Date: Tuesday Feb 25, 2025
Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)
Title: Introduction to Khovanov Homology
Speaker: Ipsa Bezbarua
Abstract: In 2002, Bar-Natan published his expository paper on Mikhail Khovanov's revolutionary new homology theory for knots. He also listed two interesting phenomenological conjectures, which remained open for a few years. In 2005, Eun Soo Lee tackled these conjectures, fully solving one and partially the other. (The latter, called the "knight move conjecture", was shown to be false in its full generality much later.) In her quest to solve the knight move conjecture, Lee developed an endomorphism on the Khovanov homology by redefining the boundary maps on the Khovanov chain complex. This new homology theory later led to fascinating topological implications, like Rasmussen's s-invariant. In this expository talk, we will discuss the latter half of Lee's paper, which describes the construction of Lee homology and how it is used to prove a special case of the knight move conjecture for alternating knots.
References:
- (Background) Dror Bar-Natan, "On Khovanov's categorification of the Jones polynomial", Algebraic & Geometric Topology 2 (2002)
- Eun Soo Lee, "An endomorphism of the Khovanov invariant", Advances in Mathematics 197 (2005)
- Ciprian Manolescu and Marco Marengon, "The Knight Move Conjecture is false", Proceedings of the American Mathematical Society, Vol. 148 (2020)
Date: Tuesday April 1st, 2025
Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)
Title: A Survey of Heegaard-Floer Homology - a non-expert's view
Speaker: Ipsa Bezbarua
Abstract: Floer homology was adapted from Morse homology by Andreas Floer to study symplectic manifolds. In 2003, Ozsvath and Szabo formulated a technique to adapt Floer homology for the study of closed 3-manifolds and knots, by cleverly making use of Heegaard splittings. Named "Heegaard-Floer homology", this new machinery has revolutionized the study of 3-manifolds and revealed a wealth of information which was previously inaccessible. In this expository talk, we go over the basics of Morse homology and Floer theory and then see how the extension to 3-manifolds and knot complements works. Time permitting, we shall also look at a remarkable application of this theory to 4-manifolds.
References:
- Ozsvath, Peter and Szabo, Zoltan. (2006). "An Introduction to Heegaard Floer Homology". In Clay Mathematics Proceedings (Vol 5. pp 3-28). American Mathematics Society.
- Dusa McDuff, Floer Theory and Low Dimensional Topology, Bull. Amer. Math. Soc. (N. S.), October 6, 2005.
- Joshua Greene, Heegaard Floer homology, AMS Notices Vol 86, No 1 (2001) pp 19-33.
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