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 classical Euclidean problem studied by Bonnet in the 19th century was to determine whether, and in how many ways, a Riemannian surface can be isometrically embedded into Euclidean 3-space so that its mean curvature is a prescribed function. He found that, generically, specifying a metric and mean curvature admitted no solution but that there are special cases in which, not only are there solutions, but there are even 1-parameter families of distinct (i.e., mutually noncongruent) solutions. Much later, these ‘Bonnet surfaces’ were found to be intimately connected with integrable systems and Lax pairs. In this talk, I will consider the analogous problem in affine geometry: To determine whether, and in how many ways, a surface endowed with a Riemannian metric g and a function H can be immersed into affine 3-space in such a way that the induced Blaschke metric is g and the induced affine mean curvature is H. This affine problem is, in many ways, richer and more interesting than the corresponding Euclidean problem. I will classify the pairs (g,H) that display the greatest flexibility in their solution space and explain what is known about the (suspected) links with integrable systems and Lax pairs.

We will discuss Witten’s gauge theory approach to Jones polynomial by counting solutions to the Kapustin-Witten (KW) equations with singular boundary conditions over 4-manifolds. We will give a classification of solutions to the KW equations over $S^1\times\Sigma\times R^+$. We prove that all solutions to the KW equations over $S^1\times\Sigma\times R^+$ are $S^1$ direction invariant and we give a classification of the KW monopole over $\Sigma\times R^+$ based on the Hermitian-Yang-Mills type structure of KW monopole equation. This is based on joint works with Rafe Mazzeo.

In the 1980s, Bill Goldman used intersection theory to define a Lie algebra structure on the free Z module L(X) generated by the closed geodesics on a hyperbolic surface X. This bracket is related to a formula for the Poisson bracket of functions on the variety of flat G-bundles over X. In related work (1970s and 1990s), Vladimir Turaev (with contributions by Kawazumi and Kuno in the 2000s) constructed a cobracket on L(X) that depends on the choice of a framing. In this talk, I will review the definition of the Goldman-Turaev Lie bialgebra of a framed surface and discuss its relevance to questions in other areas of mathematics. I'll discuss how Hodge theory can be applied to these questions. I may also discuss some related questions, such as the classification of mapping class group orbits of framings of a punctured surface.

For each K¨ahler class on a compact K¨ahler manifold there is a lower bound of the Calabi functional, which we call the ``potential energy''. Fixing the volume and letting the K¨ahler classes vary, the energy defines a functional which may be studied in it?s own right. Any critical point of the energy functional is then a K¨ahler class whose extremal K¨ahler metrics (if any) are so-called strongly extremal metrics. We take the well-studied case of Hirzebruch surfaces and generalize it in two different directions; along the dimension of the base and along the genus of the base. In the latter situation we are able to give a very concrete description of the corresponding dynamical system (as defined first by S. Simanca and L. Stelling). The talk is based on work in progress with Santiago Simanca.

We introduce periodic knots and discuss two classical results concerning their geometry, namely Murasugi's condition on the Alexander polynomial and Edmonds' condition on the genus. We then show how spectral sequences in Heegaard Floer link homology can be used to give a generalization of these two results in the case of doubly-periodic knots.

In this talk we consider the moduli space of Yang-Mills instantons on the family of hyperkahler 4 manifolds known as multi-center TaubNUT spaces. We describe the Nahm transform for flat manifolds. Then we sketch its extension to the above hyperkahler family, where it defines an isometry between the moduli space of instantons on the multi-center TaubNUT and the moduli space of solutions of a rococo system of ordinary differential equations. This is joint work with Sergey Cherkis and Andres Larrain Hubach

Our numerical computation group has studied several problems in computational finance over the past decade. One that we've looked at recently is the pricing of "collateralized debt obligations" (CDOs). The market for CDOs has grown rapidly to over US$1 trillion annually in 2006, since the appearance of JP Morgan's Bistro deal, the first synthetic CDO, in December 1997. Much of the turmoil in the financial markets recently has been due to such credit derivatives. As this suggests, there are still many open problems associated with the pricing and hedging of these complex financial instruments. I will talk briefly about some work that we have done recently in this area.

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.

We discuss the DGA map induced by an exact Lagrangian cobordism, and an analytic strategy to lift the map to integer coefficients, introduced by Fukaya, Oh, Ohta and Ono and further adapted by Ekholm, Etnyre, and Sullivan and Karlsson respectively. We then explain how this strategy can be applied to find a concrete combinatorial formula for a mini-dipped pinch move, thereby completely determining the integral DGA maps for all decomposable, orientable Lagrangian cobordisms. If time permits, we will show how to obtain this formula in a model case. We will also go into future potential work, including applications to Heegaard Floer Homology and nonorientable cobordisms.

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.

