Jo Nelson : Contact Invariants and Reeb Dynamics
- Geometry and Topology ( 100 Views )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.
Lan-Hsuan Huang : Hypersurfaces with nonnegative scalar curvature and a positive mass theorem
- Geometry and Topology ( 111 Views )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.
Katrin Wehrheim : The symplectic category: correspondences, quilts, and topological applications
- Geometry and Topology ( 101 Views )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
Zhou Zhang : Volume Form and Scalar Curvature for K\ahler-Ricci Flow over General Type Manifold
- Geometry and Topology ( 151 Views )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.
David Duncan : The Chern-Simons invariants for general compact Lie groups
- Geometry and Topology ( 146 Views )The Chern-Simons invariants are 3-manifold invariants arising from representations of the fundamental group into a Lie group G. These have been well-studied for G = SU(2), but much less is known about them for more general G. In this talk, I will review the definition of these invariants and discuss results that extend to arbitrary compact G several well-known SU(2)-computations. These extensions all have the flavor of "if you know the invariants for SU(2), then you know the invariants for general compact G". This is joint work with Kevin Fournier.
Henri Roesch : Proof of a Null Penrose Conjecture using a new Quasi-local Mass
- Geometry and Topology ( 109 Views )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.
Dan Lee : Black hole uniqueness and Penrose inequalities
- Geometry and Topology ( 113 Views )I will tell two stories. The first is the story of static spacetimes with black hole boundaries and the attempt to classify them. The second is the story of the Penrose inequality. I will then weave these two stories together in the setting of negative curvature. This last part is a report on joint work-in-progress with A. Neves.
Jimmy Petean : On the Yamabe invariant of Riemannian products
- Geometry and Topology ( 198 Views )The Yamabe invariant of a closed manifold appears naturally when studying the total scalar curvature functional on the space of Riemannian metrics on the manifold. Computations are difficult, in particular in the positive case (when the manifold admits metrics of positive scalar curvarture, and there is no unicity of metrics of constant scalar curvature on a conformal class). In this talk I will review a little of what is known about the computation of the invariant and discuss some recent joint work with K. Akutagawa and L. Florit on the Yamabe constants of Riemannian products.
Siqi He : Classification of Nahm Pole Solutions to the KW Equations on $S^1\times\Sigma\times R^+$
- Geometry and Topology ( 221 Views )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.
Richard Hain : Hodge theory and the Goldman-Turaev Lie bialgebra
- Geometry and Topology ( 178 Views )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.
Richard Hain : The Lie Algebra of the Mapping Class Group, Part 2
- Geometry and Topology ( 197 Views )In this talk I will review the construction of the Lie algebra associated to the mapping class group of a (possibly decorated) surface and explain how this generalizes the Lie algebra associated to the pure braid group. I will also explain the analogue of the KZ-equation in the mapping class group case. In the second talk I will discuss filtrations of this Lie algebra associated to curve systems on the surface and their relation to handlebody groups.
Adam Levine : Concordance of knots in homology spheres
- Geometry and Topology ( 123 Views )Knot concordance concerns the classification of knots in the 3-sphere that occur as the boundaries of embedded disks in the 4-ball. Unlike in higher dimensions, one obtains vastly different results depending on whether the disks are required to be smoothly embedded or merely locally flat (i.e. continuously embedded with a topological normal bundle); many tools arising from gauge theory and symplectic geometry can be used to illustrate this distinction. After surveying some of the recent progress in this area, I will discuss the extension of these questions to knots in 3-manifolds other than S^3. I will show how to use invariants coming from Heegaard Floer homology to obstruct not only smoothly embedded disks but also non-locally-flat piecewise-linear disks; this answers questions from the 1970s posed by Akbulut and Matsumoto. I will also discuss more recent results (joint with Jennifer Hom and Tye Lidman) giving infinitely many knots that are distinct up to non-locally-flat piecewise-linear concordance.
Jeremy Marzuola : Nonlinear Bound States on manifolds
- Geometry and Topology ( 121 Views )We will discuss the results of several joint ongoing projects (with subsets of collaborators Pierre Albin, Hans Christianson, Colin Guillarmou, Jason Metcalfe, Laurent Thomann and Michael Taylor), which explore the existence, stability and dynamics of nonlinear bound states and quasimodes on manifolds of both positive and negative curvature with various symmetry properties.
Alex Pieloch : Moduli Spaces of Real Algebraic Curves
- Geometry and Topology ( 129 Views )There is a natural relationship between moduli spaces of Riemann surfaces, mapping class groups of surfaces, and intersection patterns of simple closed curves on surfaces. In this talk, we describe an analogous relationship between moduli spaces of real algebraic curves, mapping class groups of surfaces with orientation reversing involutions, and intersection patterns of involution invariant simple closed curves on surfaces. After establishing these relationships, we obtain that the homology and cohomology groups of mapping class groups of surfaces with orientation reversing involutions satisfy duality relationships analogous to those for compact manifolds. We also obtain that higher homotopy groups associated to the moduli spaces of real algebraic curves relative to its boundary vanish in all degrees less than a determinable constant.
Thalia Jeffres : K\{a}hler-Einstein Metrics with Edge Singularities
- Geometry and Topology ( 125 Views )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.
Lorenzo Foscolo : New G2-holonomy cones and exotic nearly Kähler structures on the 6-sphere and the product of two 3-spheres.
- Geometry and Topology ( 163 Views )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.
Curtis Porter : Straightening out degeneracy in CR geometry: When can it be done?
- Geometry and Topology ( 199 Views )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).
