College of Liberal Arts and Sciences

# Department of Mathematics

## Analysis and Probability Seminar

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Title: Embeddings of the Heisenberg group and the Sparsest Cut problem
Speaker: Robert Young (Courant Institute)
Time: Friday, January 19, 2018 at 1:30 pm
Place: MONT 414Abstract: The Heisenberg group $\mathbb{H}$ is a sub-Riemannian manifold that is hard to embed in $\mathbb{R}^n$. Cheeger and Kleiner introduced a new notion of differentiation that they used to show that it does not embed nicely into $L_1$. This notion is based on surfaces in $\mathbb{H}$, and in this talk, we will describe new techniques that let us quantify the "roughness" of such surfaces, find sharp bounds on the distortion of embeddings of $\mathbb{H}$, and estimate the accuracy of an approximate algorithm for the Sparsest Cut Problem. (Joint work with Assaf Naor)

Title: Spectral heat content for Levy processes
Speaker: Hyunchul Park (SUNY at New Paltz)
Time: Friday, February 16, 2018 at 1:30 pm
Place: MONT 414Abstract: In this talk, we study a short time asymptotic behavior of spectral heat content for Levy processes. The spectral heat content of a domain $D$ can be interpreted as the amount of heat if the initial temperature on $D$ is 1 and temperature outside $D$ is identically 0 and the motion of heat particle is governed by underlying Levy processes. We study spectral heat content for arbitrary open sets with finite Lebesgue measure in a real line under some growth condition on the characteristic exponents of the Levy processes. We observe that the behavior is very different from the classical heat content for Brownian motions. We also study the spectral heat content in general dimensions when the processes are of bounded variation. Finally we prove that asymptotic expansion of spectral heat content is stable under integrable perturbation when heat loss is sufficiently large. This is joint work with Renming Song and Tomasz Grzywny.

Title: A Functional Analytic Approach to Self-improvement Properties
Speaker: Simon Bortz (University of Minnesota)
Time: Friday, March 9, 2018 at 1:30 pm
Place: MONT 414Abstract: In 1963, Norman Meyers showed that $W^{1,2}_{loc}$ solutions to divergence form elliptic operators have a gradient in $L^p$ for some $p > 2$. Modern proofs of this fact utilize the Gehring Lemma (1973). This property can be observed in fractional integro-differential equations. In fact, one can show that solutions to these fractional equations improve in differentiability as well; this was recently investigated by Kuusi, Mingione and Sire. Led by this, my co-authors and I sought to exploit hidden' non-local structure in parabolic equations to prove that solutions are locally HÃ¶lder in time with values in $L^p$ for some $p > 2$. While the Gehring' approach was effective, we found an alternative functional analytic proof that is quite simple. I will show how to apply this functional analysis approach to parabolic equations in detail, then I will indicate how this method applies to other equations. This is joint work with P. Auscher, M. Egert and O. Saari.

Title: Densely defined non-closable curl on the Mackay-Tyson-Wildrick carpets
Speaker: Alexander Teplyaev (University of Connecticut)
Time: Friday, March 23, 2018 at 1:30 pm
Place: MONT 414Abstract: The talk will discuss the possibly degenerate behavior of the exterior derivative operator defined on 1-forms on metric measure spaces. The main examples we consider are the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one-dimensional, they have positive two-dimensional Lebesgue measure and carry nontrivial 2-forms. We prove that in this case the curl operator (and therefore also the exterior derivative on 1-forms) is not closable, and that its adjoint operator has a trivial domain. This is a joint work with Michael Hinz.

Title: Gradient Bounds for Kolmogorov Type Diffusions
Speaker: Phanuel Mariano (University of Connecticut)
Time: Friday, March 30, 2018 at 1:30 pm
Place: MONT 414Abstract: The Kolmogorov diffusion is the joint process of Brownian motion together with integrated Brownian motio. The generator of this process is known to be hypoelliptic (diffusions driven by vector fields whose Lie algebra span the whole tangent space) rather than elliptic. We study gradient bounds and other functional inequalities for the diffusion semigroup generated by Kolmogorov type operators. The focus is on two different methods: coupling techniques and generalized $\Gamma$-calculus techniques. For the coupling technique, we use a coupling by parallel translation to induce a coupling on the Kolmogorov type diffusions. In the $\Gamma$-calculus approach, we will prove a new generalized curvature dimension inequality to study various functional inequalities. The class of processes we study are general and includes Kolmogorov diffusions where the Brownian motion lives on a Riemannian manifold. This talk is based on joint work with Fabrice Baudoin and Maria Gordina.

