Khovanov homology and knot Floer homology are two knot invariants which are defined using very different techniques, with Khovanov homology having its roots in representation theory and knot Floer homology in symplectic geometry. However, they seem to contain a lot of the same topological data about knots. Rasmussen conjectured that this similarity stems from a spectral sequence from Khovanov homology to knot Floer homology. In this talk I will give a construction of this spectral sequence. The construction utilizes a recently defined knot homology theory HFK_2 which provides a framework in which the two theories can be related.

Knot invariants can be defined using Legendrian isotopy invariants of the knot conormal. There are two types of invariants raised in this way: one is the knot contact differential graded algebra together with augmentations associated to this dga, and the other one is the category of simple sheaves microsupported along the knot conormal. The Nadler-Zaslow correspondence suggests a connection between the two types of invariants. In this talk, I will manifest an explicit map between augmentations and simple sheaves.

In his fundamental paper on the projective line minus three points, Deligne constructed certain extensions of mixed Tate motives arising from the fundamental group of the projective line minus three points. Since then, motivic structures on homotopy groups have been studied by many authors. In this talk, we will construct a motivic structure on the (nilpotent completion of) *n*-th homotopy group of *P**n* minus *n*+2 hyperplanes in general position.

Which 3-manifolds smoothly embed in the 4-sphere? This seemingly simple question turns out to be rather subtle. Using Donaldson's theorem, we derive strong restrictions to embedding a Seifert fibered space over an orientable base surface, which in particular gives a complete classification when e > k/2, where k is the number of exceptional fibers and e is the normalized central weight. Our results point towards an interesting conjecture which I'll discuss. This is joint work with Duncan McCoy.

We will discuss recent work on Legendrian and transverse links in universally tight contact lens spaces. There is a relationship between such links and the Berge Conjecture. The surgery duals to Berge knots in the corresponding lens space are all bindings of a rational open book decomposition. We will discuss whether these Berge duals support the universally tight contact structure on that lens space.

In this talk, I will report some recent joint work with Sun-Yung Alice Chang in which we partially generalize the Aleksandrov-Fenchel inequalities for quermassintegrals from convex domains in the Euclidean space to a class of non-convex domains.

To each three-component link in the 3-dimensional sphere we associate a characteristic map from the 3-torus to the 2-sphere, and establish a correspondence between the pairwise and Milnor triple linking numbers of the link and the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two- component link is the degree of its associated Gauss map from the 2- torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere.

In recent years, there have been more and more connections between cubic 4folds and hyperkahler manifolds. The first instance of this was noticed by Beauville-Donagi, who showed that the Fano varieties of lines on a cubic 4folds X is holomorphic symplectic. This talk aims to describe another instance of this phenomenon, which is carried out in joint work with R. Laza and C. Voisin: given a general cubic 4fold X, Donagi and Markman showed in 1995 that the family of intermediate Jacobians of smooth hyperplane sections of X has a holomorphic symplectic form. I will present a proof of this conjecture, which uses relative compactified Prym varieties.

After reviewing the proof of the Riemannian positive mass theorem in dimensions less than 8, I will briefly explain how to generalize the proof to slices of spacetime that are not time-symmetric. The basic idea is to replace minimal hypersurfaces by marginally outer-trapped hypersurfaces, and the main difficulty is to avoid using any minimization process. This is joint work with Eichmair, Huang, and Schoen.

I will describe an upcoming paper with Goncalo Oliveira investigating the properties of Yang-Mills flow on base manifolds with restricted holonomy, generalizing known results from the 4-d and Kahler cases. We show that finite-time blowup is governed by the F^7 component of the curvature in the G_2 and Spin(7) cases, and by the appropriate curvature component in the remaining cases on Berger's list. Assuming that this component remains bounded along the flow, we show that the infinite-time bubbling set is calibrated by the defining (n-4)-form.

Heegaard Floer homology is a powerful 3-manifold invariant developed by Oszvath and Szabo. Bordered Heegaard Floer homology is an extension of the Heegaard Floer theory to 3-manifolds with boundary, which lets us compute the "hat" version of Heegaard Floer for complicated manifolds by cutting them into simpler pieces. Graph manifolds are an important class of 3-manifolds which decompose in a particularly nice way; all the components of their JSJ decomposition are Seifert fibered. The majority of the talk will be devoted to introducing the terms above, starting with a brief overview of Heegaard Floer homology. At the end we see how to use bordered Heegaard Floer to compute HF-hat for any graph manifold.

