Mathematics have an essential role to play at the EPFL because of the application range of its ideas and methods, which is leading to an increasing use of mathematical models by scientists and engineers.
The mathematical fields of potential interest at EPFL are diverse, thus underlining the need to maintain strong research groups across the range of fundamental and applicable mathematics; the resulting interactions can stimulate fundamental and applied mathematical research, as well as being of direct benefit to other domains.
The goal of this workshop is to discuss the wide set of practical and theoretical applications of the computability of partition functions and address the challenges that arise.
I will talk about a project to import trace methods, usually reserved for algebraic K-theory computations, into the study of periodic orbits of continuous dynamical systems (and vice-versa). Our main result so far is that a certain fixed-point invariant built using equivariant spectra can be ``unwound'' into a more classical invariant that detects periodic orbits. As a simple consequence, periodic-point problems (i.e. finding a homotopy of a continuous map that removes its n-periodic orbits) can be reduced to equivariant fixed-point problems. This answers a conjecture of Klein and Williams, and allows us to interpret their invariant as a class in topological restriction homology (TR), coinciding with a class defined earlier in the thesis of Iwashita and separately by Luck. This is joint work with Kate Ponto.
By: Cary Malkiewich
In this talk we consider two connected problems:
First, we study the classical problem of the first passage hitting density of an Ornstein-Uhlenbeck process. We give two complementary (forward and backward) formulations of this problem and provide semi-analytical solutions for both. The corresponding problems are comparable in complexity. By using the method of heat potentials, we show how to reduce these problems to linear Volterra integral equations of the second kind. For small values of t we solve these equations analytically by using Abel equation approximation; for larger t we solve them numerically. We also provide a comparison with other known methods for finding the hitting density of interest, and argue that our method has considerable advantages and provides additional valuable insights.
Second, we study the non-linear diffusion equation associated with a particle system where the common drift depends on the rate of absorption of particles at a boundary. We provide an interpretation as a structural credit risk model with default contagion in a large interconnected banking system. Using the method of heat potentials, we derive a coupled system of Volterra integral equations for the transition density and for the loss through absorption. An approximation by expansion is given for a small interaction parameter. We also present a numerical solution algorithm and conduct computational tests.
By: Alexander LIPTON, SilaMoney, MIT & EPFL
The periods of a modular function f are integrals of f along geodesics in the hyperbolic plane joining a real irrational quadratic number with its Galois conjugate. When f is the well-known j-function, its periods have been the object of various recent works of Duke, Imamoglu and Toth, and have been viewed as analogs of singular moduli for real quadratic fields. In this talk we address two conjectures of Kaneko that predict some specific behaviours of the periods of j around geodesics that correspond to Markov quadratics. Markov quadratics are those which can be worse approximated by rationals; they give the beginning of the Lagrange spectrum in Diophantine approximation. This is joint work with O. Imamoglu.
By: Bengoechea Duro Paloma (ETH Zurich)
The following is joint work with Rachel Newton. In the spirit of work by Yongqi Liang, we relate the arithmetic of rational points to that of zero-cycles for the class of Kummer varieties over number fields. In particular, if X is any Kummer variety over a number field k, we show that if the Brauer-Manin obstruction is the only obstruction to the existence of rational points on X over all finite extensions of k, then the Brauer-Manin obstruction is the only obstruction to the existence of a zero-cycle of any odd degree on X. Building on this result and on some other recent results by Ieronymou, Skorobogatov and Zarhin, we further prove a similar Liang-type result for products of Kummer varieties and K3 surfaces over k.
By: Francesca Balestrieri (Max Planck Institute, Bonn)