Title: Sufficient conditions for $C^{1,\alpha}$ parametrization and rectifiability
Speaker: Silvia Ghinassi (Stony Brook University)
Time: Friday, April 6, 2018 at 1:30 pm
Place: MONT 414Abstract: We provide sufficient conditions for a set or measure in $\mathbb{R}^n$ to be $C^{1,\alpha}$ $d$-rectifiable, with $\alpha \in [0,1]$. The conditions use a Bishop-Jones type square function and all statements are quantitative in that the $C^{1,\alpha}$ constants depend on such a function. Key tools for the proof come from Guy David and Tatiana Toro's parametrization of Reifenberg flat sets (with holes) in the Holder and Lipschitz categories.

Title: Extending Sobolev functions from Euclidean domains
Speaker: Tapio Rajala (University of Jyvaskyla)
Time: Friday, April 13, 2018 at 1:30 pm
Place: MONT 414Abstract: When are Sobolev functions defined on a subset of the Euclidean space extendable to the whole space without increasing the Sobolev norm by more than a constant multiplicative factor? Partial answers to this question have been given by various people. For example, from the works of Calderon (1961) and Stein (1970) we know that Lipschitz domains have this property. More generally, this holds also for all uniform domains as was later shown by Jones in 1981. In this talk, we will look at how the geometry of the domain affects the extendability of Sobolev functions. We will concentrate on the Euclidean plane where in the case of bounded simply connected domains a full characterization of the domains can be given.

Title: Sub-Riemannian interpolation inequalities
Speaker: Luca Rizzi (Grenoble University)
Time: Friday, April 20, 2018 at 1:30 pm
Place: MONT 414Abstract: Sub-Riemannian structures can be described as limits of Riemannian ones with $\mathrm{Ric} \to -\infty$ and they represent, in a certain sense, the most singular case among the three great classes of geometries (Riemannian, Finlser, and sub-Riemannian ones). In this talk, we discuss how, under generic assumptions, these structures support interpolation inequalities a la Cordero-Erasquin-McCann-Schmuckenschlager. As a byproduct, we characterize the sub-Riemannian cut locus as the set of points where the squared sub-Riemannian distance fails to be semiconvex. Specifying our results to the case of the Heisenberg groups, we recover in an intrinsic way the inequalities obtained by Balogh, Kristaly and Sipos. As a further application, we obtain new and sharp results on the measure contraction properties of the standard Grushin structure. The techniques are based on optimal transport and sub-Riemannian Jacobi fields.

Title: Some recent work on multi-linear and multi-parameter singular integral operators
Speaker: Lu Zhang (SUNY at Binghampton)
Time: Friday, April 27, 2018 at 1:30 pm
Place: MONT 414Abstract: The study of multi-linear and multi-parameter singular integral operators plays an important role in modern harmonic analysis, and I will introduce some of my recent works that are related to this area. It includes some multi-linear and multi-parameter Fourier multiplier type operators, pseudo-differential operators, Fourier integral operators, and some generalized Calderon-Zygmund operators. Our work is on the boundedness of these operators on $L^p$ and multi-parameter $H^p$ spaces.

Title: Simulation of Gaussian Operator-scaling random fields
Speaker: Celine Lacaux (Avignon University)
Time: Tuesday, May 1, 2018 at 12:30 pm
Place: MONT 214Abstract: Operator-scaling random fields satisfy an anisotropic self-similarity property, which extends the classical self-similarity property. Hence they generalize the fractional Brownian field, which is the most famous isotropic Gaussian self-similar random field. In this talk we focus on operator Gaussian random fields with stationary increments and with variograms defined as anisotropic deformations of the fractional Brownian field variogram. Stein has proposed a fast and exact method of simulation of fractional Brownian fields, which is based on a locally stationary periodic representation and uses the fast Fourier transform. We adapt this method to get a fast and exact synthesis of the class of operator-scaling fields we consider. This talk is based on a joint work with Hermine BiermÃ© (Poitiers, France).

Organizer: Vasileios Chousionis and Oleksii Mostovyi