Motivated by G2-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed 3-sphere with three asymptotically cylindrical ends in X \times R^3, where X is an A2-type ALE hyperkähler manifold. We prove this conjecture by solving a real Monge-Ampère equation with singular right hand side. The method produces many other asymptotically cylindrical U(1)-invariant special Lagrangians in X \times R^2, where X arises from the Gibbons-Hawking construction. This is joint work in progress with Saman Habibi Esfahani.

The ADM mass of an isolated gravitational system is a geometric invariant measuring the total mass due to matter and other fields. In a previous work, we showed how to compute this invariant (in 3 spatial dimensions) by studying harmonic functions. Now I'll use this formula to consider the question: How flat is an asymptotically flat manifold with very little total mass? In the presence of a lower bound on Ricci curvature, we make progress on this question and confirm special cases of conjectures made by Ilmanen and Sormani.

Though the study of isoperimetric problems for Laplacian eigenvalues dates back to the 19th century, the subject has undergone a renaissance in recent decades, due in part to the discovery of connections with harmonic maps and minimal surfaces. By the combined work of several authors, we now know that unit-area metrics maximizing the first nonzero Laplace eigenvalue exist on any closed surface, and are realized by minimal surfaces in spheres. At the same time, work of Fraser-Schoen, Matthiesen-Petrides and others yields analogous results for the first eigenvalue of the Dirichlet-to-Neumann map on surfaces with boundary, with maximizing metrics induced by free boundary minimal immersions into Euclidean balls. In this talk, I'll describe a series of recent results characterizing the (perhaps surprising) asymptotic behavior of these free boundary minimal immersions (and associated Steklov-maximizing metrics) as the number of boundary components becomes large. (Based on joint work with Mikhail Karpukhin.)

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.

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).

To a Legendrian knot, one can associate an $A_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.

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.

A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes manifolds with pointwise quarter-pinched sectional curvatures and manifolds with positive curvature operator. We will give a survey of results on PIC and discuss recent joint work with J. Wolfson on fundamental groups of manifolds of PIC. The techniques used involve minimal surfaces.

If you take a surface in Euclidean space that is locally the graph of a C2 function, this induces a local coordinate system in which the metric tensor is merely C1. Geodesic flows are well defined when the metric tensor is C2, but there are lots of examples of metric tensors of class C^(2-epsilon) for which geodesics branch. Nevertheless, for the C2 surface mentioned above, the geodesic flow is well defined. This result has been noted several times. It has several proofs. One uses the fact that geodesic flows are well defined whenever the Ricci tensor is bounded. An important class of Gromov-Hausdorff limits of smooth Riemannian manifolds studied by Anderson and Cheeger puts a lower bound on the Ricci tensor (and the injectivity radius), and obtains a limiting manifold whose metric tensor is not quite C1. We will explore the question of whether the geodesic flow is well defined on such a limit, and also look at some other limits of smooth manifolds, with wilder behavior.

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.

The Chern-Simons invariants are 3-manifold invariants arising from representations of the fundamental group into a Lie group G. These have been well-studied for G = SU(2), but much less is known about them for more general G. In this talk, I will review the definition of these invariants and discuss results that extend to arbitrary compact G several well-known SU(2)-computations. These extensions all have the flavor of "if you know the invariants for SU(2), then you know the invariants for general compact G". This is joint work with Kevin Fournier.

The problem of when and how one symplectic manifold can be symplectically embedded into another is notoriously subtle, even when the spaces in question are relatively simple. Gromov's non-squeezing theorem and McDuff's Fibonacci staircase are examples of this phenomenon. One can interpret these results as realizing the principle that "variations of quantitative symplectic parameters alter the topology of symplectic embedding spaces." In this talk, we explain recent work (joint with Mihai Munteanu) showing that certain n-torus families of symplectic embeddings between 2n-d ellipsoids become homologically essential if certain quantitative invariants are close enough. We will also discuss work in progress in which we use similar methods to study Lagrangian embeddings.

A Calabi-Yau orbifold is locally modeled on C^n/G with G a finite subgroup of SU(n). If the singularity is isolated, then the crepant resolution (if it exists) is an ALE manifold, for which index-type results are well known. However, most of the time the singularity is not isolated, and for the corresponding crepant resolution there is no index theorem so far. In this talk, I present the first step towards obtaining such a result: I will introduce the class of iterated cone-edge singular manifolds and the corresponding quasi-asymptotically conical spaces (of which orbifolds and their resolutions of singularities are examples), and build-up the general set-up for studying Fredholm properties of geometrical elliptic operators on these spaces. This is joint work with Rafe Mazzeo.

The Beauville-Mukai integrable system is a well-known Lagrangian fibration, i.e., a holomorphic symplectic manifold fibred by Lagrangian complex tori. It is constructed by beginning with a complete linear system of curves on a K3 surface, and then taking the compactified relative Jacobian of the family of curves. One may ask whether other families of curves yield Lagrangian fibrations in this way. Markushevich showed that this is not the case in genus two: a Lagrangian fibration by Jacobians of genus two curves must be a Beauville-Mukai system. We generalize his result to genus three curves, and also to non-hyperelliptic curves of genus four and five.

