Nan Wu : Length of a shortest closed geodesic in manifolds of dimension 4
- Geometry and Topology ( 93 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.
Natasa Sesum : On the extension of the mean curvature flow and the Ricci flow
- Geometry and Topology ( 105 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.
Greg Galloway : Rigidity and positivity of mass for asymptotically hyperbolic manifolds
- Geometry and Topology ( 144 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.
Simon Brendle : Minimal Lagrangian diffeomorphisms between domains in the hyperbolic plane
- Geometry and Topology ( 148 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.
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.
Jesse Madnick : The Mean Curvature of (Co)Associative Submanifolds
- Geometry and Topology ( 130 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.
Aleksander Doan : Seiberg-Witten multi-monopoles on Riemann surfaces
- Geometry and Topology ( 97 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.
Tristan Collins : Sasaki-Einstein metrics and K-stability
- Geometry and Topology ( 101 Views )I will discuss the connection between Sasaki-Einstein metrics and algebraic geometry in the guise of K-stability. In particular, I will give a differential geometric perspective on K-stability which arises from the Sasakian view point, and use K-stability to find infinitely many non-isometric Sasaki-Einstein metrics on the 5-sphere. This is joint work with G. Szekelyhidi.
Daniel Halpern-Leistner : Equivariant geometry and Calabi-Yau manifolds
- Geometry and Topology ( 91 Views )Developments in high energy physics, specifically the theory of mirror symmetry, have led to deep conjectures regarding the geometry of a special class of complex manifolds called Calabi-Yau manifolds. One of the most intriguing of these conjectures states that various geometric invariants, some classical and some more homological in nature, agree for any two Calabi-Yau manifolds which are "birationally equivalent" to one another. I will discuss how new methods in equivariant geometry have shed light on this conjecture over the past few years, leading to the first substantial progress for compact Calabi-Yau manifolds of dimension greater than three. The key technique is the new theory of "Theta-stratifications," which allows one to bring ideas from equivariant Morse theory into the setting of algebraic geometry.
Henry Segerman : Connectivity of the set of triangulations of a 3- or 4-manifold
- Geometry and Topology ( 101 Views )This is joint work with Hyam Rubinstein. Matveev and Piergallini independently show that the set of triangulations of a three-manifold is connected under 2-3 and 3-2 Pachner moves, excepting triangulations with only one tetrahedron. We give a more direct proof of their result which (in work in progress) allows us to extend the result to triangulations of four-manifolds.
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.
Joanna Nelson : An integral lift of cylindrical contact homology
- Geometry and Topology ( 101 Views )I will discuss joint work with Hutchings which gives a rigorous construction of cylindrical contact homology via geometric methods. This talk will highlight our use of non-equivariant constructions, automatic transversality, and obstruction bundle gluing. Together these yield a nonequivariant homological contact invariant which is expected to be isomorphic to SH^+ under suitable assumptions. By making use of family Floer theory we obtain an S^1-equivariant theory defined over Z coefficients, which when tensored with Q recovers the classical cylindrical contact homology, now with the guarantee of well-definedness and invariance. This integral lift of contact homology also contains interesting torsion information.
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.
Ken Jackson : Numerical Methods for the Valuation of Synthetic Collateralized Debt Obligations (CDOs)
- Geometry and Topology ( 156 Views )Our numerical computation group has studied several problems in computational finance over the past decade. One that we've looked at recently is the pricing of "collateralized debt obligations" (CDOs). The market for CDOs has grown rapidly to over US$1 trillion annually in 2006, since the appearance of JP Morgan's Bistro deal, the first synthetic CDO, in December 1997. Much of the turmoil in the financial markets recently has been due to such credit derivatives. As this suggests, there are still many open problems associated with the pricing and hedging of these complex financial instruments. I will talk briefly about some work that we have done recently in this area.
Michael Abel : HOMFLY-PT homology of general link diagrams and its decategorification
- Geometry and Topology ( 123 Views )In the construction of HOMFLY-PT homology, one must start with a link presented as a braid closure. This restriction was expected by Khovanov and Rozansky to be required for the homology to be an isotopy invariant. In this talk, after reviewing the construction of the HOMFLY-PT polynomial and homology, we explore the consequences of dropping this requirement and allowing general link diagrams. We explicitly show that the Reidemeister IIb move (where the strands have opposite orientations) fails. Finally we will show that the Euler characteristic of this homology theory is a deformed version of the HOMFLY-PT polynomial which detects "braidlike" isotopy of tangles and links. This new polynomial agrees with the HOMFLY-PT polynomial on link diagrams which are presented as closed braid diagrams.
