Steven Sivek : A contact invariant in sutured monopole homology
- Geometry and Topology ( 110 Views )Kronheimer and Mrowka recently used monopole Floer homology to define an invariant of sutured manifolds, following work of Juhász in Heegaard Floer homology. Contact 3-manifolds with boundary are natural examples of such manifolds. In this talk, I will construct an invariant of a contact structure as an element of the associated sutured monopole homology group. I will discuss several interesting properties of this invariant, including gluing maps which are analogous to the Heegaard Floer sutured gluing maps of Honda, Kazez, and Matić and applications to Legendrian knots. This is joint work with John Baldwin.
Richard Hain : Fundamental groups of branched coverings of certain Kaehler manifolds
- Geometry and Topology ( 118 Views )In this talk I will discuss a result that describes (under certain conditions) the fundamental group of certain branched coverings of quasi-projective varieties that admits a complete Kaehler metric with non-positive curvature. When applied to the period map for genus 3 curves, it implies that the Torelli group in genus 3 is finitely generated. Combined with recent work of Putman and Hatcher-Margalit, it gives a new proof of Dennis Johnson's result that the Torelli group is genus g is finitely generated for all g > 2.
Jeremy Marzuola : Nonlinear Bound States on manifolds
- Geometry and Topology ( 121 Views )We will discuss the results of several joint ongoing projects (with subsets of collaborators Pierre Albin, Hans Christianson, Colin Guillarmou, Jason Metcalfe, Laurent Thomann and Michael Taylor), which explore the existence, stability and dynamics of nonlinear bound states and quasimodes on manifolds of both positive and negative curvature with various symmetry properties.
Adam Jacob : The Yang-Mills flow and the Atiyah-Bott formula on compact Kahler manifolds
- Geometry and Topology ( 117 Views )In this talk I will describe the limiting properties Yang-Mills flow on a holomorphic vector bundle E, in the case where the flow does not converge. In particular I will describe how to determine the L^2 limit of the curvature endomorphism along the flow. This proves a sharp lower bound for the Hermitian-Yang-Mills functional and thus the Yang-Mills functional, generalizing to arbitrary dimension a formula of Atiyah and Bott first proven on Riemann surfaces. I will then explain how to use this result to identify the limiting bundle along the flow, which turns out to be independent of metric and uniquely determined by the isomorphism class of E.
Justin Sawon : Lagrangian fibrations by Jacobians of low genus curves
- Geometry and Topology ( 138 Views )The Beauville-Mukai integrable system is a well-known Lagrangian fibration, i.e., a holomorphic symplectic manifold fibred by Lagrangian complex tori. It is constructed by beginning with a complete linear system of curves on a K3 surface, and then taking the compactified relative Jacobian of the family of curves. One may ask whether other families of curves yield Lagrangian fibrations in this way. Markushevich showed that this is not the case in genus two: a Lagrangian fibration by Jacobians of genus two curves must be a Beauville-Mukai system. We generalize his result to genus three curves, and also to non-hyperelliptic curves of genus four and five.
Luca Di Cerbo : Finite volume complex hyperbolic surfaces and their compactifications
- Geometry and Topology ( 136 Views )In this talk, I will discuss the geometry of finite volume complex hyperbolic surfaces and their compactifications. Finally, applications at the common edge between Riemannian and complex algebraic geometry are given.
Claude LeBrun : Four-Dimensional Einstein Manifolds, and Beyond
- Geometry and Topology ( 156 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.
Douglas LaFountain : Deligne-Mumford and the space of filtered screens
- Geometry and Topology ( 102 Views )For a genus g surface with s > 0 punctures and 2g+s > 2, decorated Teichmuller space (DTeich) is a trivial R_+^s-bundle over the usual Teichmuller space, where the fiber corresponds to families of horocycles peripheral to each puncture. As proved by R. Penner, DTeich admits a mapping class group-invariant cell decomposition, which then descends to a cell decomposition of Riemann's moduli space. In this talk we introduce a new cellular bordification of DTeich which is also MCG-invariant, namely the space of filtered screens. After an appropriate quotient, we obtain a cell decomposition for a new compactification of moduli space, which is shown to be homotopy equivalent to the Deligne-Mumford compactification. This work is joint with R. Penner.