Legendrian torus knots were classified by Etnyre and Honda. In particular, for any smooth torus knot we know the mountain range that lists all of its Legendrian representatives. I will discuss the classification of Legendrian torus links. In this classification, a natural realization question arises: what n-tuples of points on the mountain range of a (p,q)-torus knot can occur in a Legendrian (np, nq)-torus link? Another part of the classification is to understand how many different ways an n-tuple on the mountain range can be realized as an ordered link. In particular, for Legendrian representatives of an (np,nq)-torus link is it possible to do invariant preserving permutations of the components? This is joint work with Jennifer Dalton and John Etnyre.

The link between meromorphic connections on a Riemann surface and their monodromy is a very classical one, indeed so classical that it was the subject of one of Hilbert?s problems. The deformation theory of these connections, and when these deformations preserve the monodromy, is almost equally ancient. I will give an overview of some results in the area, some ancient, and some quite recent.

In the early 1980s Atiyah and Bott described a new approach to studying the cohomology of the moduli space of stable bundles: the equivariant Morse theory of the Yang-Mills functional. There are many other interesting moduli spaces that fit into a similar framework, however the catch is that the total space is singular, and it is not obvious how to construct the Morse theory of the appropriate functional. In this talk I will describe how to get around these difficulties for the moduli space of stable pairs, for which we prove a Kirwan surjectivity theorem and give a Morse-theoretic interpretation of the change in cohomology due to a flip. This builds upon earlier work with George Daskalopoulos, Jonathan Weitsman and Richard Wentworth for rank 2 Higgs bundles.

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.

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.

Every knot in S^3 bounds a PL disk in the four ball. But this is no longer true for knots in other three manifolds, as demonstrated first by Akbulut, who constructed a knot which does not bound any PL disk in a specific contractible four manifold. Then Levine showed that there exist knots that do not bound a PL disk in any homology four ball. What happens if we relax the condition of bounding PL disk to bounding a PL surface with some given genus? In the joint work with Hom and Stoffregen, we proved that for each n, there exists a knot K_n in an integer homology sphere that does not bound a PL surface of genus n in any homology four ball. The proof uses Heegaard Floer homology. More specifically, the obstruction comes from knot cobordism maps by Zemke and the construction uses recent filtered mapping cone formula for cables of the knot meridian.

I shall discuss a variational problem arising from the study of quasilocal energy in general relativity. Given a spacelike 2-surface in spacetime, the Euler-Lagrange equation for the quasilocal energy is the isometric embedding equation into the Minkowski space coupled with a fourth order nonlinear elliptic equation for the time function. This equation is important in that it gives the ground configuration in GR. In joint work with PoNing Chen and Shing-Tung Yau, we solved this system in the cases of large and small sphere limits.

I will discuss recent advances in the theory of Fano varieties over nonclosed fields (joint with B. Hassett, A. Kresch, and A. Pirutka).

The Reshetikhin-Turaev construction for the standard representation of the quantum group gl(1|1) sends tangles to C(q)-linear maps in such a way that a knot is sent to its Alexander polynomial. After a brief review of this construction, I will give an introduction to tangle Floer homology — a combinatorial generalization of knot Floer homology which sends tangles to (homotopy equivalence classes of) bigraded dg bimodules. Finally, I will discuss how to see tangle Floer homology as a categorification of the Reshetikhin-Turaev invariant. This is joint work with Alexander Ellis and Vera Vertesi.

In this talk, we describe how to obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere, that blow up along a great circle. These are (complete) constant scalar curvature metrics on the complement of S^1 inside S^m, m ≥ 5, that are conformal to the (incomplete) round metric and periodic in the sense of being invariant under a discrete group of conformal transformations. These solutions come from bifurcating branches of constant scalar curvature metrics on compact quotients of S^m \ S^1. This is a joint work with R. Bettiol (University of Notre Dame) and P. Piccione (USP-Brazil).

In this talk we will exhibit many new phenomena in the structure of Legendrian and transverse knots by giving a complete classification of all cables of the positive torus knots. We will also provide two structural theorems to ensure when cable of a Legendrian simple knot type is also Legendrian simple. Part of the results are joint work with John Etnyre and Douglas LaFountain

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.

We exhibit examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact 1-forms \lambda_1^* on rational homology spheres which admit irreducible solutions together with a version of the Selberg trace formula relating the spectrum of the Laplacian on coexact 1-forms with the volume and complex length spectrum of a hyperbolic three-manifold. Using these relationships, we also provide precise certified numerical bounds on \lambda_1^* for several hyperbolic rational homology spheres.

On a Fano manifold M we study the supremum of the possible t such that there is a Kähler metric in c_1(M) with Ricci curvature bounded below by t. We relate this to Aubin's continuity method for finding Kähler-Einstein metrics and we give bounds on it for certain manifolds.

I'll describe a new combinatorial method for computing the delta-graded knot Floer homology of a link in S^3. Our construction comes from iterating an unoriented skein exact triangle discovered by Manolescu, and yields a chain complex for knot Floer homology which is reminiscent of that of Khovanov homology, but is generated (roughly) by spanning trees of the black graph of the link. This is joint work with Adam Levine.

Starting with a surface with negative constant Gauss curvature in the Euclidean 3-space, it is possible to (nontrivially) generate an infinitude of such surfaces by solving ODE systems alone. This fact, established by Bäcklund in the late 19th century, is one of the earliest-known examples of a Bäcklund transformation. The past century has seen rich interplay between Bäcklund transformations, integrable systems and soliton theory. However, a classification of Bäcklund transformations remains largely unknown. In this talk, I will discuss some recent progress on the classification of certain types Bäcklund transformations.