Date: Tuesday Sept 12 2023
Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)
Title: Why are alternating knots interesting ?
Speaker: Ilya Kofman (CSI & GC)
Abstract: We will discuss the combinatorics, geometry and topology of alternating knots and links.
Date: Tuesday Sept 26 2023
Tuesday, Note different time and place - 4:15 - 5:15 pm in Room 5417 (Math Thesis Room)
Title: Surgery on 3-manifolds
Speaker: Susan Rutter
Abstract: In this talk I will present a proof of Lickorish's
theorem that any 3-manifold can be obtained via 1-surgery on S^3 -
where copies of solid tori are removed and glued back in - using
framed links.
Date: Tuesday Oct 3 2023
Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)
Title: Most alternating knots are hyperbolic
Speaker: Megha Bhat
Abstract: Alternating knots and links are special because they are both easier to study using geometric techniques as well as more numerous among knots with low crossing numbers. In this talk, we will prove a theorem by Menasco which states that any knot or link with a prime alternating diagram, that does not belong to a specific family of torus knots, must have a hyperbolic complement.
Date: Tuesday Oct 3 2023
Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)
Title: Quasi-isometric Rigidity and the Seifert Fibered Space Conjecture
Speaker: Yushan Jiang
Abstract: Let G be a finitely generated group with a word
metric. If G is quasi-isometric to hyperbolic space H^n, then G can be
realized as a discrete cocompact subgroup of Isom(H^n) up to take a
finite quotient (i.e. there exists a finite normal subgroup F in G, so
that G/F is a uniform lattice of Isom(H^n)). For n=3, this is a
combination of works by Sullivan, Tukia, and Cannon-Cooper. For
general n>=3, this is proved by Tukia. However, for n=2, one of
Tukia’s methods fails and need more information. It is first proved by
Gabai and Casson-Jungreis in 90s (actually, one can also use the
Geometrization Theorem to prove it). As the corollaries, they obtain
the famous Seifert Fibered Space Conjecture (part of Thurston’s
Geometrization Conjecture), and a new proof of the Nielsen Realization
Theorem. I will try to introduce quasi-isometric rigidity, the
Seifert Fibered Space Conjecture and the relationship between them.
Date: Tuesday Oct 31st
Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)
Title: Random Walks on Knots and Braids
Speaker: Joe Boninger (Boston College)
Abstract: I’ll discuss a model of random walks on a given knot or braid diagram, with applications to topology and representation theory. Given time, I will show how the Alexander polynomial and (colored) Jones polynomial—two well-known knot invariants—fit into this model.
Date: Tuesday Nov 7th & Nov 14th
Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)
Title: Simplicial Complexes on Seifert Surfaces of Links - A Tale of Two Structures
Speaker: Ipsa Bezbarua (GC)
Abstract: One of the fundamental structures studied in knot
theory is a compact connected surface whose boundary is the knot
under consideration, called a Seifert surface. In this talk, we will
learn about two simplicial complexes, constructed by Osamu Kakimizu
using the Seifert surfaces of links - the incompressible complex and
the Kakimizu complex. We will also learn some classic properties of
the Kakimizu complex, the more popular of the two, like
connectedness, contractibility and local infiniteness. We will then
try to extend some of these properties to the incompressible complex.
Date: Tuesday Dec 5th 2023
Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)
Title: Continued Fractions and Lens Spaces
Speaker: Susan Rutter (GC)
Abstract: Lens spaces L(p,q) provide interesting examples of
spaces with the same fundamental group and homology groups, but which
are not homotopic. As p and q define a rational number, we may find
various continued fractions that evaluate to p/q. In this talk I will
demonstrate the equivalence of five definitions of lens spaces,
including two that make use of continued fraction expansions.