MIT Geometry and Topology Seminar

The MIT Geometry and Topology Seminar is scheduled for Mondays at 3PM (Eastern US time) and will be held over Zoom. Schedule change: the November 30 talk will be at 10AM (Eastern US time). The link for the seminar can be found here. The password is "".

September 14 (Duncan McCoy): Double slicing for links

Abstract: A knot in S3 is said to be doubly slice if it arises as the intersection of an unknotted 2-sphere in S4 with the equatorial S3. There are several ways this definition can be generalized to links. I will discuss these competing definitions and give some interesting examples. This is joint work with Clayton McDonald.

September 21 (Yi Ni): Seifert fibered surgeries on hyperbolic fibered knots

Let K be a hyperbolic fibered knot such that p/q-surgery on K is either reducible or an atoroidal Seifert fibered space. We prove that if the monodromy of K is right-veering, then 0≤ p/q ≤ 4g(K). The upper bound 4g(K) cannot be attained if K is an L-space knot. As a consequence, we get new results on the characterizing slopes of torus knots.

September 28 (Mark Powell): Stable diffeomorphism of 4-manifolds

Abstract: I will survey the relationships between some popular equivalence relations on 4-manifolds, before focussing on the classification of 4-manifolds up to stable diffeomorphism, meaning up to connected sum with copies of S2 x S2.

October 5 (Ian Zemke): A few refinements of Heegaard Floer genus and clasp number bounds

In this talk, we will describe several new knot Floer concordance invariants which give improvements over the known Heegaard Floer 4-ball genus and clasp number bounds, and also describe how to obtain new bounds from familiar concordance invariants. This is joint work with Andras Juhasz.

October 19 (Patrick Orson): Topologically embedding spheres in knot traces

Given a compact topological 4-manifold, it is a basic question to determine when a second homology class can be represented by an embedded 2-sphere. I will discuss the question for knot traces; these are manifolds with boundary, that are homotopic to the 2-sphere, and obtained by attaching a 2-handle to the 4-ball along a framed knot in the 3-sphere. I will describe recent joint work, where we completely characterised when a generator of second homology can be represented by a locally flat embedded 2-sphere with abelian exterior fundamental group. The answer is in terms of classical and computable invariants of the knot.

October 26 (JungHwan Park): On rationally slice knots

A knot in the three-sphere is called slice if it bounds a smooth disk in the four-ball. If one only requires the disk to be in a rational homology four-ball, then we say that the knot is rationally slice. We present a rationally slice knot which is not slice even after its connected self-sum. This is joint work with Jennifer Hom, Sungkyung Kang, and Matthew Stoffregen.

November 2 (Patrick Naylor): Trisections and non-orientable 4-manifolds

Trisections provide a new perspective on 4-manifolds that may be used to construct potentially different handle decompositions. I'll discuss some basics of trisections, and how to extend certain classical theorems (Laudenbach-Poenaru, Waldhausen) to obtain Kirby diagrams and trisection diagrams for non-orientable 4-manifolds. This is joint work with Maggie Miller (MIT) and Gabriel Islambouli (University of Waterloo).

November 9 (Shaoyun Bai): Equivariant Cerf theory and SU(n) Casson invariants

I will present joint work with Boyu Zhang on defining perturbative SU(n) Casson invariants for integer homology 3-spheres. Ideas from the studies of J-holomorphic curves in symplectic manifolds play important roles in the construction, in particular issues related to equivariant transversalities. Time permitting, I will discuss possible generalizations and topological applications.

November 16 (Aliakbar Daemi): Chern-Simons functional, singular instantons, and the four-dimensional clasp number

Any homology class of degree 2 in a simply connected 4-manifold can be represented by an oriented embedded surface as well as a normally immersed sphere. In order to measure the complexity of one such homology class, one can look for the minimum genus among representatives given by embedded surfaces. Similarly, the minimum number of double points among representatives given by immersed spheres provides another measure for complexity of the homology class. It is natural to ask how these two quantities are related to each other. In my talk, I'll discuss some tools which could be useful to study the difference between these two measures on the complexity of homology classes. In particular, I'll explain that how they can be used to show that positive clasp number of a knot can be arbitrarily larger than its slice genus. If time permits, I'll also talk about some evidence towards an extension of the slice-ribbon conjecture to torus knots. This talk is based on a joint work with Chris Scaduto.

November 30 (Kouki Sato): Filtered instanton Floer homology and the homology cobordism group

Today's seminar will occur at 10AM EST.

We introduce a family of real-valued homology cobordism invariants r_s(Y) of oriented homology 3-spheres. The invariants r_s(Y) are based on a quantitative construction of filtered instanton Floer homology. Using our invariants, we give several new constraints of the set of smooth boundings of homology 3-spheres. As one of the corollaries, we give infinitely many homology 3-spheres which cannot bound any definite 4-manifold. As another corollary, we show that if the 1-surgery of a knot has negative Froyshov invariant, then the 1/n-surgeries (n>0) of the knot are linearly independent in the homology cobordism group. This is a joint work with Yuta Nozaki and Masaki Taniguchi.

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