Department of

Mathematics


Seminar Calendar
for Analysis Seminar events the next 12 months of Sunday, January 1, 2017.

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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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Thursday, January 19, 2017

2:00 pm in 243 Altgeld Hall,Thursday, January 19, 2017

The Aviles Giga functional. A history, a survey and some new results

Andrew Lorent (University of Cincinnati)

Abstract: The Aviles-Giga functional $I_{\epsilon}(u)=\int_{\Omega} \frac{\left|1-\left|\nabla u\right|^2\right|^2}{\epsilon}+\epsilon \left|\nabla^2 u\right|^2 \; dx$ is a well known second order functional that models phenomena from blistering to liquid crystals. The zero energy states of the Aviles-Giga functional have been characterized by Jabin, Otto, Perthame. Among other results they showed that if $\lim_{n\rightarrow \infty} I_{\epsilon_n}(u_n)=0$ for some sequence $u_n\in W^{2,2}_0(\Omega)$ and $u=\lim_{n\rightarrow \infty} u_n$ then $\nabla u$ is Lipschitz continuous outside a locally finite set. This is essentially a corollary to their theorem that if $u$ is a solution to the Eikonal equation $\left|\nabla u\right|=1$ a.e. and if for every "entropy" $\Phi$ function $u$ satisfies $\nabla\cdot\left[\Phi(\nabla u^{\perp})\right]=0$ distributionally in $\Omega$ then $\nabla u$ is locally Lipschitz continuous outside a locally finite set. In recent work with Guanying Peng we generalized this result by showing that if $\Omega$ is bounded and simply connected and $u$ satisfies the Eikonal equation and if $$ \nabla\cdot\left(\Sigma_{e_1 e_2}(\nabla u^{\perp})\right)=0\text{ and }\nabla\cdot\left(\Sigma_{\epsilon_1 \epsilon_2}(\nabla u^{\perp})\right)=0\text{ distributionally in }\Omega, $$ where $\Sigma_{e_1 e_2}$ and $\Sigma_{\epsilon_1 \epsilon_2}$ are the entropies introduced by Ambrosio, DeLellis, Mantegazza, Jin, Kohn, then $\nabla u$ is locally Lipschitz continuous outside a locally finite set. Most of the talk will be an elementary introduction to the Aviles Giga functional, why it is important, why the $\Gamma$-convergence conjecture is so interesting. The final third will motivate and very briefly indicate some of the methods used in the proof of the above result. We will finish with some open problems.

Tuesday, January 24, 2017

4:00 pm in 131 English Building,Tuesday, January 24, 2017

Organizational Meeting

(UIUC Math)

Abstract: A brief meeting to schedule speakers for the semester.

Tuesday, January 31, 2017

4:00 pm in 131 English Building,Tuesday, January 31, 2017

Well-posedness for the "Good" Boussinesq on the Half Line

Erin Compaan   [email] (Erin Compaan)

Abstract: I'll present some recent results on well-posedness for the "good" Boussinesq equation on the half line at low regularities. The method is one introduced by Erdogan and Tzirakis, which involves extending the problem to the full line and solving it there with Bourgain space methods. A forcing term ensures that the boundary condition is enforced. I'll introduce the method and talk about the estimates required to close the argument. This is joint work with N. Tzirakis.

Thursday, February 9, 2017

2:00 pm in 243 Altgeld Hall,Thursday, February 9, 2017

Garling sequence spaces

Ben Wallis (Northern Illinois University)

Abstract: By generalizing a construction of Garling, for each $1\leqslant p<\infty$ and each normalized, nonincreasing sequence of positive numbers $w\in c_0\setminus\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous Banach space $g(w,p)$ related to the Lorentz sequence space $d(w,p)$. Using methods originally developed for studying $d(w,p)$, we show that $g(w,p)$ admits a unique (up to equivalence) subsymmetric basis, although when the weight $w$ satisfies a certain bi-regularity condition, it does not admit a symmetric basis. We then discuss some additional properties of $g(w,p)$ related to uniform convexity and superreflexivity. Joint work with Fernando Albiac and J. L. Ansorena.

Tuesday, February 14, 2017

4:00 pm in 131 English Building,Tuesday, February 14, 2017

Banach Lattices

Chris Gartland   [email] (UIUC Math)

Abstract: Many classical spaces such as $C(K)$ and $L^p(\mu)$ carry not only a normed linear structure, but also a lattice structure which behaves well with respect to the norm and vector space operations. The abstraction of this structure gives rise to objects known as Banach lattices, and classical theorems from point-set topology and measure theory can be proved in this purely abstract setting. We'll define the category of Banach lattices and concentrate on the specific subcategories of M-spaces and abstract $L^p$-spaces. We'll outline the Kakutani representation theorems for the spaces, in the former case establishing a duality between the category of M-spaces and the category of compact Hausdorff spaces. Nutter Butters will be provided.