Valentino Tosatti : Collapsing of Calabi-Yau manifolds
- Geometry and Topology ( 99 Views )We will discuss the problem of understanding the collapsing of Ricci-flat Kahler metric on Calabi-Yau manifolds. If time permits we will also discuss the Strominger-Yau-Zaslow picture of mirror symmetry for hyperkahler manifolds. Joint work with Mark Gross and Yuguang Zhang.
Christopher R. Cornwell : Bindings of open book decompositions and lens spaces
- Geometry and Topology ( 97 Views )We will discuss recent work on Legendrian and transverse links in universally tight contact lens spaces. There is a relationship between such links and the Berge Conjecture. The surgery duals to Berge knots in the corresponding lens space are all bindings of a rational open book decomposition. We will discuss whether these Berge duals support the universally tight contact structure on that lens space.
Thalia Jeffres : K\{a}hler-Einstein Metrics with Edge Singularities
- Geometry and Topology ( 125 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.
David Rose : A categorification of quantum sl_3 projectors and the sl_3 Reshetikhin-Turaev invariants of tangles
- Geometry and Topology ( 104 Views )We discuss a recent result of the speaker giving a categorification of quantum sl_3 projectors and the sl_3 Reshetikhin-Turaev invariants of framed tangles. In more detail, we will review Kuperberg's diagrammatic description of the category of representations of quantum sl_3 (which gives a combinatorial method for computing the quantum sl_3 invariant of links) as well as Morrison and Nieh's geometric categorification of this structure. We then show that there exist elements in Morrison and Nieh's categorification which correspond to projection onto highest weight irreducible summands and use these elements to construct a categorification of the sl_3 Reshetikhin-Turaev invariant, that is, a link homology theory from which the sl_3 invariant can be obtained by taking the graded Euler characteristic. No previous knowledge of categorification or quantum groups is assumed.
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.
Lan-Hsuan Huang : Hypersurfaces with nonnegative scalar curvature and a positive mass theorem
- Geometry and Topology ( 111 Views )Since the time of Gauss, geometers have been interested in the interplay between the intrinsic metric structure of hypersurfaces and their extrinsic geometry from the ambient space. For example, a result of Sacksteder tells us that if a complete hypersurface has non-negative sectional curvature, then its second fundamental form in Euclidean space must be positive semi-definite. In a joint work with Damin Wu, we study hypersurfaces under a much weaker curvature condition. We prove that closed hypersurfaces with non-negative scalar curvature must be weakly mean convex. The proof relies on a new geometric inequality which relates the scalar curvature and mean curvature of the hypersurface to the geometry of the level sets of a height function. This result is optimal in the sense that the scalar curvature cannot be replaced by other k-th mean curvatures. The result and argument have applications to the mean curvature flow, positive mass theorem, and rigidity theorems.
Yi Wang : The Aleksandrov-Fenchel inequalities of k+1 convex domains
- Geometry and Topology ( 109 Views )In this talk, I will report some recent joint work with Sun-Yung Alice Chang in which we partially generalize the Aleksandrov-Fenchel inequalities for quermassintegrals from convex domains in the Euclidean space to a class of non-convex domains.
Carla Cederbaum : The Newtonian Limit of Geometrostatics
- Geometry and Topology ( 98 Views )Geometrostatics is an important subdomain of Einstein's General Relativity. It describes the mathematical and physical properties of static isolated relativistic systems such as stars, galaxies or black holes. For example, geometrostatic systems have a well-defined ADM-mass (Chrusciel, Bartnik) and (if this is nonzero) also a center of mass (Huisken-Yau, Metzger) induced by a CMC-foliation at infinity. We will present surface integral formulae for these physical properties in general geometrostatic systems. Together with an asymptotic analysis, these can be used to prove that ADM-mass and center of mass 'converge' to the Newtonian mass and center of mass in the Newtonian limit $c\to\infty$ (using Ehler's frame theory). We will discuss geometric similarities of geometrostatic and classical static Newtonian systems along the way.
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.
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
Luca Di Cerbo : Seiberg-Witten equations on manifolds with cusps and geometric applications.
- Geometry and Topology ( 193 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.
John Baldwin : A combinatorial spanning tree model for delta-graded knot Floer homology
- Geometry and Topology ( 106 Views )I'll describe a new combinatorial method for computing the delta-graded knot Floer homology of a link in S^3. Our construction comes from iterating an unoriented skein exact triangle discovered by Manolescu, and yields a chain complex for knot Floer homology which is reminiscent of that of Khovanov homology, but is generated (roughly) by spanning trees of the black graph of the link. This is joint work with Adam Levine.
