Aleksander Doan : Seiberg-Witten multi-monopoles on Riemann surfaces
- Geometry and Topology ( 99 Views )I will discuss a generalization of the Seiberg-Witten equations on 3-manifolds and its relation to higher-dimensional gauge theory. The main new feature is the non-compactness of the moduli space of solutions. I will explain how to tackle this problem and count the solutions when the 3-manifold is the product of a surface and a circle. In this case, the problem of compactness reduces to studying degenerations of solutions to a non-linear scalar PDE resembling the Kazdan-Warner equation.
Josh Sabloff : Topologically Distinct Lagrangian Fillings and the Generating Family Homology Number
- Geometry and Topology ( 112 Views )We construct Legendrian submanifolds with arbitrarily many topologically distinct Lagrangian fillings, thereby (secretly) answering a question about intersections of complex curves with the 4-ball asked by Boileau and Fourrier. These fillings are then combined with a TQFT-like theory for Lagrangian cobordisms between Legendrian submanifolds to produce interesting consequences for some non-classical invariants of the Legendrian submanifolds with topologically distinct fillings. Various parts of this talk are joint work with Traynor, Bourgeois-Traynor, and Cao-Gallup-Hayden.
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.
Renato Bettiol : Positive biorthogonal curvature in dimension 4
- Geometry and Topology ( 354 Views )A 4-manifold is said to have positive biorthogonal curvature if the average of sectional curvatures of any pair of orthogonal planes is positive. In this talk, I will describe a construction of metrics with positive biorthogonal curvature on the product of spheres, and then combine it with recent surgery stability results of Hoelzel to classify (up to homeomorphism) the closed simply-connected 4-manifolds that admit a metric with positive biorthogonal curvature.
Erik Van Erp : Index theory on contact manifolds and noncommutative topology
- Geometry and Topology ( 101 Views )In the early 1960s Atiyah and Singer derived a cohomological formula that computes the Fredholm index of an elliptic differential operator. The subsequent development of analytic K-theory of noncommutative C*-algebras greatly clarified the proof of the index formula, leading to many further generalizations. As a recent application of these techniques I will discuss the solution of the index problem for certain hypoelliptic operators on contact manifolds, first proposed by Epstein and Melrose. The final topological formula is quite easy to state, but the proof relies heavily on noncommutative techniques.
Mark Stern : Nahm transforms and ALF Spaces
- Geometry and Topology ( 157 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
John Berman : Measuring Ramification with Topological Hochschild Homology
- Geometry and Topology ( 180 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.
Claude LeBrun : Four-Dimensional Einstein Manifolds, and Beyond
- Geometry and Topology ( 161 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.
Bianca Santoro : Complete Kahler metrics on crepant resolutions of singular Calabi-Yau spaces
- Geometry and Topology ( 114 Views )In this talk, we plan to explain some existence results for complete Ricci-flat \kahler metrics on crepant resolutions of singularities. The method allows us to provide a wider class of examples of complete Ricci-flat Kahler metrics with richer topoplogy at infinity.
Daniel Stern : Spectral shape optimization and new behaviors for free boundary minimal surfaces
- Geometry and Topology ( 243 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.)
Subhankar Dey : Cable knots are not thin
- Geometry and Topology ( 239 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.
Christopher Cornwell : Polynomials, grid diagrams, and Legendrian knots in lens spaces
- Geometry and Topology ( 103 Views )We discuss a HOMFLY polynomial invariant for links in lens spaces. We then show how this polynomial is related to the contact invariants of Legendrian and transverse links in lens spaces having a certain tight contact structure. In fact, we generalize a result of Ng, casting Bennequin-type inequalities in such contact lens spaces into a general framework.
Simon Brendle : Minimal Lagrangian diffeomorphisms between domains in the hyperbolic plane
- Geometry and Topology ( 149 Views )Let $\Omega$ and $\tilde{\Omega}$ be domains in the hyperbolic plane with smooth boundary. Assume that both domains are uniformly convex, and have the same area. We show that there exists an area-preserving, orientation-preserving diffeomorphism $f: \Omega \to \tilde{\Omega}$ such that the graph of $f$ is a minimal surface in $\mathbb{H}^2 \times \mathbb{H}^2$. Moreover, we show that the set of all such diffeomorphisms is parametrized by the circle.
Dmitry Khavinson : From the Fundamental Theorem of Algebra to Astrophysics: a Harmonious Path
- Geometry and Topology ( 115 Views )The Fundamental Theorem of Algebra first rigorously proved by Gauss states that each complex polynomial of degree n has precisely n complex roots. In recent years various extensions of this celebrated result have been considered. We shall discuss the extension of the FTA to harmonic polynomials of degree n. In particular, the 2003 theorem of D. Khavinson and G. Swiatek that shows that the harmonic polynomial z − p(z); deg p = n > 1 has at most 3n − 2 zeros as was conjectured in the early 90's by T. Sheil-Small and A. Wilmshurst. More recently L. Geyer was able to show that the result is sharp for all n.
