Jesse Madnick : The Mean Curvature of (Co)Associative Submanifolds
- Geometry and Topology ( 133 Views )In flat R^7, two classes of submanifolds stand out: the associative 3-folds and coassociative 4-folds, which enjoy the remarkable property of being area-minimizing in their homology class. In fact, these submanifolds make sense in any 7-manifold with a G2-structure, and it is natural to ask: Under what conditions to do they continue to be minimal? We answer this question by deriving pleasantly simple formulas for their mean curvature. Time permitting, we will explain how these formulas suggest new avenues for the construction of minimal submanifolds of high codimension. This is joint work with Gavin Ball.
Dan Rutherford : Augmentations and immersed Lagrangian fillings
- Geometry and Topology ( 133 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.
Ahmad Issa : Embedding Seifert fibered spaces in the 4-sphere
- Geometry and Topology ( 132 Views )Which 3-manifolds smoothly embed in the 4-sphere? This seemingly simple question turns out to be rather subtle. Using Donaldson's theorem, we derive strong restrictions to embedding a Seifert fibered space over an orientable base surface, which in particular gives a complete classification when e > k/2, where k is the number of exceptional fibers and e is the normalized central weight. Our results point towards an interesting conjecture which I'll discuss. This is joint work with Duncan McCoy.
Gonçalo Oliveira : Gauge theory on Aloff-Wallach spaces
- Geometry and Topology ( 190 Views )I will describe joint work with Gavin Ball where we classify certain G2-Instantons on Aloff-Wallach spaces. This classification can be used to test ideas and explicitly observe various interesting phenomena. For instance, we can: (1) Vary the underlying structure and find out what happens to the G2-instantons along the way; (2) Distinguish certain G2-structures (called nearly parallel) using G2-Instantons; (3) Find G2-Instantons, with respect to these structures, which are not absolute minima of the Yang-Mills functional.
Catherine Searle : Torus actions, maximality, and non-negative curvature
- Geometry and Topology ( 152 Views )The classification of compact Riemannian manifolds with positive or non-negative sectional curvature is a long-standing problem in Riemannian geometry. One successful approach has been the introduction of symmetries, and an important first case to understand is that of continuous abelian symmetries. In recent work with Escher, we obtained an equivariant diffeomorphism classification of closed, simply-connected non-negatively curved Riemannian manifolds admitting an isotropy-maximal torus action, with implications for the Maximal Symmetry Rank Conjecture for non-negatively curved manifolds. I will discuss joint work with Escher and Dong, that builds on this work to extend the classification to those manifolds admitting an almost isotropy-maximal action.
Curtis Porter : Spinning Black Holes and CR 3-Folds
- Geometry and Topology ( 253 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.
Lisa Piccirillo : The Conway knot is not slice
- Geometry and Topology ( 139 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.
Ina Petkova : Knot Floer homology and the gl(1|1) link invariant
- Geometry and Topology ( 114 Views )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.
David Duncan : The Chern-Simons invariants for general compact Lie groups
- Geometry and Topology ( 151 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.
Hannah Schwartz : Using 2-torsion to obstruct topological isotopy
- Geometry and Topology ( 122 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.
Honghao Gao : Augmentations and sheaves for knot conormals
- Geometry and Topology ( 104 Views )Knot invariants can be defined using Legendrian isotopy invariants of the knot conormal. There are two types of invariants raised in this way: one is the knot contact differential graded algebra together with augmentations associated to this dga, and the other one is the category of simple sheaves microsupported along the knot conormal. The Nadler-Zaslow correspondence suggests a connection between the two types of invariants. In this talk, I will manifest an explicit map between augmentations and simple sheaves.
Sahana Balasubramanya : Hyperbolic structures on wreath products
- Geometry and Topology ( 115 Views )The poset of hyperbolic structures on a group G is still very far from being understood and several questions remain unanswered. In this talk, I will speak about some new results that describe hyperbolic structures on the wreath product Gwr Z, for any group G. As a consequence, I answer two open questions regarding quasi-parabolic structures: I will give an example of a group G with an uncountable chain of quasi-parabolic structures and give examples of groups that have finitely many quasi-parabolic structures.
