T he talk is going to be on progress made in approximation theory of Lie
group-valued periodic functions (loops) by so-called polynomial loops.
This
is a relatively unexplored topic within the larger area of nonlinearly
constrained approximation, which includes the study of H\"{o}lder

We will discuss complex tori, and explain the role that a special class of functions, the theta functions, play in their study. I will also outline connections between theta functions and other special functions.

For over two decades we have been proving
identities involving plethystic operators
(vertex operators for some people) by
manipulations which politely could
be called ``heuristic''. But deep down I
felt them to be quite ``fishy''. But referees accepted
them and we felt nevertheless confident
since we always got the right answer,
as amply confirmed by computer experimentations.
But suddenly this summer an example popped up
where our manipulations yielded a patently
false answer. Panic? Yes ... for a while.
In this talk we will present how in the end
all of this finally, and belatedly
could be made completely rigorous. \\

This talk will be a continuation of the talk from last week.

Categorification is big business in representation theory these days,
and much of the inspiration for categorification comes from geometric
representation theory. We'll try to explain some of the geometric
inspiration for sl(2) categorification. As an application, we
describe an interesting equivalence of categories between the derived
categories of coherent sheaves on the cotangent bundle of dual
Grassmanians. \\

Joint with Sabin Cautis and Joel Kamnitzer.

The Peter-Weyl theorem is one of the results that made me decide to study representation theory. In a few words it tells you how to describe the space $L^2(G)$ in terms of the representation theory of a compact group $G$.

The idea of this talk is to informally develop enough theory to state and understand this theorem and some of its consequences, and in this way motivate the study of Lie groups and their representations. No previews knowledge of the subject is assumed.

Single-molecule biophysical tools permit measurements of the mechanical response of individual biomolecules to external load, revealing details that are typically lost when studied by ensemble methods. Kramers theory of diffusive barrier crossing in one dimension has been used to derive analytical solutions for the observables in such experiments, in particular, for the force dependent lifetimes. We propose a minimalist model that captures the effects of multidimensionality of the free energy landscape on the kinetics of a single-molecule system under constant applied force. The model predicts a rich spectrum of scenarios for the response of the system to the applied force. Among the scenarios is the conventional decrease in the lifetime with the force, as well as a remarkable rollover in the lifetime with a seemingly counterintuitive increase of the lifetime at low force followed by a decrease in the lifetime at higher forces. Realizations of each of the predicted scenario are discussed in various biological contexts. Our model demonstrates that the rollover in the lifetime does not necessarily imply a discrete switch between two coexisting pathways on the free energy landscape, and that the rollover can also be realized for a dynamics as simple as that on a single pathway with a single bound state. Our model leads to an analytical solution that reproduces the entire spectrum of scenarios, including the rollover, in the force-dependent lifetime, in terms of the microscopic parameters of the system.