It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties of a contact structure are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact (r) surgery is symplectically/Stein fillable for r in (0,1]. This is joint work with James Conway and John Etnyre.

The talk presents an explicit classification of the ALE Ricci flat K\"ahler surfaces, generalizing previous classification results of Kronheimer. The manifolds are related to a special class of deformations of quotient singularities of type $\mathbb C^2/G$, with $G$ a finite subgroup of $U(2)$. I will also explain the relation with the Tian-Yau construction of complete Ricci flat Kahler manifolds.

I will describe how representation theory and the geometry of homogeneous spaces may be used to determine the degenerations of a given Hodge structure. This work is part of a larger program to understand the degenerations of a smooth variety that is being pursued, in various subset of collaboration, by Mark Green, Phillip Griffiths, Matt Kerr, Greg Pearlstein and me.

We establish an analog of the sphere theorem in the setting of contact geometry. Specifically, if a given three dimensional contact manifold admits a compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure is tight. The proof is a blend of topological and geometric techniques. A necessary technical result is a lower bound for the radius of a tight ball in a contact 3-manifold. We will also discuss geometric conditions in dimension three for a contact structure to be universally tight in the nonpositive curvature setting. This is joint work with Rafal Komendarczyk and Patrick Massot.

Real-world data often consists of the superposition of multiple informative signals. Examples include an image of the night sky containing both stars and galaxies; a communications message with impulsive noise; and a low rank matrix obscured by sparse corruptions. Demixing is the problem of determining the constituent signals from the observed superposition. Convex optimization offers a natural framework for solving demixing problems. This talk describes a geometric characterization of success in this framework that, when coupled with a natural incoherence model, leads into the realm of random geometry. A powerful result from spherical integral geometry then provides an exact formula for the probability that the convex demixing approach succeeds. Analysis of this formula reveals sharp phase transitions between success and failure for a large class of demixing methods. We apply our results to demixing the superposition of sparse vectors in random bases, a stylized robust communications protocol, and determining a low rank matrix corrupted by a matrix that is sparse in a random basis. Empirical results closely match our theoretical bounds. Joint work with Joel A. Tropp.

I will describe joint work with E. Giroux in which we show that every Weinstein domain admits a Lefschetz fibration over the disk (that is, a singular fibration with Weinstein fibers and Morse singularities). We also prove an analogous result for Stein domains in the complex analytic setting. The main tool used to prove these results is Donaldson's quantitative transversality.

Suppose that (d_1, ..., d_n) is an n-tuple of integers satisfying sum_j d_j = 0. Eliashberg posed the problem of computing the class of the locus in the moduli space of n-pointed, genus g curves [C;x_1,...,x_n] where sum d_j x_j = 0 in the jacobian of C. In this talk I will give the solution and sketch the proof, which uses known facts about the structure of mapping class groups.

Embedded contact homology (ECH) is an invariant of a closed contact 3-manifold, but proving its invariance is not so straightforward: the only known proof (due to Taubes) is to show that it is isomorphic to monopole Floer homology, which only depends on the underlying manifold. Colin, Ghiggini, Honda, and Hutchings defined a version of ECH for contact 3-manifolds with boundary, which are naturally sutured manifolds, and conjectured that this is also an invariant of the underlying sutured manifold. In this talk I will show that sutured ECH is indeed an invariant and discuss exactly what kind of invariant it is. This is joint work with Cagatay Kutluhan.

We first review the relation between wrapped Floer cohomology of co-core disks after Lagrangian handle attachment and the Legendrian DGA of the corresponding attaching spheres. Then we discuss a generalization of this result to the partially wrapped setting where the Legendrian dga should be enriched with loop space coefficients, and describe several cases when explicit calculations are possible via parallel copies or local coefficient systems. We also discuss applications of these ideas to the topology of Lagrangian fillings of Legendrian submanifolds. The talk reports on joint work with Y. Lekili.

Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4-manifold X with the homology of S^1 x S^3. Specifically, we show that for any smoothly embedded 3-manifold Y representing a generator of H_3(X), a suitable version of the Heegaard Floer d invariant of Y, defined using twisted coefficients, is a diffeomorphism invariant of X. We show how this invariant can be used to obstruct embeddings of certain types of 3-manifolds, including those obtained as a connected sum of a rational homology 3-sphere and any number of copies of S^1 x S^2. We also give similar obstructions to embeddings in certain open 4-manifolds, including exotic R^4s. This is joint work with Danny Ruberman.

Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. In this talk, we discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme-Guizon, Przytycki and Rong. An explicit formula for the rank of the third chromatic homology group on the main diagonal is given and used to compute the corresponding Khovanov homology group and the fourth coefficient of the Jones polynomial for links with certain diagrams.