Before the summer school
- Please watch the first two lectures from each mini-course. Here are the lectures for Piccirillo, Ray, and Stipsicz. (Probably best to watch the first Stipsicz lecture before the others.) On the first day of the workshop there will be time to ask the speakers questions about these talks and the homework they suggested in the talks. The remainder of the mini-courses will be held synchronously during the summer school. We have split the office hours for each course into a morning and after noon session to hopefully accommodate people form a variety of time zones.
- Please watch the lightning talks. You can find them here. During two of the lunch sessions we will have breakout rooms so you can discuss with the lightning speakers.
You can find videos of the research talks here.
Details about the mini-courses can be found here. Titles and abstracts for the research talk and the lightning talks can be found below.
During the lunch sessions, from 12:30 to 1:00 we will have several breakout rooms set up where people can go to chat and discuss the talks and mini-courses. On Tuesday and Wednesday they will focus on the lightning talks, and on other days on either the mini-courses or research talks. So please update your zoom application to the latest version so you will be able to freely move between the breakout rooms!
Monday
| 10:00-10:10 | Welcome & Tea | ||
| 10:10-11:10 | Piccirillo, Ray, and Stipsicz | Question and Problems from first 2 lectures |
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| 11:10-11:20 | Break | ||
| 11:20-12:20 | András Stipsicz | Mini-course 3 | |
| 12:20-1:20 | Lunch Break: Discuss mini-courses | ||
| 1:20-1:50 | Allison Miller | Embedding spheres in simple 4-manifolds Notes from talk |
Tuesday
| 9:30-10:00 | Morning TAs | Office Hours | |
| 10:00-10:10 | Tea | ||
| 10:10-11:10 | Aru Ray | Mini-course 3 | |
| 11:10-11:20 | Break | ||
| 11:20-12:20 | Lisa Piccirillo | Mini-course 3 | |
| 12:20-1:20 | Lunch Break: Discuss lighting talks | ||
| 1:20-1:50 | Sümeyra Sakalli | Symplectic 4-manifolds on the Noether line and between the Noether and half Noether lines Slides from this talk |
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| 2:00-2:30 | Evening TAs | Office Hours |
Wednesday
| 9:30-10:00 | Morning TAs | Office Hours | |
| 10:00-10:10 | Tea | ||
| 10:10-11:10 | András Stipsicz | Mini-course 4 | |
| 11:10-11:20 | Break | ||
| 11:20-12:20 | Lisa Piccirillo | Mini-course 4 | |
| 12:20-12:25 | Group Photo | ||
| 12:25-1:20 | Lunch Break: Discus lighting talks | ||
| 1:20-1:50 | Paul Melvin | Ample 4-Manifolds | |
| 2:00-2:30 | Evening TAs | Office Hours |
Thursday
| 9:30-10:00 | Morning TAs | Office Hours | |
| 10:00-10:10 | Tea | ||
| 10:10-11:10 | András Stipsicz | Mini-course 5 | |
| 11:10-11:20 | Break | ||
| 11:20-12:20 | Aru Ray | Mini-course 4 | |
| 12:20-1:20 | Lunch Break: Discuss research talks | ||
| 1:20-1:50 | Maggie Miller | Exotic Brunnian surface links | |
| 2:00-2:30 | Evening TAs | Office Hours |
Friday
| 9:30-10:00 | Morning TAs | Office Hours | |
| 10:00-10:10 | Tea | ||
| 10:10-11:10 | Aru Ray | Mini-course 5 | |
| 11:10-11:20 | Break | ||
| 11:20-12:20 | Lisa Piccirillo | Mini-course 5 | |
| 12:20-1:20 | Lunch Break | ||
| 1:20-1:50 | Hannah Schwartz | Regular homotopy and Gluck twists |
Titles and Abstracts:
Paul Melvin
Title: Ample 4-Manifolds
Abstract: This talk will begin with a review of some standard notions of stabilization of 4-manifolds, two celebrated results of Wall from the 1960s, and Freedman’s topological classification of closed, oriented simply-connected 4-manifolds from the 1980s. We will then introduce what it means for a topological 4-manifold to be ample – meaning roughly that it has loads of smooth structures – and for a smooth 4-manifold to be homologically ample – meaning its primitive ordinary homology classes can be represented by lots of smoothly embedded spheres. Finally we’ll explain how gauge theory and symplectic topology show that most smoothable simply-connected 4-manifolds are ample, and why this implies that they have stable homologically ample smoothings (this is joint work with Auckly, Kim and Ruberman).
