

Title: Localizing the E_2 page of the Adams spectral sequence

Date: 02/28/2019

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

The Adams spectral sequence is one of the central tools for calculating the stable homotopy groups of spheres, one of the motivating problems in stable homotopy theory. This talk focuses on the E_2 page, which can be calculated algorithmically in a finite range but whose large-scale structure is too complicated to be understood in full. I will give an introduction to some features of the Adams E_2 page for the sphere at p = 3, and discuss an approach for calculating it in an infinite region. This approach relies on computing an analogue of the Adams spectral sequence in Palmieri's stable category of comodules, which can be regarded as an algebraic analogue of stable homotopy theory.

Title: Cluster structure on moduli spaces of local systems for general groups

Date: 02/28/2019

Time: 3:00 PM - 4:00 PM

Place: C204A Wells Hall

There have been several references in the literature devoted to the study of the cluster structures on moduli spaces of G-local systems, all of which are based on case by case study. In this talk, we present a systematic construction that works for all groups at once. As an application, we will investigate the principal series representations of quantum groups from the perspective of cluster theory.