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.

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.

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.

The first cohomology of a hyperbolic 3-manifold has two natural norms: the Thurston norm, which measure topological complexity of surfaces representing the dual homology class, and the harmonic norm, which is just the L^2 norm on the corresponding space of harmonic 1-forms. Bergeron-Sengun-Venkatesh recently showed that these two norms are closely related, at least when the injectivity radius is bounded below. Their work was motivated by the connection of the harmonic norm to the Ray-Singer analytic torsion and issues of torsion growth. After carefully introducing both norms and the connection to torsion growth, I will discuss new results that refine and clarify the precise relationship between them; one tool here will be a third norm based on least-area surfaces. This is joint work with Jeff Brock.

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.

I will discuss a conjecture of Conner and Raymond that any aspherical manifold whose fundamental group has center possesses a circle action, and put it into the context of earlier work and conjectures of Borel and others.

We introduce a new second order parabolic evolution equation where the speed is given by the reciprocal of the null mean curvature. This flow is a generalisation of inverse mean curvature flow and it is motivated by the study of black holes and mass/energy inequalities in general relativity. We present a theory of weak solutions using level-set methods and an appropriate variational principle, and outline a natural application of the flow as a variational approach to constructing marginally outer trapped surfaces (MOTS), which play the role of quasi-local black hole boundaries in general relativity.

I will introduce a product construction in contact topology for Legendrian submanifolds, focusing on products of Legendrian knots. I will then discuss ongoing work to compute a product formula for the Legendrian contact homology invariant and some of the geometric and analytic difficulties involved. In particular, I will describe Ekholm's Morse-theoretic approach to counting holomorphic curves and how to apply it to compute invariants of products of Legendrian knots.

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.

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, I will present some new topological obstructions for solving the Einstein equations (in Riemannian signature) on a large class of closed four-manifolds. Finally, I will show that complex-hyperbolic Einstein Dehn filling compactification cannot possibly performed in dimension four. This is in striking contrast with the real-hyperbolic case, and it answers (negatively) a fifteen years old question of Michael Anderson. If time permits, I will conclude with some tantalizing open problems both in dimension four and in higher dimensions. Part of this work is joint with M. Golla (Universit\’e de Nantes).

It is an interesting project guided by Tian's conjecture to use K\"ahler-Ricci flow with changing cohomology class in the study of general type manifold. The locally smooth convergence leaves quite some freedom for the global geometry. Meanwhile, volume form and scalar curvature have shown different behavior in infinite and finite time cases.

In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of any torus immersed in Euclidean three-space is at least 2\pi^2. In this talk we will describe a solution to the Willmore conjecture based on the min-max theory of minimal surfaces. This is joint work with Andre Neves (Imperial College, UK).

A 'correspondence' between two manifolds is a submanifold in the product. This generalizes the notion of a map (whose graph is a correspondence) ... and is of little use in general since the composition of correspondences, though naturally defined, can be highly singular.

Lagrangian correspondences between symplectic manifolds however are highly useful (and will be defined carefully). They were introduced by Weinstein in an attempt to build a symplectic category that has morphisms between any pair of symplectic manifolds (not just symplectomorphic pairs).

In joint work with Chris Woodward we define such a cateory, in which all Lagrangian correspondences are composable morphisms. We extend it to a 2-category by constructing a Floer homology for generalized Lagrangian correspondences. One of the applications is a general prescription for constructing topological invariants. We consider e.g. 3-manifolds or links as morphisms (cobordisms or tangles) in a topological category. In order to obtain a topological invariant from our generalized Floer homology, it suffices to

(i) decompose morphisms into simple morphisms (e.g. by cutting between critical levels of a Morse function)

(ii) associate to the objects and simple morphisms smooth symplectic manifolds and Lagrangian correspondences between them (e.g. using moduli spaces of bundles or representations)

(iii) check that the moves between different decompositions are associated to 'good' geometric composition of Lagrangian correspondences