Thursday, February 23, 2017

2:00 pm in Altgeld Hall,Thursday, February 23, 2017

Inequalities for products of power sums and the classical moment problem.

Bruce Reznick (UIUC)

Abstract: This is a partial repeat of a seminar I gave here in the early 1980s. For $x = (x_1,\dots x_n) \in \mathbb R^n$ and $r \in \mathbb N$, define the $r$-th power sum $M_r(x) = \sum_{i=1}^n x_i^r$. Upper bounds for many products of power sums come from the Hölder and Jensen inequalities. I will discuss some other cases: for example $M_1M_3/(nM_4)>-\frac 18$, where the lower bound is best possible, and the maximum and minimum values of $M_1M_3/M_2^2$ are $\pm \frac{3\sqrt 3}{16}n^{1/2} + \frac 58 + \mathcal O(n^{-1/2})$. In the first case, the classical Hamburger moment problem gives a particularly illuminating explanation. Most of this can be found in my paper: Some inequalities for products of power sums, Pacific J. Math., 104 (1983), 443-463 (MR 84g.26015), available at https://projecteuclid.org/euclid.pjm/1102723674

Tuesday, February 28, 2017

4:00 pm in 131 English,Tuesday, February 28, 2017

John's ellipsoid theorem and applications

Matthew Romney   [email] (UIUC Math)

Abstract: We will discuss and prove a beautiful piece of classical mathematics, a theorem of Fritz John (1948) which characterizes the ellipsoid of maximal volume contained in a convex body in Euclidean space. Among many other applications, it has proven useful in my area of research, quasiconformal mappings.

Thursday, March 2, 2017

2:00 pm in 243 Altgeld Hall,Thursday, March 2, 2017

Singular integrals on Heisenberg curves

Sean Li (U Chicago)

Abstract: In 1977, Calderon proved that the Cauchy transform is bounded as a singular integral operator on the L_2 space of Lipschitz graphs in the complex plane. This subsequently sparked much work on singular integral operators on subsets of Euclidean space. It is now known that the boundedness of singular integrals of certain odd kernels is intricately linked to a rectifiability structure of the underlying sets. We study this connection between singular integrals and geometry for 1-dimensional subsets of the Heisenberg group where we find a similar connection. However, the kernels studied turn out to be positive and even, in stark contrast with the Euclidean setting. Joint work with V. Chousionis.

Tuesday, March 7, 2017

4:00 pm in 131 English Building,Tuesday, March 7, 2017

A variant of Gromov's H\"older equivalence problem for small step Carnot groups

Derek Jung   [email] (UIUC Math)

Abstract: This is the second part of a talk I gave last semester in the Graduate Geometry/Topology Seminar. A Carnot group is a Lie group that may be identified with its Lie algebra via the exponential map. This allows one to view a Carnot group as both a sub-Riemannian manifold and a geodesic metric space. It is then natural to ask the following general question: When are two Carnot groups equivalent? In this spirit, Gromov studied the problem of considering for which $k$ and $\alpha$ there exists a locally $\alpha$-H\"older homeomorphism $f:\mathbb{R}^k\to G$. Very little is known about this problem, even for the Heisenberg groups. By tweaking the class of H\"older maps, I will discuss a variant of Gromov's problem for Carnot groups of step at most three. This talk is based on a recently submitted paper. Some knowledge of differential geometry and Lie groups will be helpful.

Thursday, March 16, 2017

2:00 pm in 243 Altgeld Hall,Thursday, March 16, 2017

Regularity and transversality for Sobolev hypersurfaces

Valentino Magnani (University of Pisa)

Abstract: We show how the regularity of a Sobolev hypersurface implies a measure theoretic transversality with respect to a nonintegrable smooth distribution of possibly lower dimensional subspaces. We consider the model case where the distribution generates a stratified Lie group. These results have been obtained in collaboration with Aleksandra Zapadinskaya.

