Yang Li : On the Donaldson-Scaduto conjecture
- Geometry and Topology ( 708 Views )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.
Demetre Kazaras : If Ricci is bounded below, then mass is in control!
- Geometry and Topology ( 553 Views )The ADM mass of an isolated gravitational system is a geometric invariant measuring the total mass due to matter and other fields. In a previous work, we showed how to compute this invariant (in 3 spatial dimensions) by studying harmonic functions. Now I'll use this formula to consider the question: How flat is an asymptotically flat manifold with very little total mass? In the presence of a lower bound on Ricci curvature, we make progress on this question and confirm special cases of conjectures made by Ilmanen and Sormani.
Oguz Savk : Bridging the gaps between homology planes and Mazur manifolds.
- Geometry and Topology,Uploaded Videos ( 315 Views )We call a non-trivial homology 3-sphere a Kirby-Ramanujam sphere if it bounds a homology plane, an algebraic complex smooth surface with the same homology groups of the complex plane. In this talk, we present several infinite families of Kirby-Ramanujam spheres bounding Mazur type 4-manifolds, compact contractible smooth 4-manifolds built with only 0-, 1-, and 2-handles. Such an interplay between complex surfaces and 4-manifolds was first observed by Ramanujam and Kirby around nineteen-eighties. This is upcoming joint work with Rodolfo Aguilar Aguilar.
Luca Di Cerbo : Extended Graph Manifolds, Dehn Fillings, and Einstein Metrics
- Geometry and Topology ( 301 Views )In this talk, I will present some new topological obstructions for solving the Einstein equations (in Riemannian signature) on a large class of closed four-manifolds. Finally, I will show that complex-hyperbolic Einstein Dehn filling compactification cannot possibly performed in dimension four. This is in striking contrast with the real-hyperbolic case, and it answers (negatively) a fifteen years old question of Michael Anderson. If time permits, I will conclude with some tantalizing open problems both in dimension four and in higher dimensions. Part of this work is joint with M. Golla (Universit\’e de Nantes).
Curtis Porter : Spinning Black Holes and CR 3-Folds
- Geometry and Topology ( 276 Views )Some physically significant solutions to Einstein's field equations are spacetimes which are foliated by a family of curves called a shear-free null geodesic congruence (SFNGC). Examples include models of gravitational waves that were recently detected, and rotating black holes. The properties of a SFNGC induce a CR structure on the 3-dimensional leaf space of the foliation. The Kerr Theorem says that the family of metrics associated to a SFNGC contains a conformally flat representative iff the corresponding CR structure is embeddable in a real hyperquadric. Using Cartan's method of moving frames, we can classify which Levi-nondegenerate CR 3-folds are embeddable in the hyperquadric.
Nelia Charalambous : On the $L^p$ Spectrum of the Hodge Laplacian on Non-Compact Manifolds
- Geometry and Topology ( 271 Views )One of the central questions in Geometric Analysis is the interplay between the curvature of the manifold and the spectrum of an operator. In this talk, we will be considering the Hodge Laplacian on differential forms of any order $k$ in the Banach Space $L^p$. In particular, under sufficient curvature conditions, it will be demonstrated that the $L^p\,$ spectrum is independent of $p$ for $1\!\leq\!p\!\leq\! \infty.$ The underlying space is a $C^{\infty}$-smooth non-compact manifold $M^n$ with a lower bound on its Ricci Curvature and the Weitzenb\"ock Tensor. The further assumption on subexponential growth of the manifold is also necessary. We will see that in the case of Hyperbolic space the $L^p$ spectrum does in fact depend on $p.$ As an application, we will show that the spectrum of the Laplacian on one-forms has no gaps on certain manifolds with a pole and on manifolds that are in a warped product form. This will be done under weaker curvature restrictions than what have been used previously; it will be achieved by finding the $L^1$ spectrum of the Laplacian.
Daniel Stern : Spectral shape optimization and new behaviors for free boundary minimal surfaces
- Geometry and Topology ( 259 Views )Though the study of isoperimetric problems for Laplacian eigenvalues dates back to the 19th century, the subject has undergone a renaissance in recent decades, due in part to the discovery of connections with harmonic maps and minimal surfaces. By the combined work of several authors, we now know that unit-area metrics maximizing the first nonzero Laplace eigenvalue exist on any closed surface, and are realized by minimal surfaces in spheres. At the same time, work of Fraser-Schoen, Matthiesen-Petrides and others yields analogous results for the first eigenvalue of the Dirichlet-to-Neumann map on surfaces with boundary, with maximizing metrics induced by free boundary minimal immersions into Euclidean balls. In this talk, I'll describe a series of recent results characterizing the (perhaps surprising) asymptotic behavior of these free boundary minimal immersions (and associated Steklov-maximizing metrics) as the number of boundary components becomes large. (Based on joint work with Mikhail Karpukhin.)
