We define an explicit quasi-local mass functional which is nondecreasing along all foliations of a null cone (satisfying a convexity assumption). We use this new functional to prove the Null Penrose Conjecture under fairly generic conditions.

A central challenge in soft matter and materials science is the microscopic engineering of functional materials. Incorporating anisotropy here is of general interest, for example in actin networks, clay platelets, and polymer nanocomposites where geometry, ordering, and kinetics all play important roles in determining their properties. Nevertheless, forming a general connection between microstructure and macroscopic properties is not trivial. Here, I focus on the self-assembly and mechanics of colloidal materials with an emphasis on how shape anisotropy and interaction potential can be used to guide their design. I will first discuss the relevance of the physical interactions that give rise to a general class of colloidal gels, followed by how shape anisotropy can introduce metastable gelled states. I will also show that the slowed rotational dynamics caused by surface roughness and friction can lead to enhanced shear thickening that is not seen with smooth colloids. These results collectively show that particle-level interactions provide a powerful means to design soft materials at multiple length scales.

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.

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.

Based on the work of Gross and Sheng-Zuo, Friedman and Laza have classified variations of real Hodge structure of Calabi-Yau type over Hermitian symmetric domains. In particular, over every irreducible Hermitian symmetric domain there exists a canonical variation of real Hodge structure of Calabi-Yau type. In this talk, we wil review Friedman and Laza’s classification. A natural question to ask is whether the canonical Hermitian variations of Hodge structure of Calabi-Yau type come from families of Calabi-Yau manifolds (geometric realization). In general, this is very difficult and is still open for small dimensional domains. We will discuss an intermediate question, namely does the canonical variations occur in algebraic geometry as sub-variations of Hodge structure of those coming from families of algebraic varieties (motivic realization). In particular, we will give motivic realizations for the canonical variations of Calabi-Yau type over irreducible tube domains of type A using abelian varieties of Weil type.

In the construction of HOMFLY-PT homology, one must start with a link presented as a braid closure. This restriction was expected by Khovanov and Rozansky to be required for the homology to be an isotopy invariant. In this talk, after reviewing the construction of the HOMFLY-PT polynomial and homology, we explore the consequences of dropping this requirement and allowing general link diagrams. We explicitly show that the Reidemeister IIb move (where the strands have opposite orientations) fails. Finally we will show that the Euler characteristic of this homology theory is a deformed version of the HOMFLY-PT polynomial which detects "braidlike" isotopy of tangles and links. This new polynomial agrees with the HOMFLY-PT polynomial on link diagrams which are presented as closed braid diagrams.

I will describe a partial compactification of the moduli space, M_k, of SU(2) magnetic monopoles on R^3, wherein monopoles of charge k decompose into widely separated `monopole clusters' of lower charge going off to infinity at comparable rates. The hyperkahler metric on M_k has a complete asymptotic expansion, the leading terms of which generalize the asymptotic metric discovered by Bielawski, Gibbons and Manton in the case that the monopoles are all widely separated. This is joint work with M. Singer, and is part of a larger work in progress with R. Melrose and K. Fritzsch to fully compactify the M_k as manifolds with corners and determine their L^2 cohomology.

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.

Period integrals of an algebraic variety are transcendental objects that describe, among other things, deformations of the variety. They were originally studied by Euler, Gauss and Riemann, who were interested in analytic continuation of these objects. In this lecture, we will discuss a number of problems on period integrals in connection with mirror symmetry and Calabi-Yau geometry. We will see how the theory of D-modules have led us to solutions and insights into some of these problems.

In a Calabi-Yau manifold, mean curvature flow--the downward gradient for the area functional--preserves the Lagrangian condition. Thus Lagrangian mean curvature flow suggests a way to find minimal Lagrangian submanifolds of a CY manifold, provided the flow lasts for all time. However, finite-time singularities are expected along the flow; in fact, ill-behaved singularities are generic in some sense. In this talk we will discuss two main results: one, that type I (mild) finite-time singularities can be predicted by looking the cohomology of the initial Lagrangian submanifold, and two, that type II (ill-behaved) singularities can be modeled as unions of special Lagrangian cones. We will also discuss what these results say about using mean curvature flow to understand the topology of Lagrangian submanifolds.

I will give a brief introduction to Hwang and Mok's program to study the geometry of uniruled projective manifolds via their varieties of minimal rational tangents (VMRT). The working seminar is motivated by the idea that there may be an analogous program to study variations of Hodge structure via the characteristic varieties introduced by Sheng and Zuo. As evidence for the proposed program's viability I will show how characteristic varieties may be used to characterize the families of Calabi-Yau manifolds that solve Gross's geometric realization problem for Hermitian symmetric domains.

CR geometry studies boundaries of domains in C^n and their generalizations. A central role is played by the Levi form L of a CR manifold M, which measures the failure of the CR bundle to be integrable, so that when L has a nontrivial kernel of constant rank, M is foliated by complex manifolds. If the local transverse structure to this foliation still determines a CR manifold N, then we say M is CR-straightenable, and the Tanaka-Chern-Moser classification of CR hypersurfaces with nondegenerate Levi form can be applied to N. It remains to classify those M for which L is degenerate and no such straightening exists. This was accomplished in dimension 5 by Ebenfelt, Isaev-Zaitzev, and Medori-Spiro. I will discuss their results as well as my recent progress on the problem in dimension 7 (http://arxiv.org/abs/1511.04019).

A 4-manifold is said to have positive biorthogonal curvature if the average of sectional curvatures of any pair of orthogonal planes is positive. In this talk, I will describe a construction of metrics with positive biorthogonal curvature on the product of spheres, and then combine it with recent surgery stability results of Hoelzel to classify (up to homeomorphism) the closed simply-connected 4-manifolds that admit a metric with positive biorthogonal curvature.

