Oberseminar Differentialgeometrie: Arthur Bartels (Münster), Vortrag: Long thin covers for flows
Montag, 30.11.2020 16:15 im Raum Zoom
Abstract: Farrell?Jones pioneered a dynamic approach to K-theory computations of group rings.
One of the tool they introduced are long-thin cell structures for flows on manifolds.
Closely related are long thin covers of flows.
I will discuss what they are, why they come up in K-theory of group rings, what we know about them and what I would like to know about them.
Abstract: We will study asymptotic behavior of a simple random walk in random environments. As a driving example, we will look at percolation clusters of various sorts (including iid Bernoulli percolation as well as models carrying long-range correlations like random interlacements, its vacant set, the FK random cluster model, level sets of Gaussian free field etc.). We will sketch a method for proving an almost sure (i..e quenched) large deviation principle for the SRW and allude to the equivalence of this LD behavior to that of a homogenization of a random Hamilton-Jacobi PDE. Similar questions can be asked for the averaged (annealed) behavior of the SRW and time permitting, we will try to address a simple question: how much impact does the inherent impurity (disorder) of the environment have on the rate function(s), resp. on the effective homogenized equation(s)?
Abstract: Consider the function field F of a smooth curve over a finite field of odd characteristic.
L-functions of automorphic representations of GL(2) over F are important objects for studying the arithmetic properties of the field F. Unfortunately, they can be defined in two different ways: one by Godement-Jacquet, and one by Jacquet-Langlands. Classically, one shows that the resulting L-functions coincide using a complicated computation.
Each of these L-functions is the GCD of a family of zeta integrals associated to test data. I will categorify the question, by showing that there is a correspondence between the two families of zeta integrals, instead of just their L-functions. The resulting comparison of test data will induce an exotic symmetric monoidal structure on the category of representations of GL(2).
It turns out that an appropriate space of automorphic functions is a commutative algebra with respect to this symmetric monoidal structure. I will outline this construction, and show how it can be used to construct a category of automorphic representations.