Tuesday, April 4, 2017

4:00 pm in 131 English Building,Tuesday, April 4, 2017

The Waist Inequality and Quantitative Topology

Hadrian Quan   [email] (UIUC Math)

Abstract: If F is a continuous map from the unit n-Sphere to $\mathbb{R}^q$, then one of its fibers has (n-q)-measure at least that of an (n-q)-dimensional equator. This estimate joins other results like the Isoperimetric Inequality for being simple to state and much harder to prove than at first glance. We’ll discuss the history of this result and some of its relations to topology, geometry, and combinatorics. Time permitting, we shall also sketch a proof of Gromov’s with non-optimal constant.

Tuesday, April 11, 2017

3:00 pm in 241 Altgeld Hall,Tuesday, April 11, 2017

Equivalence of Quasiconvexity and Rank-One Convexity

Terry Harris   [email] (UIUC Math)

Abstract: In 1952 Morrey conjectured that quasiconvexity and rank-one convexity are not equivalent, for functions defined on m by n matrices. For two by two matrices this conjecture is still open. I will outline a proof that equivalence holds on the subspace of two by two upper-triangular matrices, which extends the result on diagonal matrices due to Müller. This is joint work with Bernd Kirchheim and Chun-Chi Lin.

Thursday, April 13, 2017

2:00 pm in 243 Altgeld Hall,Thursday, April 13, 2017

Metric characterization of the Radon-Nikodým property in Banach spaces

Mikhail Ostrovskii (St. John's University)

Abstract: The Radon-Nikodým property (RNP) can be characterized in many different analytic, geometric, and probabilistic ways. The RNP plays an important role in the theory of metric embeddings (works of Cheeger, Kleiner, Lee, and Naor (2006-2009)). In this connection Johnson (2009) suggested the problem of metric characterization of the RNP. The main goal of the talk is to explain the speaker's solution of this problem in terms of thick families of geodesics.

Tuesday, April 18, 2017

4:00 pm in 131 English Building,Tuesday, April 18, 2017

Free rods under tension and compression: cascading and phantom spectral lines

Jooyeon Chung (UIUC Math)

Abstract: In this talk, I will consider the spectrum of the one-dimensional vibrating free rod equation $u'''' − \tau u'' = \mu u$ under tension ($\tau > 0$) or compression ($\tau < 0$). The eigenvalues $\mu$ as functions of the tension/compression parameter $\tau$ are shown to exhibit three distinct types of behavior. In particular, eigenvalue branches in the lower half-plane exhibit a cascading pattern of barely-avoided crossings. I will graphically illustrate properties of the eigenvalue curves such as monotonicity, crossings, asymptotic growth, cascading and phantom spectral lines.

Thursday, April 20, 2017

2:00 pm in 243 Altgeld Hall,Thursday, April 20, 2017

A Hilbert bundle description of differential K-theory

Alexander Gorokhovsky (University of Colorado Boulder)

Abstract: We give a description of differential K-theory in terms of infinite dimensional Hilbert bundles. As an application we propose a construction of twisted differential K-theory. This is a joint work with J. Lott.

Tuesday, May 2, 2017

4:00 pm in 131 English Building,Tuesday, May 2, 2017

Partial synchronization in dynamical systems

Lan Wang (UIUC Math)

Abstract: In nature, synchronization phenomena are ubiquitous. In this talk, we will focus on a specific type of partial synchronization. Basically, partial synchronization occurs due to two reasons: the inhomogeneity of oscillators and the inhomogeneity of coupling strength. For the first reason, a famous representative model is the Kuramoto Model. A sufficient condition of partial synchronization will be given for this model. For the second reason, we consider a slightly different model and analyze its partial synchronized state. Since this state is so fascinating and unexpected, it is specially named as "Chimera state". In the end, some open questions will be discussed. This talk should be approachable to all math graduate students.

Thursday, May 11, 2017

2:00 pm in 241 Altgeld Hall,Thursday, May 11, 2017

Noncommutative random walks and classification of compact quantum groups of Lie type

Sergey Neshveyev (University of Oslo)

Abstract: Compact quantum groups are noncommutative generalizations of compact groups. A natural problem, studied by a number of people and explicitly formulated by Woronowicz already in the 80s, is how to classify quantum groups with representation theory "looking like" that of a compact connected Lie group. I will explain a classification result for a large class of such quantum groups. The main goal, however, will be to discuss noncommutative random walks and their Poisson boundaries, which played a crucial role in obtaining that classification.

