## Date: Tuesday Feb 6 2024

** Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room) **
**Title:** Perfect Forms over Imaginary Quadratic Fields

**Speaker: ** Ajmain Yamin (CUNY GC)

**Abstract:**
Talk is based on the following

paper by
Kristen Scheckelhoff, Kalani Thalagoda, Dan Yasaki

## Date: Tuesday Feb 27, 2024

** Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room) **
**Title: ** Orientation-preserving homeomorphisms of Euclidean space are commutators

**Speaker: ** Megha Bhat (CUNY GC)

**Abstract: ** A uniformly perfect group has commutator width p if
every element can be expressed as a product of p
commutators. Questions about commutator width have been asked and
answered for various groups such as the alternating group and the
symmetric group. I will talk about this question for homeomorphism
groups of spheres, annuli and Euclidean space, and show that each of
these has commutator width one.

## Date: Tuesday Mar 12, 2024

** Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room) **
**Title: ** The Gromov boundary of hyperbolic spaces and its generalization

**Speaker: ** Zhihao Mu (CUNY GC)

**Abstract: ** To each Gromov hyperbolic space, one can associate the space at infinity, also called Gromov boundary. The Gromov boundary has been extensively studied and turns out to be a powerful tool in geometric group theory and low dimensional topology. It has many generalizations, such as the contracting boundary in CAT(0) spaces by Charney-Sultan and the Morse boundary in metric spaces by Cordes. In this talk, I will introduce the definitions, basic properties and some relations between these boundaries by using right-angled Artin groups and mapping class groups as examples.

## Date: Tuesday Mar 19, 2024

** Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room) **
**Title: ** Contractibility of the knot complex of incompressible spanning surfaces

**Speaker: ** Ipsa Bezbarua (CUNY GC)

**Abstract: ** One of the fundamental structures studied in knot theory is a compact surface whose boundary is the link under consideration, called a spanning surface. Osamu Kakimizu constructed two very closely related simplicial complexes using the spanning surfaces of a given link - the incompressible complex and the Kakimizu complex - to study the properties of the link. In 2012, Piotr Przytycki and Jennifer Schultens showed that the Kakimizu complex is contractible for any link. In this talk, we will see that their arguments can be modified to show contractibility of the incompressible complex as well.

## Date: Tuesday March 26, 2024

** Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room) **
**Title: ** The Gordon Litherland Pairing

**Speaker: ** Susan Rutter (CUNY GC)

**Abstract: ** In 1978, Gordon and Litherland introduced a pairing
on a spanning surface of a knot, generalising the Goeritz matrix for
unorientable spanning surfaces. In 2022, Boden, Chrisman, and Karimi
introduced a generalisation of the GL pairing for links in thickened
surfaces. Moreover, the relation of the GL form to the intersection
form of the double branched cover extends to a similar theorem for the
case of thickened surfaces. I will present the classical and recent
work.

## Date: Tuesday April 2nd, 2024

** Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room) **
**Title: ** A Proof of Masur's Criterion for Unique Ergodicity of the Vertical Flow on a Translation Surface

**Speaker: ** Yash Chandra (CUNY GC)

**Abstract: ** Masur showed that on a translation surface X of genus g, if the vertical flow is minimal and the Teichmuller geodesic on the moduli space of Riemann surfaces of genus g is non-divergent, then the vertical flow on X is uniquely ergodic. In this talk, we will introduce this theorem and give a proof of it.

## Date: Tuesday April 9th, 2024

** Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room) **
**Title: ** Trisections of 4-Manifolds Through the Lens of Heegaard Splittings

**Speaker: ** Evan Scott (CUNY GC)

**Abstract: ** A trisection of a 4-manifold is a decomposition into a union of handlebodies, introduced by Gay and Kirby in 2012. In the decade since, the theory of trisections has been used to understand almost every aspect of 4-manifolds, from knotted surfaces to symplectic structures. In this talk, we introduce trisections with several examples and investigate the aphorism that "Trisections are like Heegaard splittings for 4-manifolds." The only original material is a few remarks from joint work with Jeffrey Meier. Basic knowledge of Morse functions and handle decompositions will be assumed, but is not required to understand the definition or examples.

## Date: Tuesday April 16th, 2024

** Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room) **
**Title: ** An Introduction to Ptolemy Varieties of Cusped 3-Manifolds

**Speaker: ** Michael Marinelli (CUNY GC)

**Abstract: ** Ptolemy varieties were first formulated by Christian Zickert in 2009 in order to study Chen-Simons invariants of 3-manifolds. An important quality of these varieties is that they can be used to produce parabolic representations of the fundamental group. In this talk, we will show how Ptolemy varieties are constructed and used to find parabolic representations. We will then see some theorems which are proved using these varieties. Finally, we will look at how the Ptolemy variety has been implemented computationally in SnapPy.