Talk titles and abstracts

• Title: Euler systems: what they are and where to find them
• Abstract:  The talk will give an overview of recent developments in the theory of Euler systems, and explain their applications to some of the deepest conjectures in number theory, the Birch-Swinnerton-Dyer and the Bloch-Kato conjectures.

Invited Talk: Gissell Estrada-Rodríguez (UPC i CRM).

• Title: Beyond the mean-field limit: finite-size effects in interacting swarms.
• Abstract:Continuum PDE approximations of interacting particle systems are valid only in the limit of infinitely many particles, yet real applications always involve a finite number N. In this talk, we present ongoing work aimed at quantifying this finite-size discrepancy for a system of interacting active Brownian particles with short-range repulsion, whose mean-field limit is a nonlinear, nonlocal drift-diffusion equation obtained via propagation of chaos. Starting from finite particle trajectories, we construct a mollified empirical density using Gaussian kernel smoothing and propose measuring its distance to the PDE solution with the Wasserstein metric. Unlike pointwise density comparisons, this metric captures geometric discrepancies such as shifts, spreading and clustering. We numerically study how this error depends on the particle number $N$, the mollification scale $\varepsilon$ and the interaction range $\ell$. We will also discuss our first steps towards testing the framework on physical Kilobot swarm experiments.

Invited Talk: Maria Bras Amorós (UPC)

• Title: Numerical semigroups, conjectures, algorithms, and applications.
• Abstract:  A numerical semigroup is a subset of N containing zero, with finite
   complement, and such it is closed under addition. We will introduce
   the main results on the topic, the connection with algebraic geometry,
   and the construction of different trees for their classification by
   different parameters.  Some open conjectures will be presented relating the parameters of a
   numerical semigroup as well as concerning their growth with respect to
  the maximum gap, i.e. the Frobenius number, and with respect to the
   number of gaps, i.e., the genus. We will expose the partial known
   results and the state of the art of the algorithms for counting
   semigroups.  We will finally get a glimpse into the role that numerical semigroups
   have in the area of error-correcting
   codes and in music theory.

IMTech – Number Theory Joint Seminar: David Loeffler (UniDistance Suisse)

• Title:Euler systems via ultrafilters
• Abstract:In order to attack the Bloch—Kato conjecture and the Iwasawa Main Conjecture, one needs to control the size of Selmer groups (subspaces of Galois cohomology defined by local conditions). The basic machinery of Kolyvagin shows that any Euler system for a p-adic representation gives a bound on the ”strict” Selmer group, i.e. the cohomology classes whose localization at p is 0. However, in practice one wants to bound a different, larger Selmer group with a less strict local condition; this is well understood for p-ordinary representations, but the general case is difficult, since it is very unclear how Kolyvagin’s ”derivative” operator interacts with local conditions. Using a reinterpretation of Kolyvagin’s construction via ultrafilters, we give a simple criterion for which local conditions interact well with the derivative construction. This has applications to the Bloch–Kato conjecture and to Pottharst’s formulation of the Iwasawa Main Conjecture.Euler systems: what they are and where to find them