Michael Henry : Connections between existing Legendrian knot invariants
- Geometry and Topology ( 124 Views )In this talk, we will investigate existing Legendrian knot invariants and discuss new connections between the theory of generating families and the Chekanov-Eliashberg differential graded algebra (CE-DGA). The geometric origins of the CE-DGA are Floer theoretic in nature and come out of the Symplectic Field Theory developed by Eliashberg and Hofer. On the other hand, Legendrian invariants derived from the study of 1-parameter families of smooth functions (called generating families) are Morse theoretic in nature. In the last decade, connections have been found between the Legendrian invariants derived using these two methods. In this talk, I will try to provide a clearer picture of the relationship between generating families and the CE-DGA.
Luca Di Cerbo : Seiberg-Witten equations on manifolds with cusps and geometric applications.
- Geometry and Topology ( 193 Views )In this talk, I will discuss the Seiberg-Witten equations on finite volume Riemannian manifolds which are diffeomorphic to the product of two hyperbolic Riemann surfaces of finite topological type. Finally, using a Seiberg-Witten scalar curvature estimate I will present several results concerning the Riemannian geometry of these spaces.
Claude LeBrun : Four-Dimensional Einstein Manifolds, and Beyond
- Geometry and Topology ( 156 Views )An Einstein metric is a Riemannian metric of constant Ricci curvature. One of the central problems of modern Riemannian geometry is to determine which smooth compact manifolds admit Einstein metrics. This lecture will explain some recent results concerning the 4-dimensional case of the problem, and then compare and contrast these results with our current understanding of the problem in other dimensions.
Kristen Hendricks : Periodic Knots and Heegaard Floer Homology
- Geometry and Topology ( 173 Views )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.
Andrew Cooper : Singularities of Lagrangian Mean Curvature Flow
- Geometry and Topology ( 148 Views )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.
Graeme Wilkin : Morse theory and stable pairs
- Geometry and Topology ( 133 Views )In the early 1980s Atiyah and Bott described a new approach to studying the cohomology of the moduli space of stable bundles: the equivariant Morse theory of the Yang-Mills functional. There are many other interesting moduli spaces that fit into a similar framework, however the catch is that the total space is singular, and it is not obvious how to construct the Morse theory of the appropriate functional. In this talk I will describe how to get around these difficulties for the moduli space of stable pairs, for which we prove a Kirwan surjectivity theorem and give a Morse-theoretic interpretation of the change in cohomology due to a flip. This builds upon earlier work with George Daskalopoulos, Jonathan Weitsman and Richard Wentworth for rank 2 Higgs bundles.
Dave Penneys : Planar algebras and evaluation algorithms
- Geometry and Topology ( 121 Views )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.
Ziva Myer : Product Structures for Legendrian Submanifolds with Generating Families
- Geometry and Topology ( 147 Views )In contact topology, invariants of Legendrian submanifolds in 1-jet spaces have been obtained through a variety of techniques. I will discuss how I am enriching one Morse-theoretic invariant, Generating Family Cohomology, to an A-infinity algebra by constructing product maps. The construction uses moduli spaces of Morse flow trees: spaces of intersecting gradient trajectories of functions whose critical points encode Reeb chords of the Legendrian submanifold. I will focus my talk on the construction of a 2-to-1 product and discuss how it lays the foundation for the A-infinity algebra.
Dmitri Burago : Math Mozaic
- Geometry and Topology ( 168 Views )The lecture includes the main part (to be chosen on the spot) and a few mini-talks with just definitions, motivations, some ideas of proofs, and open problems. I will discuss some (hardly all) of the following topics. 1. A survival guide for feeble fish. How fish can get from A to B in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water flow. This is related to homogenization of G-equation which is believed to govern many combustion processes. Based on a joint work with S. Ivanov and A. Novikov. 2. One of the greatest achievements in Dynamics in the XX century is the KAM Theory. It says that a small perturbation of a non-degenerate completely integrable system still has an overwhelming measure of invariant tori with quasi-periodic dynamics. What happens outside KAM tori has been remaining a great mystery. The main quantitate invariants so far are entropies. It is easy, by modern standards, to show that topological entropy can be positive. It lives, however, on a zero measure set. We are now able to show that metric entropy can become infinite too, under arbitrarily small C^{infty} perturbations, answering an old-standing problem of Kolmogorov.. Furthermore, a slightly modified construction resolves another longstanding problem of the existence of entropy non-expansive systems. In these modified examples positive positive metric entropy is generated in arbitrarily small tubular neighborhood of one trajectory. Join with S. Ivanov and Dong. Chen. 3. What is inside? Imagine a body with some intrinsic structure, which, as usual, can be thought of as a metric. One knows distances between boundary points (say, by sending waves and measuring how long it takes them to reach specific points on the boundary). One may think of medical imaging or geophysics. This topic is related to minimal fillings and surfaces in normed spaces. Joint work with S. Ivanov. 4. How well can we approximate an (unbounded) space by a metric graph whose parameters (degree of vertices, length of edges, density of vertices etc) are uniformly bounded? We want to control the ADDITIVE error. Some answers (the most difficult one is for $\R^2$) are given using dynamics and Fourier series. Joint with Ivanov. 5.How can one discretize elliptic PDEs without using finite elements, triangulations and such? On manifolds and even reasonably nice mmspaces. A notion of \rho-Laplacian and its stability. Joint with S. Ivanov and Kurylev. 6. A solution of Busemanns problem on minimality of surface area in normed spaces for 2-D surfaces (including a new formula for the area of a convex polygon). Joint with S. Ivanov.
Jason Lotay : Hyperkaehler metrics on a 4-manifold with boundary
- Geometry and Topology ( 143 Views )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.