Department of

Mathematics


Seminar Calendar
for Analysis Seminar events the year of Saturday, November 9, 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.
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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.

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

Thursday, September 19, 2019

2:00 pm in 243 Altgeld Hall,Thursday, September 19, 2019

Inductive limits of C*-algebras and compact quantum metric spaces

Konrad Aguilar (Arizona State University)

Abstract: In this talk, we will place quantum metrics, in the sense of Rieffel, on certain unital inductive limits of C*-algebras built from quantum metrics on the terms of the given inductive sequence with certain compatibility conditions. One of these conditions is that the inductive sequence forms a Cauchy sequence of quantum metric spaces in the dual Gromov-Hausdorff propinquity of Latremoliere. Since the dual propinquity is complete, this will produce a limit quantum metric space. Based on our assumptions, we then show that the C*-algebra of this limit quantum metric space is isomorphic to the given inductive limit, which finally places a quantum metric on the inductive limit. This then immediately allows us to establish a metric convergence of the inductive sequence to the inductive limit. Another consequence to our construction is that we place new quantum metrics on all unital AF algebras that extend our previous work with Latremoliere on unital AF algebras with faithful tracial state.

Friday, September 20, 2019

2:00 pm in 147 Altgeld Hall,Friday, September 20, 2019

Escaping nontangentiality: Towards a controlled tangential amortized Julia-Carathéodory theory

Meredith Sargent (University of Arkansas)

Abstract: Let $f: D \rightarrow \Omega$ be a complex analytic function. The Julia quotient is given by the ratio between the distance of $f(z)$ to the boundary of $\Omega$ and the distance of $z$ to the boundary of $D.$ A classical Julia-Carathéodory type theorem states that if there is a sequence tending to $\tau$ in the boundary of $D$ along which the Julia quotient is bounded, then the function $f$ can be extended to $\tau$ such that $f$ is nontangentially continuous and differentiable at $\tau$ and $f(\tau)$ is in the boundary of $\Omega.$ We develop an extended theory when $D$ and $\Omega$ are taken to be the upper half plane which corresponds to amortized boundedness of the Julia quotient on sets of controlled tangential approach, so-called $\lambda$-Stolz regions, and higher order regularity, including but not limited to higher order differentiability, which we measure using $\gamma$-regularity. I will discuss the proof, along with some applications, including moment theory and the fractional Laplacian. This is joint work with J.E. Pascoe and Ryan Tully-Doyle.

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.

Friday, September 27, 2019

3:00 pm in 341 Altgeld Hall,Friday, September 27, 2019

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.

Friday, October 4, 2019

3:00 pm in 341 Altgeld Hall,Friday, October 4, 2019

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).

Friday, October 11, 2019

3:00 pm in 341 Altgeld Hall,Friday, October 11, 2019

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.

Friday, October 18, 2019

3:00 pm in 341 Altgeld Hall,Friday, October 18, 2019

The geometry of real hypersurfaces in complex space

Martino Fassina (UIUC Math)

Abstract: Real hypersurfaces are the boundaries of complex domains, and their geometry is therefore crucial in understanding the theory of functions of several complex variables. I will focus in particular on the following question: does the hypersurface contain some local analytic structure? More generally, how closely ambient complex analytic varieties can contact the hypersurface? The talk will be elementary, and with plenty of pictures.

Friday, October 25, 2019

3:00 pm in 341 Altgeld Hall,Friday, October 25, 2019

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.

Thursday, October 31, 2019

2:00 pm in 243 Altgeld Hall,Thursday, October 31, 2019

Matrix Convex Sets, Tensor Products, and Noncommutative Choquet Boundaries

Roy Araiza (Purdue )

Abstract: I will discuss tensor products in the category of matrix convex sets and discuss how we may relate the study of their Choquet theory to questions about tensor product nuclearity. Based on joint work with Adam Dor-On and Thomas Sinclair.

Friday, November 1, 2019

3:00 pm in 341 Altgeld Hall,Friday, November 1, 2019

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.

Thursday, November 7, 2019

2:00 pm in 243 Altgeld Hall,Thursday, November 7, 2019

Noncommutative strong maximals and almost uniform convergence in several directions

Jose Conde Alonso (Universidad Autónoma de Madrid)

Abstract: We revisit Miguel de Guzmán's proof of the strong maximal theorem and we give an argument than can be generalized to noncommutative probability spaces. As is usual in this context, one difficulty that we must face is the definition of the operator itself: in principle, one cannot make sense of the supremum of a sequence of positive operators, something that can be done pointwise in the case of functions. We shall also discuss applications to almost everywhere convergence of sequences of averaging operators in von Neumann algebras.

Friday, November 15, 2019

3:00 pm in 341 Altgeld Hall,Friday, November 15, 2019

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

Friday, November 22, 2019

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

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.