Thomas Mark : 3-manifolds not obtained by surgery on a knot
- Geometry and Topology ( 116 Views )A well-known theorem of Lickorish and Wallace states that any closed orientable 3-manifold can be obtained by surgery on a link in the 3-sphere. For a given 3-manifold one can ask how "simple" a link can be used to obtain it, e.g., whether a manifold satisfying certain obvious necessary conditions on its fundamental group always arises by surgery on a knot. This question turns out to be rather subtle, and progress has been limited, but in general the answer is known to be "no." Here Ill summarize some recent results including joint work with Matt Hedden, Min Hoon Kim, and Kyungbae Park that give the first examples of 3-manifolds with the homology of S^1 x S^2 and having fundamental group of weight 1 that do not arise by surgery on a knot in the 3-sphere.
Alex Waldron : Yang-Mills flow on special holonomy manifolds
- Geometry and Topology ( 197 Views )I will describe an upcoming paper with Goncalo Oliveira investigating the properties of Yang-Mills flow on base manifolds with restricted holonomy, generalizing known results from the 4-d and Kahler cases. We show that finite-time blowup is governed by the F^7 component of the curvature in the G_2 and Spin(7) cases, and by the appropriate curvature component in the remaining cases on Berger's list. Assuming that this component remains bounded along the flow, we show that the infinite-time bubbling set is calibrated by the defining (n-4)-form.
Laura Fredrickson : The asymptotic geometry of the Hitchin moduli space
- Geometry and Topology ( 111 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.
Michael Lipnowski : The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds
- Geometry and Topology ( 108 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.
Gavin Ball : Quadratic closed G2-structures
- Geometry and Topology ( 142 Views )A closed G2-structure is a certain type of geometric structure on a 7-manifold M, given by a 'non-degenerate' closed 3-form. The local geometry of closed G2-structures is non-trivial, in contrast to the perhaps more familiar case of symplectic structures (where we instead have a non-degenerate closed 2-form). In particular, any closed G2-structure automatically induces a Riemannian metric on M. I will talk about a special class of closed G2-structures, those satisfying a further 'quadratic' condition. This is a second order PDE system first written down by Bryant that can be interpreted as a condition on the Ricci curvature of the induced metric. I will focus mainly on the case where the G2-structure is 'extremally Ricci-pinched', giving new examples and describing an unexpected relationship with maximal submanifolds in a certain negatively curved pseudo-Riemannian symmetric space.
Richard Hain : Hodge theory and the Goldman-Turaev Lie bialgebra
- Geometry and Topology ( 185 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.
Faramarz Vafaee : Floer homology and Dehn surgery
- Geometry and Topology ( 145 Views )The past thirty years have witnessed the birth of a beautiful array of approaches to the field of low dimensional topology, drawing on diverse tools from algebra, analysis, and combinatorics. One particular tool that has made a dramatic impact on the field is the Heegaard Floer theory of Ozsvath and Szabo. Defined 17 years ago, this theory has produced an encompassing package of invariants, which have significantly impacted the study of many areas of low dimensional topology, including Dehn surgery. In this talk, we will focus on two questions: a) which 3-manifolds do arise by Dehn surgery along a knot in the 3-sphere? b) what are all ways to obtain a fixed 3-manifold by Dehn surgery along a knot in the 3-sphere?
Steven Rayan : Asymptotic geometry of hyperpolygons
- Geometry and Topology ( 105 Views )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.
Jason Parsley : Petal Links
- Geometry and Topology ( 137 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.
Akram Alishahi : Trivial tangles, compressible surfaces and Floer homology
- Geometry and Topology ( 110 Views )Heegaard Floer homology has different extensions for 3-manifolds with boundary. In this talk, we will recall some basics of these extensions and explain how they can be used to give a computationally effective way for detecting boundary parallel components of tangles, and existence of homologically essential compressing disks. The fact that these are checkable by computer, is based on the factoring algorithm of Lipshitz-Ozsvath-Thurston for computing bordered Floer homology, and our extension of it to compute bordered-sutured Floer homology. This is joint work with Robert Lipshitz.
Melissa Zhang : Annular Khovanov homology and 2-periodic links
- Geometry and Topology ( 101 Views )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.
Adam Saltz : Link homology and Floer homology in pictures by cobordisms
- Geometry and Topology ( 118 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!
