Mathematics

Fourier transform of Radon measures on locally compact groups

Fernando Roman Garcia (UIUC Math)

Abstract: In Euclidean space, the Fourier transform of a compactly supported Radon measure is a bounded Lipschitz function. Properties of this function can translate into properties of the measure. In this talk we will see how one can develop corresponding theory for a general class of locally compact groups. If time permits, we will discuss applications of some of these results to geometric set theory in this class of groups.

CANCELED- A short proof of the Schwartz Kernel Theorem

Hadrian Quan (UIUC Math)

Abstract: Schwartz’ kernel theorem is a foundational result in the theory of distributions, going on to inspire many further techniques in analysis, e.g. Pseudodifferential Operators. And, like many other inspiring results, much is made of the statement and its consequences without considering much detail of the proof. In this talk I’ll give a proof of the theorem suggested in lecture notes of Richard Melrose.

(Non)-Removability of the Sierpiński Gasket

Dimitrios Ntalampekos (UCLA)

Abstract: Removability of sets for quasiconformal maps and Sobolev functions has applications in Complex Dynamics, in Conformal Welding, and in other problems that require``gluing" of functions to obtain a new function of the same class. We, therefore, seek geometric conditions on sets which guarantee their removability. In this talk, I will discuss some very recent results on the (non)-removability of the Sierpiński gasket. A first result is that the Sierpiński gasket is removable for continuous functions of the class $W^{1,p}$ for $p>2$. The method used applies to more general fractals that resemble the Sierpiński gasket, such as Apollonian gaskets and generalized Sierpiński gasket Julia sets. Then, I will sketch a proof that the Sierpiński gasket is non-removable for quasiconformal maps and thus for $W^{1,p}$ functions, for $1\leq p\leq 2$. The argument involves the construction of a non-Euclidean sphere, and then the use of the Bonk-Kleiner theorem to embed it quasisymmetrically to the plane.

Symmetrization Techniques in Functional Analysis

Derek Kielty (UIUC Math)

Abstract: Optimization problems are of great importance in analysis. Often times an optimization problem has many symmetries built into it. It is a natural and important question to determine if the optimizers inherit all of the symmetries of the optimization problem itself. Symmetrization techniques play an important role in answering this question. In this talk I will give a basic introduction to symmetrization techniques and discuss their applications to functional analysis. The prerequisites for this talk are strong calculus muscles and a bit of Math 540 notation.

Endomorphisms of B(H)

Chris Linden (UIUC Math)

Abstract: We will discuss a connection between the representation theory of Cuntz algebras and the classification of endomorphisms of B(H). No background in operator algebras will be assumed.

Basic, order basic, and bibasic sequences in Banach lattices

Vladimir Troitsky (University of Alberta)

Abstract: Recall that sequence in a Banach is a (Schauder) basis if every vector admits a unique series expansion which converges to the vector in norm. In a Banach lattice, one may replace norm convergence with order or uniform convergence. This leads to several types of order bases and order basic sequences. We will discuss connections between these types of sequences. This is a joint project with M.Taylor.

The Ribe Program, or, Nonlinearizing linear properties of Banach Spaces

Chris Gartland

Abstract: CANCELED BECAUSE OF STRIKE

Variations on the Mean Value Property

Jose Gonzalez-Llorente (Universidad Autonoma de Barcelona)

Abstract: The fruitful interplay between Geometric Function Theory, Potential Theory and Probability relies on the well known Mean Value Property for harmonic functions. In the last years substantial efforts have been made to clarify the probabilistic framework associated to some relevant nonlinear differential operators (such as the $p$-laplacian or the infinity laplacian) by means of appropriate (nonlinear) mean value properties. In the talk we will review some classical facts about the converse mean value property and also some more recent results about nonlinear mean value properties related to the $p$-laplacian.

The Ribe Program, or, Nonlinearizing linear properties of Banach Spaces

Chris Gartland (Illinois Math)

Abstract: I'll give an overview of a research program in geometric functional analysis named after Martin Ribe. The program is so named because of his important result in 1978 stating that two uniformly homeomorphic Banach spaces are mutually finitely representable. The aim of the program is to reformulate linear, local properties of Banach spaces into (nonlinear) metric properties. This talk is based off the survey "An Introduction to the Ribe Program" by Assaf Naor.