Daniel Ruberman : Periodic-end Dirac operators and Seiberg-Witten theory
- Geometry and Topology ( 104 Views )We study the Seiberg-Witten equations on a 4-manifold X with the homology of S1 x S3. The count of such solutions, called the Seiberg-Witten invariant, depends on choices of Riemannian metric and perturbation of the equations to make a smooth moduli space. A similar issue is resolved in dimension 3 by relating the jumps in the Seiberg-Witten invariant to the spectral flow of the Dirac operator. In dimension 4, we use Taubes' theory of periodic-end operators to relate the jumps in the Seiberg-Witten invariant to the index theory of the Dirac operator on the infinite cyclic cover of X. This circle of ideas has applications to classification of smooth manifolds homeomorphic to S1 x S3, to questions about positive scalar curvature, and to homology cobordisms. This is joint work with Tomasz Mrowka and Nikolai Saveliev.
Marcus Khuri : On the Penrose Inequality
- Geometry and Topology ( 160 Views )The cosmic censorship conjecture roughly states that singularities in the evolution of spacetime are always hidden from the outside world by event horizons. As a test for this conjecture Penrose proposed the inequality M >= (A/16pi)^1/2, relating the total ADM mass M of a spacetime to the area A of an event horizon. For time symmetric initial data sets of Einstein's equations this inequality has been confirmed, independently by Huisken and Ilmanen (for one black hole) and by Bray (for multiple black holes). The purpose of this talk is to show how the time symmetric proofs can be generalized to apply to general initial data, assuming existence for a canonical degenerate elliptic system of equations. This is joint work with Hubert Bray.
Vladimir Matveev : Geodesically equivalent metrics in the large: Beltrami and Schouten problems.
- Geometry and Topology ( 139 Views )Two metrics are geodesically equivalent if they have the same (unparameterized) geodesics. During my talk I describe geodesically equivalent metrics on closed manifolds (which is an answer to Beltrami's question) and explain the proof of Lichnerowicz-Obata conjecture (which is an answer on the infinitesimal version of the Beltrami question known as Schouten problem).
Dan Rutherford : Augmentations and immersed Lagrangian fillings
- Geometry and Topology ( 132 Views )This is joint work with Y. Pan that applies previous joint work with M. Sullivan. Let $\Lambda \subset \mathbb{R}^{3}$ be a Legendrian knot with respect to the standard contact structure. The Legendrian contact homology (LCH) DG-algebra, $\mathcal{A}(\Lambda)$, of $\Lambda$ is functorial for exact Lagrangian cobordisms in the symplectization of $\mathbb{R}^3$, i.e. a cobordism $L \subset \mathit{Symp}(\mathbb{R}^3)$ from $\Lambda_-$ to $\Lambda_+$ induces a DG-algebra map, $f_L:\mathcal{A}(\Lambda_+) \rightarrow \mathcal{A}(\Lambda_-).$ In particular, if $L$ is an exact Lagrangian filling ($\Lambda_-= \emptyset$) the induced map is an augmentation $\epsilon_L: \mathcal{A}(\Lambda_+) \rightarrow \mathbb{Z}/2.$ In this talk, I will discuss an extension of this construction to the case of immersed, exact Lagrangian cobordisms based on considering the Legendrian lift $\Sigma$ of $L$. When $L$ is an immersed, exact Lagrangian filling a choice of augmentation $\alpha$ for $\Sigma$ produces an induced augmentation $\epsilon_{(L, \alpha)}$ for $\Lambda_+$. Using the cellular formulation of LCH, we are able to show that any augmentation of $\Lambda$ may be induced by such a filling.
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.
Faramarz Vafaee : Floer homology and Dehn surgery
- Geometry and Topology ( 142 Views )The past thirty years have witnessed the birth of a beautiful array of approaches to the field of low dimensional topology, drawing on diverse tools from algebra, analysis, and combinatorics. One particular tool that has made a dramatic impact on the field is the Heegaard Floer theory of Ozsvath and Szabo. Defined 17 years ago, this theory has produced an encompassing package of invariants, which have significantly impacted the study of many areas of low dimensional topology, including Dehn surgery. In this talk, we will focus on two questions: a) which 3-manifolds do arise by Dehn surgery along a knot in the 3-sphere? b) what are all ways to obtain a fixed 3-manifold by Dehn surgery along a knot in the 3-sphere?
