Tag - Monodromy groups

Federico Binda: Motivic monodromy and p-adic cohomologies 

16th May, 2023

In this talk, I will discuss some recent advances in the theory of motives in the context of rigid analytic geometry. Building on work of Ayoub, Bondarko, we provide an equivalence between the category of “unipotent” rigid analytic motives over a non-archimedean field and the category of “monodromy maps” M → M (−1) of algebraic motives over the residue field. This allows us to build a unified framework for the study of monodromy operators and weight filtrations of cohomology theories for varieties over a local field. As an application, we give a streamlined definition of Hyodo–Kato cohomology without recourse to log-geometry, as predicted by Fontaine, and we produce an induced Clemens–Schmid chain complex.

Claus Hertling: Upper triangular matrices and induced structures: vanishing cycles, monodromy groups, distinguished bases, braid group orbits, moduli spaces

13th March, 2023

Upper triangular matrices with ones on the diagonal and entries which are integers (or algebraic integers) arise in many contexts, e.g. as Stokes matrices in the theory of meromorphic connections with irregular poles, in many situations in algebraic geometry (often related to Stokes matrices), especially in quantum cohomology and the theory of isolated hypersurface singularities, but also in the theory of Coxeter groups.

Concepts from singularity theory like vanishing cycles, monodromy groups, Seifert forms, tuples of (pseudo-)reflections and distinguished bases can be derived from upper triangular
matrices in cases beyond singularity theory and are interesting to study.

Additionally, always braid group actions on the matrices and on the distinguished bases are in the background. They give rise to certain covering spaces of the classifying space of the braid group. These are interesting natural global manifolds. Some are well known, others are new.

The talk presents concepts and old and new results. It puts emphasis on some cases from singularity theory and some 3x3 cases.

Danny Neftin: Reducible fibres and monodromy of polynomial maps

28th October, 2021

For a polynomial f∈ℚ[x], Hilbert's irreducibility theorem asserts that the fibre f-1(a) is irreducible over ℚ for all values a∈ℚ outside a "thin" set of exceptions Rf. The problem of describing Rf is closely related to determining the monodromy group of f, and has consequences to arithmetic dynamics, the Davenport-Lewis-Schinzel problem, and to the polynomial version of the question: "can you hear the shape of the drum?". We shall discuss recent progress on describing Rf and its consequences to the above topics.

Based on joint work with Joachim König.

Tasho Kaletha: An explicit supercuspidal local Langlands correspondence

29th October, 2020

We will give an explicit construction and description of a supercuspidal local Langlands correspondence for any p-adic group G that splits over a tame extension, provided p does not divide the order of the Weyl group. This construction matches any discrete Langlands parameters with trivial monodromy to an L-packet consisting of supercuspidal representations, and describes the internal structure of these L-packets.

The construction has two parts. The depth-zero part involves generalizing to disconnected groups results of Lusztig on the decomposition of a non-singular Deligne-Lusztig induction. Higher multiplicities occur in this decomposition and are handled using work of Bonnafé-Dat-Rouquier. The positive-depth part involves functorial transfer from a twisted Levi subgroup, which is made possible by an improvement of Yu's construction of supercuspidal representations obtained in recent joint work with Fintzen and Spice, and consideration of Harish Chandra characters.

We will also discuss ongoing work towards related conjectures: Shahidi's generic L-packet conjecture, Hiraga-Ichino-Ikeda formal degree conjecture,  stability and endoscopic transfer.

Harry Hyungryul Baik: Normal generators for mapping class groups are abundant in the fibered cone

17th July, 2020

We show that for almost all primitive integral cohomology classes in the fibred cone of a closed fibred hyperbolic 3-manifold, the monodromy normally generates the mapping class group of the fibre. The key idea of the proof is to use Fried’s theory of suspension flow and dynamic blow-up of Mosher. If the time permits, we also discuss the non-existence of the analogue of Fried’s continuous extension of the normalized entropy over the fibered face in the case of asymptotic translation lengths on the curve complex.

Ananth Shankar: The p-curvature conjecture and monodromy about simple closed loops

11th May, 2017

The Grothendieck-Katz p-curvature conjecture is an analogue of the Hasse principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its p-curvature vanishes modulo p, for almost all primes p. We prove that if the variety is a generic curve, then every simple closed loop has finite monodromy.