3:00 pm in 243 Altgeld,Thursday, May 11, 2017

The Complexity of Classifying Unitaries

Aristotelis Panagiotopoulos   [email] (UIUC Math)

Abstract: Classification problems occur in all areas of mathematics. Descriptive set theory provides methods for measuring the complexity of such problems. For example, using a technique developed by Hjorth, Kechris and Sofronidis proved that the problem of classifying all unitary operators of an infinite dimensional Hilbert space up to unitary equivalence is strictly more difficult than classifying graph structures on domain N up to isomorphism. In this talk I will review the basics from descriptive set theory and explain why the problem of classifying unitaries is so hard. Part of my talk will be based on recent joint work with Martino Lupini.

Friday, September 1, 2017

12:00 pm in 243 Altgeld Hall,Friday, September 1, 2017

Organizational meeting

All A. Vus (UIUC)

Abstract: We will compare our schedules and find a regular time for the seminar. Cookies will be provided.

Friday, September 8, 2017

12:00 pm in 243 Altgeld Hall,Friday, September 8, 2017

Life’s a blur, live on the edge

Derek Jung (UIUC)

Abstract: A focal problem in image processing is deblurring images and a common model for blurring images is convolution with a fixed kernel. I consider an integral operator arising from taking the blurred image of the edge of an opaque wall. I prove that the Tikhonov regularized problem for this operator is well-posed, which essentially means that one can invert the blurred image to the kernel continuously. This is joint work with Dr. Aaron Luttman and Dr. Kevin Joyce and was performed during a summer internship with National Security Technologies in Las Vegas. No background is necessary, but some knowledge of measure theory would be helpful.

Friday, September 15, 2017

12:00 pm in 243 Altgeld Hall,Friday, September 15, 2017

Fun with Heisenberg

Matt Romney (UIUC)

Abstract: This is an expository talk on a fascinating mathematical object, the Heisenberg group. It is the simplest example of what could be termed a "non-commutative vector space". Yet despite its simple definition, many basic questions about its geometry are still open. The Heisenberg group arises in a variety of mathematical contexts, most notably quantum mechanics and complex analysis of several variables. This talk will give a hands-on introduction to the Heisenberg group, and indicate some of its recent applications.

Friday, September 22, 2017

12:00 pm in 243 Altgeld Hall,Friday, September 22, 2017

Measurable Differentiable Structures and Nonbeddability into RNP spaces

Chris Gartland (UIUC)

Abstract: In 1999, Cheeger proved that doubling metric measure spaces admitting a Poincare inequality carry a 'measurable differentiable structure' with respect to which Lipschitz functions could be differentiated almost everywhere. A major consequence of this theorem is that if such a space were to biLipschitz embed into a finite dimensional normed space (or as was later proved, any RNP space), a generic point would have all of its tangent cones biLipschitz equivalent to some finite dimensional normed space. We'll outline the proof of this consequence and discuss its application to Carnot groups and inverse limits of graphs.

Friday, September 29, 2017

12:00 pm in 243 Altgeld Hall,Friday, September 29, 2017

Unrectifiability Via Projections

Fernando Roman Garcia (UIUC)

Abstract: In the study of analysis on metric spaces, rectifiable sets are the appropriate analogue of smooth manifolds. Due to a celebrated theorem of Rademacher, we know rectifiable sets have a "sort of" almost-everywhere-differentiable structure with respect to which one can define approximate tangent planes and do calculus at almost every point. Classifying rectifiable or, on the other end of the spectrum, purely unrectifiable sets is not an easy task. In this talk we will see a certain classification of purely unrectifiable sets via orthogonal projections, will see how Fourier analysis plays a key role in this subject and will talk about how (if at all) similar classification can be apply to more general metric spaces, specifically the Heisenberg Group. Note: This talk will be expository, introductory and for the most part self-contained. No knowledge beyond a basic understanding of metric spaces and measure theory will be needed to follow along.

Friday, October 6, 2017

12:00 pm in 243 Altgeld Hall,Friday, October 6, 2017

Regularity Lemmata via Graph Theory and Analysis

Aubrey Laskowski (UIUC)

Abstract: I will be exploring some interesting connections between graph theory and analysis through versions of Szemerédi's regularity lemma. Szemerédi's lemma states that every large enough graph can be divided into near equipartitions such that edges between the partitions behave almost randomly. In the style of a 2007 paper by Lovász and Szegedy, I generalize this problem to the setting of stepfunctions on a Hilbert space then dive into the world of graphons, analytic objects which act as the limit to a sequence of dense graphs. No particular background needed in graph theory or analysis.

