Mathematics

A potential theoretic approach to box counting and packing dimensions

Fernando Roman-Garcia (Illinois Math)

Abstract: In 1968 Robert Kaufman introduced a potential theoretic approach to Hausdorff dimension. This approach allowed the use of Fourier analytic tools to answer questions about fractal Hausdorff dimension. In the late 90's Kenneth Falconer introduced a similar approach to packing and box counting dimensions. This allowed further developments on this area of geometric analysis such as Marstrand-Mattila type projection theorems for these different notions of fractal dimension. In this talk we will go through the development of this approach and (if time permits) go over the proof of the projection theorem for box and packing dimensions.

A Heat Trace Anomaly on Polygons

Hadrian Quan (Illinois Math)

Abstract: Given a planar domain with smooth boundary, one can associate its heat kernel, a time dependent operator whose trace admits an asymptotic expansion in t. The coefficients in this expansion turn out to all be geometric/topological invariants of the domain. However, by considering a smooth family of domains converging to a polygon, one can conclude that these heat trace coefficients are not continuous under such domain deformation. In this talk I’ll describe work of Mazzeo-Rowlett which recasts this apparent anomaly using renormalized invariants. I’ll also use it as an excuse to talk about uncommon but useful techniques in the study of linear PDEs e.g.: domain blow-ups, polyhomogeneous expansions, and more.

Convex geometry and the Mahler conjecture

Derek Kielty (Illinois Math)

Abstract: In this talk we will give an introduction to convex geometry and discuss the Mahler conjecture. This conjecture asserts that the product of the volume of a centrally symmetric convex set and the volume of its dual is minimized on a certain family of polytopes. We will also discuss a PDE analog of this conjecture.

Lipschitz Free Spaces

Christoper Gartland (Illinois Math)

Abstract: This will be a introduction to Lipschitz free spaces. The Lipschitz free space of a metric space $M$ is a Banach space LF$(M)$ containing $M$ so that for any Banach space $B$ and contractive map $M \to B$, there exists a unique linear contraction LF$(M) \to B$ extending the original map. We'll look at some examples, and discuss current results and open problems.

Poisson equation, its approximation, and error analysis

Amir Taghvaei (Illinois MechSE)

Abstract: In this talk, I discuss the computational problem of approximating the solution of a probability weighted Poisson equation, in terms of finite number of particles sampled from the probability distribution. The poisson equation arises in the theory of nonlinear filtering and optimal transportation. I present an approximation procedure based on the stochastic viewpoint of the problem. Then, I present the error analysis of the approximation using the Lyapunov stability theory in stochastic analysis.

On generic monothetic subgroups of Polish groups

Dakota Thor Ihli (Illinois Math)

Abstract: Given a topological group $G$, we ask whether the group $\overline{\left\langle g \right\rangle}$ has the same isomorphism type for "most" $g \in G$. More precisely, is there a group $H$ such that the set $\left\{ g \in G : \overline{\left\langle g \right\rangle} \cong H \right\}$ is dense? Comeagre? If so, can we identify this $H$? In this expository talk I will discuss known results and conjectures for certain Polish groups. Emphasis will be given to the case when $G$ is the group of Lebesgue-measure preserving automorphisms of the unit interval.

Universality in Operator Spaces

Mary Angelica Gramcko-Tursi (Illinois Math)

Abstract: Given a class $\mathcal{C}$ of spaces, When does there exist a space $\mathcal{U}$ that is injectively or projectively universal for $\mathcal{C}$ under the appropriate operation-preserving mappings?  Furthermore, when is $\mathcal{U}$ in $\mathcal{C}$ ?  The question has been answered under certain conditions using tools both in analysis and logic. We will look at both classical and recent results, as well as some of the techniques used to arrive at them. If time permits, we will end with some open questions.

Supnorm estimates for $\bar\partial$ in $\mathbb{C}^n$

Martino Fassina (Illinois Math)

Abstract: Let $\Omega$ be a domain in $\mathbb{C}^n$ and $f$ a $\bar\partial$-closed form on $\Omega$. A fundamental question in complex analysis is to establish the existence of solutions to the inhomogeneous Cauchy-Riemann equations $\bar\partial u=f$ that satisfy a norm estimate in $\Omega$. Whether such solutions exist depends both on the geometry of $\Omega$ and the regularity of $f$. In this talk, we consider the case where $\Omega$ is a polydisc. We establish the existence of weak solutions to $\bar\partial$ satisfying an $L^{\infty}$ estimate on $\Omega$ whenever the datum $f$ is in $L^{\infty}(\Omega)$, thus answering an old question of Kerzman and Stein. The talk is based on joint work with Yifei Pan.

