Boundary amenability of groups via ultrapowers

Speaker: 

Isaac Goldbring

Institution: 

UC Irvine

Time: 

Tuesday, October 24, 2017 - 1:00pm to 2:00pm

Location: 

RH 440R

A (countable discrete) group $\Gamma$ acting on a compact space $X$ is said to act \emph{amenably} if there is a continuous net $(\mu_n^x)$ of probability measures indexed by the points of $X$ that are almost invariant under the action of $\Gamma$. For example, $\Gamma$ is amenable if and only if it acts amenably on a one-point space. The protoypical example of a boundary amenable non-amenable group is a non-abelian free group. More generally, if acts properly, isometrically, and transitively on a tree, then $\Gamma$ is boundary amenable. In this talk, I will present a construction of the Stone-Cech compactification of a locally compact space using C*-algebra ultrapowers that allows one to give a slick proof of the aforementioned result. This construction is motivated by the open question as to whether or not Thompson’s group is boundary amenable and I will also discuss the optimistic thought that this construction could be used to settle that problem.

On self-similar sets with overlaps and inverse theorems for entropy III

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Tuesday, May 2, 2017 - 1:00pm to 2:00pm

Location: 

RH 440R

We discuss an inverse theorem on the structure of pairs of discrete probability measures which has small amount of growth under convolution, and apply this result to self-similar sets with overlaps to show that if the dimension is less than the generic bound, then there are superexponentially close cylinders at all small enough scales. The results are by M.Hochman. 

Interacting Polya urns (on joint works with Christian Hirsch and Mark Holmes)

Speaker: 

Victor Kleptsyn

Institution: 

Universite Rennes 1, CNRS

Time: 

Tuesday, April 11, 2017 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

In the classical Polya’s urn process, there are balls of different colors in the urn, and one step of the process consists of taking out a random ball and it putting back together with one more ball of the same color. ("Ask a friend whether he’s using is A or B, and buy the same".)

It also can be modified by saying that the reinforcement probability is proportional not to the number of balls of a given color, but to its power $\alpha$, and the asymptotic behaviour for $\alpha=1$, $\alpha>1$ and for $\alpha<1$ are quite different.

The talk will be devoted to the model of interacting urns : at each moment, there is a competition for reinforcement between randomly chosen subset of colours; for a real-life analogue, one can consider companies competing on different markets (one company produces toys and computers, another sells computers and cars, etc.).

We will describe possible types of the limit behaviour of such model for different values of $\alpha$; it turns out that what happens for $\alpha>>1$ is quite different from $\alpha=1$, and both are quite interesting (this is a joint work with Christian Hirsch and Mark Holmes).

Domino tilings and determinantal formulas

Speaker: 

Victor Kleptsyn

Institution: 

Universite Rennes 1, CNRS

Time: 

Tuesday, April 4, 2017 - 10:00am to 11:00am

Host: 

Location: 

NS2 1201

Given a planar domain on the rectangular grid, how many ways are there of tiling it by dominos (that is, by 1x2 rectangles)? And how does a generic tiling of a given domain look like?

It turns out that these questions are related to the determinants-based formulas, and that likewise formulas appear in many similar situations. In this direction, one obtains the famous arctic circle theorem, describing the behaviour of a generic domino tiling of an aztec diamond, and a statement for the lozenges tilings on the hexagonal lattice, giving the shape of a corner of a cubic crystal.

On self-similar sets with overlaps and inverse theorems for entropy II

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Tuesday, April 25, 2017 - 1:00pm to 2:00pm

Location: 

RH 440R

We discuss an inverse theorem on the structure of pairs of discrete probability measures which has small amount of growth under convolution, and apply this result to self-similar sets with overlaps to show that if the dimension is less than the generic bound, then there are superexponentially close cylinders at all small enough scales. The results are by M.Hochman. 

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