**Program:**

16:00 - Guido De Philippis

"*The Plateau problem*"

In its classical formulation, the Plateau problem asks to find the surface of minimal area spanning a given boundary. Understanding the proper notion of surface, area and of "spanning a boundary" has lead since the pioneering works of Almgren-De Giorgi/Federer-Fleming/Reifenberg in the 50's to the development of beautiful and powerful tools in Geometric Measure Theory.

17:00 - Antonio Lerario

"*Statistical real algebraic **geometry*"

If we pick a real polynomial of degree d, the number of its real zeroes can range anywhere from zero to d (but the number of *complex* zeroes, counted with multiplicity is exactly d). Similarly, the *complex* zero set of a generic real polynomial in two variables is a Riemann surface (a complex curve!) whose topology depends only on the degree of the polynomial, but the number of components of its real part (a curve in the real plane) strongly depends on the choice of the coefficients. Hilbert's Sixteenth Problem asks to study the "number, shape and position" of these components -- a complete answer is known only for curves of degree up to 8! An exciting point of view comes by replacing the word "generic" with "random", asking for the typical situation. In this seminar I will present a probabilistic approach to this problem, discussing recent developments on the topology of real random manifolds...

A coffee break will be offered between the two colloquia.