Daniel Scofield : Patterns in Khovanov link and chromatic graph homology
- Geometry and Topology ( 99 Views )Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. In this talk, we discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme-Guizon, Przytycki and Rong. An explicit formula for the rank of the third chromatic homology group on the main diagonal is given and used to compute the corresponding Khovanov homology group and the fourth coefficient of the Jones polynomial for links with certain diagrams.
Bahar Acu : Foliations of contact manifolds by planar J-holomorphic curves and the Weinstein conjecture
- Geometry and Topology ( 122 Views )In this talk, we will describe foliations of high dimensional ''iterated planar" contact manifolds by J-holomorphic curves and show that, by using this kind of technology, one can prove the long-standing Weinstein conjecture for iterated planar contact manifolds.
Marcos Jardim : On the spectrum of the Dirac operator and the Dolbeault Laplacian on Kahler manifolds
- Geometry and Topology ( 121 Views )There exists a large literature on lower bounds for the spectrum of the Dirac operator and the de Rham Laplacian on Riemannian manifolds. In this talk, we will consider the twisted Dirac operator and the twisted Dolbeault Laplacian on Kahler manifolds, and study how the spectrum changes with the coupling connection. We give lower bounds for their spectrum, showing that they are attained in the case of Riemann surfaces. This is joint work with Rafael F. Leao. The talk will be based on those two preprints, 0807.0813 and 0706.0878
Michael Singer : A new approach to monopole metrics
- Geometry and Topology ( 109 Views )The moduli space of non-abelian magnetic euclidean monopoles is known to be a smooth manifold of dimension $4k$, and carries a natural complete riemannian metric. Here $k$, a positive integer, is a topological invariant of the monopole, its magnetic charge. The metric is hyperKaehler, and in particular Ricci-flat, and this is one of the reasons why these moduli spaces are popular with geometers and physicists. In this talk, I shall explain a new approach to the analysis of monopole metrics and some new results about their asymptotic behaviour. This will be a report on joint work with Richard Melrose and Chris Kottke.
Mauro Maggioni : On estimating intrinsic dimensionality of noisy high-dimensional data sets
- Geometry and Topology ( 95 Views )We discuss recent and ongoing work, joint with A. Little, on estimating the intrinsic dimensionality of data sets assumed to be sampled from a low-dimensional manifold in high dimensions and perturbed by high-dimensional noise. This work is motivated by several applications, including machine learning and dynamical systems, and by the limitations of existing algorithms. Our approach is based on a simple tool such as principal component analysis, used in a multiscale fashion, a strategy which has its roots in geometric measure theory. The theoretical analysis of the algorithm uses tools from random matrix theory and exploits concentration of measure phenomena in high-dimensions. The talk will have a tutorial flavour: no previous knowledge of what mentioned above will be required, and several toy examples to build intuition about some measure-geometric phenomena in high-dimensions will be presented.
Dan Rutherford : Generating families and invariants of Legendrian knots
- Geometry and Topology ( 137 Views )Legendrian knots in standard contact R3 have in addition to their topological knot type two classical invariants known as the Thurston-Bennequin and rotation numbers. Over the past decade several invariants have been developed which are capable of distinguishing between knots with identical classical invariants. The purpose of this talk is to describe interesting relationships between some of these new invariants. Major players in this talk are the Chekanov-Eliashberg DGA (Legendrian contact homology) and related objects, as well as combinatorial structures on front diagrams and homological invariants arising from the theory of generating families (due to Chekanov-Pushkar, Fuchs, and Traynor). The main new result (joint with Fuchs) is that, when a Legendrian knot is defined by a generating family, homology groups obtained by linearizing the Chekanov-Eliashberg DGA are isomorphic to the homology of a pair of spaces associated with the generating family.
Valentino Tosatti : The evolution of a Hermitian metric by its Chern-Ricci curvature
- Geometry and Topology ( 101 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.
Natalia Kolokolnikova : Thom polynomial and its K-theoretic generalization
- Geometry and Topology ( 165 Views )Global singularity theory originates from problems in obstruction theory. Consider the following question: is there an immersion in a given homotopy class of maps between two smooth compact manifolds M and N? We can reformulate this question as "is the set of points, where a generic smooth map between M and N is not an immersion, empty"? This set is the simplest example of a singularity. Alternatively, we can ask a question whether the cohomology class of this set is 0 or not. Turns out, there is a universal polynomial depending only on the dimensions of M and N and on the type of singularity, that, when evaluated in the corresponding characteristic classes of M and N, computes the cohomology class of a singularity. This polynomial is called the Thom polynomial, and it is the central notion of singularity theory. In my talk I will give an introduction to singularity theory, define the classic Thom polynomial and talk about different approaches to its K-theoretic generalization.