I will discuss some examples of minimal graphs with jump discontinuities in their boundaries. Robert Huff and I constructed these examples in response to a question of John Urbas: Is it possible for a minimal graph over a smooth annular domain to have an isolated jump discontinuity on the inner boundary component? I will also give a brief overview of the boundary consistency problem for Di Giorgi's generalized solutions of the minimal surface equation and discuss this question in that context. The construction of the examples uses the Weierstrass representation and the developing map introduced by Huff in the study of capillary problems.

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 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.

In this talk, I will discuss the Seiberg-Witten equations on finite volume Riemannian manifolds which are diffeomorphic to the product of two hyperbolic Riemann surfaces of finite topological type. Finally, using a Seiberg-Witten scalar curvature estimate I will present several results concerning the Riemannian geometry of these spaces.

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).

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.

The helicity of a vector field in R^3, an analog to linking number, measures the extent to which its flowlines coil and wrap around one another. Helicity turns out to be invariant under volume-preserving diffeomorphisms that are isotopic to the identity. Motivated by Bott-Taubes integration, we provide a new proof of this invariance using configuration spaces. We then present a new topological explanation for helicity, as a characteristic class. Among other results, this point of view allows us to completely characterize the diffeomorphisms under which helicity is invariant and give an explicit formula for the change in helicity under a diffeomorphism under which helicity is not invariant. (joint work with Jason Cantarella, U. of Georgia)

In this talk I will introduce some new ideas to the existence theory for a class of non-variational existence problems arising naturally in geometry and analysis. I will discuss some applications (and potential applications) to positive mass-type and Penrose-type theorems in general relativity.

An Einstein metric is a Riemannian metric of constant Ricci curvature. One of the central problems of modern Riemannian geometry is to determine which smooth compact manifolds admit Einstein metrics. This lecture will explain some recent results concerning the 4-dimensional case of the problem, and then compare and contrast these results with our current understanding of the problem in other dimensions.

Global singularity theory originates from problems in obstruction theory. Consider the following question: is there an immersion in a given homotopy class of maps between two smooth compact manifolds M and N? We can reformulate this question as "is the set of points, where a generic smooth map between M and N is not an immersion, empty"? This set is the simplest example of a singularity. Alternatively, we can ask a question whether the cohomology class of this set is 0 or not. Turns out, there is a universal polynomial depending only on the dimensions of M and N and on the type of singularity, that, when evaluated in the corresponding characteristic classes of M and N, computes the cohomology class of a singularity. This polynomial is called the Thom polynomial, and it is the central notion of singularity theory. In my talk I will give an introduction to singularity theory, define the classic Thom polynomial and talk about different approaches to its K-theoretic generalization.

On a compact 4-manifold, every self-dual connection and every anti-self-dual connection minimizes the Yang-Mills energy. In this talk, I will answer the converse question for compact homogeneous 4-manifolds. I will also survey related stability results in other dimensions.

In contact topology, invariants of Legendrian submanifolds in 1-jet spaces have been obtained through a variety of techniques. I will discuss how I am enriching one Morse-theoretic invariant, Generating Family Cohomology, to an A-infinity algebra by constructing product maps. The construction uses moduli spaces of Morse flow trees: spaces of intersecting gradient trajectories of functions whose critical points encode Reeb chords of the Legendrian submanifold. I will focus my talk on the construction of a 2-to-1 product and discuss how it lays the foundation for the A-infinity algebra.

The classification of compact Riemannian manifolds with positive or non-negative sectional curvature is a long-standing problem in Riemannian geometry. One successful approach has been the introduction of symmetries, and an important first case to understand is that of continuous abelian symmetries. In recent work with Escher, we obtained an equivariant diffeomorphism classification of closed, simply-connected non-negatively curved Riemannian manifolds admitting an isotropy-maximal torus action, with implications for the Maximal Symmetry Rank Conjecture for non-negatively curved manifolds. I will discuss joint work with Escher and Dong, that builds on this work to extend the classification to those manifolds admitting an almost isotropy-maximal action.

