Department of

Mathematics


Seminar Calendar
for Graduate Analysis Seminar events the year of Thursday, October 11, 2018.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, January 19, 2018

12:00 pm in 443 Altgeld Hall,Friday, January 19, 2018

Organizational meeting

Anna Lysts (UIUC Math)

Abstract: We will find a regular seminar time for the semester and people will volunteer for dates to give talks. Cookies will be provided of course.

Friday, January 26, 2018

12:00 pm in 443 Altgeld Hall,Friday, January 26, 2018

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.

Friday, February 2, 2018

12:00 pm in 443 Altgeld Hall,Friday, February 2, 2018

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.

Friday, February 9, 2018

12:00 pm in 443 Altgeld Hall,Friday, February 9, 2018

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.

Friday, February 16, 2018

12:00 pm in 443 Altgeld Hall,Friday, February 16, 2018

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.

Friday, March 2, 2018

12:00 pm in 120 Wohlers Hall,Friday, March 2, 2018

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

Chris Gartland

Abstract: CANCELED BECAUSE OF STRIKE

Friday, March 9, 2018

12:00 pm in 443 Altgeld Hall,Friday, March 9, 2018

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.

Friday, March 16, 2018

12:00 pm in 443 Altgeld Hall,Friday, March 16, 2018

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 $\mathcal U$.

Friday, March 30, 2018

12:00 pm in 443 Altgeld Hall,Friday, March 30, 2018

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.

Friday, April 6, 2018

12:00 pm in 443 Altgeld Hall,Friday, April 6, 2018

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.

Friday, April 13, 2018

12:00 pm in 443 Altgeld Hall,Friday, April 13, 2018

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/

Friday, April 20, 2018

12:00 pm in 443 Altgeld Hall,Friday, April 20, 2018

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.

Friday, August 31, 2018

12:00 pm in 147 Altgeld Hall,Friday, August 31, 2018

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.

Friday, September 7, 2018

3:00 pm in 145 Altgeld Hall,Friday, September 7, 2018

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.

Friday, September 14, 2018

2:00 pm in 343 Altgeld Hall,Friday, September 14, 2018

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.

Friday, September 21, 2018

3:00 pm in 145 Altgeld Hall,Friday, September 21, 2018

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.

Friday, September 28, 2018

3:00 pm in 145 Altgeld Hall,Friday, September 28, 2018

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.

Friday, October 5, 2018

3:00 pm in 145 Altgeld Hall,Friday, October 5, 2018

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.

Friday, October 12, 2018

3:00 pm in 145 Altgeld Hall,Friday, October 12, 2018

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.

Friday, October 19, 2018

3:00 pm in 145 Altgeld Hall,Friday, October 19, 2018

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.

Friday, November 2, 2018

3:00 pm in 145 Altgeld Hall,Friday, November 2, 2018

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.

Friday, November 30, 2018

3:00 pm in Altgeld Hall 145,Friday, November 30, 2018

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.

Friday, December 7, 2018

3:00 pm in 145 Altgeld Hall,Friday, December 7, 2018

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.