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Binomial coefficients have been studied since the tenth century and appear in many places. They are integers, but from some of the ways of defining them this property is not evident! We will present various proofs of the integrality of the binomial coefficients using techniques such as a recursion, combinatorics, calculus, and group theory. Time permitting, we will discuss a generalization called a $$q$$-analogue, where the non-obvious property of being an integer is replaced by the non-obvious property of being a polynomial.

Suppose someone asks you to distribute 4 points on the sphere in a nice and regular way. You would probably pick vertices of a tetrahedron. A cube for 8, a dodecahedron for 12 and an icosahedron for 20. But what if someone asks you to distribute 21 points? We are out of Platonic Bodies! I will discuss an old idea of Sobolev who proposed to put them in such a way that the average of polynomials in these points coincides with the average of polynomials on the sphere for as many polynomials as possible. This recovers the Platonic Bodies and suggests many questions. I will discuss some of them, make a quick detour into Analytic Number Theory and then tell you how to do it on Graphs. This allows us, for any Graph, to define "Platonic Bodies in the Graph". These are absolutely gorgeous, very beautiful sets that even have some surprising applications -- I promise you many nice pictures and many open problems!

From the title article by Á. Lozano-Robledo and H.B. Daniels(AMS Notices, 2017) to the current "The Ubiquity of Elliptic Curves" by E.H. Goins (AMS Notices, 2019), what is the reason for 'The unreasonable effectiveness of elliptic curves'? This talk will illustrate the role of elliptic curves, and more generally Algebraic Geometry, in Mathematical Physics, Information Theory and Cryptography, and Machine Learning, presenting some of the speaker's results and open questions for future research.

Abstract: Inspired by the recent developments in differential geometry and calculus of variations, there have been several approaches to identifying a suitable notion of local (Ricci) curvature on non-smooth spaces, such as graphs and Markov chains. I will describe some of these approaches and review a few recent developments in this topic on discrete curvature. Some of the consequences include a tight Cheeger inequality in abelian Cayley graphs, and diameter bounds on the spectral gap of the graph Laplacian. Several open questions remain.

Abstract: The notion of asymptotically hyperbolic manifolds arises largely from two different contexts: as a compactifiable manifold by conformal deformation (like the Poincarè ball), or as an initial data set in general relativity (like a hyperboloid in the Minkowski space). This talk will mainly focus on the latter case. Beginning with the Einstein Field Equations, I am going to introduce an initial data set, Einstein constraint equations, and how the concept of asymptotically hyperbolic manifolds arises naturally. From there, I will talk about the positive mass theorem for asymptotically hyperbolic manifolds, and relevant result of the joint work with Huang and Martin, which concerns about the rigidity property of the mass.

I will also briefly mention the context of conformally compactifiable manifolds and some interesting results.

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Learning seminar in geometric flow

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