Department of

# Mathematics

Seminar Calendar
for Graduate Analysis Seminar events the year of Wednesday, September 11, 2019.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     August 2019           September 2019          October 2019
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Friday, January 18, 2019

11:00 am in 145 Altgeld Hall,Friday, January 18, 2019

#### Organizational Meeting

###### Derek Kielty (Illinois Math)

Friday, January 25, 2019

2:00 pm in 141 Altgeld,Friday, January 25, 2019

#### 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.

Friday, February 1, 2019

2:00 pm in 141 Altgeld Hall,Friday, February 1, 2019

#### A Heat Trace Anomaly on Polygons

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.

Friday, February 15, 2019

2:00 pm in 141 Altgeld Hall,Friday, February 15, 2019

#### 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.

Friday, February 22, 2019

3:00 pm in 341 Altgeld Hall,Friday, February 22, 2019

#### 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.

Friday, March 1, 2019

2:00 pm in 141 Altgeld Hall,Friday, March 1, 2019

#### 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.

Friday, March 8, 2019

2:00 pm in 141 Altgeld Hall,Friday, March 8, 2019

#### 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.

Friday, April 19, 2019

2:00 pm in 141 Altgeld Hall,Friday, April 19, 2019

#### 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.

Thursday, May 2, 2019

2:00 pm in 147 Altgeld Hall,Thursday, May 2, 2019

#### 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.

Friday, August 30, 2019

1:00 pm in 141 Altgeld Hall,Friday, August 30, 2019

#### 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

Friday, September 6, 2019

3:00 pm in 341 Altgeld Hall,Friday, September 6, 2019

#### 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.

Friday, September 13, 2019

3:00 pm in 341 Altgeld Hall,Friday, September 13, 2019

#### 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

Friday, September 20, 2019

3:00 pm in 341 Altgeld Hall,Friday, September 20, 2019

#### 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.