**MIT Meeting**

Saturday, September 29, 9:30AM-12:15PM

Math Department,
4th Floor Lecture Room (Building
2, Room 449)

Title: Counting minimal size generating sets for groups coming from geometric topology

Abstract: The rank of a (finitely generated) group is the minimal number of elements needed to generate it. Now, an infinite group G usually has infinitely many minimal size generating sets; for instance, if G=<g,h> then {g,gh} also generates, as does {g^{-1},h}. We say that two generating sets that differ by a sequence of these moves are Nielsen equivalent. In this talk, we’ll discuss the rank and the number of Nielsen equivalence classes of minimal size generating sets in a number of groups coming from geometry and low-dimensional topology. We’ll spend much of the time working with the fundamental groups of 3-dimensional mapping tori, in particular mapping tori where the fiber is a torus.

Title: The Conway knot is not slice

Abstract: A knot in S^3 is trivial if it bounds an embedded disk in S^3. Concordance, defined by Fox in 1962, is a 4-dimensional extension of this idea; a knot in S^3 is slice if it bounds a smooth embedded disk in B^4. Some concordance phenomena are particularly subtle to detect. For example it can be difficult to obstruct the sliceness of a knot which bounds a locally flat disk in B^4, or which is a positive mutant of a slice knot, or for which all known concordance invariants vanish. The 11-crossing Conway knot, discovered by Conway in 1969, is a perfect storm of these phenomena. I’ll use 4-manifold topology, particularly the study of knot traces, to show that the Conway knot is not slice.

Title: On Seiberg-Witten Casson invariant of 4-manifolds

Abstract: By studying the Seiberg-Witten equations, topologists have achieved huge success in distinguishing smooth 4-manifolds whose intersection forms have

nontrivial positive part. Much less is known about 4-manifolds with vanishing second Betti number. In the case that a 4-manifold has identical homology

as S^1×S^3, Mrowka-Ruberman-Saveliev defined a Seiberg-Witten-Casson invariant. In this talk, I will discuss a series of results surrounding this invariant:

how to use it to obstruct metric with positive scalar curvature; how to express it in terms of Floer-theoretic data; how to use it to study the action of 3-dimensional

mapping class group on monopole Floer homology. En route, I will discuss various applications in gauge theory, knot theory, contact geometry and 4-dimensional

topology. (This is a joint project with Danny Ruberman and Nikolai Saveliev)