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Title: Density square functions, uniform rectifiability and Wolff potentials

Speaker: Vasileios Chousionis (University of Connecticut)

Time: Friday, September 18, 2015 at 3:15 pm

Place: MSB 109AAbstract: We characterize uniform rectifiability via density square functions, providing a square functions analogue of Preiss' theorem which characterizes rectifiability in terms of the existence of densities. We also discuss how density square functions of fractional homogeneity are related to Wolff potentials, answering a question of F. Nazarov.

Title: Two problems on long range percolation

Speaker: Roger Silva (Universidade Federal de Minas Gerais)

Time: Friday, September 25, 2015 at 4:00 pm

Place: MSB 109AAbstract: Consider an independent site percolation model on $Z^d$, with parameter $p\in (0,1)$, where all long-range connections in the axis directions are allowed. In this work we show that, given any parameter $p$, there exists an integer $K(p)$ such that all binary sequences (words) $\xi \in \{0,1\}^{\mathbb N}$ can be seen simultaneously, almost surely, even if all connections with length larger than $K(p)$ are suppressed. We also show some results concerning how $K(p)$ should scale with $p$ as $p$ goes to $0$. Related results are also obtained for the question of whether or not almost all words are seen. In another direction, suppose now that only nearest neighbor bonds and long range bonds of length $k$ parallel to each coordinate axis are allowed. We show that the percolation threshold of such a model converges to $p_c({\mathbb Z}^{2d})$ when $k$ goes to infinity, the percolation threshold for ordinary (nearest neighbor) percolation on ${\mathbb Z}^{2d}$. We also generalize this result for models whose long range bonds have several lengths.

Title: Random walks in hyperbolic spaces

Speaker: Giulio Tiozzo (Yale University)

Time: Friday, October 9, 2015 at 3:15 pm

Place: MSB 109AAbstract: Let us consider a group G of isometries of a delta-hyperbolic metric space X, which is not necessarily proper (e.g. it could be a locally infinite graph). We can define a random walk on G by picking random products of elements of G, and projecting this sample path to X. We show that such a random walk converges almost surely to the Gromov boundary of X, and with positive speed. We will discuss applications of these techniques to geometric group theory (the mapping class group and Out(F_n)) as well as to complex analysis (the Cremona group). This is joint work with J. Maher.

Title: Conformal dimension of boundaries of planar domains

Speaker: Kyle Kinneberg (Rice University)

Time: Friday, October 16, 2015 at 3:15 pm

Place: MSB 109AAbstract: The conformal dimension of a metric space is a quasisymmetric invariant that is important in hyperbolic geometry. Motivated by quasiconformal conjugation and equivalence problems for Kleinian limit sets, Julia sets, and SLE curves, we investigate the conformal dimension of boundaries of certain planar domains. We will focus primarily on the boundaries of John domains and prove, building on some ideas developed by M. Carrasco, that these have conformal dimension equal to 1.

Title: Transformed random walks on groups

Speaker: Behrang Forghani (University of Connecticut)

Time: Friday, October 23, 2015 at 3:15 pm

Place: MSB 109AAbstract: Given a countable group $G$ equipped with a probability measure $\mu$, we will propose a machinery way to produce probability measures on group $G$ with the same Poisson boundaries. Moreover, we will show how the asymptotic behaviors (for instance, asymptotic entropy) of random walks generated by these probability measures are related.

Title: Poincare inequalities and uniqueness of solutions for the subelliptic heat equation

Speaker: Bumsik Kim (University of Connecticut)

Time: Wednesday, November 4, 2015 at 3:15 pm

Place: MSB 109AAbstract: Let $L$ be a second order diffusion operator defined on a smooth manifold $M$ (or $\mathbb R^n$) satisfying the subelliptic estimate $ \| f \|_\epsilon ^2 \leq C(\left| \left< f , Lf BREAK ight> BREAK ight| + \| f\|_2^2 ) $ for all $f\in C^\infty_0(M)$, where $\|f \|_\epsilon=\left(\int |\hat{u}(\xi)|^2 (1+| \xi|^2 )^\epsilon d\xi BREAK ight)^{1/2}$ is the Sobolev norm of order $0<\epsilon<1$. In a local sense, Poincar\'e inequalities and volume doubling properties for balls with respect to $L$ are well-known. In this talk, we will discuss similar global statements when we add certain conditions on $L$. With the conditions, we will have a class of sub-Riemannian manifolds including CR Sasakian manifolds and carnot groups of step 2, but we will not rely on geometry here.

Title: Asymptotic complexity of fractal graphs

Speaker: Konstantinos Tsougkas (Uppsala Universitat)

Time: Friday, November 6, 2015 at 3:15 pm

Place: MSB 109AAbstract: The number of spanning trees of a graph can be evaluated by Kirchhoff's well known matrix tree theorem by having knowledge of the spectrum of the graph Laplacian. In fully symmetric self similar fractal graphs, such knowledge of the spectrum is obtained by the process of spectral decimation. We will apply this information to a sequence of fractal graphs approximating a fully symmetric finitely ramified fractal and establish existence and bounds of their asymptotic complexity constant.

Title: Current large deviations in the boundary-driven symmetric simple exclusion process on the Sierpinski gasket

Speaker: Joe Chen (University of Connecticut)

Time: Friday, November 13, 2015 at 3:15 pm

Place: MSB 109AAbstract: We study the symmetric simple exclusion process on the Sierpinski gasket ($SG$) driven by the action of particle reservoirs attached to boundary vertices of $SG$. We establish three hydrodynamic limit theorems for the empirical current: the law of large numbers, the large deviations principle, and the large deviations principle for the mean current on a long-time interval. On $\mathbb{Z}^d$ these results were established assuming translational invariance and Gaussian space-time diffusive estimates. But on $SG$ neither assumption is valid, and it is unclear how to make sense of the resulting reaction-diffusion PDE. In this work we overcome all the aforementioned obstacles. First, we establish the ``moving particle lemma'' on weighted graphs, using the ``octopus inequality'' of Caputo, Liggett, and Richthammer in their seminal proof of Aldous' spectral gap conjecture. This enables us to prove a local version of the two-blocks estimate on $SG$, thereby answering a question posed by Jara. Second, we actively use the theory of differential $1$-forms on $SG$ developed by the Teplyaev and collaborators, which allows us to characterize the speed of convergence of discrete $1$-forms on $SG$, and prove uniqueness of solutions to the resulting reaction-diffusion PDE. This is joint work with Alexander Teplyaev.

Title: Rough approximation theory

Speaker: Michael Barnsley (Australian National University)

Time: Friday, November 20, 2015 at 3:15 pm

Place: MSB 109A

Title: Convergence of discrete holomorphic functions on non-uniform lattices

Speaker: Brent Werness (University of Washington)

Time: Friday, December 4, 2015 at 3:15 pm

Place: MSB 109AAbstract: The theory of discrete holomorphic functions has been studied by researchers from a diverse set of fields from classical complex analysts to applied computer scientists. In the field of conformally invariant random processes, discrete analyticity has found a particularly central role as the convergence of discrete analytic functions to their continuum counterparts is the key step in the showing convergence of discrete random processes to Schramm--Loewner Evolutions. In this talk, we will discuss recent work that proves that discrete analytic functions converge to their continuum counterparts on lattices with only local control on the geometry. We will then discuss potential applications of this result to the study conformally invariant random processes on random surface models.

Title: TBA

Speaker: Guozhen Lu

Time: Wednesday, December 9, 2015 at 1:30 pm

Place: MONT 414

Organizer: Vasileios Chousionis and Oleksii Mostovyi