Colloquium, Dept. of Mathematics and Applied Math. -- Γενικό Σεμινάριο

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Colloquium

Abstract: Ramsey theory is a branch of combinatorics that seeks to find patterns in disorganized situations. One of its main achievements, Szemeredi’s theorem on arithmetic progressions, got an ergodic theoretic proof in 1977 when Furstenberg devised a Correspondence Principle to encode combinatorial information about sets of integers into a dynamical system. Since then ergodic methods have been very successful in obtaining new Ramsey theoretic results, some of which still have no purely combinatorial proof. I will survey some of the history of how ergodic theory and Ramsey theory are interconnected, and explore some recent developments.

Abstract: Topological recursion is a recursive algorithm discovered around 2007. In this talk, our goal is threefold: introduce the algorithm, provide recent results and their motivations, and present the current interests of the field. We will begin by introducing the initial data of topological recursion, which includes a Riemann surface, along with a differential and a bidifferential defined on the surface. Its output is correlation functions that encode the information of interest. Then, we will discuss the example of intersection numbers and new results on this direction. We will conclude with a general overview of the field.

Abstract: The isoperimetric problem has a long histroy, dating back to the ancient time. The problem can be formulated as a calculus of variations problem. The main theme of the talk is a geometric flow approach to this problem. We introduce a generalized mean curvature flow as a path for the optimal solution to the problem. We will discuss why such flow is natural and we will also discuss how this approach can be generalized to solve the isoperimetric problem for Riemannian manifolds with special geometric structures.

Abstract: I will survey recent results and open questions on 'Coarse Graph Theory', an emerging area that combines ideas from graph theory and coarse geometry.

Minimal surface and hypersurface doublings (in Greek: Διπλασιασμοί ελαχιστικών επιφανειών και υπερεπιφανειών)

Abstract: I will start with a general discussion of minimal surfaces and of the earlier results for minimal surface doublings. Eventually I will concentrate on more recent results, ongoing work, and open problems.

(In Greek) Θα ξεκινήσω με μία γενική συζἠτηση των ελαχιστικών επιφανειών και των πρώτων αποτελεσμάτων για διπλασιασμούς ελαχιστικών επιφάνειών. Κατόπιν θα παρουσιάσω τα πιο πρόσφατα αποτελέσματα, τρέχουσα έρευνα, και ανοιχτά προβλήματα.

Abstract: This talk introduces the technology of Distributed Acoustics Sensing (DAS), which uses Fibre-optic (FO) cable as an acoustic array to sense the environment. DAS has wide range of applications due to high spatial and temporal resolution and economic benefit by using existing dark or telecom FO cables on land, along railways and on the seafloor. The applications of DAS technology are presented. Theory on interface waves and the dispersion are presented. An example on using DAS data to estimate shear wave velocity profile in the upper part of the sediments is given, which involves data processing, estimating interface wave dispersion curves and geoacoustic inversion.

Abstract: Formulating a theory of quantum gravity is one of the biggest open problems in mathematical physics. Some of the core technical and epistemological difficulties come from the fact that General Relativity (GR) is ‘generally covariant’, i.e. invariant under change of coordinates by the arbitrary diffeomorphism of the ambient manifold. The Problem of Observables is a famous instance of the difficulties that general covariance brings into quantization: no non-trivial diffeomorphism-invariant quantity has ever been reported on the collection of all spacetimes. It turns out that there is a good reason for this. In this talk, I will present my recent joint work with Marios Christodoulou and George Sparling, where we employ methods from Descriptive Set Theory in order to show that, even in the space of all vacuum solutions, no complete observables can be Borel definable. That is, the problem of observables is to ‘analysis’ what the Delian problem is to `straightedge and compass’.

Abstract: We will discuss some recent connections between model theory (a branch of mathematical logic) and extremal graph theory (a branch of combinatorics). More precisely, we will look at the combinatorial Zarankiewicz problem, which asks for an upper bound on the number of edges of a hypergraph, assuming it has no complete sub-hypergraphs of a given size. We will see how the upper bounds can be improved if we further assume that the class of hypergraphs we consider is "definable" in certain o-minimal structures.

Abstract: Regularisation by noise refers to the phenomenon of certain non-linear dynamical systems behaving better in the presence of a noisy (stochastic) perturbation compared to their deterministic counterpart. In this talk, we will discuss such phenomena for differential equations of the form $$ dX_t=f(X_t) \, dt, \qquad X_0 =x. $$ It is well known that the above equation admits a unique solution if $f$ is Lipschitz continuous, and this is essential sharp: if $f$ is only $\alpha$-Hölder continuous for some $\alpha \in (0,1)$, one might have infinitely many solutions and if $f$ is not even continuous, it might not have solutions at all. We will consider equations of the form \begin{equation} dX_t=f(X_t) \, dt+ \sigma(X_t) \, dB^H_t, \qquad X_0 =x, \end{equation} where $B^H$ is a fractional Brownian motion of Hurst parameter $H \in (1/3,1/2)$. We will see that this equation admits a unique solution for quite irregular $f$, provided that $\sigma$ is bounded away from zero. More precisely, $f$ does not even need to be a function but merely a Schwartz distribution of regularity $\alpha > 1-(1/2H)$ (notice that $\alpha$ can be negative) in the Besov scale $\mathcal{B}^\alpha_{\infty, \infty}$. We will discuss what is meant by a solution in this case and we will present the main ideas which rely on the theory of rough paths, Malliavin calculus, and the stochastic sewing lemma. Our result provides a multiplicative noise analogue to a result of Catellier-Gubinelli in 2016.