Seminar Calendar
for Analysis Seminar events the year of Friday, April 21, 2017.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, January 19, 2017

Analysis Seminar
2:00 pm   in 243 Altgeld Hall,  Thursday, January 19, 2017
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Submitted by tumanov.
Andrew Lorent (University of Cincinnati)
The Aviles Giga functional. A history, a survey and some new results
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

Graduate Student Analysis Seminar
4:00 pm   in 131 English Building,  Tuesday, January 24, 2017
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Submitted by compaan2.
(UIUC Math)
Organizational Meeting
Abstract: A brief meeting to schedule speakers for the semester.

Tuesday, January 31, 2017

Graduate Student Analysis Seminar
4:00 pm   in 131 English Building,  Tuesday, January 31, 2017
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Submitted by compaan2.
Erin Compaan   [email] (Erin Compaan)
Well-posedness for the "Good" Boussinesq on the Half Line
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

Analysis Seminar
2:00 pm   in 243 Altgeld Hall,  Thursday, February 9, 2017
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Submitted by tumanov.
Ben Wallis (Northern Illinois University)
Garling sequence spaces
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

Graduate Student Analysis Seminar
4:00 pm   in 131 English Building,  Tuesday, February 14, 2017
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Submitted by compaan2.
Chris Gartland   [email] (UIUC Math)
Banach Lattices
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

Analysis Seminar
2:00 pm   in Altgeld Hall,  Thursday, February 23, 2017
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Submitted by tumanov.
Bruce Reznick (UIUC)
Inequalities for products of power sums and the classical moment problem.
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

Graduate Student Analysis Seminar
4:00 pm   in 131 English,  Tuesday, February 28, 2017
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Submitted by compaan2.
Matthew Romney   [email] (UIUC Math)
John's ellipsoid theorem and applications
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

Analysis Seminar
2:00 pm   in 243 Altgeld Hall,  Thursday, March 2, 2017
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Submitted by tumanov.
Sean Li (U Chicago)
Singular integrals on Heisenberg curves
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

Graduate Student Analysis Seminar
4:00 pm   in 131 English Building,  Tuesday, March 7, 2017
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Submitted by compaan2.
Derek Jung   [email] (UIUC Math)
A variant of Gromov's H\"older equivalence problem for small step Carnot groups
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

Analysis Seminar
2:00 pm   in 243 Altgeld Hall,  Thursday, March 16, 2017
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Submitted by tumanov.
Valentino Magnani (University of Pisa)
Regularity and transversality for Sobolev hypersurfaces
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

Graduate Student Analysis Seminar
4:00 pm   in 131 English Building,  Tuesday, April 4, 2017
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Submitted by compaan2.
Hadrian Quan   [email] (UIUC Math)
The Waist Inequality and Quantitative Topology
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

Graduate Analysis Seminar
3:00 pm   in 241 Altgeld Hall,  Tuesday, April 11, 2017
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Submitted by compaan2.
Terry Harris   [email] (UIUC Math)
Equivalence of Quasiconvexity and Rank-One Convexity
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

Analysis Seminar
2:00 pm   in 243 Altgeld Hall,  Thursday, April 13, 2017
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Submitted by tumanov.
Mikhail Ostrovskii (St. John's University)
Metric characterization of the Radon-Nikodım property in Banach spaces
Abstract: The Radon-Nikodım 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

Graduate Student Analysis Seminar
4:00 pm   in 131 English Building,  Tuesday, April 18, 2017
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Submitted by compaan2.
Jooyeon Chung (UIUC Math)
Free rods under tension and compression: cascading and phantom spectral lines
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

Analysis Seminar
2:00 pm   in 243 Altgeld Hall,  Thursday, April 20, 2017
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Submitted by fboca.
Alexander Gorokhovsky (University of Colorado Boulder)
A Hilbert bundle description of differential K-theory
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.

Thursday, May 11, 2017

Analysis Seminar
2:00 pm   in 241 Altgeld Hall,  Thursday, May 11, 2017
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Submitted by fboca.
Sergey Neshveyev (University of Oslo)
TBA