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• September 14, 2020: Sylvester Eriksson-Bique (University of Jyväskylä)

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​Splitting Theorem with Modulus​

Abstract: This is a talk on conformal dimension, modulus and bi-Lipschitz embeddings. It is classical, that product spaces, C x I, for I an interval and C any Q-Ahlfors regular space, attain their conformal dimension. Certainly, spaces exist which attain their conformal dimension, and do not exhibit this product structure. However, they are non-Euclidean in the sense that they do not bi-Lipschitz embed into Euclidean space. In this talk, I would like to explain how this phenomenon is a consequence of a general fact: A subset of Euclidean space admitting positive modulus must admit some form of infinitesimal product structure after a blow up. I will present how this relates to many other concepts lately central to analysis on metric spaces. The implication of these is that non-bilipschitz embedding results and rigidity can often be obtained by much less than say by a Poincaré inequality. This is joint work with Guy C. David. The talk will aim to define all relevant notions, but familiarity certainly is helpful.

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• September 21, 2020: Olli Tapiola (University of Jyväskylä)

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Uniform rectifiability, Carleson measure estimates and Varopoulos extensions

Abstract: Connections between boundary geometry and PDE estimates have been studied for a very long time but the work is still on-going and active. In the last couple of years, a lot of progress has been made particularly in domains with co-dimension 1 Ahlfors-David regular or uniformly rectifiable (UR) boundaries. In these types of domains, many results related to geometry, harmonic analysis and PDE walk hand in hand. Recently, Azzam, Hofmann, Martell, Mourgoglou and Tolsa and Hofmann and Le characterized the L^p-solvability and BMO-solvability of the Dirichlet problem with respect to uniform rectifiability and a minimal quantitative connectivity assumption. In this talk, we discuss analogues of these results in possibly disconnected spaces with co-dimension 1 UR boundaries and challenges related to this "1-quantitative" theory. Our results are inspired by earlier work of Varopoulos and they rely strongly on recent advances in epsilon-approximability theory. This is joint work with Steve Hofmann.

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• September 28, 2020: Miguel Garcia Bravo (University of Jyväskylä)

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Lusin properties for subdifferentiable functions

Abstract: The Lusin’s classical theorem states that for every Lebesgue measurable function f:R^n-->R and every ε>0 there exists a continuous function g:R^n-->R such that it agrees with the initial function f except on a set of Lebesgue measure less than ε. Several authors have shown that one can take g to be of class C^k, provided that f has some regularity properties of order k. We will start the talk by showing a bit of the history of this property. We will make special emphasis on the case of Sobolev functions, where the exceptional sets can be taken in terms of adequate Bessel capacities or Hausdorff contents. We will also mention Liu and Tai’s characterization of this property, relating it with the approximate differentiability. After this, it will be shown a new result that reads as follows: every function which has a nonempty Fréchet subdifferential (or proximal subdifferential) at almost every point also has the Lusin property of class C^1 (or C^2). It will also be provided an example showing that this result is no longer true for Taylor subexpansions of higher order.

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• October 5, 2020: Esko Heinonen (University of Jyväskylä)

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Translating solitons over Cartan-Hadamard manifolds

Abstract: The mean curvature flow (MCF) is already a classical and well studied subject but recently the translating solitons of the mean curvature flow have gained a lot of new interest. These translating solitons (or translators) are eternal solutions to the MCF that evolve by translations and (at least in some cases) arise from Type II singularities after a blow up procedure.

I will discuss about the existence of entire graphical translators on Cartan-Hadamard manifolds and we will see how the asymptotic behaviour of these solutions depends heavily on the sectional curvature of the manifold. The talk is based on a joint work with J.-B. Casteras, I. Holopainen and J. de Lira.

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• October 12, 2020: Michele Villa (University of Jyväskylä)

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Higher dimensional Jordan curves

Abstract: In the first part of the talk, I will try to convince you that the title is not as silly as it might sound by sketching a little of the history of the so-called Analyst’s travelling salesman theorem for higher dimensional sets. In particular, I will discuss some (distinct) results of Azzam, Hyde, Schul and myself which provide a full higher dimensional analogue to a famous theorem of Peter Jones from the nineties. In the second part of the talk, I will sketch the proof of some of these results. If time permits, I will give some first applications and conclude with some open questions.

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• October 19, 2020: Daniela Di Donato (University of Jyväskylä) online talk 14:15-15:15

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C^1 submanifolds and Lipschitz graphs in Carnot groups

Abstract: In Euclidean spaces, rectifiable sets are defined as being essentially contained in the countable union of C^1 submanifolds or, equivalently, of Lipschitz graphs. Hence, in Carnot groups, the corresponding notions of C^1_H-regular surfaces and intrinsic Lipschitz graphs are important to develop a satisfactory theory of intrinsic rectifiable sets. Firstly, I present their definitions and some basic properties and then I give a characterization of  C^1_H-regular surfaces. The talk is based on a joint work with Antonelli, Don and Le Donne.

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• October 26, 2020: Giovanni Canarecci (Helsinki) online talk 14:15-15:15

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Orientability and Slicing for Surfaces and Currents in the Heisenberg group

Abstract: When talking about surfaces in the Heisenberg group, it comes natural to ask how flexible they are and how much one can expect from them. When looking at currents in the Heisenberg group, one wants to know how well they can fit surfaces and how their slices behave. And mostly, one may want to know how all these pieces connect together.

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• November 9, 2020: Aapo Kauranen (University of Jyväskylä)

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Characterization of strict limits of BV homeomorphisms

Abstract: There have been recently several advances in smooth approximation of Sobolev homeomorphisms and characterization of closures of sets of Sobolev homeomorphisms in two dimensions.  In this talk I will discuss properties of possible limits of planar BV homeomorphisms under different notions of convergence. I will also present recent  work on characterization of strict limits  of BV homeomorphisms in the plane. We show that these limits is exactly the set of BV mappings satisfying a certain monotonicity condition. This condition is related to a condition used in a recent characterization of limits W^{1,p} homeomorphisms  by De Philippis and Pratelli. Talk is based on joint works with D. Campbell, S. Hencl and E. Radici.

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• November 16, 2020: Sebastiano Nicolussi Golo (University of Jyväskylä)

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From Peano to Bernstein in one seminar

Abstract: I will give a somewhat broad introduction to a vast theory concerning flow of vector fields (Peano!), the transport equation and the continuity equation. The game is all about proving some standard facts starting from the most exotic hypothesis. I will then seize the opportunity to tell you about the very cumbersome hypothesis Ambrosio, Serra Cassano and I came up with. All of this has surprising applications to the study of minimal surfaces in the Heisenberg group (Bernstein!).

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• November 30, 2020: Vesa Julin (University of Jyväskylä)

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Quantifiable approximation of BV functions using neural networks

Abstract: In my talk I will first discuss about neural networks in general and the mathematical formulation of the related optimization problem. Then I will talk about my recent project with Benny Avelin (Uppsala) where we study the convergence of the first order methods for two layer neural networks with ReLU activation function. Our viewpoint is that of function approximation and the goal is to prove convergence rates independent of the number of data-points and nodes. Specifically we prove that gradient descent with elliptic regularization (isotropic noise) converges exponentially fast to equilibrium, in the sense of distributions.