Seminars from until

Convex extensions are natural ways to extend functions defined on the extreme points of a convex set to the whole set. They rely on Choquet theory, which aims at representing points of convex sets by probability measures over their extreme points.

We focus on the case of a product of convex sets, where we find a correspondence between convex extensions and Kantorovich optimal transport problems. In particular, "classical" Kantorovich costs between probability measures are recovered as convex extensions in the simplex case, and entanglement-free quantum Wasserstein distances in the (non-simplex) case of a quantum state space. Conversely, convex extensions can themselves be recovered as suitable quotients of classical Kantorovich costs.

This correspondence provides a geometric interpretation of optimal transport and allows for transferring tools between this field and the theory of convex extensions.

Baxter's classical theorem for orthogonal polynomials on the unit circle establishes that the linear Fourier coefficients of a measure are in $\ell^1$ if and only if the nonlinear coefficients, i.e., the so-called Verblunsky coefficients, are in $\ell^1$. We prove an analog of Baxter's theorem for the SU(2)-valued nonlinear Fourier transform. The proof of Baxter's theorem is qualitative in nature. We explore possible formulations of such quantitative theorems with norm and Lipschitz estimates. In particular, we construct a counterexample for certain such estimates

How to define an ellipse; its foci and excentricities; the construction of osculating circles; trajectories of billiards; Kepler’s laws about planets; linkages drawing ellipses.

Lattices in C; doubly periodic meromorphic functions; Liouville’s theorem; Weierstrass p-function; poles and zeros; functional algebraic equation between p(z) and p′(z).

Cubic bivariate polynomials; normal forms; singular points; projective curves; tangents; addition law; examples.

Associativity of group law; Mordell’s theorem about finite generation; the rank of an elliptic curve; world records; theorem of Nagel-Lutz; automorphisms; j-invariant.

Integer and rational points on elliptic curves; congruent number problem; reduction modulo primes; applications to cryptography; modular forms; the conjecture of Birch-Swinnerton-Dyer, the ABC-conjecture.

In this talk, we will give an overview of several methods and ideas on how to find regular subgraphs in graphs with some average degree.

We will discuss the still widely opened Erdos-Rado Sunflower Conjecture and we will give a sketch of a recent proof of the Erdos-Szemeredi Conjecture. More precisely, we will show that any graph on $n$ vertices and average degree $C(k)\log(\log(n))$ contains a $k$-regular induced subgraph, which is tight up to the implied constant as shown by a classical construction of Erdos, Pyber and Szemeredi.