(http://www.ma.huji.ac.il/~drorbn/Talks/Lehigh-0006/) It is well known that when the Sun rises on midsummer's morning over the "Heel Stone" at Stonehenge, its first rays shine right through the open arms of the horseshoe arrangement. Thus astrological lineups, one of the pillars of modern thought, are much older than the famed Gaussian linking number of two knots. Recall that the latter is itself an astrological construct: one of the standard ways to compute the Gaussian linking number is to place the two knots in space and then count (with signs) the number of shade points cast on one of the knots by the other knot, with the only lighting coming from some fixed distant star.
Mathematicians can only digest stuff when it is made absolutely general. We therefore pick a knot, given as a specific embedding of S1 in \mathbbR3 , and count the number of all "Stonehenge-inspired chopstick towers" that can be built upon it; namely, the number of delicate arrangements of chopsticks whose ends are lying on the knot or are supporting each other in trivalent corners joining three chopsticks each, so that each chopstick is pointing at a different pre chosen point in heaven that has a high mythical meaning.
Quite amazingly, when these stellar webs are counted correctly, the result is a knot invariant valued in some space of diagrams, deeply related to certain aspects of Lie theory and of the theory of Hopf algebras. We will touch on the former and dwelve into the latter, finding that if the Stonehengians had taken themselves seriously some 4,000 years ago, they would have been forced to discover quasi-Hopf algebras.
The space of representations of the fundamental group of a surface M into a Lie group G is a natural object which exhibits rich geometric structure and rich symmetry. In particular these moduli spaces are interesting examples of symplectic geometry. For certain compact G, the action is ergodic. For certain noncompact G, the action is ergodic on certain components and properly discontinuous on others (which arise from uniformization). A recurrent theme is that nontrivial dynamics accompanies nontrivial topology. Uniformizations by convex domains in the real projective plane, as well as complex hyperbolic Kleinian groups furnish components - generalizing Teichmuller space - where the action is proper. Using singular hyperbolic structures and the theory of Higgs bundles developed by Hitchin, Simpson and others, we interpret arbitrary surface group homomorphisms as uniformizations, which provides a geometric framework for dynamical questions. We give a complete analysis of the topology and the dynamics for the SL(2,R)- and SU(2)- moduli spaces for the case of a once-punctured torus.
Let f be a C1 action of a finitely generated group G on a smooth, compact manifold without boundary. We show that if H0(G, D0(M)) = 0 and H1(G, D0(M)) = 0 then there is a neighbourhood in the C1 actions of G on M where every G action is conjugate to f via a homeomorphism. Actions satisfying these conditions are defined to be Anosov-like and we show that a diffeomorphism of a smooth manifold M is Anosov if and only if the corresponding \Z action is Anosov-like. Actions of groups satisfying the Strong Vanishing Condition, having dense periodic points and have no invariant vectors in the tangent space at each periodic point are Anosov-like and it would appear that the manifold must be a nilmanifold if this holds.
Multiple zeta values (MZVs) first became well-known in connection with Kontsevich's invariant in knot theory. To understand the relations of MZVs, the author reformulated them as images of a homomorphism z:\frak H0®\bold R, where \frak H0 is a subalgebra of the quasi-symmetric functions. The homomorphism z has a natural extension to the full algebra of quasi-symmetric functions, and this extension appears in a multiplicative sequence defined by A. Libgober in connection with mirror symmetry. We shall explain this, and also discuss some newly proved relations among MZVs.
This is a report on recent work of Joachim Kock, who has created a theory of "tangential quantum cohomology" for smooth projective varieties. The theory specializes to the ordinary quantum cohomology. For a homogeneous variety, it encodes all the characteristic numbers of rational curves (defined by conditions of incidence and tangency to subvarieties). Thus it greatly generalizes my work with Lars Ernstrom, in which we made a similar construction for the projective plane. Unlike in that earlier work, Kock eschews auxiliary moduli spaces, working directly in the space of maps of rational curves into the target variety and employing "modified gravitational descendants."
Foe a general complete toric variety, we give a formula of equivariant Todd class in terms of the combinatoric data.