Lev Rozansky : A categorification of the stable Witten-Reshetikhin-Turaev invariant of links in S2 x S1
- Geometry and Topology ( 161 Views )This work was done in close collaboration with M. Khovanov. The Witten-Reshetikhin-Turaev invariant Z(M,L;r) of a link L in a 3-manifold M is a seemingly random function of an integer r. However, for a small class of 3-manifolds constructed by identical gluing of two handlebodies (e.g., for S3 and for S2 x S1) and for sufficiently large values of r the ratio Z(M,L;r)/Z(M;r) is equal to a rational function J(M,L;q) of q evaluated at the first 2r-th root of unity. If M = S3, then J is the Jones polynomial. Khovanov categorified J(S3,L), that is, to a link L in S3 he assigned a homology H(L) with an extra Z-grading such that its graded Euler characteristic equals J(S3,L). We extend Khovanov's construction to links in S2 x S1 thus categorifying J(S2xS1,L). In his work on categorification of the Jones polynomial, Khovanov introduced special algebras H_n and assigned a H_m x H_n module to every (2m,2n)-tangle. We show that if a link L in S2 x S1 is presented as a closure of a (2n,2n)-tangle, then the Hochschild homology of its H_n bimodule is determined by the link itself and serves as a categorificaiton of J(S2xS1,L). Moreover, we show that this Hochschild homology can be approximated by Khovanov homology of the circular closure of the tangle within S3 by a high twist torus braid, thus providing a practical method of its computation.
Mark Stern : Stability, dynamics, and the quantum Hodge theory of vector bundles
- Geometry and Topology ( 172 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.
Erik Van Erp : Index theory on contact manifolds and noncommutative topology
- Geometry and Topology ( 99 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.
Christopher Cornwell : Polynomials, grid diagrams, and Legendrian knots in lens spaces
- Geometry and Topology ( 102 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.
Chi Li : Construction of rotationally symmetric Kahler-Ricci solitons
- Geometry and Topology ( 105 Views )Using Calabi's method, I will construct rotationally symmetric Kahler- Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao and Feldman-Ilmanen-Knopf.
Lenhard Ng : Cotangent bundles and applying symplectic techniques to topology
- Geometry and Topology ( 100 Views )I'll discuss how one can use cotangent and conormal bundles to translate some basic questions in topology into questions in symplectic geometry. This symplectic viewpoint allows one, for instance, to (re)prove that certain smooth structures on spheres are exotic, and to define new knot invariants via holomorphic curves. I'll describe properties of the knot invariant and some recent applications to transverse knots in contact geometry.
Valentino Tosatti : The Calabi-Yau equation on symplectic four-manifolds
- Geometry and Topology ( 170 Views )Abstract: The Calabi conjecture, proved by Yau thirty years ago, says that on a compact Kahler manifold one can find a unique Kahler metric in every Kahler class with prescribed volume form. Donaldson recently conjectured that this theorem can be extended to symplectic forms with a compatible almost complex structure in 4 dimensions, and gave possible applications to the symplectic topology of 4-manifolds. I will discuss Donaldson's conjecture and some recents developments (joint work with B. Weinkove and partly with S.-T. Yau).
William Allard : Currents in metric spaces
- Geometry and Topology ( 94 Views )Motivated by the need to formulate and solve Plateau type problems in higher dimensions and codimensions, normal and integral currents were introduced by Federer and Fleming around 1960; their work was, to some extent a generalization of earlier work by DeGeorgi in codimension one as well as the work of Reifenberg in arbitrary codimensions. Since then a great deal of work has been done on the Plateau problem and related variational problems. This work has always been based on geometric measure theory. The so-called closure theorem for integral currents and the boundary rectifiability theorem are essential ingredients in all of this work; these theorems depend on the Besicovitch-Federer structure theory for set of finite Hausdorff measure in Euclidean space. More recently, in the work of Ambrosio and others, a useful theory of Sobolev spaces for functions with values in an arbitrary metric space has been developed and applied to a variety of problems. Ambrosio and Kirchheim have developed a theory of currents in metric spaces in which they are able to give geometrically appealing proofs of generalizations of the aforementioned closure and rectifiability theorems using some ideas of Almgren and DeGiorgi and avoiding the use of the Besicovitch-Federer structure theory. In this talk I will describe how they do it.