The simplest model of vortex filament motion in an ideal fluid leads to an integrable nonlinear evolution equation, known as the localized induction approximation or the vortex filament flow, closely related to the cubic focusing nonlinear Schroedinger equation. For closed finite-gap solutions of this flow, certain geometric and topological features of the evolving curves appear to be correlated with the algebro-geometric data used to construct them. In this talk, I will briefly discuss this construction, and some low-genus examples (in particular, Kirchhoff elastic rod centerlines) where this correlation is well understood. I will mainly discuss recent joint work with Annalisa Calini, describing how to generate a family of closed finite-gap solutions of increasingly higher genus via a sequence of deformations of the multiply covered circle. We prove that every step in this sequence corresponds to constructing a cable on previous filament; moreover, the cable's knot type (which is invariant under the evolution) can be read off from the deformation sequence.

A petal diagram of a knot or link consists of a center point surrounded by n non-nested loops; it represents n strands of the link at various heights which all project onto the same center point. Though every knot has a petal diagram, extremely few links have petal diagrams. The goal of this project is to characterize and enumerate which links do. First, we tabulate all petal links of 2-5 components. We then show all petal links arise as circle graphs -- the intersection graph of a set of chords of a circle. This establishes lower bounds on the number of petal links and allows us to conjecture upper bounds. We then discuss using petal diagrams to model certain classes of knots and links.

Heegaard Floer homology has different extensions for 3-manifolds with boundary. In this talk, we will recall some basics of these extensions and explain how they can be used to give a computationally effective way for detecting boundary parallel components of tangles, and existence of homologically essential compressing disks. The fact that these are checkable by computer, is based on the factoring algorithm of Lipshitz-Ozsvath-Thurston for computing bordered Floer homology, and our extension of it to compute bordered-sutured Floer homology. This is joint work with Robert Lipshitz.

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.

Geometrostatics is an important subdomain of Einstein's General Relativity. It describes the mathematical and physical properties of static isolated relativistic systems such as stars, galaxies or black holes. For example, geometrostatic systems have a well-defined ADM-mass (Chrusciel, Bartnik) and (if this is nonzero) also a center of mass (Huisken-Yau, Metzger) induced by a CMC-foliation at infinity. We will present surface integral formulae for these physical properties in general geometrostatic systems. Together with an asymptotic analysis, these can be used to prove that ADM-mass and center of mass 'converge' to the Newtonian mass and center of mass in the Newtonian limit $c\to\infty$ (using Ehler's frame theory). We will discuss geometric similarities of geometrostatic and classical static Newtonian systems along the way.

Among the first significant results of algebraic topology is the computation, by Thom, Milnor, Novikov, and Wall among others, of the bordism groups of stably complex and oriented manifolds. After reviewing these results, I will discuss the notion of derived orbifolds, and briefly indicate how the bordism groups of these objects appear as universal recipients of invariants arising in Gromov-Witten theory and symplectic topology. Finally, I will state what is known about them, as well as some conjectures about the structure of these groups.

Holomorphic symplectic manifolds (aka hyperkahler manifolds) are complex analogues of real symplectic manifolds. They have a rich geometric structure, though few compact examples are known. In this talk I will describe attempts to construct and classify holomorphic symplectic manifolds that also admit a holomorphic fibration. In particular, we will consider examples in four dimensions that are fibred by abelian surfaces known as Prym varieties.

Weight Filtrations are mysterious: they record some shadow of how a variety might be recovered from smooth and projective ones. Some of the information recorded by weight filtrations can be understood via the motivic measures they define, i.e. group homomorphisms from the Grothendieck ring of varieties. With Zakharevich’s discovery of the higher K groups of the category of varieties, there is an ongoing project to understand these groups by lifting motivic measures (on the level of K_0) to so-called "derived" ones, i.e. on the level of K_i for all i. I will describe some of this work, which shows that if one closely studies how the Gillet-Soulé weight complex is constructed, then one can also obtain derived motivic measures to non-additive categories as well, such as the compact objects in the category of motivic spaces, along with that of compact objects in the motivic stable homotopy category. These new derived motivic measures allow us to answer questions in the literature, providing new ways to understand the higher K groups of varieties, and relating them to other interesting algebro-geometric objects in the literature.

The Chekanov-Eliashberg dga is a powerful invariant for Legendrian links. Using augmentations of this dga, one can truncate its differential to produce linearized contact homology. About two decades ago, Sabloff established a duality in this setting, closely linked to the Poincaré duality of Lagrangian fillings. This truncation has since been generalized into a unital A-infinity category, Aug_+. In this talk, I will present new results that extend Sabloff duality from the level of cochain complexes to A-infinity bimodules over Aug_+. The key tool in this extension is Ng's LSFT algebra, which enlarges the Chekanov-Eliashberg dga. If time permits, I will also discuss how the LSFT algebra encodes additional homotopy coherent data, providing further insights into Sabloff duality.