## Discrete Geometry & Combinatorics Seminar

A weekly seminar on topics in discrete geometry with a combinatorial flavor.

We meet on Wednesdays 2:00-3:00pm in South Hall, 4607B.

Past schedules:

#### New examples and non-examples of pseudomodular groups

Carmen Galaz-García | April 3

• The cusp set of a discrete subgroup $$\Gamma$$ of $$PSL(2,\mathbb{R})$$ is the set of points fixed by parabolic elements of $$\Gamma$$. It can be checked that the cusp set of $$PSL(2,\mathbb{Z})$$ is $$\mathbb{Q}\cup\{\infty\}$$. The question that arises then is: how strong is the cusp set as an invariant of discrete subgroups of $$PSL(2,\mathbb{R})$$? For example, if $$G$$ also has cusp set equal to $$\mathbb{Q}\cup\{\infty\}$$, is it commensurable with $$PSL(2,\mathbb{Z})$$? The answer on the negative was provided by D. Long and A. Reid in 2001 with finitely many examples, calling them pseudomodular groups. On 2016 Lou, Tan and Vo produced two infinite families of pseudomodular groups, and called them jigsaw groups. In this talk, I'll examine a third family of pseudomodular groups obtained with the jigsaw construction as well as some new family non-examples.

#### Rendering 3 manifolds in Virtual Reality

Steve Trettel | May 1

• Over the past two years, in collaboration with Kenny Kim and Dennis Adderton in the department of Media Arts and Technology, I have been working on rendering geometrically correct views of the interior of constant curvature manifolds in virtual reality. Because of how busy this year has been with the job market and graduation, I realize I have not gotten to show a lot of you our work, and am excited to take an opportunity to do so now!

The Allosphere is a 40ft diameter spherical screen designed for immersive virtual reality experiences on campus by the department of Media Arts and Technology. We will meet at 2pm in SH 4607B and I will give a brief 15min overview of the mathematics involved, and then we will head over together to Eilings hall for a virtual reality experience. We will look at the six 4 dimensional regular polytopes, the Hopf fibration and Seifert fibrations of the 3 sphere, and then time - permitting explore some manifolds with more complicated topology, including Poincare Dodecahedral space and the Figure 8 Knot complement.

#### Solving the Word Problem for Artin Groups

Ashlee Kalauli | May 8

• A finitely presented group G has a solvable word problem if there exists an algorithm to decide if a product of generators and their inverses represents the identity. In 1926, Artin showed that braid groups have such an algorithm. The computation time of this algorithm, however, was exponential in the length of the word. In this talk, we will discuss how Frank Garside’s solution provides a more efficient algorithm for solving the word problem for braid groups. We then discuss plans to modify this solution to provide a new solution to the word problem for Euclidean Artin groups.

#### Algebraic descriptions of modular curves and applications

• A modular curve is a quotient of the upper half-plane by a finite-index subgroup of SL(2,$$\mathbb{Z}$$). They arise in number theory and algebraic geometry as moduli spaces of elliptic curves (with some extra data). Using the moduli space interpretation, I will show how to find explicit descriptions of modular curves as varieties in $$\mathbb{P}^2$$. Time permitting, I will show how these explicit descriptions lead to important results in number theory and string theory.