Organizational Meeting

Kesav Krishnan (Illinois Math)

Abstract: This meeting will be to decide an optimal time for weekly meetings, as well as discuss a tentative schedule for speakers

Introduction to Metric Embeddings into Banach Spaces

Chris Gartland (UIUC)

Abstract: This talk will survey some results on the existence or nonexistence of embeddings of certain metric spaces into Banach spaces. Some proofs will be provided whenever sufficiently elementary, but most results will only be stated. There is a wealth of tools used in this field, and we'll encounter results whose proofs could include graph-theoretical combinatorics or abstract harmonic analysis. Topics to be covered (time permitting of course) range from: elementary facts - every finite metric space isometrically embeds into $\ell^\infty$, to graduate level analysis - almost everywhere differentiation of absolutely continuous functions, and finally to a recent, deep application in computer science - a full solution to the Goemans-Linial conjecture. Families of expander graphs and the Heisenberg group are metric spaces that play a special role in this story.

Probabilistic Methods for PDE

Kesav Krishnan (UIUC Math)

Abstract: In this talk I will introduce some aspects of Markov Processes, in particular diffusions and their connection to elliptic operators. In particular I will discuss the link between Brownian motion and the Laplacian, the probabilistic interpretation of properties of harmonic functions, such as the mean value theorem, and probabilistic solutions to linear elliptic and parabolic PDE. Time permitting, methods for non-linear PDE will be discussed

Introduction to Quasiconformal and Quasisymmetric maps on metric spaces

Stathis Chrontsios (UIUC Math)

Abstract: The talk will be a quick introduction to quasiconformal and quasisymmetric maps on metric spaces. I will start by describing how quasiconformal maps first appeared as generalizations of conformal maps on the complex plane and how they were generalized in arbitrary metric spaces. In addition, I will present how they gave rise to quasisymmetric maps on the real line and their later generalization in metric spaces. Moreover, I will discuss interesting quasisymmetric invariants and the definition of the conformal gauge. Last but not least, I will mention some applications this theory has had in Geometric Group Theory and some open problems.

Iwaniec's conjecture

Terence Lee John Harris (UIUC Math)

Abstract: The Beurling-Ahlfors transform is given by $(Sf)(z) = \frac{-1}{\pi} \int_{\mathbb{C}} \frac{f(\zeta) }{(z-\zeta)^2} \, d\zeta$. Finding the exact norm of $S$ on $L^p(\mathbb{C})$ is an open problem for $p \neq 2$. In this talk, I will outline some of the basic connections between this problem and other areas, such as martingales, quasiconformal mappings and the calculus of variations.

A sharp eigenvalue inequality for signed graph Laplacians

Derek Kielty (UIUC Math)

Abstract: Abstract: The collection of eigenvalues of a graph Laplacian matrix carries information about the topology the graph. Graph Laplacians are also closely related to discretizations of the Laplacian differential operator. Signed graph Laplacians are a generalization that encode attractions or repulsions between the vertices of a graph by assigning weights to the edges of the graph. In this talk I give an introduction to graph Laplacians and will discuss a sharp inequality on the eigenvalues of signed graph Laplacians (based on joint work with Ikemefuna Agbanusi and Jared Bronski).

An Introduction to Groupoids in Operator Algebras

Vincent Villalobos (UIUC Math)

Abstract: This talk will introduce and define groupoids while motivating their importance in operator algebras. We will discuss the various structures that arise within groupoids and then explore the construction of the groupoid $C^\ast$-algebra. Finally, we will discuss the specific case of a Lie groupoid and how the added smooth structure affects the groupoid $C^\ast$-algebra.

Continued Fractions and Ergodic theory

Maria Siskaki (UIUC Math)

Abstract: I will talk about how continued fractions arise. Continued fractions have had various applications in transcendental number theory and diophantine approximation . I will explain how tools from ergodic theory can be used to solve problems involving continued fractions. In particular, I will talk about the ergodic properties of the Gauss and Farey maps. The talk will be introductory.

Connections of Fixed-Point Theorems with Complexity

Basilis Livanos (UIUC CS)

Abstract: In this talk, we study how fixed-point theorems arise in the field of complexity theory and their deep connections with complexity classes like TFNP which contain problems who are guaranteed to have a solution. In the process, we provide an introduction to complexity theory and also a generalization of Bessaga's and Meyers's converse theorems to Banach's fixed-point theorem. The talk will be introductory and no prior knowledge of complexity theory will be needed.

Heat Kernel Proof of the Lefschetz Fixed Point Theorem

Gayana Jayasinghe (UIUC Math)

Abstract: I'll motivate the theorem and present a proof based on the heat kernel, which showcases some nice ideas in spectral geometry

Eigenvalues in Euclidean and Hyperbolic Geometry

Xiaolong Hans (UIUC Math)

Abstract: In this talk, we will explore the beauty and power of eigenvalues Laplacian operator. We start by talking about basic properties of Laplacian, Rayleigh quotient, monotonicity, how the lowest eigenvalues detect the difference of a geometric object from being a ball, Cheeger's inequality and how it distinguishes two different classes of infinite volume hyperbolic manifolds. The talk will emphasize intuition with no background in Laplacian assumed.