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.
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).
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.)
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.
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.
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.
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.
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.
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.
Igor Zelenko (Texas A&M U) : Gromovs h-principle for corank two distribution of odd rank with maximal first Kronecker index
- Geometry and Topology ( 150 Views )While establishing various versions of the h-principle for contact distributions (Eliashberg (1989) in dimension 3, Borman-Eliashberg-Murphy (2015) in arbitrary dimension, and even-contact contact (D. McDuff, 1987) distributions are among the most remarkable advances in differential topology in the last four decades, very little is known about analogous results for other classes of distributions, e.g. generic distributions of corank 2 or higher. The smallest dimensional nontrivial case of corank 2 distributions is Engel distributions, i.e. the maximally nonholonomic rank 2 distributions on $4$-manifolds. This case is highly nontrivial and was treated recently by Casals-Pérez-del Pino-Presas (2017) and Casals-Pérez-Presas (2017). In my talk, I will show how to use the method of convex integration in order to establish all versions of the h-principle for corank 2 distributions of arbitrary odd rank satisfying a natural generic assumption on the associated pencil of skew-symmetric forms. During the talk, I will try to give all the necessary background related to the method of convex integration in principle. This is the joint work with Milan Jovanovic, Javier Martinez-Aguinaga, and Alvaro del Pino.
Jason Parsley : Petal Links
- Geometry and Topology ( 144 Views )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.
Carla Cederbaum : From Newton to Einstein: a guided tour through space and time
- Geometry and Topology ( 144 Views )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.
Adam Levine : Concordance of knots in homology spheres
- Geometry and Topology ( 143 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.
Matt Kerr : Normal Functions over Locally Symmetric Varieties
- Geometry and Topology ( 142 Views )
An algebraic cycle homologous to zero on a variety leads to an extension of Hodge-theoretic data. In a variational context, the resulting section of a bundle of complex tori is called a normal function, and is used to study cycles modulo rational or algebraic equivalence.
The archetype for interesting normal functions arises from the Ceresa cycle, consisting of the difference of two copies of a curve in its Jacobian. The profound geometric consequences of its existence are evidenced in work of Nori, Hain and (most recently) Totaro. In contrast, a theorem of Green and Voisin demonstrates the *absence* of normal functions arising from cycles on very general projective hypersurfaces of large enough degree.
Inspired by recent work of Friedman-Laza on Hermitian variation of Hodge structure and Oort's conjecture on special subvarieties in the Torelli locus, R. Keast and I wondered about the existence of normal functions over etale neighborhoods of Shimura varieties. In this talk I will explain our classification of the cases where a Green-Voisin analogue does *not* hold, and where one expects interesting cycles (and generalized cycles) to occur. I will also give evidence that these predictions might be "sharp", and draw some geometric consequences.
Michael Henry : Connections between existing Legendrian knot invariants
- Geometry and Topology ( 141 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.
Thalia Jeffres : K\{a}hler-Einstein Metrics with Edge Singularities
- Geometry and Topology ( 140 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.
Dan Rutherford : Augmentations and immersed Lagrangian fillings
- Geometry and Topology ( 140 Views )This is joint work with Y. Pan that applies previous joint work with M. Sullivan. Let $\Lambda \subset \mathbb{R}^{3}$ be a Legendrian knot with respect to the standard contact structure. The Legendrian contact homology (LCH) DG-algebra, $\mathcal{A}(\Lambda)$, of $\Lambda$ is functorial for exact Lagrangian cobordisms in the symplectization of $\mathbb{R}^3$, i.e. a cobordism $L \subset \mathit{Symp}(\mathbb{R}^3)$ from $\Lambda_-$ to $\Lambda_+$ induces a DG-algebra map, $f_L:\mathcal{A}(\Lambda_+) \rightarrow \mathcal{A}(\Lambda_-).$ In particular, if $L$ is an exact Lagrangian filling ($\Lambda_-= \emptyset$) the induced map is an augmentation $\epsilon_L: \mathcal{A}(\Lambda_+) \rightarrow \mathbb{Z}/2.$ In this talk, I will discuss an extension of this construction to the case of immersed, exact Lagrangian cobordisms based on considering the Legendrian lift $\Sigma$ of $L$. When $L$ is an immersed, exact Lagrangian filling a choice of augmentation $\alpha$ for $\Sigma$ produces an induced augmentation $\epsilon_{(L, \alpha)}$ for $\Lambda_+$. Using the cellular formulation of LCH, we are able to show that any augmentation of $\Lambda$ may be induced by such a filling.
Ana-Maria Brecan : On the intersection pairing between cycles in SU(p,q)-flag domains and maximally real Schubert varieties
- Geometry and Topology ( 139 Views )An SU(p, q)-flag domain is an open orbit of the real Lie group SU(p, q) acting on the complex flag manifold associated to its complexification SL(p + q, C). Any such flag domain contains certain compact complex submanifolds, called cycles, which encode much of the topological, complex geometric and repre- sentation theoretical properties of the flag domain. This talk is concerned with the description of these cycles in homology using a specific type of Schubert varieties. They are defined by the condition that the fixed point of the Borel group in question is in the closed SU(p,q)-orbit in the ambient manifold. We consider the Schubert varieties of this type which are of com- plementary dimension to the cycles. It is known that if such a variety has non-empty intersection with a certain base cycle, then it does so transversally (in finitely many points). With the goal of understanding this duality, we describe these points of intersection in terms of flags as well as in terms of fixed points of a given maximal torus. The relevant Schubert varieties are described in terms of Weyl group elements.