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.
[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.
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.