2015-2016

Wednesday May 18th. 2:30-3:30pm, French Hall Seminar Rm 4206. Jim Gill (St Louis University, MO).

Almost Never Conditioning

Abstract: Conditional probability's elementary definition is simple to understand. A deeper definition allows us to condition on events which will almost surely never happen. Of course, uncountably many of these almost surely never happening events will coalesce to almost surety. How do we understand this? This talk we be of no interest to actual probabilists.

Wednesday April 27th. 2:30-3:30pm, Room 125 WCharlton. Andrea Pinamonti (University of Bologna).

Porosity, Differentiability and Pansu's Theorem

Abstract: We use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a σ-porous set. The second result states that irregular points of a Lipschitz function form a σ-porous set. We use these observations to give a new proof of Pansu's theorem for Lipschitz maps from a general Carnot group to a Euclidean space. Based on joint work with Gareth Speight.

Thursday April 21st. 12-1pm, Rm 120 WCharlton. Anton Lukyanenko (University of Michigan).

Quasi-crystals in non-commutative spaces

Abstract: The integer lattice in R^3 is a standard example of a separated net, but other nets also arise in applications (most famously the quasi-crystals in chemistry), and one would like to know whether they are simply perturbations of lattices. In Euclidean space, a criterion of Laczkovich allows one to easily make nets that are not a bounded-distance perturbation of any lattice (not BD rectifiable), and an intricate construction due to McMullen and Burago-Kleiner provides a net that is not even bi-Lipschitz to any lattice (not BL rectifiable).

We study nets and quasi-crystals in the Heisenberg group and more generally (rational) Carnot groups. Lattices in these groups are quite tame, and by a theorem of Malcev may even be viewed as the integer points in appropriate coordinates. We show that a generic net need not be well-behaved: in addition to nets that are not BD or BL rectifiable, there exist BD-rectifiable ``exotic nets'' that are neither coarsely dense nor uniformly discrete in Malcev coordinates.

On the other hand, in applications, a natural construction of quasi-crystals yields easily-understood nets whose BD rectifiability is based on a certain Diophantine condition, showing that almost every Heisenberg quasi-crystal is a BD perturbation of a lattice.

Wednesday April 13th. 2:30-3:30pm, Room 119 WCharlton. Scott Zimmerman (University of Pittsburgh).

Sobolev extensions of Lipschitz mappings into metric spaces

Abstract: A metric space is Lipschitz (n-1)-connected if there is a constant C>0 so that any L-Lipschitz map from the Euclidean (n-1)-sphere into the space can be extended to a CL-Lipschitz map on the n-ball. In particular, Wenger and Young showed in 2010 that the nth sub-Riemannian Heisenberg group is Lipschitz (n-1)-connected.

For any Lipschitz continuous mapping from a closed set in Euclidean space into the Heisenberg group, we will construct an extension which is a Sobolev mapping into the Euclidean space and whose weak derivatives satisfy the horizontality condition almost everywhere. Moreover, if the co-domain is replaced by any Lipschitz (n-1)-connected metric space, then the same construction may be used to build an extension which is in the Reshetnyak-Sobolev class. No prior knowledge of the Heisenberg group is necessary.

Thursday April 7th. 12-1pm, Room 120 WCharlton. Matthew Badger (University of Connecticut).

Geometry of measures: rectifiability

Abstract: A fundamental concept in geometric measure theory is a decomposition of sets and measures into its rectifiable ("regular") and purely unrectifiable ("irregular") parts. The qualitative theory of rectifiability in Euclidean space, developed across the last century, beginning with the groundbreaking work of Besicovitch (1928, 1938) and later generalized and improved upon in a series of papers by Morse and Randolph (1944), Federer (1947), Moore (1950), Marstrand (1961, 1964), Mattila (1975), and Preiss (1987). In particular, in the presence of "absolute continuity", these investigations revealed a deep connection between the rectifiability of a measure and the asymptotic behavior of the measure on small balls. A quantitative counterpart to this theory emerged in the 1990s, with major contributions by Jones (1990), David and Semmes (1991, 1993), Okikiolu (1992) and Pajot (1997).

