The workshop focused on recent developments in contact geometry and
low-dimensional topology, with a particular emphasis on modern
homological invariants. The workshop was open to all; the following
captures most of the participants.
Speakers: Akram Alishahi, Irving Dai, Nathan Dowlin,
Chris Gerig, Kyle Hayden, Kristen Hendricks, Jennifer Hom, Jianfeng
Lin, Marco Marengon, Jo Nelson, Jacob Rooney, Xiaolin Danny Shi,
Matthew Stoffregen, Linh Truong, Michael Willis, Ian Zemke.
[09:30-11:00] Dai-Hom-Stoffregen-Truong Homology cobordism and knot
concordance The set of homology 3-spheres, up to
an equivalence relation called homology cobordism, forms
an abelian group. We show that this group contains an
infinite rank summand. The proof uses an algebraic
modification of the involutive Heegaard Floer package of
Hendricks-Manolescu and Hendricks-Manolescu-Zemke. We will
also describe an infinite family of integer-valued
concordance homomorphisms defined using knot Floer
homology. These invariants have topological applications
to concordance genus, concordance unknotting number, and
bridge index.
[11:00-13:00] Lunch break
[13:00-14:00] Hayden Knot trace invariants and exotic
4-manifolds We show that the concordance
invariant "nu" from knot Floer homology is a
diffeomorphism invariant of the "trace" of a knot K,
i.e. the 4-manifold obtained by attaching a 2-handle to
the 4-ball along K. We'll use this computable invariant to
distinguish exotic pairs of knot traces and other simple
4-manifolds. We'll also present examples which show that
the related concordance invariants "tau" and "epsilon" are
*not* knot trace invariants. This is joint work with Tom
Mark and Lisa Piccirillo.
[14:15-15:15] Zemke Surgery exact triangles in involutive
Heegaard Floer homologyWe will describe work in
progress to construct surgery exact triangles in Hendricks
and Manolescu's involutive Heegaard Floer homology. This
project is joint with K. Hendricks, J. Hom and
M. Stoffregen. Our model uses a conceptually simple model
for the involution. As an application, we obtain a "knot
surgery formula", which allows the local equivalence class
of the involutive Heegaard Floer homology of certain
3-manifolds to be computed.
[15:30-16:30] Hendricks Involutive bordered Floer
homologyWe give a bordered extension of the hat
version of involutive Heegaard Floer homology and use it
to give an algorithm to compute the invariant for general
3-manifolds. As an application, we prove that the hat
version of involutive HF satisfies (various versions of) a
surgery exact triangle. Most of this talk is joint with
R. Lipshitz (and relies on technical results first
recorded by R. Lipshitz, P. Ozsvath, and D. Thurston); a
small portion at the end is joint with J. Hom,
M. Stoffregen, and I. Zemke.
[16:30-17:30] Tea
Friday, Jan 3
[08:30-09:30] Breakfast
[09:30-11:00] Alishahi-Dowlin Holomorphic motivations for an
algebraically defined knot Floer type invariant
(Dowlin) I will define a link invariant HFK_2
which is isomorphic to (delta-graded) knot Floer
homology. Like knot Floer homology, HFK_2 is defined as a
Lagrangian Floer homology, so the differential is a count
of pseudo-holomorphic curves. By making certain
simplifications in the chain complex, I will provide a
guess for the homology HFK_2 which has a combinatorial
description. The result is an algebraically defined link
invariant which is conjecturally isomorphic to HFK_2. The
algebraically defined complex also comes equipped with a
cube filtration -- it turns out that the E_2 page of the
spectral sequence induced by this filtration is Khovanov
homology.An algebraic braid invariant vs
holomorphic invariants (Alishahi) I will
continue with describing a local approach to prove our
conjecture. First, I will sketch a tangle refinement of
our algebraically defined HFK_2, and then I will compare
it with the Ozsvath-Szabo's tangle
invariants. Specifically, our local invariants have the
structure of a DA bimodule, defined over an algebra
isomorphic to the Ozsvath-Szabo's algebra, and our DA
bimodule for any open braid is chain homotopic to their
braid invariant.
