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.