Let Y be a hypersurface in a 2n-dimensional holomorphic symplectic manifold X. The restriction $\sigma|_Y$ of the holomorphic symplectic form induces a rank one foliation on Y. If this "characteristic foliation" has compact leaves, then the space of leaves Y/F is a holomorphic symplectic manifold of dimension 2n-2. This construction also works when Y is a coisotropic submanifold of higher codimension, and is known as "coisotropic reduction". In this talk we will consider when the characteristic foliation has compact leaves, and look at some applications of coisotropic reduction.

In this talk we will describe some results about Betti, Hodge, and characteristic numbers of compact hyperkahler manifolds. In (complex) dimension four one can find universal bounds for all of these invariants (Beauville, Guan); in higher dimensions it is still possible to find some bounds. We also describe how these bounds are related to the question: are there finitely many hyperkahler manifolds in each dimension, up to deformation?

An oriented hypersurface in a hyperkaehler 4-manifold naturally inherits a coclosed coframing. Bryant showed that, in the real analytic case, any oriented 3-manifold with a coclosed coframing can always be locally �?thickened�? to a hyperkaehler 4-manifold, in an essentially unique way. This raises the natural question: when can these 3-manifolds with this structure arise as the boundary of a hyperkaehler 4-manifold? In particular, starting from a compact hyperkaehler 4-manifold with boundary, which deformations of the boundary structure can be extended to a hyperkaehler deformation of the interior? I will discuss recent progress on this problem, which is joint work with Joel Fine and Michael Singer.

The past thirty years have witnessed the birth of a beautiful array of approaches to the field of low dimensional topology, drawing on diverse tools from algebra, analysis, and combinatorics. One particular tool that has made a dramatic impact on the field is the Heegaard Floer theory of Ozsvath and Szabo. Defined 17 years ago, this theory has produced an encompassing package of invariants, which have significantly impacted the study of many areas of low dimensional topology, including Dehn surgery. In this talk, we will focus on two questions: a) which 3-manifolds do arise by Dehn surgery along a knot in the 3-sphere? b) what are all ways to obtain a fixed 3-manifold by Dehn surgery along a knot in the 3-sphere?

A closed G2-structure is a certain type of geometric structure on a 7-manifold M, given by a 'non-degenerate' closed 3-form. The local geometry of closed G2-structures is non-trivial, in contrast to the perhaps more familiar case of symplectic structures (where we instead have a non-degenerate closed 2-form). In particular, any closed G2-structure automatically induces a Riemannian metric on M. I will talk about a special class of closed G2-structures, those satisfying a further 'quadratic' condition. This is a second order PDE system first written down by Bryant that can be interpreted as a condition on the Ricci curvature of the induced metric. I will focus mainly on the case where the G2-structure is 'extremally Ricci-pinched', giving new examples and describing an unexpected relationship with maximal submanifolds in a certain negatively curved pseudo-Riemannian symmetric space.

Surgery-theoretic classifications fail for 4-manifolds because many 4-manifolds have second homology classes not representable by smoothly embedded spheres. Knot traces are the prototypical example of 4-manifolds with such classes. I�?ll give a flexible technique for constructing pairs of distinct knots with diffeomorphic traces. Using this construction, I will show that there are knot traces where the minimal genus smooth surface generating second homology is not the obvious one, resolving question 1.41 on the Kirby problem list. I will also use this construction to show that Conway knot does not bound a smooth disk in the four ball, which completes the classification of slice knots under 13 crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.

In this talk, I will discuss the geometry of finite volume complex hyperbolic surfaces and their compactifications. Finally, applications at the common edge between Riemannian and complex algebraic geometry are given.