[11:00-13:00] Lunch break
[13:00-14:00] Nelson S^1-equivariant and nonequivariant contact
homologyI will discuss joint work with Hutchings
which constructs nonequivariant and a family Floer
equivariant version of contact homology. Both theories are
generated by two copies of each Reeb orbit over Z and
capture interesting torsion information. I will then
explain how one can recover the original cylindrical
theory proposed by Eliashberg-Givental-Hofer via our
construction.
[14:15-15:15] Gerig Probing smooth 4-manifolds with
near-symplectic formsMost 4-manifolds do not
admit symplectic forms, but most near-symplectic , certain
closed 2-forms which are symplectic outside of a
collection of circles. Just like the Seiberg-Witten (SW)
invariants, there are invariants in terms of J-holomorphic
curves that are compatible with the near-symplectic
form. Although the SW invariants don't apply to
(potentially exotic) 4-spheres, nor do these spheres admit
near-symplectic forms, there is still a way to bring in
near-symplectic techniques and various contact Floer
homologies.
[15:30-16:30] Rooney Cobordism maps in embedded contact
homology An outstanding problem in embedded
contact homology has been a self-contained construction of
cobordism maps via a count of holomorphic curves. We give
such a construction for contact manifolds with no elliptic
Reeb orbits up to a certain action. The maps are defined
by counting both curves in the cobordism and new objects
that we call ECH buildings.
[16:30-17:30] Tea
Saturday, Jan 4
[08:30-09:30] Breakfast
[09:30-11:00] Lin-Shi The geography problem of 4-manifolds:
10/8+4A fundamental problem in 4-dimensional
topology is the following geography question: "which
simply connected topological 4-manifolds admit a smooth
structure?" After the celebrated work of Kirby-Siebenmann,
Freedman, and Donaldson, the last uncharted territory of
this geography question is the "11/8-Conjecture". This
conjecture, proposed by Matsumoto, states that for any
smooth spin 4-manifold, the ratio of its second-Betti
number and signature is least 11/8. Furuta proved the
"10/8+2"-Theorem by studying the existence of certain
Pin(2)-equivariant stable maps between representation
spheres. In this talk, we will present a complete solution
to Furuta's problem by analyzing the Pin(2)-equivariant
Mahowald invariants. In particular, we improve Furuta's
result into a ''10/8+4''-Theorem. Furthermore, we show
that within the current existing framework, this is the
limit. This talk is based on joint work by Mike Hopkins,
Jianfeng Lin, XiaoLin Danny Shi and Zhouli Xu.
[11:15-12:45] Marengon-Willis A generalization of Rasmussen's invariant,
with applications to surfaces in some
four-manifolds Building on previous work of
Rozansky and Willis, we generalise Rasmussen's s-invariant
to connected sums of $S^1 \times S^2$. Such an invariant
can be computed by approximating the Khovanov-Lee complex
of a link in $\#^r S^1 \times S^2$ with that of
appropriate links in $S^3$. We use the approximation
result to compute the s-invariant of a family of links in
$S^3$ which seems otherwise inaccessible, and use this
computation to deduce an adjunction inequality for
null-homologous surfaces in a (punctured) connected sum of
$\bar{CP^2}$. This inequality has several consequences:
first, the s-invariant of a knot in the three-sphere does
not increase under the operation of adding a
null-homologous full twist. Second, the s-invariant cannot
be used to distinguish $S^4$ from homotopy 4-spheres
obtained by Gluck twist on $S^4$. We also prove a
connected sum formula for the s-invariant, improving a
previous result of Beliakova and Wehrli. We define two
s-invariants for links in $\#^r S^1 \times S^2$. One of
them gives a lower bound to the slice genus in $\natural^r
S^1 \times B^3$ and the other one to the slice genus in
$\natural^r D^2 \times S^2$ . Lastly, we give a
combinatorial proof of the slice Bennequin inequality in
$\#^r S^1 \times S^2$. This is joint work of the speakers
with Ciprian Manolescu and Sucharit Sarkar.
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.
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.
Taxi (around $50), Uber or Lyft (around $20, surge pricing
around $50). Must be boarded from designated pick-up lot
called LAXit, which is walkable from Terminals 1,2,7,8;
otherwise, there are green shuttles between the terminals and
the LAXit lot.
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.