Conference: Gauge Theory and Low Dimensional Topology

Linh Truong, University of Michigan: Four-genus bounds from the 10/8+4 theorem

Donald and Vafaee constructed a knot slicing obstruction for knots in the three-sphere by producing a bound relating the signature and second Betti number of a spin 4-manifold whose boundary is zero-surgery on the knot. Their bound relies on Furuta's 10/8 theorem and can be improved with the 10/8 + 4 theorem of Hopkins, Lin, Shi, and Xu. I will explain how to expand on this technique to obtain four-ball genus bounds and compute the bounds for some satellite knots. This is joint work in progress with Sashka Kjuchukova and Gordana Matic.  Video

Anubhav Mukherjee, Princeton University  Video

Duncan McCoy, l'Université du Québec à Montréal: Characterizing slopes for alternating knots

Given a knot K in the 3-sphere, we say that a rational number p/q is a characterizing slope for K if the oriented homeomorphism type of p/q-surgery on K determines K uniquely amongst all knots in the 3-sphere. Conjecturally, all but finitely non-integer slopes should be characterizing for a given knot. I will explain some context for this conjecture and describe some recent progress. This will include discussion of how one can prove this conjecture for various classes of knots, including alternating knots.  Video

Matt Stoffregen, Michigan State University: An Exact Triangle in Monopole Floer Spectra

We will describe an exact triangle for monopole Floer spectra.  Using the exact triangle, we'll calculate the monopole Floer spectra for almost-rational plumbings, and indicate some possible directions for further calculations. This is all joint work with Hirofumi Sasahira, and parts are also joint with Irving Dai.  Video

Ian Montague, Brandeis University: Equivariant Rokhlin, Eta, and Kappa Invariants of Seifert-Fibered Homology Spheres

For every integer m > 1, I will introduce a Z/m-equivariant homology cobordism invariant called the equivariant Rokhlin invariant, which agrees with the mod 2 reduction of (a variant of) the equivariant Seiberg-Witten Floer correction term appearing in previous work of mine. I will then explain how to calculate this equivariant correction term explicitly for Seifert-fibered homology spheres with respect to cyclic group actions contained in the standard S^1-action, which involves (among other things) computing equivariant eta-invariants of the Dirac operator. Using these calculations, we obstruct the existence of free Z/m-equivariant homology cobordisms between free Z/m-equivariant Brieskorn homology spheres, as well as provide obstructions to extending cyclic group actions on Brieskorn homology spheres over small spin fillings via equivariant relative 10/8-ths type inequalities.  Video

Jennifer Hom, Georgia Tech: PL-surfaces in homology 4-balls

We consider manifold-knot pairs (Y, K) where Y is a homology 3-sphere that bounds a homology 4-ball. Adam Levine proved that there exists pairs (Y, K) such that K does not bound a PL-disk in any bounding homology ball. We show that the minimum genus of a PL surface S in any bounding homology ball can be arbitrarily large. The proof relies on Heegaard Floer homology. This is joint work with Matthew Stoffregen and Hugo Zhou.  Video

Matt Hedden, Michigan State University: Rational slice genus bounds from knot Floer homology

We use relative adjunction inequalities for properly embedded surfaces in smooth 4-manifolds to study the "rational slice genus" of knots in a rational homology sphere.  This quantity is a 4-d analogue of the rational Seifert genus introduced by Calegari and Gordon.  Unlike the latter quantity, however, determining the rational slice genus of a given knot involves solving an infinite number of minimal genus problems in the 3-manifold times an interval.  It is therefore surprising that we can compute it for certain classes, including Floer simple knots.  This is joint work with Katherine Raoux.  Video

Tom Mrowka, Massachusetts Institute of Technology  Video

Tom Mark, University of Virginia: Fillable contact structures from positive surgery

For a Legendrian knot K in a closed contact 3-manifold, we describe a necessary and sufficient condition for contact n-surgery along K to yield a weakly symplectically fillable contact manifold, for some integer n>0. When specialized to knots in the standard 3-sphere this gives an effective criterion for the existence of a fillable positive surgery, along with various obstructions. These are sufficient to determine, for example, whether such a surgery exists for all knots of up to 10 crossings. The result also has certain purely topological consequences, such as the fact that a knot admitting a lens space surgery must have slice genus equal to its 4-dimensional clasp number. We will mainly explore these topologically-flavored aspects, but will give some hints of the general proof if time allows.  Video

Boyu Zhang, University of Maryland: The smooth closing lemma for area-preserving diffeomorphisms of surfaces 

In this talk, I will introduce a proof of the smooth closing lemma for area-preserving diffeomorphisms on surfaces. The proof is based on a Weyl formula for Periodic Floer Homology spectral invariants and a non-vanishing result of twisted Seiberg-Witten Floer homology. This is joint work with Dan Cristofaro-Gardiner and Rohil Prasad.  Video

Paul Feehan, Rutgers University Slides Video

Vinicius Ramos, Instituto de Matemática Pura e Aplicada: The Toda lattice, billiards and the Viterbo conjecture  

