Title: A Survey on Analog Models of Computation

by Amaury Pouly (CNRS) as part of Computability theory and applications

Abstract: TBA

Title: Topological Groups Seminar One-Week Hiatus

by Break (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract: TBA

Title: Groups Admitting Proper Actions by Affine Isometries on Lp Spaces

by Indira Chatterji (Laboratoire J.A. Dieudonné de l’Université de Nice) as part of Topological Groups

Lecture held in Elysium.

Abstract

Introduction, known results, and open questions regarding groups admitting a proper action by affine isometries on an $L_p$ space.

Title: Sacks’ Splitting Theorem Re-examined (again)

by Rod Downey (Victoria University of Wellington) as part of Computability theory and applications

Abstract: TBA

Title: Uncertainty Principles on Locally Compact Groups

by Ajay Kumar (University of Delhi) as part of Topological Groups

Lecture held in Elysium.

Abstract

Some of the uncertainty principles on $ mathbb{R}^n $ are as follows:

Qualitative Uncertainty Principle: Let $f$ be a non-zero function in $L^1(mathbb{R}^n)$. Then the Lebesgue measures of the sets ${x: f(x)

eq 0 }$ and $ {xi : widehat{f}(xi)

eq 0}$ cannot both be finite.

Hardy’s Theorem: Let $ a,b,c $ be three real positive numbers and let $f: mathbb{R}^n to mathbb{C}$ be a measurable function such that

(i) $|f(x)| leq cexp{(-api |x|^2)}$, for all $ x in mathbb{R}^n$

(ii) $|widehat{f}(xi)| leq cexp{(-bpi |xi|^2)}$, for all $xi in mathbb{R}^n $.

Then following holds:

If $ab>1$, then $f=0$ a.e.

If $ab =1$, then $f(x)= alpha exp{(-api |x|^2)}$ for some constant $alpha$.

If $ab< 1$, then there are infinitely many linear independent functions satisfying above conditions.

Heisenberg Inequality: If $f in L^2(mathbb{R}^n)$ and $a,b in mathbb{R}^n$, then

$$

left( int_{mathbb{R}^n}|x-a|^2|f(x)|^2 dx right) left( int_{mathbb{R}^n}|xi-b|^2|widehat{f}(xi)|^2 dxi right) geq frac{n^2|f|^4}{16pi^2}.

$$

Beurling's Theorem: Let $f in L^1(mathbb{R}^n) $ and for some $ k(1leq kleq n) $ satisfies

$$

int_{mathbb{R}^{2n}} |f(x_1, x_2, dots , x_n)||widehat{f}(xi_1, xi_2, dots , xi_n)|e^{2pi |x_kxi_k|} dx_1dots dx_n dxi_1dots dxi_n< infty.

$$

Then $f = 0$ a.e.

We investigate these principles on locally compact groups, in particular Type I

groups and nilpotent Lie groups for Fourier transform and Gabor transform.

Title: PA relative to an enumeration oracle

by Mariya Soskova (University of Wisconsin-Madison) as part of Computability theory and applications

Abstract: TBA

Title: Statistical Chaos — a new barrier in the prediction/simulation of physical systems

by Cristóbal Rojas (Universidad Andrés Bello) as part of Computability theory and applications

Abstract

It is well known that for systems exhibiting “sensitivity to initial conditions”, it is practically impossible to predict individual trajectories beyond a very limited time horizon. To overcome this difficulty, a statistical approach was developed — while the computed trajectories are not individually meaningful, when regarded as an ensemble, their average represents a statistical distribution that can be used to make meaningful probabilistic predictions about the system. This statistical paradigm is ubiquitous in modern applications. In this talk we present a new obstacle in applying the statistical approach. We show that the statistical behavior of a parametrized system may exhibit “sensitivity to parameters”, and that this may lead to non computability of the limiting, meaningful, statistical distribution. We will explain all this in the simplest nonlinear class of systems: quadratic maps of the interval [0,1]. This is joint work with M. Yampolsky.

Title: Probability Measures and Structure of Locally Compact Groups

by C.R.E. Raja (Indian Statistical Instititute) as part of Topological Groups

Lecture held in Elysium.

Abstract

We will have an overview of how existence of certain types of

probability measures forces locally compact groups to have particular

structures and vice versa. Examples are Choquet-Deny measures, recurrent

measures etc., and groups of the kind amenable, polynomial growth, etc.