The Complexity of Isomorphism Classes of Banach Spaces

Mary Angelica Tursi (UIUC Math)

Abstract: It is commonly known that separable Banach spaces embed isometrically into the separable space $C(\Delta)$, where $\Delta$ is the Cantor set. Taking the Effros-Borel structure $\mathcal F(C(\Delta))$, we can then view the collection of separable Banach spaces as a Borel subset $\mathcal B \subseteq \mathcal F(C(\Delta))$ and consider the existence of an isomorphism between Banach spaces to be an equivalence relation on $\mathcal B$. For this expository talk, I will present some basic descriptive set theoretic techniques used to determine the complexity of isomorphism equivalence classes, in particular the Borel case of the class for $\ell_2$, and a non-Borel analytic case with Pelczynski’s universal space $\mathcal U$.

Bi-Lipschitz reflections of the plane

Terry Harris (UIUC Math)

Abstract: I will talk about a problem concerning the differentiability of a class of bi-Lipschitz reflections of the plane, which is still open.

Lipschitz differentiability and rigidity for convex-cocompact actions on rank-one symmetric spaces

Guy C. David (Ball State University)

Abstract: We discuss a recent theorem of the speaker and Kyle Kinneberg concerning rigidity for convex-cocompact actions on non-compact rank-one symmetric spaces, which generalizes a result of Bonk and Kleiner from real hyperbolic space. A key part of the proof concerns analysis on some non-Euclidean metric spaces (Cheeger's "Lipschitz differentiability spaces" and Carnot groups), and this will be the main focus of the talk.

Uncertainty in Fourier Analysis

Aubrey Laskowski (UIUC Math)

Abstract: The foundational idea behind the Heisenberg Uncertainty Principle is that it is not possible to localize both a function and its Fourier transform simultaneously. I will be discussing some applications of uncertainty in Fourier analysis and speaking about some generalizations which are useful, specifically in how uncertainty can be used in a proof of the Malgrange-Ehrenpreis theorem.

Fractal solutions of dispersive PDE on the torus

George Shakan (UIUC Math)

Abstract: I will discuss cancellation in exponential sums and how this leads to bounds for the fractal dimension of solutions to certain PDE, the ultimate “square root cancellation” implying exact knowledge of the dimension. In Schrodinger's equation, I provide bounds for the fractal dimension of the graph of the solution when restricted to a line on the torus. This is joint work with Burak Erdogan. More information can be found on my blog at https://gshakan.wordpress.com/2018/03/05/844/

Compactness of the branch set for quasiregular mappings and mappings of finite distortion

Rami Luisto (Charles University, Prague)

Abstract: Quasiregular mappings and mappings of finite distortion are natural generalizations of holomorphic mappings to higher dimensions. Whereas the pointwise derivatives of holomorphic mappings map circles to circles, QR-maps and MFD are defined by requiring that the differential maps balls to ellipsoids with controlled eccentricity. Under certain mild integrability conditions, mappings of finite distortion are continuous, open and discrete, as are all quasiregular mappings by the Reshetnyak theorem. For continuous, open and discrete mappings between Euclidean n-domains the branch set, i.e. the set of points where the mapping fails to be a local homeomorphism, has topological dimension of at most $n-2$ by the Cernavskii-Vaisala theorem. For quasiregular mappings more properties for the branch set are known, but several important questions remain open. In this talk we show that an entire mappings of finite distortion cannot have a compact branch set when its distortion is locally finite and satisfies a certain asymptotic growth condition; $K(x) < o(\log (|x|))$. In particular this implies that the branch set of entire quasiregular mappings is either non-compact or empty. We furthermore show that the growth bound is asymptotically strict by constructing a continuous, open and discrete mapping of finite distortion from the Euclidean $n$-space to itself which is piecewise smooth, has a branch set homeomorphic to the$(n-2)$ torus and distortion arbitrarily close to the asymptotic bound $\log (|x|)$. The talk is based on joint work with Aapo Kauranen and Ville Tengvall.

Dimensions results for mappings of jet spaces

Derek Jung (Illinois Math)

Abstract: In 1954, Marstrand partially answered the question: If you project a set in Euclidean space onto a plane, how does the size of the projection compare to that of the original set? I will continue work done in the past decade by Tyson with others to study this question in the sub-Riemannian setting. I will define analogues of horizontal and vertical projections in jet space Carnot groups. I will then explore how these maps affect Hausdorff dimension. About the first half of this talk will be spent defining and describing properties of these groups, which are simultaneously sub-Riemannian manifolds and Lie groups. This is recent research of the speaker.