John McCuan : Minimal graphs with jump discontinuities
- Geometry and Topology ( 254 Views )I will discuss some examples of minimal graphs with jump discontinuities in their boundaries. Robert Huff and I constructed these examples in response to a question of John Urbas: Is it possible for a minimal graph over a smooth annular domain to have an isolated jump discontinuity on the inner boundary component? I will also give a brief overview of the boundary consistency problem for Di Giorgi's generalized solutions of the minimal surface equation and discuss this question in that context. The construction of the examples uses the Weierstrass representation and the developing map introduced by Huff in the study of capillary problems.
Siqi He : Classification of Nahm Pole Solutions to the KW Equations on $S^1\times\Sigma\times R^+$
- Geometry and Topology ( 251 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.
Subhankar Dey : Cable knots are not thin
- Geometry and Topology ( 246 Views )Thurston's geometrization conjecture and its subsequent proof for Haken manifolds distinguish knots in S^3 by the geometries in the complement of the knots. While the definition of alternating knots make use of nice knot diagrams, Knot Floer homology, a knot invariant toolbox, defined by Ozsvath-Szabo and Rasumussen, generalizes the definition of alternating knots in the context of knot Floer homology and defines family of quasi-alternating knots which contains all alternating knots. Using Lipshitz-Ozsvath-Thurston's bordered Floer homology, we prove a partial affirmation of a folklore conjecture in knot Floer theory, which bridges these two viewpoints of looking at knots.
Tye Lidman : Homology cobordisms with no 3-handles
- Geometry and Topology ( 243 Views )Homology cobordisms are a special type of manifold which are relevant to a variety of areas in geometric topology, including knot theory and triangulability. We study the behavior of a variety of invariants under a particular family of four-dimensional homology cobordisms which naturally arise from Stein manifolds. This is joint work with Ali Daemi, Jen Hom, Shea Vela-Vick, and Mike Wong.
Daniel Stern : Scalar curvature and circle-valued harmonic maps
- Geometry and Topology ( 242 Views )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.
Mark Stern : Instanton Decay
- Geometry and Topology ( 242 Views )The inverse square law is fundamental to our understanding of electromagnetism. The question of the decay of the fields associated to nonabelian gauge theories is more difficult because the equations determining these fields are nonlinear. In this talk, I will discuss recent progress in establishing decay rates for instantons.
Isaac Sundberg : The Khovanov homology of slice disks
- Geometry and Topology ( 236 Views )To a cobordism between links, Khovanov homology assigns a linear map that is invariant under boundary-preserving isotopy of the cobordism. In this talk, we study those maps arising from surfaces in the 4-ball and apply our findings to existence and uniqueness questions regarding slice disks bounding a given knot. This reflects joint works with Jonah Swann and Kyle Hayden.
Richard Hain : The Lie Algebra of the Mapping Class Group, Part 1
- Geometry and Topology ( 223 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.
Amit Einav : Entropic Inequality on the Sphere
- Geometry and Topology ( 222 Views )It is an interesting well known fact that the relative entropy with respect to the Gaussian measure on $\mathbb{R}^N$ satisfies a simple subadditivity property. Namely, if $\Pi_1^{(i)}(F_N)$ is the first marginal of the density function F_N in the i-th variable then \begin{equation} \sum_{i=1}^N H(\Pi_1^{(i)}(F_N) | \gamma_1) \leq H(F_N | \gamma_N), \end{equation} where $\gamma_k$ is the standard Gaussian on $\mathbb{R}^k$. Surprisingly enough, when one tries to achieve a similar result on $\mathbb{S}^{N-1}(\sqrt{N})$ a factor of 2 appears in the right hand side of the inequality (a result due to Carlen, Lieb and Loss), and the constant is sharp. Besides a deviation from the simple equivalence of ensembles principle in equilibrium Statistical Mechanics, this entropic inequality on the sphere has interesting ramifications in other fields, such as Kinetic Theory. In this talk we will present conditions on the density function F_N, on the sphere, under which we can get an almost subaditivity property; i.e. the factor 2 can be replaced with a factor of $1+\epsilon_N$, with $\epsilon_N$ given explicitly and going to zero. The main tools to be used in order to proved this result are an entropy conservation extension of F_N to $\mathbb{R}^N$ together with comparison of appropriate transportation distances such as the entropy, Fisher information and Wasserstein distance between the marginal of the original density and that of the extension. Time permitting, we will give an example, one that arises naturally in the investigation of the so-called Kac Model, to many families of functions that satisfy these conditions.