In this talk, I will describe my recent work with Raanan Schul. We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in Euclidean space, without assuming the "absolute continuity" regularity assumptions of previous investigations. In particular, we characterize locally finite measures that give full measure to a countable family of finite length curves. Our characterization is in terms of a geometric square function that is built from "snapshots" of the measure at multiple locations and scales.

This talk will be aimed at a general mathematical audience.

Wednesday April 6th. 2:30pm, Room 119 WCharlton. Estibalitz Durand-Cartagena (UNED, Madrid).

Rectifiability of self-contracted curves in the Euclidean space and applications.

Abstract: Self-contracted curves were introduced by Daniilidis-Ley-Sabourau (2010) to provide a unified framework for the study of convex and quasiconvex gradient dynamical system. A curve gamma:I->R^n is self-contracted, if for every t1<=t2<=t3 in I we have d(gamma(t1),gamma(t3))>= d(gamma(t2),gamma(t3)). We prove that in Euclidean spaces of finite dimension, bounded self-contracted curves have finite length. Applications to continuous and discrete dynamical systems will be also discussed.

The talk is based on a joint work with A. Daniiliidis, G. David and A. Lemenant.

Wednesday March 30th. 2:30-3:30pm, Room 119 WCharlton. Professor Miodrag Mateljevic (University of Belgrade, Serbia).

Distortion of quasiconformal and harmonic mappings

Abstract: We study quasiconformal (qc) mappings in the plane and in space. In particular, we investigate the Lipschitz-continuity of mappings which satisfy in addition a certain partial differential equation (or inequality). Some of the obtained results can be considered as versions of a Kellogg-Warshawski-type theorem for qc-mappings. Among the other tools, we use interior estimate for a Poisson-type inequality.

Wednesday March 16th. 2:30-3:30pm, Room 119 WCharlton. Thomas Bieske (University of South Florida).

Geometry and Partial Differential Equations in Sub-Riemannian Spaces.

Abstract: Sub-Riemannian spaces are metric spaces where travel in certain directions is restricted. We will discuss the interplay between this geometry and solutions to partial differential equations. This talk will be accessible to graduate students.

Thursday March 10th. 12-1pm, Room 120 WCharlton. Xiangdong Xie (Bowling Green State University).

Rigidity of fiber-preserving quasisymmetric maps.

Abstract: We study the rigidity property of quasisymmetric maps that preserve a foliation. We show that, under quite general conditions, such quasisymmetric maps are biLipschitz. We then give an application to quasisymmetric maps between Carnot groups. This is a joint work with Enrico Le Donne.

Monday March 7th. 2:30-3:30pm, Room 119 WCharlton. Vyron Vellis (University of Jyvaskyla, Finland).

Quasisymmetric parametrization on the real line.

Abstract: A well known problem in Geometric Analysis asks whether a quasisymmetric (or bi-Lipschitz) mapping from a subset E of a space X to X extends to a quasisymmetric (or bi-Lipschitz) mapping of X into itself. In this talk we answer the question in the case that X is the real line. We also briefly discuss the case where X is the n-space R^n.

Wednesday February 24th. 2:30-3:30pm, Room 119 WCharlton. Nages Shanmugalingam.

Characterising $\infty$-Poincar\'e inequality in the metric setting: geometric, metric, and functional perspectives.

Abstract: In the metric setting, a viable theory of Sobolev spaces in terms of upper gradients exists when the metric space supports a Poincar\'e inequality. The weakest of the Poincar\'e inequalities is the $\infty$-Poincar\'e inequality. In this talk I will discuss some characterizations of $\infty$-Poincar\'e inequality. The talk is based on joint works with Estibalitz Durand-Cartagena, Jesus Jaramillo, and Alex Williams.

