The workshop focused on recent developments in low-dimensional topology, with a particular emphasis on modern Floer homological invariants such as Heegaard Floer homology. The workshop was open to all; the following captures most of the participants.

All talks were held
at Institute for Pure and
Applied Mathematics,
located here.

- Wednesday, Jan 3
- [07:30-08:30] Breakfast
- [08:30-09:30]
*András Stipsicz*(Alfréd Rényi Institute), Uspilon type invariants of knotsWe review the definition of the Upsilon invariant of a knot, show some variants of the definition and some applications. - [09:45-10:45]
*Jennifer Hom*(Georgia Tech), Heegaard Floer homology and homology cobordism, part oneWe study applications of Heegaard Floer homology to homology cobordism. In particular, to a homology sphere Y, we associate a module HF_conn(Y), called the connected Heegaard Floer homology of Y, and show that this module is invariant under homology cobordism and isomorphic to a summand of HF_red(Y). The definition of this invariant relies on involutive Heegaard Floer homology. This is joint work with Kristen Hendricks and Tye Lidman. - [11:00-12:00]
*Kristen Hendricks*(Michigan State U), Heegaard Floer homology and homology cobordism, part twoContinuing from Jen Hom's talk, we compute some basic examples of connected Heegaard Floer homology, and use it to derive various applications to homology cobordism. We show, for example, that an integer homology sphere whose connected homology has dimension one always has infinite order in the homology cobordism group; we also construct a new filtration of the homology cobordism group and use it to give a reproof of Furuta's theorem. We also study the limiting behavior of numerical invariants from involutive Heegaard Floer homology under connected sum. This is joint work with Jen Hom and Tye Lidman. - [12:00-13:45] Lunch break
- [13:45-14:45]
*John Baldwin*(Boston College), Khovanov homology detects the trefoilI'll discuss joint work with Steven Sivek in which we prove that Khovanov homology detects the trefoil. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a new bypass exact triangle in sutured instanton homology; and Kronheimer and Mrowka's spectral sequence relating Khovanov homology with singular instanton knot homology. - [15:00-16:00]
*Aliakbar Daemi*(Simons Center), Chern-Simons functional and the homology cobordism groupThe set of 3-manifolds with the same homology as the 3-dimensional sphere, modulo an equivalence relation called homology cobordance, forms a group. The additive structure of this group is given by taking connected sum. This group is called the homology cobordism group and plays a special role in low dimensional topology and knot theory. In this talk, I will explain how one can construct a family of invariants of the homology cobordism group by applying ideas from min-max theory in Floer theory. The relationship between these invariants and the Froyshov's invariant will be discussed. I will also talk about some topological applications. - [16:00-17:00] Tea