Allison Miller
Title: Embedding spheres in simple 4-manifolds
Abstract:Given a knot K and an integer n, there is an associated nice 4-manifold called the n-trace of K. This manifold is homotopy equivalent to a 2-sphere, and so it is natural to ask when it has an embedded spherical generator for second homology. In a parallel to Freedman’s famous result that all knots with trivial Alexander polynomial are algebraically slice, it turns out that the vanishing of certain knot invariants implies that such a spherical generator exists, at least in the topological category. (There’s even an “if and only if” statement). This is joint work with P. Feller, M. Nagel, P. Orson, M. Powell, and A. Ray.
Maggie Miller
Title: Exotic Brunnian surface links
Abstract: For each natural number $n\ge 2$, I will show how to construct pairs of $n$-component surface links $(S_1, S_2)$ properly embedded in $B^4$ that are Brunnian (meaning every proper sublink of $S_i$ is an unlink of surfaces) yet exotic (meaning $S_1$ and $S_2$ are topologically isotopic rel boundary but not smoothly isotopic even if we allow boundary to move). This is an example of very subtle exotic behavior: the individual components (and other proper sublinks) of $S_1$ and $S_2$ are smoothly isotopic rel boundary, yet $S_1$ and $S_2$ as a whole are not – so the operation relating $S_1$ and $S_2$ must somehow make use of all surface components simultaneously. We may choose $S_1$ and $S_2$ to be unions of disks or alternatively produce an infinite family $\{S_m\mid m\in\mathbb{N}\}$ of pairwise exotic Brunnian surface links that each have a positive-genus component.
This is joint work with Kyle Hayden, Alexandra Kjuchukova, Siddhi Krishna, Mark Powell, and Nathan Sunukjian.
Sümeyra Sakalli
Title: Symplectic 4-manifolds on the Noether line and between the Noether and half Noether lines
Abstract: We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg-Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries introduced by Karakurt and Starkston, and by using complex singularities. We also construct certain configurations of complex singularities in the rational elliptic surfaces geometrically (without using monodromies). By using these configurations, we give symplectic embeddings of star shaped plumbings inside (some blow-ups of) elliptic surfaces.
Hannah Schwartz
Title: Regular homotopy and Gluck twists
Abstract: Any 2-sphere K smoothly embedded in the 4-sphere is related to the unknotted one through a finite sequence of locally supported homotopies called finger moves and Whitney moves. In this talk, I will present joint work with Naylor that the Gluck twist of any sphere admitting a regular homotopy to the unknot consisting of only one finger and one Whitney move is diffeomorphic to the standard 4-sphere. We will also discuss some well-known families of 2-spheres (including certain roll-spun knots) for which this is the case, as well as results joint with Joseph, Klug, and Ruppik that are critical to our proof.
Lightning talks:
Fraser Binns
Title: The Botany Question for Knot Floer Homology
Abstract: Knot Floer homology is a powerful vector space valued link invariant. Given such an invariant it is reasonably natural to ask; “which links have knot Floer homology isomorphic to a given vector space?”. In this talk I will answer two special cases of this question, based on joint work in progress with Subhankar Dey.
Sarah Blackwell
Title: Triple Knot Grid Diagrams
Abstract: In this talk I will introduce a project I have been working on this year involving representing Lagrangian-like surfaces in $\mathbb{CP}^2$ by “triple knot grid diagrams.” I will describe the setup for these diagrams, which comes from the standard (genus one) trisection of $\mathbb{CP}^2$, and explain why the surfaces I am thinking about are “Lagrangian-like.”
