Melissa Zhang : Annular Khovanov homology and 2-periodic links
- Geometry and Topology ( 102 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.
Richard Hain : Hodge theory and the Goldman-Turaev Lie bialgebra
- Geometry and Topology ( 186 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.
David Shea Vela-Vick : Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links
- Geometry and Topology ( 98 Views )To each three-component link in the 3-dimensional sphere we associate a characteristic map from the 3-torus to the 2-sphere, and establish a correspondence between the pairwise and Milnor triple linking numbers of the link and the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two- component link is the degree of its associated Gauss map from the 2- torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere.
Alex Freire : Motion of networks by curvature in two and three dimensions
- Geometry and Topology ( 152 Views )The main topic is the motion of a network of embedded curves moving by curvature in a convex planar domain, with three curves meeting at each vertex making 120 degree angles, and normal intersections at the boundary. I'll discuss the origin of this flow as a sharp-interface limit, the existence and linearized stability of static solutions, and what is known regarding global existence. A similar problem can be posed for systems of surfaces moving by mean curvature- if there is time, I'll discuss local existence in the surface case.
Yi Wang : The Aleksandrov-Fenchel inequalities of k+1 convex domains
- Geometry and Topology ( 110 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.
Vera Vértesi : Knots in contact 3--manifolds
- Geometry and Topology ( 161 Views )In this talk I will give a purely combinatorial description of Knot Floer Homology for knots in the three-sphere (Manolescu-Ozsváth-Szabó-Thurston). In this homology there is a naturally associated invariant for transverse knots. This invariant gives a combinatorial but still an effective way to distinguish transverse knots (Ng-Ozsváth-Thurston). Moreover it leads to the construction of an infinite family of non-transversely simple knot-types (Vértesi).
Goncalo Oliveira : Monopoles in 3 Dimensions
- Geometry and Topology ( 110 Views )Monopoles are solutions to the Bogomolnyi equation, which is a PDE for a connection and an Higgs field (a section of an certain bundle) on a 3 dimensional Riemannian manifold. In this talk I plan to introduce these equations. Then I want to tell you some properties of its solutions on R^3. Finally, I plan to speak about monopoles on a more general class of noncompact manifolds known as asymptotically conical. My main goal is to explain the geometric meaning of the parameters needed to give coordinates on an open set of the moduli space of monopoles.
Christopher R. Cornwell : Bindings of open book decompositions and lens spaces
- Geometry and Topology ( 98 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.
David Rose : A categorification of quantum sl_3 projectors and the sl_3 Reshetikhin-Turaev invariants of tangles
- Geometry and Topology ( 105 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.
Valentino Tosatti : Collapsing of Calabi-Yau manifolds
- Geometry and Topology ( 101 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.
Fernando Schwartz : On the topology of black holes
- Geometry and Topology ( 145 Views )An important special case of the general construction of black holes translates into a problem in Riemannian geometry, since a totally geodesic slice of spacetime is an asymptotically flat Riemannian manifold with nonnegative scalar curvature, and the restriction of the event horizon to the slice is the apparent horizon in the slice. In this talk we show how to construct new examples of Riemannian manifolds with nonspherical apparent horizon, in dimensions four and above. More precisely, for any $n,m\ge 1$, we construct asymptotically flat, scalar flat Riemannian manifolds with apparent horizon that is a smooth outermost minimal hypersurface with topology $S^n\times S^{m+1}$.
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.
Tori Akin : Automorphisms of the Punctured Mapping Class Group
- Geometry and Topology ( 141 Views )We can describe the point-pushing subgroup of the mapping class group topologically as the set of maps that push a puncture around loops in the surface. However, we can characterize this topological subgroup in purely algebraic terms. Using group theoretic tools and a classical theorem of Burnside, we can recover a result of Ivanov-McCarthy establishing the triviality of Out(Mod±). To this end, well demonstrate that the point-pushing subgroup is unique in the mapping class group.
Siqi He : Classification of Nahm Pole Solutions to the KW Equations on $S^1\times\Sigma\times R^+$
- Geometry and Topology ( 228 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.
Deepam Patel : Motivic structures on higher homotopy of non-nilpotent spaces
- Geometry and Topology ( 97 Views )In his fundamental paper on the projective line minus three points, Deligne constructed certain extensions of mixed Tate motives arising from the fundamental group of the projective line minus three points. Since then, motivic structures on homotopy groups have been studied by many authors. In this talk, we will construct a motivic structure on the (nilpotent completion of) *n*-th homotopy group of *P**n* minus *n*+2 hyperplanes in general position.
