Conference 1 Schedule
Wednesday, July 12
9:30 - 10:30 Josh Greene (Boston College)
Title: From ball fillings to cube tilings
Abstract: I will describe an obstruction to a rational homology 3-sphere bounding a smooth rational homology 4-ball. It only applies in special cases, and when it does, it takes the form of a lattice embedding condition. We conjecture that the obstruction is complete, and we prove it in a certain extremal case. The proof utilizes the Hajós-Minkowski theorem that every lattice tiling of Euclidean space by cubes contains a pair of cubes that abut along a facet of each. The talk is based on joint work, part with Slaven Jabuka, part with Brendan Owens.
10:45 - 11:45 Gary Guth (University of Oregon)
Title: Bordered Floer homology and satellite concordances
Abstract: An important goal of four-dimensional topology is to tease out the differences between smooth and topological equivalence. We consider the case of surfaces: when are surfaces exotically knotted, i.e. when are surfaces related by a topological isotopy but not a smooth one? By utilizing a satellite construction, we construct new examples of exotic surfaces which can be distinguished by using Bordered Floer homology to compare their induced maps. This draws on joint work with Kyle Hayden, Sungkyung Kang, and JungHwan Park.
1:30 - 2:30 Orsola Capovilla-Searle (UC Davis)
Title: Distinguishing exact Lagrangian fillings with Newton polytopes
Abstract: In joint work with Roger Casals, we show that Newton polytopes associated to augmented values of Reeb chords can distinguish infinitely many distinct Lagrangian fillings, both for Legendrian links and higher-dimensional Legendrian spheres. The computations we perform work in finite characteristic, which significantly simplifies arguments and also allows us to show that there exist Legendrian links with infinitely many non-orientable exact Lagrangian fillings.
3:00 - 4:00 Liam Watson (University of British Columbia)
Title: Khovanov multicurves are linear
Abstract: For a given invariant the geography problem asks for a characterization of values the invariant attains. For example, it is well understood which Laurent polynomials arise as the Alexander polynomial of a knot. By contrast, very little is known about which values the Jones polynomial takes. And the situation is at least as bad for Khovanov homology. So it is a little surprising that the Khovanov homology of a tangle, which can be framed in terms of immersed curves in the 4-punctured sphere, satisfies rather strict geography—the invariants are “linear”. My talk, which is based on joint work with Artem Kotelskiy and Claudius Zibrowius, will explain what “linear” means, and tell part of the story of how this comes up.
Thursday, July 13
9:30 - 10:30 John Baldwin (Boston College)
Title: Non-fibered knot detection
Abstract: A knot is fibered if and only if its knot Floer homology has rank one in the top Alexander grading. With this as motivation, we call a knot "nearly-fibered" if its knot Floer homology has rank two in the top Alexander grading. I'll describe a complete classification of genus-1 nearly fibered knots in the 3-sphere. With this, we are able to prove for the first time that knot Floer homology and Khovanov homology can detect non-fibered knots, and that HOMFLY homology detects infinitely many knots. In addition, we show that 0-surgery detects infinitely many knots, generalizing Gabai's results from his resolution of the Property R Conjecture. This is joint work with Sivek.
10:45 - 11:45 Miriam Kuzbary (Georgia Tech)
Title: Asymptotic behavior of invariants of homology spheres
Abstract: As shown by Morita, every integral homology 3-sphere Y has a decomposition into two simple pieces (called a Heegaard splitting) glued along a surface diffeomorphism which acts trivially on the homology of the surface. These diffeomorphisms form the Torelli subgroup of the mapping class group of the surface, and the Torelli group is finitely generated for surfaces of genus 3 or higher. Though the group of integral homology spheres is infinitely generated, by fixing the genus of a Heegaard splitting we can use finite generation in the surface setting to better understand how invariants like the Rokhlin and Casson invariant change. This perspective has led to important results in the study of the Torelli group and the Casson invariant in work of Birman-Craggs-Johnson, Morita, and Broaddus-Farb-Putman. In work in progress with Santana Afton and Tye Lidman, we show that the d-invariant from Heegaard Floer homology of an integral homology sphere is bounded above by a linear function of the word length of a corresponding gluing map in the Torelli group. Moreover, we show the d-invariant is bounded for homology spheres corresponding to various large families of mapping classes. If time permits, we will discuss the case of rational homology spheres.
1:30 - 2:30 Irving Dai (UT Austin)
Title: Lattice homology and Seiberg-Witten Floer spectra
Abstract: Using tools from lattice homology, we calculate the Seiberg-Witten Floer spectra of Seifert fibered rational homology spheres. We also discuss some speculative connections with algebraic geometry. This is joint work with Hirofumi Sasahira and Matthew Stoffregen.
3:00 - 4:00 Maggie Miller (UT Austin) (previously scheduled for Friday 2pm)
Title: Splitting spheres in S^4
Abstract: A 2-component link L is split if its components lie in disjoint balls. The boundary of either of these balls is called a splitting sphere for L. In the 3-sphere, 2-component split links have unique splitting spheres, meaning any two splitting spheres for L are isotopic in S^3-L. In this talk, we’ll discuss why this fails in dimension 4: many 2-component split links of surfaces in the 4-sphere do not have unique splitting spheres. (In fact, many unlinks have non-unique splitting spheres.) This is joint work with Mark Hughes and Seungwon Kim.
6:00 - 8:00 BANQUET
Friday, July 14
9:30 - 10:30 Hokuto Konno (University of Tokyo)
Title: Exotic Dehn twists on 4-manifolds
Abstract: A self-diffeomorphism of a smooth manifold is said to be exotic if it is topologically isotopic to the identity but smoothly not. We provide the first examples of exotic diffeomorphisms (in a relative sense) of contractible 4-manifolds, more generally of definite 4-manifolds. Such examples are given as Dehn twists along certain Seifert homology 3-spheres, which also give new examples of exotic diffeomorphisms that survive after one stabilization, and the smallest closed 4-manifold known to support an exotic diffeomorphism. The proof uses families Seiberg-Witten theory over RP^2. This is joint work with Abhishek Mallick and Masaki Taniguchi.
11:00 - 12:00 Sucharit Sarkar (UCLA)
Title: Mixed invariants in Khovanov homology for unorientable cobordisms
Abstract: Using Bar-Natan's and Lee's deformations of Khovanov homology of links, we define minus, plus, and infinity versions of Khovanov homology. Given an unorientable cobordism in [0,1]\times S^3 from a link L_0 to a link L_1, we define a mixed invariant as a map from the minus version of the Khovanov homology of L_0 to the plus version of the Khovanov homology of L_1. The construction is similar to the mixed invariant in Heegaard Floer homology. This invariant can be used to distinguish exotic cobordisms, that is, two cobordisms which are topologically isotopic but not smoothly isotopic. This is joint with Robert Lipshitz.
2:00 - 3:00 Mike Miller Eismeier (Columbia University) (previously scheduled for Thursday 3pm)
Title: CS from CS
Abstract: The Cosmetic Surgery conjecture asserts that if K is a knot in a 3-manifold Y has r-surgery oriented diffeomorphic to s-surgery, then r = s. Much partial progress has been made; for instance, if Y = S^3, any potential counterexample must have r = -s and r in {2, 1, 1/2, 1/3, ...}
Using a carefully chosen cobordism, we show that a Chern-Simons invariant associates to 1/2-surgery on K in S^3 is strictly larger than the invariant associated to -1/2 surgery, thus removing r = 1/2 from the list above. The technique can be extended to cover the case r = 1/4, but other r currently appear out of reach.
This talk represents joint work with Tye Lidman.