I'll report on some recent developments in the geometry of real m-submanifolds of complex n-space, with m < n. Topics include: the local analytic geometry near points where the tangent space contains a complex line; characteristic class formulas enumerating such CR singularities on compact submanifolds; real algebraic examples.
A solution of the vortex filament flow is a curve S in three-dimensional space evolving by dS/dt = kB. These induce solutions of the focusing cubic nonlinear Schrodinger equation (NLS) in 1+1dimensions via the Hasimoto map. In joint work with A. Calini, we have discovered several instances of connections between geometrical (symmetry, planarity, self-intersection) or toplogical (knot type) features of the curve, and the spectrum of the corresponding periodic NLS potential.
Understanding the shape of surfaces with low regularity is crucial in applications ranging from problems in computer vision to the study of lipid membranes. We will discuss new existence and rigidity results for surfaces which are at most C1 and their applications in biology and computer assisted shape recognitoion and representation.
Taxicab Geometry is a non-Euclidean geometry, defined by E. F. Krause in 1975. We will define the area formulas for a circle, rectangle, triangle, trapezoid and ellipse in taxicab geometry.
This talk uses the representation theory of loop groups to construct a family of Dirac operators on homogeneous loop spaces (viewed either as loops on G/H or as (LG)/(LH) ). From an algebraic point of view, this generalizes recent work of Kostant, who constructed a Dirac operator with an interesting cubic term associated to a compact homogeneous space G/H. From a geometry/topology point of view, Witten showed that the elliptic genus can be viewed as the index of a Dirac operator on loop spaces, and Taubes has constructed such a Dirac operator locally in a neighborhood of the constant loops.
Loewner and Nirenberg found that the expression -[1/u] uij transforms as a symmetric (0,2) tensor under projective coordinate changes of a domain in Rn so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold, the section u can be regarded as a metric potential analogous to the local potential in Kähler geometry. If M is a compact locally projectively flat manifold, then the existence of such a negative section u of the dual of the tautological bundle over M such that -[1/u] uij is a Riemannian metric implies that M is projectively equivalent to a quotient of a bounded convex domain in Rn. We also discuss a natural Monge-Ampère equation which occurs in this context and draw analogies to the similar situations on complex manifolds (the Kähler-Einstein problem), affine flat manifolds (work of Cheng-Yau), and locally conformally flat manifolds (work of Schoen-Yau).
Milnor (1950) showed that the total curvature of a closed polygonal space curve was ³ 2p with equality if and only if the curve was convex and planar. For knots the total curvature was shown to be strictly greater than 4p. We expand upon these ideas and extend two theorems of Milnor and Totaro and show the following: If t denotes the curvature and k the torsion of a closed, generic, and oriented polygonal space curve X in \E3, then we show that òX Ö{k2 + t2} ds = òX k ds + òX | t | ds > 4p if t is positive. We also show that òX Ö{k2 + t2} ds ³ 2pn if no four consecutive vertices lie in a plane and X has linking number n with a straight line. The simple proofs use some pretty ideas in integral geometry by Fary, Banchoff, and Wiener.
We will also show discuss some of the history of the geometry of curves in the plane and in space, as well as some new joint work by Juan Carlos Alvarez Paiva and the speaker.
The purpose of my talk is the investigation a group of natural automorphisms of an invariant almost complex structure, some group of natural isometries, a group of special affine transformations on homogeneous spaces. Topology and analytic structure of these groups will be described.
The diameter rigidity theorem of Gromoll and Grove [1987] states that a Riemannian manifold with sectional curvature ³ 1 and diameter ³ p/2 is either homeomorphic to a sphere, locally isometric to a rank one symmetric space, or it has the cohomology ring of the Cayley plane CaP. The reason that they were only able to recognize the cohomology ring of CaP is due to an exceptional case in another theorem [Gromoll and Grove, 1988]: A Riemannian submersion s:Sm® Bb with connected fibers that is defined on the Euclidean sphere Sm is metrically congruent to a Hopf fibration unless possibly (m,b) = (15,8). We will rule out the exceptional cases in both theorems.
Our motivation of studying multiple valued functions in the setting of calculus of variation comes from Almgren's big regularity paper in 1983, where he concludes that the m dimensional mass (area counting multiplicity) minimizing integral currents are analytic except on the sets S of Hausdorff codimension at least two, i.e. Hm-2+e(S) = 0, " e > 0. The first step in his approach is to understand the regularity of Dirichlet Minimizing Multiple Valued Functions, which he uses to approximate mass minimizing integral currents.