- Thursday, Jan 4
- [07:30-08:30] Breakfast
- [08:30-09:30]
*Allison Miller*(UT Austin), Knot traces and concordanceA conjecture of Akbulut and Kirby from 1978 states that the concordance class of a knot is determined by its 0-surgery. In 2015, Yasui disproved this conjecture by providing pairs of knots which have the same 0-surgeries yet which can be distinguished in (smooth) concordance by an invariant associated to the four-dimensional traces of such surgeries. In this talk, I will discuss joint work with Lisa Piccirillo in which we construct many pairs of knots which have diffeomorphic 0-surgery traces yet some of which can be distinguished in smooth concordance by the Heegaard Floer d-invariants of their double branched covers. I will also discuss the applicability of this work to the existence of interesting invertible satellite maps on the smooth concordance group. - [09:45-10:45]
*Lisa Piccirillo*(UT Austin), Integer knot surgeries and Stein knot tracesIt is well known that there exist pairs of knots K and K' in S^3 and integers n such that the two three-manifolds obtained by performing n-surgery on K and K' are homeomorphic, however the literature contains only a few systematic methods for constructing such a K and K'. This talk will begin with survey two of these methods. Define the n-trace of a knot K to be the four manifold obtained by attaching a single n framed two-handle to the four ball along K and observe that the boundary of the n-trace on K is n-surgery on K. It is natural then to ask when the discussed methods of constructing homeomorphisms of n-surgeries extend to diffeomorphisms of the n traces; I will discuss what is known about this. Finally I will discuss recent joint work with T. Mark and F. Vafaee which uses these ideas to show that certain knot traces which are not obviously Stein do in fact admit Stein structures. - [11:00-12:00]
*John Etnyre*(Georgia Tech), Contact surgeries and symplectic fillingsIt is well known that all contact manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. What is not so well understood is what properties of a contact structure are preserved by positive contact surgeries (the case for negative contact surgeries is fairly well understood now). In this talk we will discuss some new results about positive contact surgeries. In particular we will completely characterize when contact r surgery on a knot in the standard contact 3-sphere is symplectically fillable if r is in (0,1] and in general give obstructions to fillability in terms of the tau invariant of the knot. This is joint work with James Conway and Bulent Tosun. - [12:00-13:45] Lunch break
- [13:45-14:45]
*Liam Watson*(U de Sherbrooke), Bordered Floer homology via immersed curves, part 1The Heegaard Floer homology of a manifold with torus boundary can be expressed as a collection of immersed curves (possibly decorated with local systems). This provides a geometric structure theorem, interpreting the algebraic invariants that arise in bordered Floer homology. From this point of view, the Heegaard Floer homology of a closed manifold obtained by gluing manifolds (with boundary) along a torus may be recovered as the Lagrangian intersection Floer homology of the associated curves. In practice, this reduces gluing problems to simple minimal intersection counts. This pair of talks, which are part of a joint project with Hanselman, Rasmussen, and Watson, will set up this machinery and describe some of the applications that follow. - [15:00-16:00]
*Jonathan Hanselman*(Princeton U), Bordered Floer homology via immersed curves, part 2The Heegaard Floer homology of a manifold with torus boundary can be expressed as a collection of immersed curves (possibly decorated with local systems). This provides a geometric structure theorem, interpreting the algebraic invariants that arise in bordered Floer homology. From this point of view, the Heegaard Floer homology of a closed manifold obtained by gluing manifolds (with boundary) along a torus may be recovered as the Lagrangian intersection Floer homology of the associated curves. In practice, this reduces gluing problems to simple minimal intersection counts. This pair of talks, which are part of a joint project with Hanselman, Rasmussen, and Watson, will set up this machinery and describe some of the applications that follow. - [16:00-17:00] Tea

- Friday, Jan 5
- [07:00-08:00] Breakfast
- [08:00-09:00]
*Ian Zemke*(Princeton U), Heegaard Floer mixed invariants of mapping toriThe Seiberg Witten invariants of smooth closed 4-manifolds are powerful tools for studying 4-dimensional topology. The mixed invariants of 4-manifolds from Heegaard Floer homology are conjecturally equal to the Seiberg Witten invariants. In this talk we will describe how the graph TQFT for Heegaard Floer homology can be used to compute the Heegaard Floer mixed invariants of some 4-dimensional mapping tori in terms of Lefschetz numbers on HF^+. The computation is in terms of traces, cotraces, and a "broken path" graph cobordism. - [09:10-10:10]
*Akram Alishahi*(Columbia U), Bordered Floer homology and compressibilityIn this talk, we will describe how bordered Floer homology detects homologically essential compressing disks and how bordered-sutured Floer homology detects partly boundary parallel tangles and bridges. Lipshitz-Ozsvath-Thurston have a factoring algorithm for computing bordered Floer homology. If time permits, we will describe an extension of their algorithm to compute bordered-sutured Floer homology. This is a joint work with Robert Lipshitz. - [10:20-11:20]
*Robert Lipshitz*(U of Oregon), Bordered HF^- with torus boundary: formal structureWe will outline the formal structure of bordered HF^- with torus boundary and illustrate it with a few computations. This is joint work in progress with Peter Ozsvath and Dylan Thurston. - [11:20-13:00] Lunch break
- [13:00-14:00]
*Adam Levine*(Duke U), Concordance of knots in homology spheresEvery knot in the 3-sphere bounds a non-locally flat piecewise-linear (PL) disk in the 4-ball, but Akbulut showed in 1990 that the same is not true for knots in the boundary of an arbitrary contractible 4-manifold. We strengthen this result by showing that there exists a knot K in a homology sphere Y (which is the boundary of a contractible 4-manifold) such that K does not bound a PL disk in any homology 4-ball bounded by Y. In more recent work (joint with Jen Hom and Tye Lidman), we show that the group of knots in homology spheres modulo non-locally-flat PL concordance is infinitely generated and contains an infinite cyclic subgroup. - [14:10-15:10]
*Josh Greene*(Boston College), Fibered Simple KnotsJohn Luecke and I classified the simple knots in lens spaces that fiber. The answer takes an elementary number theoretic form, which I find satisfying if rather peculiar. I will focus on aspects of the proof that I have not elaborated on in previous talks. - [15:10-16:00] Tea