The Toda lattice is one of the earliest examples of non-linear completely integrable systems. Under a large deformation, the Hamiltonian flow can be seen to converge to a billiard flow in a simplex. In the 1970s, action-angle coordinates were computed for the standard system using a non-canonical transformation and some spectral theory. In this talk, I will explain how to adapt these coordinates to the situation to a large deformation and how this leads to new examples of symplectomorphisms of Lagrangian products with toric domains. In particular, we find a sequence of Lagrangian products whose symplectic systolic ratio is one and we prove that they are symplectic balls. This is joint work with Y. Ostrover and D. Sepe.  PDF Slides  Video

Zhenkun Li, Stanford University: Instanton Floer homology and Heegaard diagrams

Instanton Floer homology was introduced by Floer in 1980s and has become a power invariants for three manifolds and knots since then. It has lead to many milestone results, such as the approval of Property P conjecture. Heegaard diagrams, on the other hand, is a combinatorial methods to describe 3-manifolds. In principle, Heegaard diagrams determine 3-manifolds and hence determine their instanton Floer homology as well. However, no explicit relations between these two objects were known before. In this talk, for a 3-manifold Y, I will talk about how to extract some information about the instanton Floer homology of Y from the Heegaard diagrams of Y. Additionally, I will explore some of the applications and future directions of this work. This is a joint work with Baldwin and Ye.  Video

Fan Ye, Harvard University: 2-torsions in singular instanton homology

Shumakovitch conjectured that the (unreduced) Khovanov homology of any nontrivial knot has 2-torsions. Inspired by the spectral sequence from Khovanov homology to singular instanton homology constructed by Kronheimer-Mrowka, we study the 2-torsions in unreduced variant of singular instanton homology for knots. When K is a fibered knot, we aim to show that the singular instanton homology has 2-torsions by comparing the homology groups with complex coefficients and Z/2 coefficients. Also, we aim to prove that the framed instanton homology of the closed 3-manifold obtained from S^3 by 1/2 surgery along any knot of genus >1 has 2-torsions. This is a joint work in progress with Deeparaj Bhat and Zhenkun Li. Slides Video

Aliakbar Daemi, Washington University in St. Louis: Instantons, suspension, and surgery: Part 1

Any oriented connected closed 3-manifold Y is obtained by (Dehn) surgery on a link in the 3-sphere, and the minimal number of connected components of any such link is called the (Dehn) surgery number of Y. In this talk, I will discuss a joint work with Miller Eismeier, where we show that there are integer homology 3-spheres with arbitrarily large Dehn surgery number. This answers a question of Auckly, who previously constructed the first example of an integer homology 3-sphere with Dehn surgery number 2. The proof uses Froyshov's invariant q_3, which is defined using mod 2 instanton homology. There are two key ingredients in the proof. The first one is the computation of q_3 for connected sums of Poincare homology spheres; this computation is based on a joint work with Miller Eismeier and Scaduto on a connected sum theorem for mod 2 instanton homology. The second ingredient is an inequality involving q_3 of a 3-manifold Y in terms of b^+ of a 4-manifold filling Y. The proof of this inequality is based on the idea of suspensions of instanton Floer complexes, and it will be discussed in more details in Mike's talk.  Video   

Mike Miller Eismeier, Columbia University: Instantons, suspension, and surgery: Part 2 

Continuing Ali's discussion, I will explain how the idea of "suspension" naturally arises when trying to associate relative invariants to manifolds with one boundary component and b^+(W) > 0. I will then discuss how to use this construction to prove an inequality for q_3(Y) when Y bounds a manifold W with H_1(W;Z) free of 2-torsion, and discuss how this construction can be used to study obstructed cobordism maps more generally.  Video

Sherry Gong, Texas A&M University: Ribbon concordances and slice obstructions: experiments and examples

We will discuss some computations of ribbon concordances between knots and talk about the methods and the results. This is a joint work with Nathan Dunfield.  Slides Video

Francesco Lin, Columbia University: Homology cobordism and the geometry of hyperbolic three-manifolds

A major challenge in the study of the structure of the three-dimensional homology cobordism group is to understand the interaction between hyperbolic geometry and homology cobordism. In this talk, I will discuss how monopole Floer homology can be used to study some basic properties of certain subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying some natural geometric constraints.  Video

Dave Auckly, Kansas State University, Subtly knotted surfaces separated by many internal stabilizations 

Smoothly knotted surfaces are now well-known objects in 4-manifolds. It is known that any two smoothly knotted surfaces will become isotopic after some number of internal stabilizations, i.e., adding some amount of genus in a coordinate chart. In this talk, we will demonstrate that it is very common for homology classes in smooth four-manifolds to be represented by infinite collections of surfaces so that there is a diffeomorphism taking any one surface in the collection to any other surface in the collection, so that the diffeomorphism is topologically isotopic to the identity, smoothly pseudoisotopic to the identity, and becomes smoothly isotopic to the identity after one external stabilization. Furthermore, the internal stabilization diameter of the collection of surfaces will be infinite.  Video