In this talk I will introduce some new ideas to the existence theory for a class of non-variational existence problems arising naturally in geometry and analysis. I will discuss some applications (and potential applications) to positive mass-type and Penrose-type theorems in general relativity.

A problem originally studied by P. Miao is whether the positive mass theorem holds on manifolds with certain singularities along a hypersurface. I will discuss an approach to this problem which uses the Ricci flow to smooth out the metric, so that one can apply the usual positive mass theorem. This allows for extending the rigidity statement in the zero mass case to higher dimensions, which was only known in the 3 dimensional case previously. This is joint work with D. McFeron.

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.

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.

The cosmos and its laws have fascinated people since the ancient times. Many scientists and philosophers have tried to describe and explain what they saw in the sky. And almost all of them have used mathematics to formulate their ideas and compute predictions for the future. Today, we have made huge progress in understanding and predicting how planets, stars, and galaxies behave. But still, the mysteries of our universe are formulated and resolved in mathematical language and always with new mathematical methods and ideas. In this lecture, you will hear about two of the most famous physicists of all times, Isaac Newton (1643-1727) and Albert Einstein (1879-1955), and about their theories of the universe. You will learn about common features and central differences in their viewpoints and in the mathematics they used to formulate their theories. In passing, you will also encounter the famous mathematician Carl Friedrich Gauß (1777-1855) and his beautiful ideas about curvature.

The k-dilation of a map measures how much the map stretches k-dimensional volumes. The 1-dilation is the usual Lipschitz constant. We consider the problem of finding the smallest k-dilation among all degree 1 maps from one rectangle to another rectangle. (These are n-dimensional rectangles.) In general the linear map is far from optimal.

We will prove that certain simply-connected four-manifolds with positive-definite intersection forms cannot admit symplectic structures. This is related to the existence of so-called perfect Morse functions. This is joint work with Jennifer Hom.

Homological Mirror Symmetry proposes an equivalence between the derived category of coherent sheaves on a complex manifold and the (derived) Fukaya category of the mirror symplectic manifold. It is natural to consider the behaviour of these categories and equivalences under deformations of the underlying spaces.

In this talk I will describe Toda's results on deformations of the category Coh(X) of coherent sheaves on a complex manifold X. They come from deformations of X as a complex manifold, non-commutative deformations, and gerby deformations. These can all be interpreted as deformations of X as a generalized complex manifold; in some instances it is possible to deform X to a symplectic manifold. Toda also described how to deform Fourier-Mukai equivalences, and I will present some examples coming from mirror SYZ fibrations.

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.

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.

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)

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.

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.

Nakajima quiver varieties lie at the interface of geometry and representation theory and provide an important class of examples of Calabi-Yau manifolds. I will discuss a particular instance, hyperpolygon space, which arises from a certain shape of quiver. The simplest of these is a noncompact complex surface admitting the structure of an "instanton", and therefore fits nicely into the Kronheimer-Nakajima classification of ALE hyperkaehler 4-manifolds, which is a geometric realization of the McKay correspondence for finite subgroups of SU(2). For more general hyperpolygon spaces, we can speculate on how this classification might be extended by studying the geometry of hyperpolygons at "infinity". This talk represents previous work with Jonathan Fisher and ongoing work with Hartmut Weiss.

We introduce a new tool for relating the scalar curvature of a Riemannian manifold to its global geometry and topology, based on the study of level sets of harmonic functions and harmonic maps to the circle. We will explain how these ideas lead to simple new proofs and improvements upon some well-known results in three-manifold geometry and general relativity, previously studied primarily via minimal surface and Dirac operator methods.

I'll discuss how one can use cotangent and conormal bundles to translate some basic questions in topology into questions in symplectic geometry. This symplectic viewpoint allows one, for instance, to (re)prove that certain smooth structures on spheres are exotic, and to define new knot invariants via holomorphic curves. I'll describe properties of the knot invariant and some recent applications to transverse knots in contact geometry.

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.

In this talk, I will examine the geometry of moduli spaces of stable bundles on Hopf and Kodaira surfaces, which are compact complex surfaces that do not admit Kaehler metrics. In particular, I will show that these moduli admit interesting geometric structures such as hypercomplex structures and strong HKT-metrics, in the case of Hopf surfaces, as well as algebraic integrable systems.

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.

The cohomology (with complex coefficients) of a compact kahler manifold M admits an action of the algebra sl(2,C), and this action plays an essential role in the analysis of the cohomology. In the case that M is a hyperkahler manifold Verbitsky and Looijenga—Lunts showed there is a family of such sl(2,C)’s generating an algebra isomorphic to so(4,b_2-2), and this algebra similarly can tell us quite a bit about the cohomology of the hyperkahler. I will describe some results of this nature for both the Hodge numbers and Nagai’s conjecture on the nilpotent logarithm of monodromy arising from a degeneration. This is joint work with Mark Green, Radu Laza and Yoonjoo Kim.

We can describe the point-pushing subgroup of the mapping class group topologically as the set of maps that push a puncture around loops in the surface. However, we can characterize this topological subgroup in purely algebraic terms. Using group theoretic tools and a classical theorem of Burnside, we can recover a result of Ivanov-McCarthy establishing the triviality of Out(Mod±). To this end, we’ll demonstrate that the point-pushing subgroup is “unique” in the mapping class group.

We will discuss the problem of understanding the collapsing of Ricci-flat Kahler metric on Calabi-Yau manifolds. If time permits we will also discuss the Strominger-Yau-Zaslow picture of mirror symmetry for hyperkahler manifolds. Joint work with Mark Gross and Yuguang Zhang.

We discuss a recent result of the speaker giving a categorification of quantum sl_3 projectors and the sl_3 Reshetikhin-Turaev invariants of framed tangles. In more detail, we will review Kuperberg's diagrammatic description of the category of representations of quantum sl_3 (which gives a combinatorial method for computing the quantum sl_3 invariant of links) as well as Morrison and Nieh's geometric categorification of this structure. We then show that there exist elements in Morrison and Nieh's categorification which correspond to projection onto highest weight irreducible summands and use these elements to construct a categorification of the sl_3 Reshetikhin-Turaev invariant, that is, a link homology theory from which the sl_3 invariant can be obtained by taking the graded Euler characteristic. No previous knowledge of categorification or quantum groups is assumed.

I will exhibit a spectral sequence from the annular Khovanov homology of a 2-periodic link to that of its quotient, which in turn proves rank inequalities and decategorifies to polynomial congruences. While previous work used heavier algebraic machinery to prove this rank inequality in a particular sl_2 weight space grading, we instead mimic Borel's construction of equivariant cohomology and employ grading considerations to give a combinatorial proof of the rank inequality for all quantum and sl_2 weight space gradings. Curiously, the same methods suggest a similar spectral sequence relating the Khovanov homology of a 2-periodic link and the annular Khovanov homology of its quotient link. We'll discuss partial results on this front.

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.

One of the central conjectures in Kähler geometry is the Yau-Tian-Donaldson conjecture relating the existence of canonical Kähler metrics to algebro-geometric stability. A natural question is to ask what happens when such a metric does not exist, and here Kähler metrics of Poincaré type are expected to play an important role. These metrics are Kähler metrics defined on the complement of a divisor in a compact complex manifold and have a cusp-like singularity near the divisor. The blow-up theorem of Arezzo-Pacard and its generalizations give sufficient conditions for the blow-up of a compact Kähler manifold admitting a canonical metric to also carry such a metric. I will describe an extension of this result to the Poincaré type setting.

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.

TBA