Yuhao Hu : Geometry of Bäcklund Transformations
- Geometry and Topology ( 102 Views )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.
Ailana Fraser : Survey of positive isotropic curvature results
- Geometry and Topology ( 157 Views )A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes manifolds with pointwise quarter-pinched sectional curvatures and manifolds with positive curvature operator. We will give a survey of results on PIC and discuss recent joint work with J. Wolfson on fundamental groups of manifolds of PIC. The techniques used involve minimal surfaces.
Fernando Schwartz : On the topology of black holes
- Geometry and Topology ( 144 Views )An important special case of the general construction of black holes translates into a problem in Riemannian geometry, since a totally geodesic slice of spacetime is an asymptotically flat Riemannian manifold with nonnegative scalar curvature, and the restriction of the event horizon to the slice is the apparent horizon in the slice. In this talk we show how to construct new examples of Riemannian manifolds with nonspherical apparent horizon, in dimensions four and above. More precisely, for any $n,m\ge 1$, we construct asymptotically flat, scalar flat Riemannian manifolds with apparent horizon that is a smooth outermost minimal hypersurface with topology $S^n\times S^{m+1}$.
Goncalo Oliveira : Monopoles in 3 Dimensions
- Geometry and Topology ( 108 Views )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.
Benoit Charbonneau : Asymptotic Hodge Theory of Vector Bundles
- Geometry and Topology ( 111 Views )In joint work with Mark Stern, we introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the bundle to admit a holomorphic structure. We study compatibility under the Chern isomorphism of these filtrations with the Hodge filtration on cohomology.
Emma Carberry : Conformal Surface Geometry: an algebro-geometric approach.
- Geometry and Topology ( 143 Views )A number of classical integrable systems, for example harmonic maps of the plane to a compact Lie group or symmetric space, can be transformed into a \{\\em linear\} flow on a complex torus. This torus is the Jacobian of an algebraic curve, called the spectral curve. Recently several authors have produced an analogous one-dimensional analytic variety for conformal 2-tori in $S4$ (which are not in general integrable!) using the geometry of the quaternions. It is hoped that this new development will lead to progress on the Willmore conjecture for reasons that I will explain. However this variety is at present quite mysterious; very little is known about it. I will discuss the simplest case, namely constant mean curvature tori in $\mathbb{R}3$. I will demonstrate that in this case the variety is not at all mysterious and interpret its points geometrically in terms of transformations generalising the classical transform of Darboux. This is joint work with Katrin Leschke and Franz Pedit.
Niky Kamran : The Penrose process and the wave equation in Kerr geometry
- Geometry and Topology ( 112 Views )We shall review the Penrose process for extracting mass and angular momentum from the Kerr black hole solution of the Einstein equations. We will show that Christodoulou's bound on the maximal energy gain by the classical Penrose process can be realized by choosing suitable wave packet initial data for the scalar wave equation in Kerr geometry, thereby putting super-radiance on a rigorous mathematical footing. This is joint work with Felix Finster, Joel Smoller and Shing-Tung Yau.
Richard Hain : On a problem of Eliashberg
- Geometry and Topology ( 106 Views )Suppose that (d_1, ..., d_n) is an n-tuple of integers satisfying sum_j d_j = 0. Eliashberg posed the problem of computing the class of the locus in the moduli space of n-pointed, genus g curves [C;x_1,...,x_n] where sum d_j x_j = 0 in the jacobian of C. In this talk I will give the solution and sketch the proof, which uses known facts about the structure of mapping class groups.
Lan-Hsuan Huang : Constant mean curvature foliations for isolated systems in general relativity
- Geometry and Topology ( 122 Views )We will discuss the existence and uniqueness of the foliation by stable spheres with constant mean curvature for asymptotically flat manifolds satisfying the Regge-Teitelboim condition at infinity. This work generalizes the earlier results of Huisken/Yau, Ye, and Metzger. We will also discuss the concept of the center of mass in general relativity.