In 2004 G. Neumann and D. Khavinson showed that the maximal number of zeros of rational harmonic functions z − r(z); deg r = n > 1 is 5n − 5. It turned out that this result conrfimed several consecutive conjectures made by astrophysicists S. Mao, A. Petters, H. Witt and, in its final form, the conjecture of S. H. Rhie that were dealing with the estimate of the maximal number of images of a star if the light from it is deflected by n co-planar masses. The first non-trivial case of one mass was already investigated by A. Einstein around 1912.
We shall also discuss the problem of gravitational lensing of a point source of light, e.g., a star, by an elliptic galaxy, more precisely the problem of the maximal number of images that one can observe. Under some more or less "natural" assumptions on the mass distribution within the galaxy one can prove that the number of visible images can never be more than four in some cases and six in the other. Interestingly, the former situation can actually occur and has been observed by astronomers. Still there are much more open questions than there are answers.
Ralph Howard : Tangent cones and regularity of real hypersurfaces
- Geometry and Topology ( 116 Views )We characterize $C^1$ embedded hypersurfaces of $R^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most $m < 3/2$. It follows any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is $C^1$. In the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface. In particular, any convex real analytic hypersurface $X$ of $R^n$ is $C^1$. Furthermore, if $X$ is real algebraic, strictly convex, and unbounded then its projective closure is a $C^1$ hypersurface as well, which shows that $X$ is the graph of a function defined over an entire hyperplane. This is joint work with Mohammad Ghomi.
Greg Galloway : Rigidity and positivity of mass for asymptotically hyperbolic manifolds
- Geometry and Topology ( 147 Views )We discuss an approach to the proof of positivity of mass without spin assumption, for asymptotically hyperbolic Riemannian manifolds, based on the general methodology of Schoen and Yau. Our approach makes use of the "BPS brane action" introduced by Witten and Yau in their work on the AdS/CFT correspondence, and takes hints from work of Lohkamp. This is joint work with Lars Andersson and Mingliang Cai.
Luca Di Cerbo : Seiberg-Witten equations on manifolds with cusps and geometric applications.
- Geometry and Topology ( 195 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.
Natasa Sesum : On the extension of the mean curvature flow and the Ricci flow
- Geometry and Topology ( 106 Views )In the talk we will discuss curvature conditions under which we can guarantee the existence of a smooth solution to the Ricci flow and the mean curvature flow equation. These are improvements of Hamilton's and Husiken's results on extending the Ricci flow and the mean curvature flow, under conditions that the norm of Riemannian curvature and the norm of the second fundamental form are uniformly bounded along the flow in finite time, respectively.
Michael Henry : Connections between existing Legendrian knot invariants
- Geometry and Topology ( 126 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.
Calvin McPhail-Snyder : Making the Jones polynomial more geometric
- Geometry and Topology ( 351 Views )The colored Jones polynomials are conjectured to detect geometric information about knot complements, such as hyperbolic volume. These relationships ("volume conjectures") are known in a number of special cases but are in general quite mysterious. In this talk I will discuss a program to better understand them by constructing holonomy invariants, which depend on both a knot K and a representation of its knot group into SL_2(C). By defining a version of the Jones polynomial that knows about geometric data, we hope to better understand why the ordinary Jones polynomial does too. Along the way we can obtain more powerful quantum invariants of knots and other topological objects.
Curtis Porter : Straightening out degeneracy in CR geometry: When can it be done?
- Geometry and Topology ( 201 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).
Jonathan Hanselman : Bordered Heegaard Floer homology and graph manifolds
- Geometry and Topology ( 108 Views )Heegaard Floer homology is a powerful 3-manifold invariant developed by Oszvath and Szabo. Bordered Heegaard Floer homology is an extension of the Heegaard Floer theory to 3-manifolds with boundary, which lets us compute the "hat" version of Heegaard Floer for complicated manifolds by cutting them into simpler pieces. Graph manifolds are an important class of 3-manifolds which decompose in a particularly nice way; all the components of their JSJ decomposition are Seifert fibered. The majority of the talk will be devoted to introducing the terms above, starting with a brief overview of Heegaard Floer homology. At the end we see how to use bordered Heegaard Floer to compute HF-hat for any graph manifold.
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.
Wenzhao Chen : Negative amphicheiral knots and the half-Alexander polynomial
- Geometry and Topology ( 86 Views )In this talk, we will study strongly negative amphicheiral knots - a class of knots with symmetry. These knots provide torsion elements in the knot concordance group, which are less understood than infinite-order elements. We will introduce the half-Alexander polynomial, an equivariant version of the Alexander polynomial for strongly negative amphicheiral knots, focusing on its applications to knot concordance. In particular, I will show how it facilitated the construction of the first examples of non-slice amphicheiral knots of determinant one. This talk is based on joint work with Keegan Boyle.