Date: Tuesday Mar 1st and 8th 2022
Title: Counting Simple Closed Geodesics on Closed Hyperbolic Surfaces I & II
Speaker: Yushan Jiang (CUNY GC)
Abstract: Just as "prime numbers" for integers, closed
geodesics play a central role for the geometry and dynamics on the
surfaces. So counting closed geodesics can be very important. There
are lots of famous works dealing with this counting problem via
different view of points: the trace formula (Selberg), statistical
property of geodesic flow (Margulis), etc. However, if we add an
additional requirement "simple", things could be very different. In
this talk, I will basically focus on the story and ideas (rather than
the details) of Mirzakhani's work on counting simple closed geodesics,
and try to illustrate how dynamics, mapping class groups and the
geometry of Teichmuller spaces come together to give a beautiful proof
of the counting problem.
Date: Tuesday Mar 22nd, 2022
Title: Groups acting on trees
Speaker: Zhihao Mu (CUNY GC)
Abstract: Given a nice action of a group on some spaces by isometries, one can deduce some algebraic properties on the group. In this talk, we will start with groups acting freely on trees, then introduce the notions of free product with amalgamation and HNN extension by examples. If time permits, we will also state and sketch the proof of group-theoretic analogs of prime decomposition for 3-manifolds, which are Grushko theorem and the accessibility for finitely generated group.
Reference: Office Hours with a Geometric Group Theorist edited by Matt Clay and Dan Margalit (OHGGT), Chapter 3.
Date: Tuesday Mar 29th, 2022
Title: Many roads to two bridges
Speaker: Michael Marinelli
Abstract: The bridge number of a knot is rich in topological information despite being a diagrammatic invariant. While a classification of knots with bridge number three and up remains elusive, two bridge knots simplify nicely enough to yield several classifications. We will describe the 4-plat, Schubert and Conway presentations of these knots and their various advantages.
Date: Tuesday April 26th, 2022
Title: Automorphisms of surfaces and related objects
Speaker: Yassin Chandran
Abstract: The curve graph has become one of the most fruitful tools used in the study of the mapping class group of finite type surfaces. In this talk, we'll introduce the curve graph and prove that they are all uniformly Gromov hyperbolic. Time permitting, we'll discuss a few combinatorial models (marking complex, pants complex) of spaces related to surfaces, how their study also factors through the study of the curve complex, and Ivanov's metaconjecture concerning the rigidity of the automorphism groups of such objects associated with surfaces.
Date: Tuesday May 3rd, 2022
Title: The Dehn Complex of a Knot
Speaker: Eden Morris
Abstract: Cube complexes have been recently used to resolve many important problems in group theory and low-dimensional topology. Square complexes and VH-complexes are special types of cube complexes. For a knot diagram D, we will define the Dehn Complex of D, and show that it is a VH-complex. This means that there are two distinct classes of edges: vertical and horizontal. We will explore interesting properties of VH-complexes by examining the links of vertices. As an example we will construct the Dehn complex of the trefoil knot and show that the hyperplane structure of the complex gives us an amalgamated product structure of the fundamental group of the knot complement.
Date: Tuesday May 10th, 2022
Title: A Modular Uniformization of the Heawood Map
Speaker: Ajmain Yamin
Abstract: In this talk I will explore connections between topological graph theory, number theory and the theory of modular forms. The main fascination of my talk will be the Heawood map, a seven color map on the torus in which where every region is adjacent to every other. I will explain how to "uniformize" the Heawood map as a quotient of the upper half plane by a finite index torsion free normal subgroup of the modular group PSL(2,Z). I will then study modular forms for this subgroup and explain how to recover the standard hexagonal-tiling model of the Heawood map by integrating a weight 2 cusp form for this "Heawood modular group".