Visualizations of Schottky GroupsOften times, to truly understand a mathematical object it must be viewed from several different perspectives, involving several different foundations. In this thesis, I will present eye-catching visualization techniques for something called Schottky groups, which are similar to general linear groups of degree n over the complex numbers. This is built on an understanding of complex numbers to explore the structure of specific kinds of linear groups. My discussions will include the mathematics needed to imagine such an object, as well as the numerics required to compute such an object. Parts of this will include discussion of programming these objects.This thesis is an exploration of a particular kind of projective linear group, Schottky groups, and their variations by understanding them geometrically. The general idea here is how we can study fractal-like sets by looking at images of circles representing group elements. This exploration will lead to the discovery and understanding of Schottky groups.Furthermore, I do this exploration using first and foremost, a background in complex analysis, and in particular, a deep understanding of Mobius maps. This leads to discovery and understanding of anti-Mobius maps, which will be our main tool in understanding reflections. Additionally, I will use Python3 to program examples and experiments of these ideas. The Python code will provide specific examples. as well as mathematical challenges of its own. These challenges will include creating Mobius and anti-Mobius classes, and using these classes to perform all of these operations.

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