Walks, Difference Equations and Elliptic Curves
Friday, 3.11.17, 10:15-11:15, Hörsaal FRIAS, Albertsstr. 19
In the recent years, the nature of the generating series of walks in\nthe quarter plane has attracted the attention of many authors in\ncombinatorics and probability. The main questions are: are they\nalgebraic, holonomic (solutions of linear differential equations) or at\nleast hyperalgebraic (solutions of algebraic differential equations)? In\nthis talk, we will show how the nature of the generating function can\nbe approached via the study of a discrete functional equation over a\ncurve E, of genus zero or one. In the first case, the functional\nequation corresponds to a so called q-difference equation and all the\nrelated generating series are differentially transcendental. For the\ngenus one case, the dynamic of the functional equation corresponds to\nthe addition by a given point P of the elliptic curve E. In that\nsituation, one can relate the nature of the generating series to the\nfact that the point P is of torsion or not.\n This is a collaboration with T. Dreyfus (Irma, Strasbourg), J. Roques\n(Institut Fourier, Grenoble) and M.F. Singer (NCSU, Raleigh).
The algebraic topology of p-adic Lie groups
Thursday, 9.11.17, 10:15-11:15, Raum 403, Eckerstr. 1
Lazard proved that the cohomology of compact p-adic Lie groups satisfies a version of Poincare duality. This indicates an analogy between p-adic Lie groups and real manifolds. I will explain some results which develop this analogy further.
From elementary number theory to string theory and back again
Friday, 10.11.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
I will describe some surprising interactions between number theory, algebraic geometry and mirror symmetry that have appeared in my recent work with Mircea Mustata and Chenyang Xu and that have led to a solution of Veys' 1999 conjecture on poles of maximal order of Igusa zeta functions. The talk will be aimed at a general audience and will emphasize some key ideas from each of the fields involved rather than the technical aspects of the proof.
Modular representations of sl_n with two-row nilpotent p-character
Wednesday, 15.11.17, 10:45-11:45, Hörsaal FRIAS
We will study the category of modular representations of the\nspecial linear Lie algebra with a central p-character given by a\nnilpotent whose Jordan type is a two-row partition. Building on work of\nCautis and Kamnitzer, we construct a categorification of the affine\ntangle calculus using these categories; the main technical tool is a\ngeometric localization-type result of Bezrukavnikov, Mirkovic and\nRumynin. Using this, we give combinatorial dimension formulae for the\nirreducible modules, composition multiplicities of the simples in the\nbaby Vermas, and a description of the Ext spaces. This Ext algebra is an\n"annular" analogues of Khovanov's arc algebra, and can be used to give\nan extension of Khovanov homology to links in the annulus. This is joint\nwith Rina Anno and David Yang.
Functorial test modules
Friday, 17.11.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
In my talk I will report on joint work with Manuel Blickle. I will explain how one can generalize the definition of test ideals \btau to so-called Cartier modules in a functorial way. We obtain several transformation rules with respect to f^! and f_* for various classes of morphisms f: X \bto Y, e.g. for f smooth one has an isomorphism f^! \btau = \btau f^!. Part of the reason for working in this generality is that one has an equivalence with constructible etale p-torsion sheaves up to nilpotence of Cartier modules and these results further support the idea that the test module construction relates to etale nearby cycles similarly to the complex situation where multiplier ideals relate to complex nearby cycles.
Local systems in motivic homotopy theory
Friday, 24.11.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
Local systems in motivic homotopy theory
Friday, 24.11.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
This talk is first devoted to explain how Voevodsky's theory of motivic complexes is built on, and provides, a natural notion of local system. Two key players involved here are homotopy sheaves and cycle modules, which are respectively the theoretical and concrete side of the same notion. The first part of the talk will recall these two notions and their fundamental link.\n The second part of the talk will move on relative motivic complexes. I will explain the construction of a natural t-structure, analogous to the perverse t-structure (but not realized to it!), obtained in collaboration with Bondarko along ideas of Ayoub. Besides describing some of its good properties, I will explain a program to compute its heart, in terms of cycle modules.\n If time allows, I will wander a little bit on the aimed tool motivating these technicalities: the delta-homotopy Leray spectral sequence.\n