Dates: April 17  21, 2023
Location: Lakeside Village Pavilion, University of Miami, Coral Gables, FL
Organizers: N. Saveliev & C. Scaduto
9:30am 
Linh Truong, University of Michigan: Fourgenus bounds from the 10/8+4 theorem Donald and Vafaee constructed a knot slicing obstruction for knots in the threesphere by producing a bound relating the signature and second Betti number of a spin 4manifold whose boundary is zerosurgery 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 fourball genus bounds and compute the bounds for some satellite knots. This is joint work in progress with Sashka Kjuchukova and Gordana Matic. Video 
11:00am 
Anubhav Mukherjee, Princeton University Video 
1:30pm 
Duncan McCoy, l'Université du Québec à Montréal: Characterizing slopes for alternating knots Given a knot K in the 3sphere, we say that a rational number p/q is a characterizing slope for K if the oriented homeomorphism type of p/qsurgery on K determines K uniquely amongst all knots in the 3sphere. Conjecturally, all but finitely noninteger 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 
3:00pm 
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 almostrational 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 
9:30am 
Ian Montague, Brandeis University: Equivariant Rokhlin, Eta, and Kappa Invariants of SeifertFibered Homology Spheres For every integer m > 1, I will introduce a Z/mequivariant homology cobordism invariant called the equivariant Rokhlin invariant, which agrees with the mod 2 reduction of (a variant of) the equivariant SeibergWitten Floer correction term appearing in previous work of mine. I will then explain how to calculate this equivariant correction term explicitly for Seifertfibered homology spheres with respect to cyclic group actions contained in the standard S^1action, which involves (among other things) computing equivariant etainvariants of the Dirac operator. Using these calculations, we obstruct the existence of free Z/mequivariant homology cobordisms between free Z/mequivariant Brieskorn homology spheres, as well as provide obstructions to extending cyclic group actions on Brieskorn homology spheres over small spin fillings via equivariant relative 10/8ths type inequalities. Video 
11:00am 
Jennifer Hom, Georgia Tech: PLsurfaces in homology 4balls We consider manifoldknot pairs (Y, K) where Y is a homology 3sphere that bounds a homology 4ball. Adam Levine proved that there exists pairs (Y, K) such that K does not bound a PLdisk 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 
1:30pm 
Matt Hedden, Michigan State University: Rational slice genus bounds from knot Floer homology We use relative adjunction inequalities for properly embedded surfaces in smooth 4manifolds to study the "rational slice genus" of knots in a rational homology sphere. This quantity is a 4d 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 3manifold 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 
3:00pm 
Tom Mrowka, Massachusetts Institute of Technology Video 
9:30am 
Tom Mark, University of Virginia: Fillable contact structures from positive surgery For a Legendrian knot K in a closed contact 3manifold, we describe a necessary and sufficient condition for contact nsurgery along K to yield a weakly symplectically fillable contact manifold, for some integer n>0. When specialized to knots in the standard 3sphere 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 4dimensional clasp number. We will mainly explore these topologicallyflavored aspects, but will give some hints of the general proof if time allows. Video 
11:00am 
Boyu Zhang, University of Maryland: The smooth closing lemma for areapreserving diffeomorphisms of surfaces In this talk, I will introduce a proof of the smooth closing lemma for areapreserving diffeomorphisms on surfaces. The proof is based on a Weyl formula for Periodic Floer Homology spectral invariants and a nonvanishing result of twisted SeibergWitten Floer homology. This is joint work with Dan CristofaroGardiner and Rohil Prasad. Video 
1:30pm  
3:00pm 
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 nonlinear 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, actionangle coordinates were computed for the standard system using a noncanonical 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 
9:30am 
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 3manifolds. In principle, Heegaard diagrams determine 3manifolds 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 3manifold 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 
11:00am 
Fan Ye, Harvard University: 2torsions in singular instanton homology Shumakovitch conjectured that the (unreduced) Khovanov homology of any nontrivial knot has 2torsions. Inspired by the spectral sequence from Khovanov homology to singular instanton homology constructed by KronheimerMrowka, we study the 2torsions 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 2torsions 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 3manifold obtained from S^3 by 1/2 surgery along any knot of genus >1 has 2torsions. This is a joint work in progress with Deeparaj Bhat and Zhenkun Li. Slides Video 
1:30pm 
Aliakbar Daemi, Washington University in St. Louis: Instantons, suspension, and surgery: Part 1 Any oriented connected closed 3manifold Y is obtained by (Dehn) surgery on a link in the 3sphere, 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 3spheres with arbitrarily large Dehn surgery number. This answers a question of Auckly, who previously constructed the first example of an integer homology 3sphere 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 3manifold Y in terms of b^+ of a 4manifold 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 
3:00pm 
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 2torsion, and discuss how this construction can be used to study obstructed cobordism maps more generally. Video 
9:30am 
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 
11:00am 
Francesco Lin, Columbia University: Homology cobordism and the geometry of hyperbolic threemanifolds A major challenge in the study of the structure of the threedimensional 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 
1:30pm 
Dave Auckly, Kansas State University, Subtly knotted surfaces separated by many internal stabilizations Smoothly knotted surfaces are now wellknown objects in 4manifolds. 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 fourmanifolds 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 
Participants
David Auckly  Kansas State University  Ken Baker  University of Miami 
Deeparaj Bhat  Massachusetts Institute of Technology  Hans Boden  McMaster University 
Aliakbar Daemi  Washington University in St. Louis  Clair Dai  Harvard University 
Josh Drouin  Kansas State University  Paul Feehan  Rutgers University 
Malcolm Gabbard

Kansas State University  Sherry Gong  Texas A&M University 
Matthew Hedden  Michigan State University  Jennifer Hom  Georgia Tech 
Alexei Kovalev  University of Cambridge  KyoungSeog Lee  University of Miami 
Thomas Leness  Florida International University  Jiakai Li  Harvard University 
Zhenkun Li  Stanford University  Francesco Lin  Columbia University 
Zedan Liu  University of Miami  Fabiola ManjarrezGutiérrez  Universidad Nacional Autónoma de México 
Thomas Mark  University of Virginia  Duncan McCoy  l'Université du Québec à Montréal 
Mike Miller Eismeier  Columbia University  Ian Montague  Brandeis University 
Tomasz Mrowka  Massachusetts Institute of Technology  Anubhav Mukherjee  Princeton University 
Minh Nguyen  Washington University in St. Louis  Isacco Nonino  University of Glasgow 
Sinem Onaran  Hacettepe University  Jessie Osnes  Kansas State University 
Vinicius Ramos 
Instituto de Matemática Pura e Aplicada  Ali Naseri Sadr  Boston College 
Mary Stelow  Massachusetts Institute of Technology  Matthew Stoffregen  Michigan State University 
Linh Truong  University of Michigan  Joshua Wang  Harvard University 
Fan Ye  Harvard University  Kevin Yeh  Boston College 
Boyu Zhang  University of Maryland 