HColl_2017

We shall give an introduction, from various perspectives, to this classical, elementary-looking conjecture formulated by M. Hochster at the beginning of the 70's. We shall then give a hint on how it was finally solved using ideas and techniques from a completely different domain, far away from the familiar noetherian world of commutative algebra, building upon G. Faltings's almost mathematics and P. Scholze's perfectoid geometry.

Laurent Desvillettes (Université Paris Diderot): Coagulation-diffusion equations: how to use results on one dimensional singular parabolic equations for infinite dimensional reaction-diffusion systems

Coagulation-diffusion equations naturally appear in the physics of polymers. They consist in a system of an infinite number of reaction-diffusion equations with infinite sums in the reaction terms.

One of the main issue in this field is the possible appearance of the so-called gelation phenomenon, in which a new phase (the gel) is produced after a finite time. This physical phenomenon corresponds to a blowup in the system of coagulation-diffusion, which can be studied thanks to the duality method and its improvements, devised for one single singular reaction-diffusion equation.

Gian Michele Graf (ETH Zürich): An overview on topological insulators

Topological insulators are materials, which are conducting at their edges, though not in the bulk. Their essential physical properties take the form of an index, often associated to the Hamiltonian."Topological" simply refers to the fact that indices remain invariant under continuous changes. Earliest examples occurred in connection with the Quantum Hall effect, which has been the source of various mathematical developments. For instance, the notion of index which was originally tied to crystalline systems or relatedly to fiber bundles, got extended to disordered systems. In the last decade the concept has been refined by conditioning it to symmetries, such as time-reversal invariance. A general property of the indices, which should be preserved by any generalization, is bulk-edge correspondence, by which they should admit dual formulations, that is in terms of either the bulk or the edge properties of the sample. Examples in dimension 1 and 2 exhibiting different symmetries will be provided.

Tony Yue Yu (CMI and Université Paris-Sud, Orsay): Counting open curves via Berkovich geometry

Motivated by mirror symmetry, we study the counting of open curves in log Calabi-Yau surfaces. Although we start with a complex surface, the counting is achieved by applying methods from Berkovich geometry (non-archimedean analytic geometry). This gives rise to new geometric invariants inaccessible by classical methods. These invariants satisfy a list of very nice properties and can be computed explicitly. If time permits, I will mention the conjectural wall-crossing formula, relations with the works of Gross-Hacking-Keel and applications towards mirror symmetry.

Robert L. Pego (Carnegie Mellon University): Merging-splitting group dynamics by Bernstein function theory (or: How to count fish using mathematics)

We study coagulation-fragmentation equations inspired by a simple model derived in fisheries science to explain data on the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no H-theorem, we are able to develop a rather complete description of equilibrium profiles and large-time behavior, based on complex function theory for Bernstein and Pick (Herglotz) functions. The generating function for discrete equilibrium profiles also generates the Fuss-Catalan numbers that count all ternary trees with (math_error_dvips_noexec): n

nodes. The structure of equilibrium profiles and other related sequences is explained through a new and elegant characterization of the generating functions of completely monotone sequences as those Pick functions analytic and nonnegative on (math_error_dvips_noexec): (-\infty,1)

. This is joint work with Jian-Guo Liu and Pierre Degond.