Representations of Toeplitz-Cuntz-Krieger algebras

Adam Dor-On (Technion)

Abstract: By a result of Glimm, we know that classifying representations of non-type-I $C^*$-algebras up to unitary equivalence is essentially impossible (at least with countable structures). Instead of this, one either restricts to a tractable subclass or weakens the invariant. In the theory of free semigroup algebras, this is done for Toeplitz-Cuntz algebras, and is achieved via two key results in the theory: the first is a theorem of Davidson, Katsoulis and Pitts on the $2\times 2$ structure of free semigroup algebras, and the second is a Lebesgue-von Neumann-Wold decomposition theorem of Kennedy. This talk is about joint work with Ken Davidson and Boyu Li, where we generalize this theory to representations of Toeplitz-Cuntz-$Krieger$ algebras associated to a directed graph $G$. We prove a structure theorem akin to that of Davidson, Katsoulis and Pitts, and provide a Lebesuge-von Neumann Wold decomposition using Kennedy's theorem. We discuss some of the difficulties and similarities when passing to the more general context of operator algebras associated to directed graphs.

Organizational Meeting

Derek Kielty

Abstract: We will have a short meeting to decide on a weekly seminar time and make a tentative schedule of speakers for the semester. All are welcome, there will be cookies.

Building sandcastles via optimal transportation

Derek Kielty (Illinois Math)

Abstract: You’re given a lump of sand and a blue print for a sandcastle. While there are many ways to rearrange the individual grains of sand into your castle, you ask yourself, “What is the optimal way?” The theory of optimal transportation was developed to make these kinds of questions precise. In the process it developed connections to probability, geometry, and partial differential equations. In this talk I will give an introduction to optimal transportation and discuss applications to some geometric inequalities.

Some results for the Chevyshev Greedy Algorithm

Eugenio Hernández (Universidad Autónoma de Madrid)

Abstract: I will present some estimates for the Lebesgue type parameter of the Chevyshev Greedy Algorithm as well as for the weak thresholding variant. Some examples will be presented showing the optimality of the results.

Covering Lemmas and Differentiation

Chris Gartland (Illinois Math)

Abstract: The classical Lebesgue density theorem states that for any Lebesgue measurable $E \subset [0,1]$ and $\mathcal{L}$-almost every $x \in E$, $\lim_{r \to 0} \frac{ \mathcal{L}(E \cap B_r(x))}{\mathcal{L}(B_r(x))} = 1$. A typical way to prove this uses a maximal inequality, which in turn uses a weak Vitali covering lemma and that fact that $\mathcal{L}$ is doubling, meaning $\sup_{x \in [0,1]} \sup_{r > 0} \frac{\mathcal{L}(B_{2r}(x))}{\mathcal{L}(B_r(x))} < \infty$. The statement of the density theorem has a clear generalization to any metric measure space and can be proven true in any doubling space by proving a stronger Vitali covering lemma. In this talk, we'll work only with measure spaces and won't consider any metric or topological structure. The sets $\{B_r(x)\}_{r >0}$ willbe generalized to nets of measurable sets $\{B_\alpha(x)\}_{\alpha \in A}$ that "converge" to $x$. We then show that the stronger Vitali covering lemma is actually equivalent to the density theorem in this setting. An application will include an alternate proof of the almost sure convergence of uniformly bounded martingales.

Examples of amenable, non-unitarizable quantum groups

Michael Brannan (Texas A&M)

Abstract: A well-known theorem of Day and Dixmier from around 1950 states that if G is an amenable locally compact group, then any uniformly bounded representation of G on a Hilbert space is similar to a unitary representation. In short, amenable groups are ``unitarizable''. In this talk, I will focus on the question of whether a version of the Day-Dixmier unitarizability theorem holds in the more general framework of locally compact quantum groups. It turns out that the answer to this question is no: We show that many amenable quantum groups (including all Drinfeld-Jimbo-Woronowicz q-deformations of classical compact groups) admit non-unitarizable uniformly bounded representations. (Joint work with Sang-Gyun Youn.)

Three and a half asymptotic properties

Ryan Causey (Miami University Ohio)

Abstract: We introduce several isomorphic and isometric properties related to asymptotic uniform smoothness. These properties are analogues of p-smoothability, martingale type p, and equal norm martingale type p. We discuss distinctness, alternative characterizations, and renorming theorems for these properties.

Decay of cone averages of the Fourier transform

Terence Harris (Illinois Math)

Abstract: I will give an introduction to the techniques of decoupling and induction on scales from harmonic analysis, and then describe how they relate to the average $L^2$ decay over the cone of the Fourier transform of fractal measures.