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.

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.

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.

We'll review the quantum sl_n knot invariants and their description via MOY calculus, as well as work of Cautis-Kamnitzer-Licata-Morrison showing how these invariants arise naturally from a duality between sl_n and sl_m called skew Howe duality. We'll then discuss work (joint with Aaron Lauda and Hoel Queffelec) categorifying this result to give elementary constructions of Khovanov and Khovanov-Rozansky knot homology. Time permitting, we'll also discuss work (joint with Daniel Tubbenhauer) relating symmetric Howe duality to the colored Jones polynomial, and giving a new diagrammatic method for computing this invariant.

In joint work with H. Sá Earp we introduced a method to construct G2–instantons over compact G2–manifolds arising as the twisted connected sum of a matching pair of building blocks. I will recall some of the background (including the twisted connected sum construction and a short discussion as to why one should care about G2–instantons), discuss our main result and explain how to interpret it in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a K3 surface. If time permits, I will discuss an idea to construct the input required by our gluing theorem.

We motivate and introduce nonlinear harmonic forms. These are de Rham representatives $z$ of cohomology classes which minimize the energy $\|z\|_{L_2}^2$ subject to a nonlinear constraint. We give basic existence results for quadratic constraints, discuss the rich Euler Lagrange equations, and ask many regularity questions.

Compact 6-dimensional nearly Kähler manifolds are the cross-sections of Riemannian cones with holonomy G2. A long-standing problem has been the question of existence of complete nearly Kähler 6-manifolds besides the four known homogeneous ones. We resolve this problem by proving the existence of exotic (inhomogeneous) nearly Kähler structures on the 6-sphere and on the product of two 3-spheres. This is joint work with Mark Haskins, Imperial College London.

Monopoles are solutions to the Bogomolnyi equation, which is a PDE for a connection and an Higgs field (a section of an certain bundle) on a 3 dimensional Riemannian manifold. In this talk I plan to introduce these equations. Then I want to tell you some properties of its solutions on R^3. Finally, I plan to speak about monopoles on a more general class of noncompact manifolds known as asymptotically conical. My main goal is to explain the geometric meaning of the parameters needed to give coordinates on an open set of the moduli space of monopoles.

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).

This is joint work with Mike Sullivan. We consider a Legendrian surface, $L$, in $R^5$ (or more generally in the 1-jet space of a surface). Such a Legendrian can be conveniently presented via its front projection which is a surface in $R^3$ that is immersed except for certain standard singularities. We associate a differential graded algebra (DGA) to $L$ by starting with a cellular decomposition of the base projection (to $R^2$) of $L$ that contains the projection of the singular set of $L$ in its 1-skeleton. A collection of generators is associated to each cell, and the differential is determined in a formulaic manner by the nature of the singular set above the boundary of a cell. Our motivation is to give a cellular computation of the Legendrian contact homology DGA of $L$. In this setting, the construction of Legendrian contact homology was carried out by Etnyre-Ekholm-Sullvan with the differential defined by counting holomorphic disks in $C^2$ with boundary on the Lagrangian projection of $L$. Equivalence of our DGA with LCH may be established using work of Ekholm on gradient flow trees. Time permitting, we will discuss constructions of augmentations of the cellular DGA from two parameter families of functions.

It is an interesting well known fact that the relative entropy with respect to the Gaussian measure on $\mathbb{R}^N$ satisfies a simple subadditivity property. Namely, if $\Pi_1^{(i)}(F_N)$ is the first marginal of the density function F_N in the i-th variable then \begin{equation} \sum_{i=1}^N H(\Pi_1^{(i)}(F_N) | \gamma_1) \leq H(F_N | \gamma_N), \end{equation} where $\gamma_k$ is the standard Gaussian on $\mathbb{R}^k$. Surprisingly enough, when one tries to achieve a similar result on $\mathbb{S}^{N-1}(\sqrt{N})$ a factor of 2 appears in the right hand side of the inequality (a result due to Carlen, Lieb and Loss), and the constant is sharp. Besides a deviation from the simple equivalence of ensembles principle in equilibrium Statistical Mechanics, this entropic inequality on the sphere has interesting ramifications in other fields, such as Kinetic Theory. In this talk we will present conditions on the density function F_N, on the sphere, under which we can get an ‘almost’ subaditivity property; i.e. the factor 2 can be replaced with a factor of $1+\epsilon_N$, with $\epsilon_N$ given explicitly and going to zero. The main tools to be used in order to proved this result are an entropy conservation extension of F_N to $\mathbb{R}^N$ together with comparison of appropriate transportation distances such as the entropy, Fisher information and Wasserstein distance between the marginal of the original density and that of the extension. Time permitting, we will give an example, one that arises naturally in the investigation of the so-called Kac Model, to many families of functions that satisfy these conditions.

The inverse square law is fundamental to our understanding of electromagnetism. The question of the decay of the fields associated to nonabelian gauge theories is more difficult because the equations determining these fields are nonlinear. In this talk, I will discuss recent progress in establishing decay rates for instantons.

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.

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.

Jones' planar algebras are a useful tool for studying and constructing fusion categories, which generalize the representation categories of (quantum) groups. Thus we think of fusion categories and planar algebras as encoding quantum symmetries. I will give an overview of Jones' planar algebras with attention to specific examples. Along the way, we will discuss several evaluation algorithms which give quantum invariants, including the Jones polynomial.

We give a new proof for the existence of mean curvature flow with surgery for 2-convex hypersurfaces. Our proof works in all dimensions, including mean convex surfaces in R^3. We also derive a-priori estimates for a more general class of flows, called (alpha,delta)-flows. This is joint work with Bruce Kleiner.

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.