Since the time of Gauss, geometers have been interested in the interplay between the intrinsic metric structure of hypersurfaces and their extrinsic geometry from the ambient space. For example, a result of Sacksteder tells us that if a complete hypersurface has non-negative sectional curvature, then its second fundamental form in Euclidean space must be positive semi-definite. In a joint work with Damin Wu, we study hypersurfaces under a much weaker curvature condition. We prove that closed hypersurfaces with non-negative scalar curvature must be weakly mean convex. The proof relies on a new geometric inequality which relates the scalar curvature and mean curvature of the hypersurface to the geometry of the level sets of a height function. This result is optimal in the sense that the scalar curvature cannot be replaced by other k-th mean curvatures. The result and argument have applications to the mean curvature flow, positive mass theorem, and rigidity theorems.

The central axis of the famous DNA double helix is often topologically constrained or even circular. The topology of this axis can influence which proteins interact with the underlying DNA. Subsequently, in all cells there are proteins whose primary function is to change the DNA axis topology -- for example converting a torus link into an unknot. Additionally, there are several protein families that change the axis topology as a by-product of their interaction with DNA. This talk will describe typical DNA conformations, and the families of proteins that change these conformations. I'll present a few examples illustrating how Dehn surgery and other low-dimensional topological methods have been useful in understanding certain DNA-protein interactions, and discuss the most common topological techniques used to attack these problems.

We discuss the following recent result of the speaker. Suppose a closed 3-manifold M has scalar curvature at least 1, and has nontrivial second homotopy group, and is not covered by the cylinder (S^2)*R. Then the pi_2-systole of M (i.e. the minimal area in the second homotopy group) is bounded by a constant that is approximately 5.44pi. If we include quotients of cylinder into consideration, then the best upper bound is weakened to 8_pi. This shows a topological gap in the pi_2-systolic inequality. We will discuss the ideas behind this theorem, as well as the proof using Huisken and Ilmanen’s weak inverse mean curvature flow.

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.

We introduce a construction that associates a Riemannian metric $g_F$ (called the \emph{Binet-Legendre} metric) to a given Finsler metric $F$ on a smooth manifold $M$. The transformation $F \mapsto g_F$ is $C^0$-stable and has good smoothness properties, in contrast to previously considered constructions. The Riemannian metric $g_F$ also behaves nicely under conformal or isometric transformations of the Finsler metric $F$ that makes it a powerful tool in Finsler geometry. We illustrate that by solving a number of named problems in Finsler geometry. In particular, we extend a classical result of Wang to all dimensions. We answer a question of Matsumoto about local conformal mapping between two Berwaldian spaces and use it to investigate essentially conformally Berwaldian manifolds. We describe all possible conformal self maps and all self similarities on a Finsler manifold, generalizing the famous result of Obata to Finslerian manifolds. We also classify all compact conformally flat Finsler manifolds. We solve a conjecture of Deng and Hou on locally symmetric Finsler spaces. We prove smoothness of isometries of Holder-continuous Finsler metrics. We construct new `easy to calculate' conformal and metric invariants of Finsler manifolds. The results are based on the papers arXiv:1104.1647, arXiv:1409.5611, arXiv:1408.6401, arXiv:1506.08935, arXiv:1406.2924 partially joint with M. Troyanov (EPF Lausanne) and Yu. Nikolayevsky (Melbourne)

I will discuss the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci curvature. This is an evolution equation which coincides with the Ricci flow if the initial metric is Kahler, and was first studied by M.Gill. I will describe the maximal existence time for the flow in terms of the initial data, and thendiscuss the behavior of the flow on complex surfaces and on some higher-dimensional manifolds. This is joint work with Ben Weinkove.

Contact geometry is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability. The associated one form is called a contact form and uniquely determines a vector field called the Reeb vector field on the manifold. I will explain how to make use of J-holomorphic curves to obtain a Floer theoretic contact invariant whose chain complex is generated by closed Reeb orbits. In particular, I will explain the pitfalls in defining contact homology and discuss my work which gives a rigorous construction of cylindrical contact homology via geometric methods. This talk will feature numerous graphics to acclimate people to the realm of contact geometry.