Thurston's geometrization conjecture and its subsequent proof for Haken manifolds distinguish knots in S^3 by the geometries in the complement of the knots. While the definition of alternating knots make use of nice knot diagrams, Knot Floer homology, a knot invariant toolbox, defined by Ozsvath-Szabo and Rasumussen, generalizes the definition of alternating knots in the context of knot Floer homology and defines family of quasi-alternating knots which contains all alternating knots. Using Lipshitz-Ozsvath-Thurston's bordered Floer homology, we prove a partial affirmation of a folklore conjecture in knot Floer theory, which bridges these two viewpoints of looking at knots.

We construct Legendrian submanifolds with arbitrarily many topologically distinct Lagrangian fillings, thereby (secretly) answering a question about intersections of complex curves with the 4-ball asked by Boileau and Fourrier. These fillings are then combined with a TQFT-like theory for Lagrangian cobordisms between Legendrian submanifolds to produce interesting consequences for some non-classical invariants of the Legendrian submanifolds with topologically distinct fillings. Various parts of this talk are joint work with Traynor, Bourgeois-Traynor, and Cao-Gallup-Hayden.

We study certain linear representations of the knot group that induce augmentations of knot contact homology. This perspective on augmentations enhances our understanding of the relationship between the augmentation polynomial and the $A$-polynomial of the knot. For example, we show that for 2-bridge knots the two polynomials agree. We also show this is never the case for (non-2-bridge) torus knots, nor for a family of 3-bridge pretzel knots. Moreover, we show that these representations provide a lower bound on the meridional rank of the knot. As a consequence, our results give another proof that torus knots have meridional rank and bridge number that are equal.

In this talk I will review the construction of the Lie algebra associated to the mapping class group of a (possibly decorated) surface and explain how this generalizes the Lie algebra associated to the pure braid group. I will also explain the analogue of the KZ-equation in the mapping class group case. In the second talk I will discuss filtrations of this Lie algebra associated to curve systems on the surface and their relation to handlebody groups.

The max-flow min-cut theorem, traditionally applied to problems of maximizing the flow of commodities along a network (e.g. oils in pipelines) and minimizing the costs of disrupting networks (e.g. damn construction), has found recent applications in information processing. In this talk, I will recast and generalize max-flow min-cut as a form of twisted Poincare Duality for spacetimes and more singular "directed spaces." Flows correspond to the top-dimensional homology, taking local coefficients and values in a sheaves of semigroups, on directed spaces. Cuts correspond to certain distinguished sections of a dualizing sheaf. Thus max-flow min-cut dualities extend to higher dimensional analogues of flows, higher dimensional analogues of directed graphs (e.g. dynamical systems), and constraints more complicated than upper bounds. I will describe the formal result, including a construction of directed sheaf homology, and some real-world applications.

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.

Homology cobordisms are a special type of manifold which are relevant to a variety of areas in geometric topology, including knot theory and triangulability. We study the behavior of a variety of invariants under a particular family of four-dimensional homology cobordisms which naturally arise from Stein manifolds. This is joint work with Ali Daemi, Jen Hom, Shea Vela-Vick, and Mike Wong.

I will describe joint work with Gavin Ball where we classify certain G2-Instantons on Aloff-Wallach spaces. This classification can be used to test ideas and explicitly observe various interesting phenomena. For instance, we can: (1) Vary the underlying structure and find out what happens to the G2-instantons along the way; (2) Distinguish certain G2-structures (called nearly parallel) using G2-Instantons; (3) Find G2-Instantons, with respect to these structures, which are not absolute minima of the Yang-Mills functional.

Using a level set formulation and elliptic regularization we define a notion of weak solutions of nonlinear mean curvature flow with Neumann boundary condition. We then outline the proof of an existence result for the weak level set flow. Finally, we discuss some geometric applications of this flow.