Richard Hain : The Lie Algebra of the Mapping Class Group, Part 2
- Geometry and Topology ( 218 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.
John Berman : Measuring Ramification with Topological Hochschild Homology
- Geometry and Topology ( 196 Views )Topological Hochschild homology (THH) has recently been popular as an approximation to algebraic K-theory, but it is also a measure of ramification in the sense of number theory. I will survey the interaction between THH and number theory, along with some surprising connections to classical algebraic topology. This will culminate in a new computation of THH of any ring of integers R, suggesting the philosophy: Spec(R) -> Spec(Z) is one point away from being etale.
Colleen Robles : A refinement of the Lefschetz decomposition for hyperkahler manifolds
- Geometry and Topology ( 191 Views )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.
Mark Stern : Stability, dynamics, and the quantum Hodge theory of vector bundles
- Geometry and Topology ( 187 Views )I will discuss various approaches to the question: When does a vector bundle admit a holomorphic structure? I will explore applications of Yang-Mills theory, geometric quantization, and discrete dynamics to this problem.
Alexander Volkmann : Nonlinear mean curvature flow with Neumann boundary condition
- Geometry and Topology ( 178 Views )Using a level set formulation and elliptic regularization we define a notion of weak solutions of nonlinear mean curvature flow with Neumann boundary condition. We then outline the proof of an existence result for the weak level set flow. Finally, we discuss some geometric applications of this flow.
Mark Stern : Nahm transforms and ALF Spaces
- Geometry and Topology ( 175 Views )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
Mark Stern : Geometry of stable Yang-Mills connections
- Geometry and Topology ( 174 Views )On a compact 4-manifold, every self-dual connection and every anti-self-dual connection minimizes the Yang-Mills energy. In this talk, I will answer the converse question for compact homogeneous 4-manifolds. I will also survey related stability results in other dimensions.
Julio Rebelo : On closed currents invariant by holomorphic foliations
- Geometry and Topology ( 168 Views )Let M be an algebraic complex surface equipped with a singular foliation F. We assume that F leaves invariant a closed current on M or, equivalently, that F possesses a transversely invariant measure. The purpose of this talk is two-fold. First we want to classify the pairs (M, F) as above, a problem that is usually regarded as a step towards developing a suitable Ergodic Theory for these foliations. On the other hand we want to explain the connection of this problem with the Kobayashi hyperbolicity of general type surfaces. In particular we shall sketch a new proof of McQuillan's theorem proving the Green-Griffiths conjecture for general type surfaces having positive Segre class.
Justin Sawon : Holomorphic coisotropic reduction
- Geometry and Topology ( 162 Views )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.
Julian Chaidez : Essential tori In spaces of symplectic embeddings
- Geometry and Topology ( 157 Views )The problem of when and how one symplectic manifold can be symplectically embedded into another is notoriously subtle, even when the spaces in question are relatively simple. Gromov's non-squeezing theorem and McDuff's Fibonacci staircase are examples of this phenomenon. One can interpret these results as realizing the principle that "variations of quantitative symplectic parameters alter the topology of symplectic embedding spaces." In this talk, we explain recent work (joint with Mihai Munteanu) showing that certain n-torus families of symplectic embeddings between 2n-d ellipsoids become homologically essential if certain quantitative invariants are close enough. We will also discuss work in progress in which we use similar methods to study Lagrangian embeddings.
Emma Carberry : Conformal Surface Geometry: an algebro-geometric approach.
- Geometry and Topology ( 156 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.
Lisa Traynor : Legendrian Torus Links
- Geometry and Topology ( 153 Views )Legendrian torus knots were classified by Etnyre and Honda. In particular, for any smooth torus knot we know the mountain range that lists all of its Legendrian representatives. I will discuss the classification of Legendrian torus links. In this classification, a natural realization question arises: what n-tuples of points on the mountain range of a (p,q)-torus knot can occur in a Legendrian (np, nq)-torus link? Another part of the classification is to understand how many different ways an n-tuple on the mountain range can be realized as an ordered link. In particular, for Legendrian representatives of an (np,nq)-torus link is it possible to do invariant preserving permutations of the components? This is joint work with Jennifer Dalton and John Etnyre.
Lisa Piccirillo : The Conway knot is not slice
- Geometry and Topology ( 153 Views )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. Ill 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.