Jacob Caudell
Title: Lens space surgeries, lattices, and the Poincaré homology sphere
Abstract: From the cut and paste perspective, lens spaces are the simplest non-trivial 3-manifolds. About a decade ago, Greene resolved the question of which lens spaces may be obtained by performing Dehn surgery on a non-trivial knot in the 3-sphere. In this lightning talk, we will present a sketch of the methodology that Greene developed to answer this question, show how to port this technique to studying (connected sums of) lens spaces arising as surgery on knots in the Poincaré homology sphere, and present some results obtained by applying these arguments.
Nicholas Cazet
Title: Measure homology
Abstract: Measure homology expands on singular homology with real coefficients. Both are functors from the category of topological spaces to the category of chain complexes of real vector spaces, and the two theories agree on spaces homotopic to CW-complexes. Measure homology uses signed measures on function spaces to give an invariant that is useful in wild (algebraic) topology. This homology theory is isometrically isomorphic to singular homology for countable CW-complexes, thus there exists a definition of simplicial volume based on signed measures. We will construct the measure chain complex and give recent results, including our progress on understanding the topological properties influencing the zeroth homology space.
Tam Cheetham-West
Title: Hyperbolic 4-manifolds with boundary
Abstract: The goal of this talk is to discuss theorems of Long and Reid concerning hyperbolic and flat 3-manifolds which occur as the boundaries of hyperbolic 4-manifolds.
Shintaro Fushida-Hardy
Title: Combinatorial proofs and genus bounds
Abstract: The advent of gauge theory in 90s lead to many new results in 4 dimensional topology. Since then, many of the results have been reproved without using gauge theory, in what are often called combinatorial proofs. I’ll describe some of the aforementioned results and the motivation behind discovering combinatorial proofs. Finally I’ll give a brief description of Peter Lambert-Cole’s recent combinatorial proof of the adjunction inequality.
Thomas Kindred
Title: Obstructing smooth multisections of n-manifolds
Abstract: Just as every 3-manifold admits a Heegaard splitting, every smooth 4-manifold admits a trisection, i.e. a decomposition into three handlebodies with simple intersections. Every smooth 5-manifold also admits a trisection. Multisections generalize these decompositions to manifolds of arbitrary dimension. In the PL category, every manifold admits a multisection. I will sketch a proof that this is not true in the smooth category.
Feride Ceren Kose
Title: Symmetric Unions and Jones Polynomial
Abstract: It is still an open question if there exist non-trivial knots with trivial Jones polynomials. Symmetric unions appear often when we construct non-trivial knots with trivial Alexander polynomials; hence, it is natural to expect to have a non-trivial symmetric union with trivial Jones polynomial. In fact, Tanaka proposes a way of constructing symmetric unions that have trivial Jones polynomials. Meanwhile, Khovanov homology, the categorification of the Jones polynomial, does detect the unknot. With the help of this result, we use Khovanov homology and show that the knots suggested by Tanaka are trivial.
Vincent Longo
Title: Infinitely many counterexamples to Batson’s conjecture
Abstract: Batson’s conjecture is a non-orientable version of Milnor’s conjecture, which states that the 4-ball genus of a torus knot T(p,q) is equal to (p-1)(q-1)/2. Batson’s conjecture states that the nonorientable 4-ball genus is equal to the pinch number of a torus knot, i.e. the number of a specific type of (nonorientable) band surgeries needed to obtain the unknot. The conjecture was recently proved to be false by Lobb. We will show that Lobb’s counterexample fits into an infinite family of counterexamples.
Clayton McDonald
Title: Doubly slice links
Abstract: A knot in the 3-sphere is doubly slice if it is the cross section of an unknotted sphere in the 4-sphere. In joint work with Duncan McCoy, I will discuss various generalizations of this concept to links. With this discussion will be a few constructions and obstructions, with a focus on Montesinos links.
