Geometry and Topology Student Seminar

Date: Tuesday Sept 9th

Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)

Title: An elementary introduction to Knot Theory

Speaker: Evan Scott, GC

Abstract: To open the student seminar, this will be an accessible, elementary talk introducing students to knot theory. We'll clearly describe the objects of study, thoroughly describe the motivations for studying the field, and then try to give a sense for what it feels like to do (one kind of) knot theory by walking through the classic example of 3-colorings. We'll conclude by showing many varied examples of interesting questions in knot theory, with which we hope to pique the interests of students with varied tastes! No background of any kind is required, though familiarity with the fundamental group and with homology may enrich the experience.

Date: Tuesday Sept 16th & Sept 30th

Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)

Title: Immersed Curves in Khovanov Homology I & II

Speaker: Susan Rutter, GC

Abstract: In 2019, Kotelskiy, Watson and Zibrowius published their paper with the above title, connecting the Khovanov homology of a link to the wrapped Floer homology of invariants called multicurves. I will cover key results from this paper and how I am using them in my research. To do this, I will discuss the structure of the Bar-Natan dotted cobordism category, so familiarity with this will be useful, but not necessary.

Date: Tuesday Oct 21st

Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)

Title: Heegaard Diagrams and the Alexander Polynomial

Speaker: Joe Boninger, Boston College

Abstract: I will give a brief introduction to the Alexander polynomial for knots, with a focus on how it can be computed from a Heegaard diagram for a knot. Given time, I’ll say a bit about how this relates to knot Floer homology. (You don’t need to know anything about knot Floer homology.)

Date: Tuesday Oct 28th

Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)

Title: Everything you wanted to know about Seifert surfaces (but didn't know where to start)

Speaker: Ipsa Bezbarua, GC

Abstract: In 1923, JW Alexander discovered the Alexander polynomial, which was the only knot invariant at that time. It was essentially algebraic/combinatorial in nature, and there were great attempts to understand it from a geometric perspective. This was resolved in 1930 by Pontrjagin and Frankl and in 1933 by Seifert; both found ways to prove the existence of Seifert surfaces, which have now become ubiquitous in discussions involving knot theory. In this talk, we will learn about Seifert surfaces and about Seifert's algorithm to construct such a surface for a given knot. We will also look at some important consequences arising from their existence, like the prime decomposition of any given knot. Finally, time permitting, we will discuss (non-)equivalence of Seifert surfaces for a given knot in 3 dimensions.

Date: Tuesday Nov 4th

Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)

Title: How I learned to love isotopies of surfaces

Speaker: Ipsa Bezbarua, GC

Abstract: It is a well-known fact that there exist knots with non-isotopic Seifert surfaces of the same genus in the 3-sphere. In 1982, Charles Livingston asked if there exist knots with non-isotopic Seifert surfaces of the same genus that are non-isotopic even when pushed into the 4-ball. In this talk, we will see how to use techniques from Khovanov homology to settle this long-standing question. Then we will switch gears and see how for alternating knots, any two Seifert surfaces of the same genus will remain isotopic even after being pushed into the 4-ball.

Date: Tuesday Nov 11th