Knot concordance can be regarded as the study of knots as boundaries of surfaces embedded in spaces of dimension 4. Specifically, two knots K_0 and K_1 are said to be smoothly concordant if there is a smooth embedding of the 2--dimensional annulus S^1 × [0, 1] into the 4--dimensional cylinder S^3 × [0, 1] that restricts to the given knots at each end. Smooth concordance is an equivalence relation, and the set of smooth concordance classes of knots, C, is an abelian group with connected sum as the binary operation. The algebraic structure of C, the concordance class of the unknot, and the set of knots that are topologically slice but not smoothly slice are much studied objects in low-dimensional topology. Gauge theoretical results on the nonexistence of certain definite smooth 4-manifolds can be used to better understand these objects. In this talk I will explain how the study of anti-self dual connections on 4--manifolds can be used to shown that (1) the group of topologically slice knots up to smooth concordance contains a subgroup isomorphic to Z^\infty, and (2) satellite operations that are similar to cables are not homomorphisms on C.

Kronheimer and Mrowka recently used monopole Floer homology to define an invariant of sutured manifolds, following work of Juhász in Heegaard Floer homology. Contact 3-manifolds with boundary are natural examples of such manifolds. In this talk, I will construct an invariant of a contact structure as an element of the associated sutured monopole homology group. I will discuss several interesting properties of this invariant, including gluing maps which are analogous to the Heegaard Floer sutured gluing maps of Honda, Kazez, and Matić and applications to Legendrian knots. This is joint work with John Baldwin.

We will start by discussing a flow of isometric $G_2$ structures. We consider the negative gradient flow of the energy functional restricted to the class of $G_2$ structures inducing a given Riemannian metric. We will discuss various analytic aspects of the flow including global and local derivative estimates, a compactness theorem and a monotonicity formula for the solutions. After defining an entropy functional we will prove that low entropy initial data lead to solutions that exist for all time and converge smoothly to a $G_2$ structure with divergence free torsion. We will also discuss finite time singularities and the singular set of the solutions. Finally, we will discuss the isometric flow "coupled” with the Ricci flow of the underlying metric, which again is a flow of $G_2$ structures, and discuss some of its properties. This is a based on two separate joint works with Panagiotis Gianniotis (University of Athens) and Spiro Karigiannis (University of Waterloo).

Heegaard Floer homology, a kind of (3+1)-dimensional field theory, associates chain complexes to 3-manifolds and chain maps to 4-manifolds with boundary. These complexes and maps are defined by counting holomorphic curves, and are hard to compute. Bordered Floer homology extends Heegaard Floer theory one dimension lower, assigning algebras to surfaces and differential modules to 3-manifolds with (parameterized) boundary. After introducing the bordered Floer framework, we will illustrate its construction in a toy case where it is explicit and combinatorial: planar grid diagrams. This is joint work with Peter Ozsvath and Dylan Thurston.

Kapustin and Witten introduced interesting boundary value problems for magnetic monopoles on a Riemann surface times an interval. They described the moduli space of such solutions in terms of Hecke modifications of holomorphic bundles over the Riemann surface. I will explain this and prove existence and uniqueness for such monopoles.

I will discuss recent joint work with Ben Elias in which we introduce a theory of diagonalization of functors. Our main application is the diagonalization of the the Rouquier complex associated to full-twist braid, acting on the category of Soergel bimodules. The ``eigenprojections'' yield categorified Young symmetrizers, which are related to the flag Hilbert scheme by a beautiful recent conjecture of Gorsky-Rasmussen. Finally, I will mention a relationship with stable homology of torus links, which was recently investigated by myself and Michael Abel.

TBA

It is known that, on closed manifolds, Einstein metrics of negative scalar curvature are critical points of the usual volume functional constrained to the space of metrics of constant scalar curvature. In this talk, I will discuss how this variational characterization of Einstein metrics can be localized to compact manifolds with boundary. I will present the critical point equation and focus on geometric properties of its general solutions. In particular, when a solution has zero scalar curvature and its boundary can be isometrically embedded into the Euclidean space, I will show that the volume of this critical metric is always greater than or equal to the Euclidean volume enclosed by the image of the isometric embedding and two volumes are the same if and only if the critical metric is isometric to the Euclidean metric on a standard round ball. This is a joint work with Prof. Luen-Fai Tam.