Kai Nakamura
Title: Trace embeddings from 0-surgery homeomorphisms
Abstract: Manolescu and Piccirillo recently proposed a construction of exotic homotopy 4-spheres using 0-surgery homeomorphisms and rasmussens s-invariant. They produced a family of knots that if any were slice, one could construct an exotic 4-sphere. I will discuss this and recent work where I show that these knots are not slice. Hence they cannot be used to construct an exotic 4-sphere. Sliceness is obstructed by showing that if any of these knots were slice, another knot which has homeomorphic 0-surgery to one of the knots would be null homologously slice in #nCP^2 by constructing a trace embedding.
Patrick Naylor
Title: Gluck twists of roll spun knots
Abstract: Given an embedded 2-sphere in the 4-sphere, a Gluck twist is a surgery which produces a 4-manifold homeomorphic, but not obviously diffeomorphic to the 4-sphere. This operation has been well studied, but it remains an open question whether all Gluck twists are standard. Many 2-spheres arising from “spinning” constructions are known to have standard Gluck twists, but roll-spun knots are notably absent from the list of such spheres. In this talk, I will discuss some recent results that show that the Gluck twist of the roll spin of an unknotting number one knot is standard. This is joint work with Hannah Schwartz.
Minh Nguyen
Title: 3/2 Spinors on 4-manifolds and finite dimensional approximation
Abstract: Rarita-Schwinger operator $Q$ was initially proposed in the 1941 paper by Rarita and Schwinger to study wave functions of particles of spin $3/2$, and there is a vast amount of physics literature on its properties. Roughly speaking, $3/2-$spinors are spinor-valued 1-forms that also happen to be in the kernel of the Clifford multiplication. Let $X$ be a Riemannian spin $4-$manifold. Associated to a fixed spin structure on $X$, we define a Seiberg-Witten-like system of non-linear PDEs using $Q$ and the Hodge-Dirac operator $d^* + d^+$ after suitable gauge-fixing. The moduli space of solutions $\mathcal{M}$ contains (3/2-spinors, purely imaginary 1-forms). Unlike in the case of Seiberg-Witten equations, solutions are hard to find or construct. However, by adapting the finite dimensional technique of Furuta, we provide a topological condition of $X$ to ensure that $\mathcal{M}$ is non-compact; and thus, contains infinitely many elements.
Seppo Niemi-Colvin
Title: Invariance of Knot Lattice Homology
Abstract: This talk will give an overview of the invariance of knot lattice homology. In particular, it will cover the knots for which knot lattice defined and what presentation of the knot it uses as input (for it to be invariant over), along with mention of the context motivating knot lattice homology.
Braeden Reinoso
Title: Capping off open books and fractional Dehn twist coefficients
Abstract: Open book decompositions offer one (of many) ways to view a 3-manifold in terms of maps on surfaces. The fractional Dehn twist coefficients of an open book are integers defined on the surface level that contain important contact geometric information on the 3-manifold level. In this brief talk, I’ll introduce open book decompositions, fractional Dehn twist coefficients, and summarize some results about how they change under different operations on open books.
Agniva Roy
Title: Constructions, and computing invariants, of Legendrian spheres in the standard S^5
Abstract: In this talk we will demonstrate constructions of Legendrian spheres in standard higher dimensional contact spheres and demonstrate isotopies between different constructions. These become useful in computing invariants of some of these constructions.
Benjamin Matthias Ruppik
Title: Ribbon 2-knots and Casson-Whitney unknotting
Abstract: In 1986, Miyazaki observed that you can unknot a ribbon 2-knot by stabilizing fusion number many times. With Jason Joseph, Michael Klug, Hannah Schwartz we found a similar statement for an unknotting number defined in terms of regular homotopies: Fusion number many finger moves followed by Whitney moves can unknot a given ribbon 2-knot.