In this paper, we study the regularity of Stationary Harmonic Multiple Valued Functions, which we would use to study Stationary Integral Currents. Our conclusion is that the Two Dimensional Stationary Harmonic Multiple Valued Functions are analytic except on the sets of Codimension at least one on the domain, i.e H1+e(S) = 0, " e > 0.
A problem which is related to intersection homology theory will be mentioned as well.
If we study how idealised soap films (minimal surfaces) intersect polyhedral obstacles then we can extend methods and results to associated spaces including 3-orbifolds and pseudo-manifolds. Associated spaces are produced by putting identifications on the boundary of the polyhedral obstacle. We can also examine minimizing surfaces and foliations by minimal surfaces in these settings.
We will prove and extend the result from the polyhedral obstacle problem that for every convex pyramid obstacle there is a set of planes intersecting the pyramid only at the apex which are minimal surfaces but not minimizers. These are the planes whose normals intersect the interior of the pyramid. This extends to the cone of any surface by the following identifications which do not affect the proof.
Start with an obstacle in the shape of a pyramid with a polygonal base. Identify edges on the polygon to give the desired surface and extend these identifications over the pyramid to give the cone of this surface. The apex of a cone of any surface other than the sphere is a non-manifold point, its link is the surface being coned. The cone of a sphere is a 3-ball. The cone of \mathbbR\mathbb P2 is a 3-orbifold, the quotient of a 3-ball by an involution. Pushing a surface onto, off, or over such a cone point can change its topology.
Orbifolds and pseudo 3-manifolds produced by the above process can contain 1-manifolds with singular curvature. The quotient of a smooth 3-manifold by a rotation has such a singularity. For all but a countable number of values of singular curvature, and all positive curvatures, minimal surfaces in general position intersect the singularity normally. Being able to work in H3 and S3 in addition to Euclidean space can give control over these singular curvatures.
A natural extension of the classical central limit theorem for probability distributions in Euclidian spaces, can be formulated in the context of complete manifolds. Here, we give a central limit theorem for extrinsic sample means from a nonfocal distribution on a closed submanifold of EN. This theorem is further specified, in the case of two equivariant embeddings of homogeneous spaces, that are of relevance in high level image analysis.For small sample of images, bootstrap approximations are used. Three concrete applications are also given. The first application is related to a recent debate in planetary astronomy on the correct denomination of Pluto. In the second application, related to reconstructive plastic surgery we determine with confidence the extrinsic mean shape of a group of eight anatomical landmarks in normal populations of children aged 9 and 14. Finally, our third application is related to recent joint work with K.V.Mardia on high level image analysis and projective shapes.
We show that the topological structure of a locally smooth orbifold is determined by its orbifold diffeomorphism group. Let \operatornameDiffr\operatorname\textupOrb(O) denote the Cr orbifold diffeomorphisms of an orbifold O. Suppose that F:\operatornameDiffr\operatorname\textupOrb(O1) ® \operatornameDiffr\operatorname\textupOrb(O2) is a group isomorphism between the the orbifold diffeomorphism groups of two orbifolds O1 and O2. We show that F is induced by a homeomorphism h:XO1 ® XO2, where XO denotes the underlying topological space of O. That is, F(f) = h f h-1 for all f Î \operatornameDiffr\operatorname\textupOrb(O1). Furthermore, if r > 0, h is a Cr manifold diffeomorphism when restricted to the complement of the singular set of each stratum.
We discuss how a number of recent results concerning braid groups and configuration spaces generalize naturally to fiber-type hyperplane arrangements and certain higher dimensional analogues. These include results on the vanishing of Whitehead groups, and results relating the Lie algebras arising (i) from the lower central series of the fundamental groups of certain spaces and (ii) from the loop space homology of related spaces.
(Joint with Bruner and Mahowald) The new Hopkins-Mahowald-Miller spectrum eo2 can be applied in obstruction theory to obtain new nonimmersions of real projective spaces. We will describe the results and method.
