Steven Sivek : Sutured embedded contact homology is an invariant
- Geometry and Topology ( 108 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.
Adam Levine : Non-orientable surfaces in homology cobordisms
- Geometry and Topology ( 105 Views )We study the minimal genus problem for embeddings of closed, non-orientable surfaces in a homology cobordism between rational homology spheres, using obstructions derived from Heegaard Floer homology and from the Atiyah-Singer index theorem. For instance, we show that if a non-orientable surface embeds essentially in the product of a lens space with an interval, its genus and normal Euler number are the same as those of a stabilization of a non-orientable surface embedded in the lens space itself. This is joint work with Danny Ruberman and Saso Strle.
Goncalo Oliveira : Monopoles in Higher Dimensions
- Geometry and Topology ( 120 Views )The Monopole (Bogomolnyi) equations are Geometric PDEs in 3 dimensions. In this talk I shall introduce a generalization of the monopole equations to both Calabi Yau and G_2 manifolds. I will motivate the possible relations of conjectural enumerative theories arising from "counting" monopoles and calibrated cycles of codimension 3. Then, I plan to state the existence of solutions and sketch how these examples are constructed.
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.
Robert Bryant : The affine Bonnet problem
- Geometry and Topology ( 246 Views )The classical Euclidean problem studied by Bonnet in the 19th century was to determine whether, and in how many ways, a Riemannian surface can be isometrically embedded into Euclidean 3-space so that its mean curvature is a prescribed function. He found that, generically, specifying a metric and mean curvature admitted no solution but that there are special cases in which, not only are there solutions, but there are even 1-parameter families of distinct (i.e., mutually noncongruent) solutions. Much later, these Bonnet surfaces were found to be intimately connected with integrable systems and Lax pairs. In this talk, I will consider the analogous problem in affine geometry: To determine whether, and in how many ways, a surface endowed with a Riemannian metric g and a function H can be immersed into affine 3-space in such a way that the induced Blaschke metric is g and the induced affine mean curvature is H. This affine problem is, in many ways, richer and more interesting than the corresponding Euclidean problem. I will classify the pairs (g,H) that display the greatest flexibility in their solution space and explain what is known about the (suspected) links with integrable systems and Lax pairs.
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.
Jeremy Van Horn-Morris : Fiber genus and the topology of symplectic fillings
- Geometry and Topology ( 119 Views )Work of Donaldson, Giroux, and many others shows how to associate a singular surface fibration to a symplectic 4-manifold, either closed or with boundary, as well as to a contact 3-manifold. These are Lefschetz pencils, fibrations and open books, resp. It was asked by Stipsicz, Ozbagci, Korkmaz and others, whether the genus (or genus and self intersection) of the fiber of these fibrations gave an a priori bound on the topological complexity of the symplectic manifold. This question is equivalent to asking for a bound on the length of a factorization of a mapping class element of the fiber into right handed Dehn twists. We will discuss some of the known conditions which can produce such a bound, as well as present examples where such a bound does not exist. This is joint work with I. Baykur.
Luca Di Cerbo : Positivity in K\ahler-Einstein theory and hyperbolic geometry
- Geometry and Topology ( 110 Views )We characterize logarithmic pairs which admit K\"ahler-Einstein metrics with negative scalar curvature and small cone-edge singularities along a simple normal crossing divisor. We show that if such K\"ahler-Einstein metrics exist for all small cone-angles then they exist for all angles in a fixed range depending on the dimension only. Remarkably, the existence of such a uniform range can be used to derive many interesting results in hyperbolic geometry. We give effective bounds on the number of cusped complex hyperbolic manifolds with given upper bound on the volume. We estimate the number of ends of such manifolds in terms of their volume. Finally, we discuss the projective algebraicity of minimal compactifications (Siu-Yau) of finite volume complex hyperbolic manifolds.
Peter Lambert-Cole : Products of Legendrian Knots and Invariants in Contact Topology
- Geometry and Topology ( 101 Views )I will introduce a product construction in contact topology for Legendrian submanifolds, focusing on products of Legendrian knots. I will then discuss ongoing work to compute a product formula for the Legendrian contact homology invariant and some of the geometric and analytic difficulties involved. In particular, I will describe Ekholm's Morse-theoretic approach to counting holomorphic curves and how to apply it to compute invariants of products of Legendrian knots.
Benoit Charbonneau : Asymptotic Hodge Theory of Vector Bundles
- Geometry and Topology ( 112 Views )In joint work with Mark Stern, we introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the bundle to admit a holomorphic structure. We study compatibility under the Chern isomorphism of these filtrations with the Hodge filtration on cohomology.