Thursday February 18th. 12-1pm, Room 120 WCharlton. Lizaveta Ihnatsyeva (Kansas State University).

Measure density and extension of Besov and Triebel-Lizorkin functions

Abstract: Recently there have been introduced several analogues for Besov spaces and Triebel-Lizorkin spaces in a quite general setting, which, in particular, includes certain topological manifolds, fractals and graphs. Employing one of the available definitions, we study extension domains for Besov-type and for Triebel-Lizorkin type functions in the setting of a metric measure space with a doubling measure; as a special case we obtain a characterization of extension domains for classical Besov spaces defined via $L^p$-modulus of smoothness.

The talk is based on joint work with Toni Heikkinen and Heli Tuominen.

Wednesday February 10th. 2.30-3.30pm, Room 119 WCharlton. Gareth Speight.

Lusin Approximation for Horizontal Curves in Carnot Groups (Part 2)

Wednesday February 3rd. 2.30-3.30pm, Room 119 WCharlton. Gareth Speight.

Lusin Approximation for Horizontal Curves in Carnot Groups

Abstract: A Carnot group is a Lie group whose tangent space admits a decomposition as a direct sum of subspaces. The first subspace, called the horizontal layer, plays a special role and generates the remaining subspaces via Lie brackets. An absolutely continuous curve is horizontal if its tangents belong to the horizontal layer. In a Carnot group, any pair of points can be connected by horizontal curves. We ask whether any horizontal curve can be approximated by a smooth horizontal curve, and discuss recent work with Enrico Le Donne for the case of step 2 Carnot groups.

Friday Jan 15th. 3.30-4.30pm, Seminar Room 4206. Professor Jesus Jaramillo. (University Complutense de Madrid, Spain).

Smooth surjections and surjective restrictions.

Abstract: Given a smooth surjective mapping $f : E \to F$ between Banach spaces, we investigate the existence of a subspace $G$ of $E$, with smaller density, such that the restriction of $f$ to $G$ remains surjective. Using metric tools, we give some positive results, and we show also some counterexamples. In particular, we stress on the connections of this problem with the regularity and openess properties of the mapping $f$.

Thursday Nov 19th. 1.00pm-1.50pm, Seminar Room 4206. Panu Lahti. Mathematical Institute, Oxford. UK.

A notion of quasicontinuity for functions of bounded variation on metric spaces.

Abstract: Sobolev functions are known to be quasicontinuous, meaning that the restriction of a Sobolev function outside a set of small capacity is continuous. The same cannot hold for functions of bounded variation, or BV functions, since they can have jump sets with large 1-capacity. On a metric space equipped with a doubling measure supporting a Poincar\'e inequality, we show a weaker notion of quasicontinuity for BV functions. More precisely, we show that given a BV function, discarding a set of small 1-capacity makes the function continuous outside its jump set and "one-sidedly" continuous in its jump set.

Thursday Nov 12th. 1.00pm-1.50pm, Seminar Room 4206. Dmitriy Stolyarov, Michigan State University and Chebyshev Lab, St. Petersburg State University.

Sobolev-type embedding theorems and isomorphic types for certain Banach spaces

Abstract: Are the spaces~$C([0,1]^2)$ and~$C^1([0,1]^2)$ isomorphic? The answer “no” is provided by Henkin's theorem. We will prove a natural generalization of Henkin's theorem to smoothnesses of higher order. The main analytic tool is a new Sobolev-type embedding theorem for vector fields. We will focus our attention on a new class of embedding theorems arising from Banach space theory questions.

Tuesday Nov 10th. 4.00pm-5.00pm, Room TBA. Dmitriy Stolyarov, Michigan State University and Chebyshev Lab, St. Petersburg State University.

Bellman functions for small mean oscillation.

