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The seminar usually starts on Mondays at 14:15 and talks take place on campus and/or via Zoom depending on the pandemic situation.

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Location for the second period (November/December): MaD380 / online

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• September 13, 2021: Carlos Mudarra (University of Jyväskylä)

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Whitney extension theorems for convex functions

Abstract: The classical Whitney extension theorem determines whether a jet (i.e., a function, along with a suitable family of polynomials) defined on a subset of R^n admits an extension to all of R^n of class C^m or C^{m,w}.

It is natural to wonder under what conditions on those jets, the extensions can be taken to be convex as well. In this talk, we will give full answers to this question for the classes C^1 and C^{1,w}. In the latter case, the solution is given by explicit formulas that are valid even when R^n is replaced with a Hilbert space. Moreover, the techniques we use allow us to obtain improved versions of the original Whitney's theorem for general functions of class C^{1,1} or C^{1,w}.

Additionally, applications to calculus of variations, differential geometry, ODEs, and functional analysis will be discussed. For instance, Kirszbraun's theorem is proven via simple and explicit formulas.

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Based on joint work with Daniel Azagra and Erwan Le Gruyer.

 

• September 20, 2021: Jiayin Liu (University of Jyväskylä)

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Existence of hyperbolic motions for Newtonian and generalized N-body problem: a geometric proof

Abstract: In a recent paper, Maderna-Venturelli established the existence of hyperbolic motions for the classical N-body problem. In this talk, we give a new geometric proof of the existence of hyperbolic motions for a wide range of potentials, including both homogenous (e.g. Newtonian potential) and non-homogenous (e.g. Lennard-Jones potential) ones.

       The method of Maderna-Venturelli is mainly based on classical results for N-body problem, including Von Zeipel (1908), Chazy (1922), and Marchal-Saari (1976). Since for general potentials we can not apply those results, we investigate new intrinsic geometric structures induced by generic potentials. It is a joint work with Duokui Yan and Yuan Zhou.

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• October 4, 2021: Elefterios Soultanis (Radboud University)

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The homotopic Plateau-Douglas problem

Abstract: The Plateau-Douglas problem generalizes Plateau's famous problem and asks to find an area minimizing (weakly conformal) map spanning k given curves (inside a given ambient space) from a surface with k boundary components and given genus. In this talk I will describe the homotopic variant of this problem, where the area minimizer is subject to further topological restrictions. I will describe the relevant topological data, namely 1-homotopy classes, and discuss the minimization problem in a metric space setting where no smooth structure is available.

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• October 11, 2021: Tuomas Sahlsten (Aalto University)

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Averaging surface geometries and Laplacian eigenfunctions

Abstract: Since the works of Brooks, Makover and Mirzakhani, averaging surface geometries has been effective in certain spectral theoretic questions on random Riemann surfaces. These include estimating the spectral gap and studying universal properties of the eigenfunctions of the Laplace-Beltrami operator. The key tool used here is a (pre-)trace formula by Selberg that connects the Laplacian to a sum over the periodic orbits on the surface involving their lengths. Randomisation allows us to average out some pathological surface geometries so that we can control the periodic loop structure in the trace formula. In this talk we will give an overview of this averaging method and how we have used it to study Laplacian eigenfunctions.

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• November 15, 2021: Daniele Cannarsa (University of Jyväskylä)

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Surfaces in 3D contact sub-Riemannian manifolds

Abstract. In this talk we consider two objects induced by a 3-dimensional contact sub-Riemannian manifold on   an embedded surface: the induced distance, defined using the infimum of lengths of curves contained in the surface, and the stochastic process defined by a certain limit of Laplace-Beltrami operators. First, we identify some global conditions for the induced distance to be finite, and we prove that it is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution. Second, we recognise that the stochastic process defined here moves along the characteristic foliation induced on the surface by the contact distribution. Thus, we show that for this stochastic process elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from the separatrices. We illustrate those results with some examples.
[Joint work with Ugo Boscain, Davide Barilari and Karen Habermann.]

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• November 22, 2021: Ville Tengvall (University of Jyväskylä)

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A self-contained proof to Martio's conjecture in the class of BLD-mappings

Abstract. We give a self-contained proof to Martio’s conjecture in the class of BLD-mappings. Unlike the earlier proofs, our proof avoids the use of modulus of a path family. Our approach is based on techniques introduced by Onninen and Zhong in 2008.

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• November 29, 2021: Borys Kuca (University of Jyväskylä)

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Fourier analytic methods in the polynomial Szemerédi theorem

Abstract: From the moment of its invention, Fourier analysis has found applications in a tremendous number of branches of mathematics, ranging from signal processing and differential equations to combinatorics and number theory. In this talk, I will describe Fourier analytic techniques utilised in additive combinatorics. Specifically, I will focus on the polynomial Szemerédi theorem of Bergelson and Leibman, which says that each sufficiently large subsets of integers contain polynomial patterns of a certain form. While the original proof of the theorem comes from ergodic theory, some of its special cases have recently been proven in a quantitative manner using Fourier analytic methods. I will discuss these recent developments with the emphasis on the techniques used in the proofs.