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.

Title: On a Class of Profinite Groups Related to a Theorem of Prodanov

by Dikran Dikranjan (University of Udine) as part of Topological Groups

Lecture held in Elysium.

Abstract

A short history of minimal groups is given, featuring illustrative examples and leading to current research:$

ewline$

$quad$ * non-compact minimal groups,$

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$quad$ * equivalence between minimality and essentiality of dense subgroups of compact groups,$

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$quad$ * equivalence between minimality and compactness in LCA, $

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$quad$ * hereditary formulations of minimality facilitate optimal statements of theorems, $

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$quad$ * a locally compact hereditarily locally minimal infinite group $G$ is $

ewline$

$quad$ $quad$ (a) $congmathbb{Z}p$, some prime $p$, when $G$ is nilpotent,$

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$quad$ $quad$ (b) a Lie group when $G$ is connected,$

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$quad$ * classification of hereditarily minimal locally compact solvable groups,$

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$quad$ * existence of classes of hereditarily non-topologizable groups: $

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$quad$ $quad$ (a) bounded infinite finitely generated,$

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$quad$ $quad$ (b) unbounded finitely generated,$

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$quad$ $quad$ (c) countable not finitely generated, $

ewline$

$quad$ $quad$ (d) uncountable.

Title: Priority arguments in descriptive set theory

by Andrew Marks (UCLA) as part of Computability theory and applications

Abstract

We give a new characterization of when a Borel set is

Sigma^0_n complete for n at at least 3. This characterization is

proved using Antonio Montalb’an’s true stages machinery for

conducting priority arguments.

As an application, we prove the decomposability conjecture in

descriptive set theory assuming projective determinacy. This

conjecture characterizes precisely which Borel functions are

decomposable into a countable union of continuous functions with

$Pi^0_n$ domains. Our proof also uses a theorem of Leo Harrington

that assuming the axiom of determinacy there is no $omega_1$ length

sequence of distinct Borel sets of bounded rank. This is joint work

with Adam Day.

Title: Genericity and randomness with ITTMs

by Benoit Monin (LACL/Créteil University) as part of Computability theory and applications

Abstract

We will talk about constructibility through the study of Infinite-Time Turing machines. The study of Infinite-Time Turing machines, ITTMs for short, goes back to a paper by Hamkins and Lewis. Informally these machines work like regular Turing machines, with in addition that the time of computation can be any ordinal. Special rules are then defined to specify what happens at a limit step of computation.

This simple computational model yields several new non-trivial classes of objects, the first one being the class of objects which are computable using some ITTM. These classes have been later well understood and characterized by Welch. ITTMs are not the first attempt of extending computability notions. This was done previously for instance with alpha-recursion theory, an extension of recursion theory to Sigma_1-definability of subsets of ordinals, within initial segments of the Godel constructible hierarchy. Even though alpha-recursion theory is defined in a rather abstract way, the specialists have a good intuition of what “compute” means in this setting, and this intuition relies on the rough idea of “some” informal machine carrying computation times through the ordinal. ITTMs appeared all the more interesting, as they consist of a precise machine model that corresponds to part of alpha-recursion theory.

Recently Carl and Schlicht used the ITTM model to extend algorithmic randomness and effective genericity notions in this setting. Genericity and randomness are two different approaches to study typical objects, that is, objects having “all the typical properties” for some notion of typicality. For randomness, a property is typical if the class of reals sharing it is of measure 1, whereas for genericity, a property is typical if the class of reals sharing it is co-meager.

We will present a general framework to study randomness and genericity within Godel’s constructible hierarchy. Using this framework, we will present various theorems about randomness and genericity with respect to ITTMs. We will then end with a few exciting open questions for which we believe Beller Jensen and Welch’s forcing technique of their book “coding the universe” should be useful.

Title: Totally disconnected locally compact groups and the scale

by George Willis (University of Newcastle) as part of Topological Groups

Lecture held in Elysium.

Abstract

The scale is a positive, integer-valued function defined on any totally disconnected, locally compact (t.d.l.c.) group that reflects the structure of the group. Following a brief overview of the main directions of current research on t.d.l.c. groups, the talk will introduce the scale and describe aspects of group structure that it reveals. In particular, the notions of tidy subgroup, contraction subgroup and flat subgroup of a t.d.l.c. will be explained and illustrated with examples.