Knot concordance concerns the classification of knots in the 3-sphere that occur as the boundaries of embedded disks in the 4-ball. Unlike in higher dimensions, one obtains vastly different results depending on whether the disks are required to be smoothly embedded or merely locally flat (i.e. continuously embedded with a topological normal bundle); many tools arising from gauge theory and symplectic geometry can be used to illustrate this distinction. After surveying some of the recent progress in this area, I will discuss the extension of these questions to knots in 3-manifolds other than S^3. I will show how to use invariants coming from Heegaard Floer homology to obstruct not only smoothly embedded disks but also non-locally-flat piecewise-linear disks; this answers questions from the 1970s posed by Akbulut and Matsumoto. I will also discuss more recent results (joint with Jennifer Hom and Tye Lidman) giving infinitely many knots that are distinct up to non-locally-flat piecewise-linear concordance.

In this talk, we will investigate existing Legendrian knot invariants and discuss new connections between the theory of generating families and the Chekanov-Eliashberg differential graded algebra (CE-DGA). The geometric origins of the CE-DGA are Floer theoretic in nature and come out of the Symplectic Field Theory developed by Eliashberg and Hofer. On the other hand, Legendrian invariants derived from the study of 1-parameter families of smooth functions (called generating families) are Morse theoretic in nature. In the last decade, connections have been found between the Legendrian invariants derived using these two methods. In this talk, I will try to provide a clearer picture of the relationship between generating families and the CE-DGA.

A quasi-local mass has been a long sought after quantity in general relativity. A recent candidate has been the Liu-Yau mass. One can show that the Liu-Yau mass of any two-surface is the maximum of the Brown-York energy for that two-surface. This means that it has significant disadvantages as a mass. It is much better interpreted as an energy and I will show one way of doing so. The Liu-Yau mass is especially interesting in spherical geometries, where mass and energy are indistinguishable. For a spherical two-surface, it equals the minimum of the amount of energy at rest that one needs to put inside the two-surface to generate the given surface geometry. Thus it gives interesting information about the interior, something no other mass or energy function does.

In this talk, I will describe some recent work carried out with Rafe Mazzeo and Yanir Rubinstein, regarding the existence and nature of solutions to the problem of K\"{a}hler-Einstein metrics of constant negative curvature with certain prescribed singularities along a divisor in a compact, complex manifold. Earlier work of Aubin and of Yau established for the smooth compact case that for negative curvature, there are no obstructions beyond the immediate observation that the first Chern class of $M$ must be positive. I will include a brief outline of the method of solution in the smooth case. Since the publication of these earlier works, study of the negative case has focused on extension to various noncompact settings. In the situation described here, we considered metrics with conical singularities along a divisor. The most prominent feature of these metrics is that they are incomplete. Solution of this problem became possible recently when Simon Donaldson achieved a breakthrough in the linear theory.

High-dimensional data can be difficult to analyze. Assume data are distributed on a low-dimensional manifold. The Vector Diffusion Mapping (VDM), introduced by Singer-Wu, is a non-linear dimension reduction technique and is shown robust to noise. It has applications in cryo-electron microscopy and image denoising and has potential application in time-frequency analysis. In this talk, I will present a theoretical analysis of the effectiveness of the VDM. Specifically, I will discuss parametrisation of the manifold and an embedding which is equivalent to the truncated VDM. In the differential geometry language, I use eigen-vector fields of the connection Laplacian operator to construct local coordinate charts that depend only on geometric properties of the manifold. Next, I use the coordinate charts to embed the entire manifold into a finite-dimensional Euclidean space. The proof of the results relies on solving the elliptic system and provide estimates for eigenvector fields and the heat kernel and their gradients.

We will discuss the existence and uniqueness of the foliation by stable spheres with constant mean curvature for asymptotically flat manifolds satisfying the Regge-Teitelboim condition at infinity. This work generalizes the earlier results of Huisken/Yau, Ye, and Metzger. We will also discuss the concept of the center of mass in general relativity.

TBA

We will discuss the results of several joint ongoing projects (with subsets of collaborators Pierre Albin, Hans Christianson, Colin Guillarmou, Jason Metcalfe, Laurent Thomann and Michael Taylor), which explore the existence, stability and dynamics of nonlinear bound states and quasimodes on manifolds of both positive and negative curvature with various symmetry properties.