Conservative Methods for Liberal ODE's

Nikolas Wojtalewicz (Illinois Math)

Abstract: A conservative method for a dynamical system is a numerical method of solving a dynamical system which preserves conserved quantities associated to that dynamical system. While many methods, such as symplectic or Runge-Kutta methods, have properties that allow them to preserve some types of conserved quantities for specific dynamical systems, few methods can preserve any type of conserved quantity for any given system. In this talk, we introduce the Multiplier method, a conservative method for solving a dynamical system which preserves any type of conserved quantity. The talk will be divided into three parts: first, discussing the basic theory and terms behind the Multiplier method; second, going over the proof on how to apply the Multiplier method; finally, if time permits, we will show some example applications of the Multiplier method, as well as compare the Multiplier method with another numerical method.

An introduction to noncommutative entropy

Christopher Linden (Illinois Math)

Abstract: I will attempt to give an accessible introduction to the theory of noncommutative entropy, focusing on examples and comparisons to the classical theory.

A hobbyists view of the mean curvature flow

Gayana Jayasinghe (Illinois Math)

Abstract: I'll introduce the mean curvature flow and talk about some nice results and ideas, sketching a few proofs along the way. There will be pictures.

Shadows of the Four Corner Cantor Set

Chi Huynh (Illinois Math)

Abstract: The set of particular interest will be $C(1/4) = C_{1/4} \times C_{1/4}$ where $C_{1/4}$ is the 1/4-Cantor set in $\mathbb{R}$. I will be presenting two proofs on the projections of $C(1/4)$ onto lines in $\mathbb{R}^2$. By utilizing the self-similar structure, these proofs present more detailed information on projections of $C(1/4)$ than the Marstrand projection theorem is able to. Due to time constraints, I will only go over one of the proofs in details, then sketch the proof of the sharper result by pointing out the necessary lemmas to obtain it.

Tensor algebras of product systems and their C*-envelopes

Elias Katsoulis (East Carolina University)

Abstract: Let $(G, P)$ be an abelian, lattice ordered group and let $X$ be a compactly aligned, $\tilde{\phi}$-injective product system over $P$. We show that the C*-envelope of the Nica tensor algebra $\mathcal{N} \mathcal{T} ^+_X$ is the Cuntz-Nica-Pimsner algebra $\mathcal{N} \mathcal{O} _X$ as defined by Sims and Yeend. We give several applications of this result. In particular, we show that the Hao-Ng isomorphism problem for generalized gauge actions of discrete groups on $C^*$-algebras of product systems has an affirmative answer in many cases, generalizing recent results of Bedos, Quigg, Kaliszewski and Robertson and of the second author.

Bases in $L^p$ spaces

Chris Gartland (Illinois Math)

Abstract: We will discuss examples of bases in $L^p$ spaces such as the Walsh and Haar systems.

Infinitesimals in Analysis, Topology, and Probability

Peter Loeb (Illinois Math)

Abstract: The notion of an infinitesimal quantity eluded rigorous treatment until the work of Abraham Robinson in 1960. Recent extensions and applications of his theory, called nonstandard analysis, have produced new results in many areas including operator theory, stochastic processes, mathematical economics and mathematical physics. Infinitely small and infinitely large quantities can play an essential role in the creative process. At the level of calculus, the integral can now be correctly defined as the nearest ordinary number to a sum of infinitesimal quantities. In Probability theory, Brownian motion can now be rigorously parameterized by a random walk with infinitesimal increments. In economics, an ideal economy can be formed from an infinite number of agents, each having an infinitesimal influence on the economy. After an introduction to this powerful method, I will discuss applications to calculus, the imbedding of topological spaces into compact spaces, and measure and probability theory. This includes the work of Y. Sun who showed that the measure spaces introduced by the present speaker can be used to finally make sense of the notion of an infinite number of equally weighted, independent random variables in probability theory and economics.

Nonsolvability of elliptic operators in the flat category

Martino Fassina (Illinois Math)

Abstract: In 1957 a ground-breaking three-page paper in the Annals marked the birth of CR geometry. There, Hans Lewy gave the first example of a locally non-solvable first-order linear partial differential equation. In this talk I will present a Lewy-type phenomenon for flat functions. That is, smooth functions whose derivatives are all equal to zero at a point. The result is elementary in nature, and no deep analytic background is required to understand this talk. I will describe some of the consequences of the result, with special attention to complex analysis. The talk is based on joint work with Yifei Pan.