Diana Davis : Periodic paths on the pentagon
- Geometry and Topology ( 171 Views )Mathematicians have long understood periodic billiard trajectories on the square table, which occur when the slope of the trajectory is rational. In this talk, I'll explain my work on periodic trajectories on the regular pentagon, describing their geometry, symbolic dynamics, and group structure. The periodic trajectories are very beautiful, and some of them exhibit a surprising "dense but not equidistributed" behavior. There will be lots of pictures. This is joint work with Samuel Lelièvre.
Benoit Charbonneau : Singular monopoles on the product of a circle and a surface
- Geometry and Topology ( 113 Views )In this talk, I will discuss work done with Jacques Hurtubise (McGill) to relate singular solutions to the Bogomolny equation on a circle times a surface to pairs [holomorphic bundle, meromorphic endomorphism] on the surface. The endomorphism is meromorphic, generically bijective, and corresponds to a return map. Its poles and zeros are related to the singularities of the corresponding solution to the Bogomolny equation. This talk is based on arXiv:0812.0221.
Justin Sawon : On the topology of compact hyperkahler manifolds
- Geometry and Topology ( 149 Views )In this talk we will describe some results about Betti, Hodge, and characteristic numbers of compact hyperkahler manifolds. In (complex) dimension four one can find universal bounds for all of these invariants (Beauville, Guan); in higher dimensions it is still possible to find some bounds. We also describe how these bounds are related to the question: are there finitely many hyperkahler manifolds in each dimension, up to deformation?
Chindu Mohanakumar : Coherent orientations of DGA maps associated to exact Lagrangian cobordisms
- Geometry and Topology ( 135 Views )We discuss the DGA map induced by an exact Lagrangian cobordism, and an analytic strategy to lift the map to integer coefficients, introduced by Fukaya, Oh, Ohta and Ono and further adapted by Ekholm, Etnyre, and Sullivan and Karlsson respectively. We then explain how this strategy can be applied to find a concrete combinatorial formula for a mini-dipped pinch move, thereby completely determining the integral DGA maps for all decomposable, orientable Lagrangian cobordisms. If time permits, we will show how to obtain this formula in a model case. We will also go into future potential work, including applications to Heegaard Floer Homology and nonorientable cobordisms.
Colleen Robles : A refinement of the Lefschetz decomposition for hyperkahler manifolds
- Geometry and Topology ( 179 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.
Igor Zelenko (Texas A&M U) : Gromovs h-principle for corank two distribution of odd rank with maximal first Kronecker index
- Geometry and Topology ( 122 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.
Steven Sivek : Sutured embedded contact homology is an invariant
- Geometry and Topology ( 109 Views )Embedded contact homology (ECH) is an invariant of a closed contact 3-manifold, but proving its invariance is not so straightforward: the only known proof (due to Taubes) is to show that it is isomorphic to monopole Floer homology, which only depends on the underlying manifold. Colin, Ghiggini, Honda, and Hutchings defined a version of ECH for contact 3-manifolds with boundary, which are naturally sutured manifolds, and conjectured that this is also an invariant of the underlying sutured manifold. In this talk I will show that sutured ECH is indeed an invariant and discuss exactly what kind of invariant it is. This is joint work with Cagatay Kutluhan.
Sergey Cherkis : Moduli Spaces of Yang-Mills Instantons on multi-Taub-NUT Spaces
- Geometry and Topology ( 110 Views )We formulate the Nahm transform producing self-dual Hermitian connections on Asymptotically Locally Flat hyperkaehler manifolds. Using this formulation we describe the moduli spaces of such connections and explicitly compute their asymptotic metrics.
Jonathan Hanselman : Satellite knots and immersed curves
- Geometry and Topology ( 110 Views )Satellite operations are an valuable method of constructing complicated knots from simpler ones, and much work has gone into understanding how knot invariants change under these operations. We describe a new way of computing the (UV=0 quotient of the) knot Floer complex using an immersed Heegaard diagram obtained from a Heegaard diagram for the pattern and the immersed curve representing the knot Floer complex of the companion. This is particularly useful for (1,1)-patterns, since in this case the resulting immersed diagram is genus one. In some cases the immersed curve representing the satellite knot Floer complex can be obtained directly by deforming the diagram, generalizing earlier work with Watson on cables. This is joint work with Wenzhao Chen.
Chi Li : Construction of rotationally symmetric Kahler-Ricci solitons
- Geometry and Topology ( 108 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.