Lectures in Monroe Hall, room 110 
3:00  3:50pm 
David Gay (Georgia) 
Reflections on the combinatorics of trisection diagrams
Trisection diagrams are a special class of Heegaard triples, i.e. surfaces with three collections of simple closed curves. They are used to describe 4manifolds. Are they any better than other descriptions of 4manifolds, e.g. Kirby diagrams? In principle, the answer is "no" because you can go back and forth between the two types of descriptions. But in practice trisection diagrams might be a little better organized, at least in terms of posing interesting questions and giving us some reasonable complexity measures with which to organize our descriptions. In particular, interesting questions about the combinatorics of curves on surfaces arise. I'll discuss some simple observations and formulate some (probably hard) questions. 
Tea/coffee 
4:10  5:00 
Daniel Ruberman (Brandeis) 
Positive scalar curvature and 4manifolds
I will discuss some work from the last few years giving obstructions to the existence of a metric of positive scalar curvature on a smooth 4manifold, and to homotopies through such metrics. Obstructions to existence come from the interplay between SeibergWitten theory and index theory; obstructions to homotopy come from index theory on endperiodic manifolds. I will describe recent work with Kazaras and Saveliev showing that these homotopy invariants are in fact bordism invariants. 
Thursday, December 13 
Morning lectures: Monroe Hall, room 130 
Tea, coffee and bagels starting at 8:30am 
9:00  9:50am 
John Baldwin (Boston College) 
Contact structures, instantons, and \(SU(2)\) representations
A strengthening of the Poincare Conjecture asks whether the fundamental group of every closed 3manifold which is not the 3sphere admits a nontrivial homomorphism to \(SU(2)\). With that as motivation, I'll describe a connection between Stein fillings of a 3manifold and \(SU(2)\) representations of its fundamental group, coming from instanton Floer homology. This connection can be used to prove the existence of nontrivial \(SU(2)\) representations for all Seifert fibered spaces (reproducing a result of Fintushel and Stern) and much more! I'll end with a discussion of how these techniques may also be useful in computing instanton Floer homology (and in proving that it agrees with other Floer homology theories in some cases). This work is joint with Steven Sivek. 
10:00  10:30 
Gabriel Islambouli (University of Virginia) 
Stabilization trees of Heegaard splittings and Trisections
We give a survey of known results regarding the structure of Heegaard splittings under stabilization as well as discuss some recent work giving insights into its relation to the tree of stabilizations of trisections. 
Tea/coffee 
10:45  11:15 
Adam Saltz (Georgia) 
Knotted surfaces, bridge trisections, and link homology
Bridge trisections are a new perspective on knotted surfaces in \(S^4\) which allow us to apply some techniques from classical knot theory. I'll describe a new invariant of knotted surfaces derived from bridge trisections and link homology (e.g. Khovanov homology). 
11:25  12:15 
Maggie Miller (Princeton) 
Fibering ribbon disk complements
I will construct singular fibrations on \(3\)manifolds with boundary, and discuss when they trace smooth fibrations (over \(S^1\)) of a \(4\)manifold \(X^4\), with particular emphasis on when \(X^4\) is the complement of a ribbon disk in \(B^4\) whose boundary is a fibered knot in \(S^3\). This is related to a question of Casson and Gordon. In particular, we can constructively show that every ribbon disk with fibered boundary and exactly two minima is itself fibered, and explicitly describe how the fibers are embedded into \(B^4\). 
Lunch break 
Afternoon lectures: Monroe Hall, room 110 
2:00  2:50 
Peter LambertCole (Georgia Tech) 
Bridge trisections and the Thom conjecture
The classical degreegenus formula computes the genus of a nonsingular algebraic curve in the complex projective plane. The wellknown Thom conjecture posits that this is a lower bound on the genus of smoothly embedded, oriented and connected surface in \({\mathbb C}P^2\). The conjecture was first proved twentyfive years ago by Kronheimer and Mrowka, using SeibergWitten invariants. In this talk, we will describe a new proof of the conjecture that combines contact geometry with the novel theory of bridge trisections of knotted surfaces. Notably, the proof completely avoids any gauge theory or pseudoholomorphic curve techniques. 
3:00  3:30 
William Olsen (Georgia) 
Trisecting Ozsváth Szabó fourmanifold invariants
Given a closed, connected, oriented, smooth fourmanifold \(X\) with \(b_{2}^{+}(X) >1\), Ozsváth and Szabó construct powerful invariants via a TQFT strategy known as Heegaard Floer homology. One popular approach to computing these invariants is the following twostep procedure: first, break apart \(X\) into two suitable pieces \(X = X_{1} \cup X_{2}\), and use the TQFT structure alluded to above to produce relative invariants associated to each piece \(X_{1}, X_{2}\); and second, use the internal structure of the Heegaard Floer homology groups to produce an invariant from the relative invariants computed in the previous step. In this lecture, we demonstrate how one can use relative trisection diagrams to compute the relative invariants associated to each of the pieces \(X_{1}\) and \(X_{2}\).

