Classical group representation theory deals with group actions on linear spaces; we consider group actions on compact convex spaces, preserving topological and convex structure. We focus on irreducible actions, and show that for a large class of groups - including connected Lie groups - these can be determined. There is a close connection between this and the theory of bounded harmonic functions on symmetric spaces and their boundary values.
Classical group representation theory deals with group actions on linear spaces; we consider group actions on compact convex spaces, preserving topological and convex structure. We focus on irreducible actions, and show that for a large class of groups - including connected Lie groups - these can be determined. There is a close connection between this and the theory of bounded harmonic functions on symmetric spaces and their boundary values.
Title: The (in)compatibility of 3 and 5 dimensional Heisenberg geometry with Lebesgue spaces
Abstract: The 3-dimensional (discrete) Heisenberg geometry is the shortest-path metric on the infinite graph whose vertex set is the integer grid $\Z^3$ and the neighbors of each integer vector $(a,b,c)$ are the four integer vectors $$(a+ 1,b,c), (a- 1,b,c), (a,b+ 1,c+ a), (a,b- 1,c- a).$$
1) Abstract of Wayne's part:
Today, in our modern world, we perceive the physical universe in mathematical terms; whether degrees on longitude and latitude on earth, or in units of space-time beyond our earthly horizons. This talk will present two ancient cuneiform tablets from Babylonia which offer a geometric impression of the physical world as experienced by ancient Babylonians. Comparisons will be made with a range of other ancient mathematical, geographic, and astronomical materials from the cuneiform Ancient Near East.
2) Abstract of Mourtaza's part:
Minimal and non-minimal automorphism groups of homogeneous structures
A Hausdorff topological group G is said minimal if G does not admit any strictly coarser Hausdorff group topology.
Examples include the isometry group of the Urysohn sphere, due to Uspenskij, and Aut(M) for M stable and w-categorical, a deep fact due to Ben Yacov and Tsankov.
Abstract: Cut and project point sets are defined by identifying a strip of a fixed n-dimensional lattice (the "cut"), and projecting the lattice points in that strip to a d-dimensional subspace (the "project"). Such sets have a rich history in the study of mathematical models of quasicrystals, and include well known examples such as the Fibonacci chain and vertex sets of Penrose tilings.
Abstract:
A group G is stable in permutations if every almost-action of G on a finite set is close to some actual action. Part of the interest in this notion comes from the observation that a non-residually finite stable group cannot be sofic.
I will show that surface groups are stable in a flexible sense, that is if one is allowed to "add a few extra points" to the action. This is the first non-trivial stability result for a non-amenable group.
Title: Title: Self maps of varieties over finite fields
Abstract: Esnault and Srinivas proved that as in Betti cohomology over the complex numbers, the value of the entropy of an automorphism of a smooth proper surface over a finite field $\F_q$ is taken in the subspace spanned by algebraic cycles inside $\ell$-adic cohomology. In this talk we will discuss some analogous questions in higher dimensions motivated by their results and techniques.
Let (V,<, >) be a finite dimensional inner product space and K a self adjoint element of End(V ). It is an axiom of physics that the expected value of A in End(V ) in equilibrium at temperature T with respect to K is
the ration Tr(A exp (-K/T))/Tr(exp (-K/T)).
Let (V,<, >) be a finite dimensional inner product space and K a self adjoint element of End(V ). It is an axiom of physics that the expected value of A in End(V ) in equilibrium at temperature T with respect to K is
the ration Tr(A exp (-K/T))/Tr(exp (-K/T)).
Combinatorial group theory began with Dehn's study of surface
groups, where he used arguments from hyperbolic geometry to solve the
word/conjugacy problems. In 1984, Cannon generalized those ideas to all
"hyperbolic groups", where he was able to give a solution to the
word/conjugacy problem, and to show that their growth function satisfies
a finite linear recursion. The key observation that led to his
discoveries is that the global geometry of a hyperbolic group is determined locally:
first, one discovers the local picture of G, then the recursive structure
