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).

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?

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 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.

Analyzing the motion of a small body is often done by making a point-particle approximation. This simplification is not entirely appropriate in general relativity since, roughly speaking, too much mass in too little space creates black holes. In place of point-particles one considers one-parameter families of space-time metrics $\gamma_\varepsilon$ in which $\varepsilon\to 0$ corresponds to a body shrinking to zero size. In addition, certain point-particle limit properties are imposed on $\gamma_\varepsilon$. While there are some examples of such metrics $\gamma_\varepsilon$ (e.g. Schwarzschild-de Sitter space-time), there is no general existence theorem for such space-times. This talk will discuss a gluing construction which produces initial data with desirable point-particle limit properties.

In this talk I will describe the limiting properties Yang-Mills flow on a holomorphic vector bundle E, in the case where the flow does not converge. In particular I will describe how to determine the L^2 limit of the curvature endomorphism along the flow. This proves a sharp lower bound for the Hermitian-Yang-Mills functional and thus the Yang-Mills functional, generalizing to arbitrary dimension a formula of Atiyah and Bott first proven on Riemann surfaces. I will then explain how to use this result to identify the limiting bundle along the flow, which turns out to be independent of metric and uniquely determined by the isomorphism class of E.

The Fundamental Theorem of Algebra first rigorously proved by Gauss states that each complex polynomial of degree n has precisely n complex roots. In recent years various extensions of this celebrated result have been considered. We shall discuss the extension of the FTA to harmonic polynomials of degree n. In particular, the 2003 theorem of D. Khavinson and G. Swiatek that shows that the harmonic polynomial z − p(z); deg p = n > 1 has at most 3n − 2 zeros as was conjectured in the early 90's by T. Sheil-Small and A. Wilmshurst. More recently L. Geyer was able to show that the result is sharp for all n.

In 2004 G. Neumann and D. Khavinson showed that the maximal number of zeros of rational harmonic functions z − r(z); deg r = n > 1 is 5n − 5. It turned out that this result conrfimed several consecutive conjectures made by astrophysicists S. Mao, A. Petters, H. Witt and, in its final form, the conjecture of S. H. Rhie that were dealing with the estimate of the maximal number of images of a star if the light from it is deflected by n co-planar masses. The first non-trivial case of one mass was already investigated by A. Einstein around 1912.

We shall also discuss the problem of gravitational lensing of a point source of light, e.g., a star, by an elliptic galaxy, more precisely the problem of the maximal number of images that one can observe. Under some more or less "natural" assumptions on the mass distribution within the galaxy one can prove that the number of visible images can never be more than four in some cases and six in the other. Interestingly, the former situation can actually occur and has been observed by astronomers. Still there are much more open questions than there are answers.

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.

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).

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.

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

Embedded contact homology (ECH) is an invariant of a closed contact 3-manifold, but proving its invariance is not so straightforward: the only known proof (due to Taubes) is to show that it is isomorphic to monopole Floer homology, which only depends on the underlying manifold. Colin, Ghiggini, Honda, and Hutchings defined a version of ECH for contact 3-manifolds with boundary, which are naturally sutured manifolds, and conjectured that this is also an invariant of the underlying sutured manifold. In this talk I will show that sutured ECH is indeed an invariant and discuss exactly what kind of invariant it is. This is joint work with Cagatay Kutluhan.

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.

In this talk, I will show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector, extending Lev Rozansky's work with infinite torus braids. I will also show a similar result for the limiting Lipshitz-Sarkar-Khovanov homotopy types of the closures of such braids. Extensions to more general infinite braids will also be considered. This is joint work with Gabriel Islambouli.

TBA

The talk will give an overview of recent results on simple one-dimensional rules started from seeds, i.e., from bounded perturbations of the quiescent state. Two phenomena, replication, and robust periodic solutions emanating from one of the edges, are of particular interest. The talk will emphasize examples and interesting open problems. (Joint work with D. Griffeath, G. Gliner and M. Pelfrey, and A. Holroyd.)

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.

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.

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.

In this talk, we show that for any closed 4-dimensional simply-connected Riemannian manifold $M$ with Ricci curvature $|Ric| \leq 3$, volume $vol(M)>v>0$ and diameter $diam(M) \leq D$, the length of a shortest closed geodesic on $M$ is bounded by a function $F(v,D)$ . The proof of this result is based on the diffeomorphism finiteness theorem for the manifolds satisfying above conditions proved by J. Cheeger and A. Naber. This talk is based on the joint work with Zhifei Zhu.