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. 

Spectral Properties of Continuum Fibonacci Schrodinger Operators

Speaker: 

May Mei

Institution: 

Denison University

Time: 

Tuesday, May 23, 2017 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

In an award winning 2014 paper, Damanik, Fillman, and Gorodetski rigorously established a framework for investigating Schrodinger operators on the real line whose potentials are generated by ergodic subshifts. In the case of the Fibonacci subshift, they also described the asymptotic behavior in the large energy and small coupling settings when the potential pieces are characteristic functions of intervals of equal length. These estimates relied on explicit formulae and calculations, and thus could not be immediately generalized. In joint work with Fillman, we show that when the potential pieces are square integrable, the Hausdorff dimension of the spectrum tends to one in the large energy and small coupling settings.

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

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Tuesday, March 14, 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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