Abstract: In 2003, L. Slavin and V. I. Vasyunin independently found sharp constants in the John--Nirenberg inequality for functions on an interval. The proof was based on a solution of a specific two-dimensional extremal boundary value problem that was dual to the initial infinite-dimensional problem, i.e. finding the exact Bellman function. Since then, their method has been transferred to many other inequalities (for example, various forms of the reverse H\"older inequality for Muckenhoupt weights).

We will give a general theory that includes the preceding development. We also give a proof of the duality theorem between the two extremal problems (the initial extremal problem on a class of functions and a finite-dimensional Bellman function problem) that lies at the heart of the theory.

Thursday Nov 5th. 1.00pm-1.50pm, Seminar Room 4206. Kostiantyn Drach. Geometry Department, School of Mathematics and Mechanical Engineering, V. N. Karazin Kharkiv National University.

Reverse isoperimetric problems in Alexandrov metric spaces.

Abstract: In this talk we will discuss a solution of the reverse isoperimetric problemin Alexandrov CBB spaces homeomorphic to a two-dimensional disc. In particular, we will .nd a unique minimizer of the area of this disc provided that the length of its boundary is fixed and the boundary itself has bounded curvature (in a weak sense). The result is obtained by combining methods from the optimal control theory and the fundamental results on gluing and isometric embeddings of Alexandrov metric spaces with curvature bounded below. We will cover the key steps of the proof, and will go into more details depending on time availability.

This is a joint work with prof. Alexander Borisenko.

Thursday Oct 27nd. 1.00pm-1.50pm, Seminar Room 4206. Pavel Zatitskiy. Chebyshev laboratory, St. Petersburg State University, Russia.

Moduli of convexity of $L^p$ spaces: classical results and a Bellman function approach.

Abstract: I will talk about classical results on uniform convexity of Lebesgue spaces $L^p$ (Clarkson's inequalities, Hanner's inequalities) and their proofs based on the Bellman function technique. This is joint work with P.Ivanisvili and D.M.Stolyarov.

Thursday Oct 22nd. 1.00pm-1.50pm, Seminar Room 4206. Micheal Goldberg.

The Stein-Tomas theorem, with an epilogue by Frank and Sabin.

Thursday Oct 15th. 1.00pm-1.50pm, Seminar Room 4206. Lukas Maly.

Self-improvement of Poincaré inequalities and Sobolev-type embeddings II.

Thursday Oct 8th. 1.00pm-1.50pm, Seminar Room 4206. Lukas Maly.

Self-improvement of Poincaré inequalities and Sobolev-type embeddings I.

Abstract: The standard toolbox for analysis on metric spaces includes various Poincaré inequalities, which undergo self-improvement under certain assumptions about the metric measure space. During the talk, I would like to discuss two possible directions of such an improvement of Poincaré inequalities in the context of rearrangement-invariant norms. One of them follows the lines of Keith–Zhong’s result on open-endedness, which was recently studied by Noel DeJarnette for the Orlicz gauges. The other direction leads to Sobolev-type embeddings for Newtonian (and Hajłasz) functions. In particular, most of the optimal Sobolev embeddings known in R^n are valid even in metric spaces that admit a 1-Poincaré inequality.

Wednesday Sept 16rd. 1.00pm-1.50pm, Seminar Room 4206. Xiaoyuw Cui

New characterizations of Sobolev spaces on Euclidean spaces and Heisenberg groups.

Abstract: Recently, many new features of Sobolev spaces were studied in many papers. This talk is devoted to give a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces to the Heisenberg group setting. Moreover, our theorems also provide different characterizations for the second order Sobolev spaces in Euclidean spaces.

Thursday Sept 3rd. 1.00pm-1.50pm, Seminar Room 4206. Gareth Speight

Rademacher’s Theorem and Differentiability in Small Sets

Abstract: Rademacher’s theorem states that Lipschitz functions between Euclidean spaces are differentiable almost everywhere. Different areas of modern research generalize Rademacher’s theorem to more exotic spaces or investigate whether it is optimal. We discuss how aspects of both areas meet in the Heisenberg group. Based on joint work with Andrea Pinamonti.