Friday, October 13, 2017

12:00 pm in 243 Altgeld Hall,Friday, October 13, 2017

The Determinant of the Laplacian in Geometric Analysis

Hadrian Quan (UIUC)

Abstract: In this talk I’ll introduce a few different types of Spectral Functions, and describe their use in geometry, culminating with an introduction to the Determinant of the Laplacian. I’ll spend some time trying to make sense of what this object could mean, especially since the eigenvalues of the laplacian accumulate at infinity. Zeta functions may be involved, but I promise the number $\frac{-1}{12}$ will be nowhere to be seen.

Thursday, October 19, 2017

2:00 pm in 243 Altgeld Hall,Thursday, October 19, 2017

The Schwarz range domain for harmonic mappings of the unit disc with boundary normalization

Józef Zajac (State School of Higher Education, Chelm, Poland)

Abstract: The family ${\mathcal F}$ of boundary normalized harmonic mappings of the unit disc into itself is considered by the author. An expression of the Schwarz type inequalities concerning the mappings in question will be presented. In particular, introduced by Partyka and the author, the Schwarz range domain $$ \bigcup_{F\in {\mathcal F}} \{ F(z) \colon |z|\leq r\},$$ where $0 \leq r < 1$, is well described within the class ${\mathcal F}$. Previously obtained result says that $|F(0)| \leq \frac{2}{3}$ for $F \in {\mathcal F}$, whereas the present one describes precisely the Schwarz range domain of $F(0)$ for $F \in {\mathcal F} $. This research has been strongly motivated by certain problems of gas flow mechanics. Boundary normalization appears to be naturally applicable to aerodynamics when studying mechanics of the jet outlet stream described by harmonic mappings. Some conjectures coming out from this research, concerning jet engine construction will be also presented.

Friday, October 20, 2017

12:00 pm in 243 Altgeld Hall,Friday, October 20, 2017

Toeplitz Operators and Bott Periodicity

Timothy Drake (UIUC)

Abstract: The Toeplitz operators are a class of bounded operators on the Hardy space $H^2$, which may be thought of as infinite-dimensional analogues of Toeplitz matrices. They form a $C^*$-algebra known as the Toeplitz algebra, which plays a central role in the theory of $C^*$-algebras as the "universal $C^*$-algebra generated by a single isometry". I will discuss the properties of this algebra and its relation to $K$-theory, and use it to sketch a proof of the Bott Periodicity Theorem for $C^*$-algebras.

Tuesday, October 24, 2017

2:00 pm in 243 Altgeld Hall,Tuesday, October 24, 2017

Greedy bases and Lebesgue-type constants

Pablo Berna (University of Murcia)

Abstract: Given a basis $\mathcal{B}:=(e_j)_{j=1}^\infty$ of a Banach space $\mathbb{X}$, the $Greedy$ $Algorithm$ $\lbrace \mathcal{G}_m\rbrace_{m=1}^\infty$ is an algorithm of approximation where the objective is to approximate each element $x\in\mathbb{X}$ by $\mathcal{G}_m(x) = \sum_{j\in A(x)}e_j^*(x)e_j$, where $A(x)$ is a set of $m$ indices associated with the largest coefficients of $x$ in absolute value. In this talk we will study the $Greedy$ $Bases$ and we will obtain new bounds of the called $Lebesgue-type$ $constants$ for the greedy approximation. These new bounds are given only in terms of the upper democracy functions of the basis and its dual and we will also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces.

Thursday, October 26, 2017

2:00 pm in 243 Altgeld Hall,Thursday, October 26, 2017

A Spider's Web of Doughnuts

Daniel Stoertz (Northern Illinois University)

Abstract: A result in complex dynamics states that, if the Julia set of a holomorphic map of polynomial type is a Cantor set, then the fast escaping set of its Poincaré Linearizer at a repelling fixed point is a spider's web. Some work has been done to generalize this result into higher-dimensional dynamics with quasiregular mappings, though this work has required that the Cantor sets be tame. We will summarize the necessary background in complex dynamics and the dynamics of quasiregular mappings to state the result for wild (and therefore all) Cantor sets. We will then present topological results justifying interest, as well as outline the strategy toward proving this more general result. For a particular type of Cantor set, the resulting fast escaping set will be built out of tori, making a spider's web of doughnuts.

Friday, October 27, 2017

12:00 pm in 243 Altgeld Hall,Friday, October 27, 2017

To Be Announced

Maria Siskaki

Friday, November 3, 2017

12:00 pm in 245 Altgeld Hall,Friday, November 3, 2017

To Be Announced

Kimberly Duran (UIUC)

Friday, December 1, 2017

12:00 pm in 243 Altgeld Hall,Friday, December 1, 2017

To Be Announced

Lan Wang (UIUC)