Tea/coffee 
4:10  5:00 
Akram Alishahi (Columbia) 
Cobordism maps for tangle Floer homology
Tangles are generalizations of classical oriented tangles such that they are in onetoone correspondence with sutured manifolds. In this talk, we first define tangles, cobordisms between them and outline the formal construction of tangle Floer homology. Then, we will discuss the construction of TQFT type cobordism maps for cobordisms between tangles, and review some applications. This is joint work with Eaman Eftekhary. 
Conference banquet 6:30pm at Afghan Kabob, 400 Emmet St. 
Friday, December 14 
Lectures in Monroe Hall, room 110 
Tea, coffee and bagels starting at 8:30am 
9:00  9:50am 
Sergei Gukov (Cal Tech) 
Hidden Algebraic Structures in Topology
To be announced.

10:00  10:30
 Miriam Kuzbary (Rice) 
A new concordance invariant of knots in sums of \(S^2 \times S^1\)
Milnor invariants of links in the 3sphere capture fundamental information about links up to concordance and can be
thought of as higher order linking numbers. In the early 80s, Turaev and Porter independently proved their longconjectured correspondence with Massey products of the link complement and in 1990, Tim Cochran introduced a beautiful construction to compute them
using intersection theory. Using this perspective, I have generalized these invariants to nullhomologous
knots K inside certain 3manifolds M with free homology groups. This
new invariant, called the Dwyer number of a knot, can be computed in many cases and provides the weight of the first nonvanishing Massey product in the knot complement in M. Moreover, the Dwyer number detects knots K in M bounding smoothly embedded disks
in specific 4manifolds with boundary M which are not concordant to the unknot which motivated the definition of a new link concordance group using the knotification construction of
Ozsváth and Szabó.
Parts of this talk are joint work with Matthew Hedden. 
Tea/coffee 
10:45  11:15 
Irving Dai (Princeton) 
Involutive Floer Homology and Applications to the Homology Cobordism Group
In this talk, we discuss some recent applications of
involutive Heegaard Floer homology (defined by Hendricks and
Manolescu) to the homology cobordism group. We establish some
nontorsion results and show that the homology cobordism group
admits an infiniterank summand. This is joint work with Jennifer
Hom, Matthew Stoffregen, and Linh Truong. 
11:25  12:15 
Juanita Pinzón Caicedo (NC State) 
Satellites of Infinite Rank in Concordance
Oriented knots are said to be concordant if they cobound an embedded cylinder in the interval times the 3sphere. This defines an equivalence relation under which the set of knots becomes an abelian group with the connected sum operation. The importance of this group lies in its strong connection with the study of 4manifolds. Indeed, many questions pertaining to 4manifolds with small topology (like the 4sphere) can be addressed in terms of concordance. A powerful tool for studying the algebraic structure of this group comes from satellite operations or the process of tying a given knot \(P\) along another knot \(K\) to produce a third knot \(P(K)\). In the talk I will describe how to use \(SO(3)\) gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank subgroup of the smooth concordance group.

Lunch break 
2:00  2:50 
Tye Lidman (NC State) 
Spineless fourmanifolds
Given two homotopy equivalent manifolds with different dimensions, it is natural to ask if the smaller one embeds in the larger one. We will discuss this problem in the case of fourmanifolds
homotopy equivalent to surfaces. This is joint work with Adam Levine.

Tea/coffee 
3:30  5:00 
Open problems and discussion 