Ákos Nagy : From instantons to vortices on spherically symmetric ALF manifolds
- Geometry and Topology ( 114 Views )Yang-Mills theory on Asymptotically Locally Flat (ALF) 4-manifolds has been intensely studied by geometers and physicists since the late 70's. The most important examples are R^3 x S^1, the (multi-)Taub-NUT spaces, and the Euclidean Schwarzschild manifold. In this talk, I will outline the correspondence between spherically symmetric Yang-Mills instantons and planar Abelian vortices (following the ideas of Witten, Taubes, and Garcia-Prada), and then apply this instanton-vortex duality to spherically symmetric ALF 4-manifolds. Finally, I will show how this construction can be used to describe the low energy instanton moduli spaces of the Euclidean Schwarzschild manifold, and its generalizations. This is a joint work with Gonçalo Oliveira (IMPA).
Juanita Pinzon Caicedo : Gauge Theory and Knot Concordance
- Geometry and Topology ( 112 Views )Knot concordance can be regarded as the study of knots as boundaries of surfaces embedded in spaces of dimension 4. Specifically, two knots K_0 and K_1 are said to be smoothly concordant if there is a smooth embedding of the 2--dimensional annulus S^1 × [0, 1] into the 4--dimensional cylinder S^3 × [0, 1] that restricts to the given knots at each end. Smooth concordance is an equivalence relation, and the set of smooth concordance classes of knots, C, is an abelian group with connected sum as the binary operation. The algebraic structure of C, the concordance class of the unknot, and the set of knots that are topologically slice but not smoothly slice are much studied objects in low-dimensional topology. Gauge theoretical results on the nonexistence of certain definite smooth 4-manifolds can be used to better understand these objects. In this talk I will explain how the study of anti-self dual connections on 4--manifolds can be used to shown that (1) the group of topologically slice knots up to smooth concordance contains a subgroup isomorphic to Z^\infty, and (2) satellite operations that are similar to cables are not homomorphisms on C.
Yu Wang : Quantitative stratification of stationary Yang-Mills and recent progress on global gauge problem
- Geometry and Topology ( 138 Views )Given a stationary Yang-Mills connection A, we are interested in studying its singular structure. In this talk we introduce a quantitative way to stratify the singular sets. Our main results include a Minkowski Volume estimate and the rectifiability of this quantitative stratification, which leads to the rectifiability of the classical stratifications S^k(A) for all integer k. To be precise, we first recall certain background preliminaries needed for this talk. After giving the statements of the main results, I will briefly describe the machinary used in the proof, and explain the new points and the major difficulty that we have faced. The main results in the talk are based on a work by myself last year. If time allows I will further discuss some open problems regarding global gauge in this field, and recent progress in those directions made jointly with Aaron Naber.
Nan Wu : Length of a shortest closed geodesic in manifolds of dimension 4
- Geometry and Topology ( 94 Views )In this talk, we show that for any closed 4-dimensional simply-connected Riemannian manifold $M$ with Ricci curvature $|Ric| \leq 3$, volume $vol(M)>v>0$ and diameter $diam(M) \leq D$, the length of a shortest closed geodesic on $M$ is bounded by a function $F(v,D)$ . The proof of this result is based on the diffeomorphism finiteness theorem for the manifolds satisfying above conditions proved by J. Cheeger and A. Naber. This talk is based on the joint work with Zhifei Zhu.
Tristan Collins : Sasaki-Einstein metrics and K-stability
- Geometry and Topology ( 102 Views )I will discuss the connection between Sasaki-Einstein metrics and algebraic geometry in the guise of K-stability. In particular, I will give a differential geometric perspective on K-stability which arises from the Sasakian view point, and use K-stability to find infinitely many non-isometric Sasaki-Einstein metrics on the 5-sphere. This is joint work with G. Szekelyhidi.
Daniel Scofield : Patterns in Khovanov link and chromatic graph homology
- Geometry and Topology ( 100 Views )Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. In this talk, we discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme-Guizon, Przytycki and Rong. An explicit formula for the rank of the third chromatic homology group on the main diagonal is given and used to compute the corresponding Khovanov homology group and the fourth coefficient of the Jones polynomial for links with certain diagrams.