Work in progress concerning the problem of deciding the numerical value of L.-S. category of spaces with three cells of positive dimension will be described.
Extended joint work with Mahowald and Ravenel has suggested that the Telescope Conjecture is most likely false. One consequence of this has been the construction of various "telescopic" or "finite" localizations that preserve many of the pleasant properties of classical Bousfield localization, but which agree with the geometrically defined telescope, in appropriate cases. Mahowald and Sadofsky have constructed a version of the Adams spectral sequence which converges to the homotopy groups of certain finite localizations. They have conjectured that the E2 term of this spectral sequence coincides with an elementary homological algebra construction on the homology of the Steenrod algebra. I will talk about recent progress toward this conjecture.
When a closed manifold admits an action of a finite group, we construct various higher order orbifold invariants through the associated family of iterated twisted free loop spaces.
These orbifold invariants based on the Euler characteristic are higher order generalizations of the orbifold Euler characteristic defined by physicists in mid 1980s using commuting pairs of elements of the group. Higher order generalization involves pairwise commuting n-tuple of elements of the group. We give generating functions for these invariants for symmetric groups.
Higher order orbifold invariants based on signature, spin index, and Hirzebruch chi-y characteristic can be constructed. We will discuss their properties including their modularity properties. Some of these orbifold invariants are known as orbifold elliptic genera.
There are many known examples of link complements in the 3-sphere that carry a hyperbolic structure, but in one dimension higher none have yet been displayed. For this purpose, the proper generalization of a ``link'' in dimension 4 is a disjoint union of tori and Klein bottles inside a closed 4-manifold. We prove that some of the examples of hyperbolic 4-manifolds constructed by Ratcliffe and Tschantz have orientable double covers that are complements of tori inside the 4-sphere.
Recently, Louis Kauffman introduced the definition of virtual knots and showed that essentially all known quantum link invariants (and their associated Vassiliev invariants) have highly non-trivial extensions to the virtual category. One motivation behind virtuals is the relationship between them and Gauss arrow diagrams. This talk will discuss joint work with Kauffman, in progress, on virtual knot theory, and in particular on the problem of extending known invariants to invariants of flat virtuals. (Flat virtual knots and links are a generalization of the category of immersed curves in the plane. Problems about knotted standard virtuals give rise to corresponding hard problems about immersed curves.)
We discuss special representations of the mapping class groups that arise from semisimple and non-semisimple TQFT-constructions in dim=3. We find combinatorial formulas for Frohman Nicas homology TQFT as well as for the Casson invaraint from Heegaard splittings.
Higher order link polynomials were defined by combining ingredients from link polynomials and Vassiliev invariants. In this talk, we will survey the following results on this topic:
1. The classification of the order 1 Homfly polynomial, done by the speaker.
2. The theorem that each nth partial derivatives of the Homfly polynomials is a higher order Homfly polynomial of order n, due to Lickorish and the speaker. This also greatly simplifies the work in (1).
3. The determination of the free part of the higher order Conway skein module, due to Andersen and Turaev.
4. An affirmative answer to the question, asked by Lickorish-Rong, whether all partial derivatives of the Homfly link polynomials are linearly independent.
5. The classification of all the higher order Conway polynomials, following the work above.
What is "the right" boundary to put on a spacetime? The 1973 construction of Geroch, Kronheimer, and Penrose, called the Causal Boundary (defined purely in terms of the causal structure of the spacetime), has long seemed to be a natural object, but not much more could be said about it. A few years ago I showed that the Future Causal Boundary has a categorically universal property, but only with respect to its causal structure, not its topology. This talk will report on the expansion of this universality to the topological category, in terms of a topology to be inferred from any set with a causal structure. (Part of what must be shown is that this causally inferred topology is the "right one" to use.)
Result: In the category of spacetimes with spacelike boundaries, the Future Causal Boundary is categorically universal; any other reasonable future boundary must be a topological quotient of the Future Causal Boundary.
Particles of spin more than 2 are unknown to exist and there is some indication they cannot be described in terms of ordinary Lie (gauge theory). Burgers, Behrends and van Dam describe the theory of such particles in terms of `field dependent' analogs of Lie algebra representations. We explain the underlying mathematics as that of an sh-Lie algebra - of a rather novel form.