Mark Stern : Introduction to nonlinear harmonic forms.
- Geometry and Topology ( 139 Views )We motivate and introduce nonlinear harmonic forms. These are de Rham representatives $z$ of cohomology classes which minimize the energy $\|z\|_{L_2}^2$ subject to a nonlinear constraint. We give basic existence results for quadratic constraints, discuss the rich Euler Lagrange equations, and ask many regularity questions.
Ana-Maria Brecan : On the intersection pairing between cycles in SU(p,q)-flag domains and maximally real Schubert varieties
- Geometry and Topology ( 132 Views )An SU(p, q)-flag domain is an open orbit of the real Lie group SU(p, q) acting on the complex flag manifold associated to its complexification SL(p + q, C). Any such flag domain contains certain compact complex submanifolds, called cycles, which encode much of the topological, complex geometric and repre- sentation theoretical properties of the flag domain. This talk is concerned with the description of these cycles in homology using a specific type of Schubert varieties. They are defined by the condition that the fixed point of the Borel group in question is in the closed SU(p,q)-orbit in the ambient manifold. We consider the Schubert varieties of this type which are of com- plementary dimension to the cycles. It is known that if such a variety has non-empty intersection with a certain base cycle, then it does so transversally (in finitely many points). With the goal of understanding this duality, we describe these points of intersection in terms of flags as well as in terms of fixed points of a given maximal torus. The relevant Schubert varieties are described in terms of Weyl group elements.
Kyle Hayden : Complex curves through a contact lens
- Geometry and Topology ( 116 Views )Every four-dimensional Stein domain has a height function whose regular level sets are contact three-manifolds. This allows us to study complex curves in the Stein domain via their intersection with these contact level sets, where we can comfortably apply three-dimensional tools. We use this perspective to characterize the links in Stein-fillable contact manifolds that bound complex curves in their Stein fillings. (Some of this is joint work with Baykur, Etnyre, Hedden, Kawamuro, and Van Horn-Morris.)
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.
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.
Thomas Walpuski : G2instantons over twisted connected sums
- Geometry and Topology ( 163 Views )In joint work with H. Sá Earp we introduced a method to construct G2instantons over compact G2manifolds arising as the twisted connected sum of a matching pair of building blocks. I will recall some of the background (including the twisted connected sum construction and a short discussion as to why one should care about G2instantons), discuss our main result and explain how to interpret it in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a K3 surface. If time permits, I will discuss an idea to construct the input required by our gluing theorem.
Daniel Stern : Min-Max Methods for Ginzburg-Landau Functionals and Connections to Geometric Measure Theory
- Geometry and Topology ( 128 Views )I will present a natural method for producing unstable critical points of the Ginzburg-Landau functionals on an arbitrary manifold, and describe results showing that a nontrivial portion of the energy must concentrate on a generalized minimal submanifold of codimension two.
Mark Stern : Geometry of stable Yang-Mills connections
- Geometry and Topology ( 155 Views )On a compact 4-manifold, every self-dual connection and every anti-self-dual connection minimizes the Yang-Mills energy. In this talk, I will answer the converse question for compact homogeneous 4-manifolds. I will also survey related stability results in other dimensions.
Gordana Matic : Contact invariant in sutured Floer homology and fillability
- Geometry and Topology ( 127 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.
Iva Stavrov : On a point-particle gluing construction
- Geometry and Topology ( 117 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.
Jan Metzger : On isoperimetric surfaces in asymptotically flat manifolds
- Geometry and Topology ( 111 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.
Fernando Marques : The space of positive scalar curvature metrics on the three-sphere
- Geometry and Topology ( 121 Views )In this talk we will discuss a proof of the path-connectedness of the space of positive scalar curvature metrics on the three-sphere. The proof uses the Ricci flow with surgery and the connected sum construction of Gromov and Lawson. The work of Perelman on Hamilton's Ricci flow is fundamental. If time permits we will also discuss an application to general relativity.
Gábor Székelyhidi : Greatest lower bounds on the RIcci curvature of Fano manifolds
- Geometry and Topology ( 110 Views )On a Fano manifold M we study the supremum of the possible t such that there is a Kähler metric in c_1(M) with Ricci curvature bounded below by t. We relate this to Aubin's continuity method for finding Kähler-Einstein metrics and we give bounds on it for certain manifolds.
Dorothy Buck : Knotted DNA
- Geometry and Topology ( 120 Views )The central axis of the famous DNA double helix is often topologically constrained or even circular. The topology of this axis can influence which proteins interact with the underlying DNA. Subsequently, in all cells there are proteins whose primary function is to change the DNA axis topology -- for example converting a torus link into an unknot. Additionally, there are several protein families that change the axis topology as a by-product of their interaction with DNA. This talk will describe typical DNA conformations, and the families of proteins that change these conformations. I'll present a few examples illustrating how Dehn surgery and other low-dimensional topological methods have been useful in understanding certain DNA-protein interactions, and discuss the most common topological techniques used to attack these problems.
Bianca Santoro : Bifurcation of periodic solutions to the singular Yamabe problem on spheres.
- Geometry and Topology ( 110 Views )In this talk, we describe how to obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere, that blow up along a great circle. These are (complete) constant scalar curvature metrics on the complement of S^1 inside S^m, m ≥ 5, that are conformal to the (incomplete) round metric and periodic in the sense of being invariant under a discrete group of conformal transformations. These solutions come from bifurcating branches of constant scalar curvature metrics on compact quotients of S^m \ S^1. This is a joint work with R. Bettiol (University of Notre Dame) and P. Piccione (USP-Brazil).
Kristen Moore : Evolving hypersurfaces by their inverse null mean curvature.
- Geometry and Topology ( 122 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.
Tomasz Mrowka : Using instantons to show that Khovanov homology detects the unknot.
- Geometry and Topology ( 114 Views )A couple of years ago Kronheimer and I revisited our work on instantons with codimension two singularities. This leads to knot invariants associated to each partial flag manifold. There invariants are related to Khovanov homology for the case of $\IP1$ and Khovanov-Rozansky homology fo $\IP^n$. We have recently understood how to use the $\IP1$ case to prove that Khovanov homology detects the unknot.
Jeff Streets : Long time existence of minimizing movement solutions of Calabi flow
- Geometry and Topology ( 116 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.
Akram Alishahi : Trivial tangles, compressible surfaces and Floer homology
- Geometry and Topology ( 108 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.
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.
Sergey Cherkis : Moduli Spaces of Yang-Mills Instantons on multi-Taub-NUT Spaces
- Geometry and Topology ( 109 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.
Justin Sawon : On the topology of compact hyperkahler manifolds
- Geometry and Topology ( 146 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?
Diana Davis : Periodic paths on the pentagon
- Geometry and Topology ( 170 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.
Alex Freire : Motion of networks by curvature in two and three dimensions
- Geometry and Topology ( 150 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.
Sahana Balasubramanya : Hyperbolic structures on wreath products
- Geometry and Topology ( 114 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.
Goncalo Oliveira : Monopoles in Higher Dimensions
- Geometry and Topology ( 119 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.
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.