**Maps**

Google maps, campus maps, and a map with some restaurants marked.**Travel**

UCLA is located in the Westwood neighborhood of Los Angeles. The closest airport is LAX. To get from LAX to Westwood, the following transportation options are available.- Flyaway Westwood Shuttle ($10, by credit card only, travel time 25 to 45 minutes). Direct bus to Westwood, departing once an hour from the front of each LAX terminal.
- Culver City Bus Line 6 or 6-Rapid ($1, travel time 50 to 90 minutes). To catch this bus, take the shuttle at LAX to the City Bus Center. You can plan your Metro trip here.
- SuperShuttle (around $25). Door-to-door van service. While they make regular rounds at LAX to pick up passengers, it is best to make a reservation online or by telephone.
- Prime Time Shuttle. Similar to SuperShuttle, but may be a few dollars cheaper.
- Taxi (around $45).
- Uber or Lyft (around $20). Must be boarded from the Departure level at LAX. So you need to go from the Arrival (lower) level to the Departure (upper) level, which could be tricky depending on the Terminal.

**Accommodation**

The following hotels are within walking distance to UCLA (or offer free car service to campus). Unless otherwise mentioned, you need to book your accommodation yourself.- UCLA Guest House, 330 Charles E. Young Dr. East, Los Angeles CA 90095, (310) 825-2923. Rates starting at $187. On-campus hotel, limited on-site parking, laundry facility, free wireless Internet, and continental breakfast.
- UCLA Tiverton House, 900 Tiverton Ave., Los Angeles CA 90024, 310-794-0151. Rates starting at $175. Free parking, continental breakfast, community kitchen, recreation room, fitness center, business center, guest library, wireless Internet in lounges, laundry room.
- Hilgard House, 927 Hilgard Ave., Los Angeles CA 90024, (310) 208-3945, (800) 826-3934. Rates starting at $198. Free parking, wireless Internet, and continental breakfast.
- Royal Palace Westwood, 1052 Tiverton Ave., Los Angeles CA 90024, (310) 208-6677. Rates starting at $179. Free parking, free wireless Internet, continental breakfast, discounts for nearby attractions.
- Claremont Hotel, 1044 Tiverton Ave., Los Angeles California 90024, (310) 208-5957. (Might be closed in January for renovation.) Rates starting at $90. Economy class hotel amenities. Complimentary coffee & tea, wireless Internet, and use of refrigerator and microwave oven. Parking lot nearby starting at $6.50 daily.
- Luskin Conference Center, 425 Westwood Plaza, Los Angeles, CA 90095, (855) 522-8252. Rates starting at $279. Brand new on-campus hotel. Restaurant, fitness room, free wi-fi.
- Hotel Palomar, 10740 Wilshire Blvd., Los Angeles CA 90024, (310) 475-8711, (800) 472-8556. Rates starting at $320. Restaurant, pool, 24-hour fitness room, shuttle to/from UCLA, pet-friendly, day-care center for kids, same day laundry/dry-cleaning service. Free wireless Internet.
- Hotel Angeleno, 170 N. Church Lane, (310) 476-6411. Rates starting at $209. It is 3 miles from UCLA, but offers a free car service to campus.

**Food**

Here is list of campus eateries. Westwood Village (15-20 min walk south of campus) has a variety of restaurants; see for example Yelp.