Department of

Mathematics


Seminar Calendar
for Analysis Seminar events the year of Tuesday, March 26, 2019.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
    February 2019            March 2019             April 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

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.

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.

Thursday, February 21, 2019

2:00 pm in 243 Altgeld Hall,Thursday, February 21, 2019

Lipschitz free spaces on finite metric spaces

Denka Kutzarova-Ford (UIUC Math)

Abstract: We prove that the Lipschitz free space on any finite metric space contains a large well-complemented subspace which is close to $\ell_1^n$. We show that Lipschitz free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell_1^n$ of the corresponding dimension. These classes contain well-known families of diamond graphs and Laakso graphs. The paper is joint with S. J. Dilworth and M. Ostrovskii.

Friday, February 22, 2019

2:00 pm in 141 Altgeld Hall,Friday, February 22, 2019

Monic representations for higher-rank graph C*-algebras

Judith Packer (University of Colorado Boulder)

Abstract: We discuss the notion of monic representations for C*-algebras associated to finite higher–rank graphs without sources, generalizing a concept first defined by D. Dutkay and P. Jorgensen for representations of Cuntz algebras. Monic representations are those that, when restricted to the commutative C*-subalgebra of continuous functions on the infinite path space associated to the graph, admit a cyclic vector. We connect these representations to earlier work on dynamical systems with C. Farsi, E. Gillaspy, and S. Kang. The results discussed are based on joint work with C. Farsi, E. Gillaspy, S. Kang, and P. Jorgensen.

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.

3:00 pm in 341 Altgeld Hall,Friday, March 8, 2019

Completely bounded analogues of the Choquet and Shilov boundaries for operator spaces

Raphael Clouatre (University of Manitoba)

Abstract: Given a unital operator algebra, it is natural to seek the smallest $C^*$-algebra generated by a completely isometric image of it, by analogy with the classical Shilov boundary of a uniform algebra. In keeping with this analogy, one method for constructing the so-called $C^*$-envelope is through a non-commutative version of the Choquet boundary. It is known that such a procedure can be also be applied to operator spaces, although in this case the envelope has less structure. In this talk, I will present a certain completely bounded version of the non-commutative Choquet boundary of an operator space that yields the structure of a $C^*$-algebra for the associated Shilov boundary. I will explain how the resulting $C^*$-algebras enjoy some of the properties expected of an envelope, but I will also highlight their shortcomings along with some outstanding questions about them. This is joint work with Christopher Ramsey.

Thursday, March 14, 2019

2:00 pm in 243 Altgeld Hall,Thursday, March 14, 2019

Generalized Derivatives

Alastair Fletcher (Northern Illinois University)

Abstract: Quasiregular mappings are only differentiable almost everywhere. There is, however, a satisfactory replacement for the derivative at points of non-diffferentiability. These are generalized derivatives and were introduced by Gutlyanskii et al in 2000. In this talk, we discuss some recent results on generalized derivatives, in particular the question of how many generalized derivatives there can be at a particular point, and explaining how versions of the Chain Rule and Inverse Function Formula hold in this setting. We also give some applications to Schroeder functional equations.

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, April 25, 2019

2:00 pm in 243 Altgeld Hall,Thursday, April 25, 2019

Classification of irreversible and reversible operator algebras

Adam Dor-On (UIUC Math)

Abstract: C*-algebras have been studied quite extensively in the literature, especially in an attempt to classify them using K-theory. One canonical example is classification of Cuntz-Krieger algebras of a directed graph where K-theory was shown to coincide with Bowen-Franks groups of the subshift associated to the graph. On the other hand, non-self-adjoint operator algebras have been used to encode one-sided processes such as continuous maps on a compact space, stochastic matrices and graphs in their own right. In this talk we will survey results from both irreversible and reversible classification, and uncover a beautiful hierarchy of classification results for irreversible and reversible operator algebras.

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.