Hannah Schwartz : Using 2-torsion to obstruct topological isotopy
- Geometry and Topology ( 120 Views )It is well known that two knots in S^3 are ambiently isotopic if and only if there is an orientation preserving automorphism of S^3 carrying one knot to the other. In this talk, we will examine a family of smooth 4-manifolds in which the analogue of this fact does not hold, i.e. each manifold contains a pair of smoothly embedded, homotopic 2-spheres that are related by a diffeomorphism, but not smoothly isotopic. In particular, the presence of 2-torsion in the fundamental groups of these 4-manifolds can be used to obstruct even a topological isotopy between the 2-spheres; this shows that Gabai's recent "4D Lightbulb Theorem" does not hold without the 2-torsion hypothesis.
Jer-Chin Chuang : Subdivisions and Transgressive Chains
- Geometry and Topology ( 171 Views )Combinatorial transgressions are secondary invariants of a space admitting triangulations. They arise from subdivisions and are analogous to transgressive forms such as those in Chern-Weil theory. In this talk, I characterize transgressions that are path-independent of subdivision sequence. The result is obtained by using a cohomology on posets that is shown to be equivalent to higher derived functors of the inverse (or projective) limit over the opposite poset.
John Etnyre : The Contact Sphere Theorem and Tightness in Contact Metric Manifolds
- Geometry and Topology ( 114 Views )We establish an analog of the sphere theorem in the setting of contact geometry. Specifically, if a given three dimensional contact manifold admits a compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure is tight. The proof is a blend of topological and geometric techniques. A necessary technical result is a lower bound for the radius of a tight ball in a contact 3-manifold. We will also discuss geometric conditions in dimension three for a contact structure to be universally tight in the nonpositive curvature setting. This is joint work with Rafal Komendarczyk and Patrick Massot.
Adam Saltz : Link homology and Floer homology in pictures by cobordisms
- Geometry and Topology ( 115 Views )There are no fewer than eight link homology theories which admit spectral sequences from Khovanov homology. These theories have very different origins -- representation theory, gauge theory, symplectic topology -- so it's natural to ask for some kind of unifying theory. I will attempt to describe this theory using Bar-Natan's pictorial formulation of link homology. This strengthens a result of Baldwin, Hedden, and Lobb and proves new functoriality results for several link homology theories. I won't assume much specific knowledge of these link homology theories, and the bulk of the talk will be accessible to graduate students!
Aaron Naber : Orbifold Regularity of Collapsed Spaces and applications to Einstein Manifolds.
- Geometry and Topology ( 142 Views )Let (M_i,g_i) be a sequence of Riemannian n-manifolds with uniformly bounded curvature such that (M_i,g_i)->(X,d), a metric space, in the Gromov Hausdorff sense. Then we show that there is a closed subset S of X with codimension at least 3 and dimension at most n-5 such that X-S is a Riemannian Orbifold. We use this and an \epsilon-regularity theorem to show that metric spaces in the closure of the moduli space of Einstein 4-manifolds are Riemannian Orbifolds away from a finite number of points. This is joint with G. Tian.
Colleen Robles : Degeneration of Hodge structure
- Geometry and Topology ( 124 Views )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.
Laura Fredrickson : The asymptotic geometry of the Hitchin moduli space
- Geometry and Topology ( 110 Views )Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmuller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperkahler metric. A conjectural description of its asymptotic structure appears in the work of Gaiotto-Moore-Neitzke and there has been a lot of progress on this recently. I will discuss some recent results.
Jeremy Van Horn-Morris : Fiber genus and the topology of symplectic fillings
- Geometry and Topology ( 117 Views )Work of Donaldson, Giroux, and many others shows how to associate a singular surface fibration to a symplectic 4-manifold, either closed or with boundary, as well as to a contact 3-manifold. These are Lefschetz pencils, fibrations and open books, resp. It was asked by Stipsicz, Ozbagci, Korkmaz and others, whether the genus (or genus and self intersection) of the fiber of these fibrations gave an a priori bound on the topological complexity of the symplectic manifold. This question is equivalent to asking for a bound on the length of a factorization of a mapping class element of the fiber into right handed Dehn twists. We will discuss some of the known conditions which can produce such a bound, as well as present examples where such a bound does not exist. This is joint work with I. Baykur.