Legendrian knots in standard contact R3 have in addition to their topological knot type two classical invariants known as the Thurston-Bennequin and rotation numbers. Over the past decade several invariants have been developed which are capable of distinguishing between knots with identical classical invariants. The purpose of this talk is to describe interesting relationships between some of these new invariants. Major players in this talk are the Chekanov-Eliashberg DGA (Legendrian contact homology) and related objects, as well as combinatorial structures on front diagrams and homological invariants arising from the theory of generating families (due to Chekanov-Pushkar, Fuchs, and Traynor). The main new result (joint with Fuchs) is that, when a Legendrian knot is defined by a generating family, homology groups obtained by linearizing the Chekanov-Eliashberg DGA are isomorphic to the homology of a pair of spaces associated with the generating family.

We consider the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed geometry in state space: the eigenvectors of the Jacobian of the flux are given. This is formulated as a system of algebraic-differential equations whose solution space is analyzed using Darboux and Cartan-K\"ahler theorems. It turns out that already the case with three equations is fairly complex. We give a complete list of possible scenarios for the general systems of two and three equations and for rich systems (i.e. when the given eigenvector fields are pairwise in involution) of arbitrary size. As an application we characterize conservative systems with the same eigencurves as compressible gas dynamics.

This is joint work with Kris Jenssen (Penn State University)

We discuss recent and ongoing work, joint with A. Little, on estimating the intrinsic dimensionality of data sets assumed to be sampled from a low-dimensional manifold in high dimensions and perturbed by high-dimensional noise. This work is motivated by several applications, including machine learning and dynamical systems, and by the limitations of existing algorithms. Our approach is based on a simple tool such as principal component analysis, used in a multiscale fashion, a strategy which has its roots in geometric measure theory. The theoretical analysis of the algorithm uses tools from random matrix theory and exploits concentration of measure phenomena in high-dimensions. The talk will have a tutorial flavour: no previous knowledge of what mentioned above will be required, and several toy examples to build intuition about some measure-geometric phenomena in high-dimensions will be presented.

The celebrated result of Lockhart-Mcowen says that on a non-compact complete manifold, an elliptic operator (with proper asymptotic conditions) is Fredholm between weighted Sobolev spaces if and only if the weight is not an indicial root. We show that a proper weighted Sobolev-theory exists even when the weight is an indicial root. We also discuss some applications to singular $G_{2}-$instantons which converges to their tangent cones in polynomial rates.

The moduli space of non-abelian magnetic euclidean monopoles is known to be a smooth manifold of dimension $4k$, and carries a natural complete riemannian metric. Here $k$, a positive integer, is a topological invariant of the monopole, its magnetic charge. The metric is hyperKaehler, and in particular Ricci-flat, and this is one of the reasons why these moduli spaces are popular with geometers and physicists. In this talk, I shall explain a new approach to the analysis of monopole metrics and some new results about their asymptotic behaviour. This will be a report on joint work with Richard Melrose and Chris Kottke.

Motivated by the need to formulate and solve Plateau type problems in higher dimensions and codimensions, normal and integral currents were introduced by Federer and Fleming around 1960; their work was, to some extent a generalization of earlier work by DeGeorgi in codimension one as well as the work of Reifenberg in arbitrary codimensions. Since then a great deal of work has been done on the Plateau problem and related variational problems. This work has always been based on geometric measure theory. The so-called closure theorem for integral currents and the boundary rectifiability theorem are essential ingredients in all of this work; these theorems depend on the Besicovitch-Federer structure theory for set of finite Hausdorff measure in Euclidean space. More recently, in the work of Ambrosio and others, a useful theory of Sobolev spaces for functions with values in an arbitrary metric space has been developed and applied to a variety of problems. Ambrosio and Kirchheim have developed a theory of currents in metric spaces in which they are able to give geometrically appealing proofs of generalizations of the aforementioned closure and rectifiability theorems using some ideas of Almgren and DeGiorgi and avoiding the use of the Besicovitch-Federer structure theory. In this talk I will describe how they do it.