Dominic Dotterrer : Algorithms for the isoperimetric problem in high dimensional cubes
- Geometry and Topology ( 117 Views )Thinking of high dimensional cubes as large cellular complexes, we pose the question of finding for each cellular cycle, a small cellular chain which bounds it. We will describe an algorithm which solves this problem. The algorithm is based on a trichotomy which describes the different ways cycles can sit in a cube. We will give examples of large topological spheres which are cellularly embedded in the cubes in a purely combinatorial way. These cycles will show that the exponent obtained by our algorithm is sharp.
Kristen Hendricks : Periodic Knots and Heegaard Floer Homology
- Geometry and Topology ( 176 Views )We introduce periodic knots and discuss two classical results concerning their geometry, namely Murasugi's condition on the Alexander polynomial and Edmonds' condition on the genus. We then show how spectral sequences in Heegaard Floer link homology can be used to give a generalization of these two results in the case of doubly-periodic knots.
Christopher R Cornwell : Knot contact homology and representations of knot groups
- Geometry and Topology ( 114 Views )We study certain linear representations of the knot group that induce augmentations of knot contact homology. This perspective on augmentations enhances our understanding of the relationship between the augmentation polynomial and the $A$-polynomial of the knot. For example, we show that for 2-bridge knots the two polynomials agree. We also show this is never the case for (non-2-bridge) torus knots, nor for a family of 3-bridge pretzel knots. Moreover, we show that these representations provide a lower bound on the meridional rank of the knot. As a consequence, our results give another proof that torus knots have meridional rank and bridge number that are equal.
Alexander Volkmann : Nonlinear mean curvature flow with Neumann boundary condition
- Geometry and Topology ( 164 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.
Ioana Suvaina : ALE Ricci flat Kahler surfaces
- Geometry and Topology ( 123 Views )The talk presents an explicit classification of the ALE Ricci flat K\"ahler surfaces, generalizing previous classification results of Kronheimer. The manifolds are related to a special class of deformations of quotient singularities of type $\mathbb C^2/G$, with $G$ a finite subgroup of $U(2)$. I will also explain the relation with the Tian-Yau construction of complete Ricci flat Kahler manifolds.
Kristen Moore : Evolving hypersurfaces by their inverse null mean curvature.
- Geometry and Topology ( 123 Views )We introduce a new second order parabolic evolution equation where the speed is given by the reciprocal of the null mean curvature. This flow is a generalisation of inverse mean curvature flow and it is motivated by the study of black holes and mass/energy inequalities in general relativity. We present a theory of weak solutions using level-set methods and an appropriate variational principle, and outline a natural application of the flow as a variational approach to constructing marginally outer trapped surfaces (MOTS), which play the role of quasi-local black hole boundaries in general relativity.
Iva Stavrov : On a point-particle gluing construction
- Geometry and Topology ( 119 Views )Analyzing the motion of a small body is often done by making a point-particle approximation. This simplification is not entirely appropriate in general relativity since, roughly speaking, too much mass in too little space creates black holes. In place of point-particles one considers one-parameter families of space-time metrics $\gamma_\varepsilon$ in which $\varepsilon\to 0$ corresponds to a body shrinking to zero size. In addition, certain point-particle limit properties are imposed on $\gamma_\varepsilon$. While there are some examples of such metrics $\gamma_\varepsilon$ (e.g. Schwarzschild-de Sitter space-time), there is no general existence theorem for such space-times. This talk will discuss a gluing construction which produces initial data with desirable point-particle limit properties.
Paul Allen : The Dirichlet problem for curve shortening flow.
- Geometry and Topology ( 104 Views )We consider the Dirichlet problem for curve shortening flow on surfaces of constant curvature and show long-time existence of the flow when the initial curve is embedded in a convex region. Furthermore, the limit curve of the flow is a geodesic. The proof relies on an adaptation of Huisken's distance comparison estimate for planar curves, a maximum principle of Angenent, and a blow-up analysis of singularities.
Gábor Székelyhidi : On the positive mass theorem for manifolds with corners.
- Geometry and Topology ( 99 Views )A problem originally studied by P. Miao is whether the positive mass theorem holds on manifolds with certain singularities along a hypersurface. I will discuss an approach to this problem which uses the Ricci flow to smooth out the metric, so that one can apply the usual positive mass theorem. This allows for extending the rigidity statement in the zero mass case to higher dimensions, which was only known in the 3 dimensional case previously. This is joint work with D. McFeron.
Jeff Streets : Long time existence of minimizing movement solutions of Calabi flow
- Geometry and Topology ( 117 Views )In 1982 Calabi proposed studying gradient flow of the L^2 norm of the scalar curvature (now called "Calabi flow") as a tool for finding canonical metrics within a given Kahler class. The main motivating conjecture behind this flow (due to Calabi-Chen) asserts the smooth long time existence of this flow with arbitrary initial data. By exploiting aspects of the Mabuchi-Semmes-Donaldson metric on the space of Kahler metrics I will construct a kind of weak solution to this flow, known as a minimizing movement, which exists for all time.