Jacques Hurtubise : Isomonodromy deformations of connections
- Geometry and Topology ( 144 Views )The link between meromorphic connections on a Riemann surface and their monodromy is a very classical one, indeed so classical that it was the subject of one of Hilbert?s problems. The deformation theory of these connections, and when these deformations preserve the monodromy, is almost equally ancient. I will give an overview of some results in the area, some ancient, and some quite recent.
Pankaj K. Agarwal : Union of geometric objects, epsilon-nets, and hitting sets
- Geometry and Topology ( 111 Views )The combinatorial complexity of the union of a set of geometric objects is the total number of faces of all dimensions that lie on the boundary of the union. We review some recent results on the complexity of the union of geometric objects in 2d and 3D satisfying various natural conditions and on computing the union. We then discuss the critical roles they play in computing an epsilon net and a hitting set of a set system.
Bulent Tosun : Legendrian and transverse knots in cabled knot types
- Geometry and Topology ( 109 Views )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
Adam Levine : Non-orientable surfaces in homology cobordisms
- Geometry and Topology ( 104 Views )We study the minimal genus problem for embeddings of closed, non-orientable surfaces in a homology cobordism between rational homology spheres, using obstructions derived from Heegaard Floer homology and from the Atiyah-Singer index theorem. For instance, we show that if a non-orientable surface embeds essentially in the product of a lens space with an interval, its genus and normal Euler number are the same as those of a stabilization of a non-orientable surface embedded in the lens space itself. This is joint work with Danny Ruberman and Saso Strle.
Michael Willis : The Khovanov homology of infinite braids
- Geometry and Topology ( 105 Views )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.
Michael Lipnowski : The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds
- Geometry and Topology ( 107 Views )We exhibit examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact 1-forms \lambda_1^* on rational homology spheres which admit irreducible solutions together with a version of the Selberg trace formula relating the spectrum of the Laplacian on coexact 1-forms with the volume and complex length spectrum of a hyperbolic three-manifold. Using these relationships, we also provide precise certified numerical bounds on \lambda_1^* for several hyperbolic rational homology spheres.
Simon Brendle : Singularity formation in geometric flows
- Geometry and Topology ( 289 Views )Geometric evolution equations like the Ricci flow and the mean curvature flow play a central role in differential geometry. The main problem is to understand singularity formation. In this talk, I will discuss recent results which give a complete picture of all the possible limit flows in 2D mean curvature flow with positive mean curvature, and in 3D Ricci flow.
Chen-Yun Lin : An embedding theorem: differential geometry behind massive data analysis
- Geometry and Topology ( 130 Views )High-dimensional data can be difficult to analyze. Assume data are distributed on a low-dimensional manifold. The Vector Diffusion Mapping (VDM), introduced by Singer-Wu, is a non-linear dimension reduction technique and is shown robust to noise. It has applications in cryo-electron microscopy and image denoising and has potential application in time-frequency analysis. In this talk, I will present a theoretical analysis of the effectiveness of the VDM. Specifically, I will discuss parametrisation of the manifold and an embedding which is equivalent to the truncated VDM. In the differential geometry language, I use eigen-vector fields of the connection Laplacian operator to construct local coordinate charts that depend only on geometric properties of the manifold. Next, I use the coordinate charts to embed the entire manifold into a finite-dimensional Euclidean space. The proof of the results relies on solving the elliptic system and provide estimates for eigenvector fields and the heat kernel and their gradients.
Yuanqi Wang : A critical elliptic theory and its applications in higher-dimensional gauge theory.
- Geometry and Topology ( 114 Views )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.
Irina Kogan : Geometry of Hyperbolic Conservation Laws
- Geometry and Topology ( 105 Views )We consider the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed geometry in state space: the eigenvectors of the Jacobian of the flux are given. This is formulated as a system of algebraic-differential equations whose solution space is analyzed using Darboux and Cartan-K\"ahler theorems. It turns out that already the case with three equations is fairly complex. We give a complete list of possible scenarios for the general systems of two and three equations and for rich systems (i.e. when the given eigenvector fields are pairwise in involution) of arbitrary size. As an application we characterize conservative systems with the same eigencurves as compressible gas dynamics.
This is joint work with Kris Jenssen (Penn State University)