Gordana Matic : Contact invariant in sutured Floer homology and fillability
- Geometry and Topology ( 128 Views )In the 70's Thurston and Winkelnkemper showed how an open book decomposition of a 3-manifold can be used to construct a contact structure. In 2000 Giroux showed that every contact structure on a 3-manifold can be obtained from that process. Ozsvath and Szabo used this fact to define an invariant of contact structures in their Heegaard Floer homology, providing an important new tool to study contact 3-manifolds. In joint work with Ko Honda and Will Kazez we describe a simple way to visualize this contact invariant and provide a generalization and some applications. When the contact manifold has boundary, we define an invariant of contact structure living in sutured Floer homology, a variant of Heegaard Floer homology for a manifold with boundary due to Andras Juhasz. We describe a natural gluing map on sutured Floer homology and show how it produces a (1+1)-dimensional TQFT leading to new obstructions to fillability.
Mu-Tao Wang : A variational problem for isometric embeddings and its applications in general relativity
- Geometry and Topology ( 116 Views )I shall discuss a variational problem arising from the study of quasilocal energy in general relativity. Given a spacelike 2-surface in spacetime, the Euler-Lagrange equation for the quasilocal energy is the isometric embedding equation into the Minkowski space coupled with a fourth order nonlinear elliptic equation for the time function. This equation is important in that it gives the ground configuration in GR. In joint work with PoNing Chen and Shing-Tung Yau, we solved this system in the cases of large and small sphere limits.
Yanir Rubinstein : Einstein metrics on Kahler manifolds
- Geometry and Topology ( 219 Views )The Uniformization Theorem implies that any compact Riemann surface has a constant curvature metric. Kahler-Einstein (KE) metrics are a natural generalization of such metrics, and the search for them has a long and rich history, going back to Schouten, Kahler (30's), Calabi (50's), Aubin, Yau (70's) and Tian (90's), among others. Yet, despite much progress, a complete picture is available only in complex dimension 2. In contrast to such smooth KE metrics, in the mid 90's Tian conjectured the existence of KE metrics with conical singularities along a divisor (i.e., for which the manifold is `bent' at some angle along a complex hypersurface), motivated by applications to algebraic geometry and Calabi-Yau manifolds. More recently, Donaldson suggested a program for constructing smooth KE metrics of positive curvature out of such singular ones, and put forward several influential conjectures. In this talk I will try to give an introduction to Kahler-Einstein geometry and briefly describe some recent work mostly joint with R. Mazzeo that resolves some of these conjectures. One key ingredient is a new C^{2,\alpha} a priori estimate and continuity method for the complex Monge-Ampere equation. It follows that many algebraic varieties that may not admit smooth KE metrics (e.g., Fano or minimal varieties) nevertheless admit KE metrics bent along a simple normal crossing divisor.
Fernando Coda Marques : Min-max theory and the Willmore conjecture
- Geometry and Topology ( 110 Views )In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of any torus immersed in Euclidean three-space is at least 2\pi^2. In this talk we will describe a solution to the Willmore conjecture based on the min-max theory of minimal surfaces. This is joint work with Andre Neves (Imperial College, UK).
Jan Metzger : On isoperimetric surfaces in asymptotically flat manifolds
- Geometry and Topology ( 112 Views )I will present joint work with Michael Eichmair on the existence of large isoperimetric regions in complete asymptotically flat manifolds of arbitrary dimension with metric asymptotic to Schwarzschild. The key idea is an effective isopermetric inequality that forces nearly optimal regions to center in the manifold.
Carla Cederbaum : From Newton to Einstein: a guided tour through space and time
- Geometry and Topology ( 135 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.
Valentino Tosatti : The evolution of a Hermitian metric by its Chern-Ricci curvature
- Geometry and Topology ( 103 Views )I will discuss the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci curvature. This is an evolution equation which coincides with the Ricci flow if the initial metric is Kahler, and was first studied by M.Gill. I will describe the maximal existence time for the flow in terms of the initial data, and thendiscuss the behavior of the flow on complex surfaces and on some higher-dimensional manifolds. This is joint work with Ben Weinkove.
John Etnyre : Surgery and Tight Contact Structures
- Geometry and Topology ( 114 Views )One of the fundamental problems in 3-dimensional contact geometry is the construction of tight contact structures on closed manifolds. Two obvious ways to try to construct such structures are via Legendrian surgery and admissible transverse surgery. It was long thought that when performed on a closed tight contact manifold these operations would yield a tight contact manifold. We show that this is not true for admissible transverse surgery. Along the way we discuss the relations between these two surgery operations and construct some contact structures with interesting properties.
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.