| Participant | Organization | Title | Abstract | Scheduling comments | |||||||||
| Agol, Ian | UIC | Ricci flow | I'll discuss various aspects of Hamilton's program and Perelman's work to prove geometrization using Ricci flow. I'll discuss an application of Perelman's work to volumes of hyperbolic 3-manifolds. | agol@math.uic.edu | |||||||||
| Bigelow, Stephen | UCSB | bigelow@math.ucsb.edu | |||||||||||
| Bonahon, Francis | USC | Quantum hyperbolic invariants of surface diffeomorphisms | A general relationship between hyperbolic geometry, based on the Lie group PSL2, and certain aspects of topological quantum field theory, related to the quantum group Uq(sl2), seems to be emerging, although still in a very imprecise way. We discuss such a connection, based on the Chekhov-Fock quantization of Teichmuller space, and construct certain invariants of surface diffeomorphisms. The main point is that finite dimensional representations of the Chekhov-Fock algebra (a completely algebraic object) are controlled by the same data as pleated surfaces in hyperbolic 3-manifolds. This is joint work with Xiaobo Liu, still very much in progress. | leaving Wed. p.m. | fbonahon@math.usc.edu | ||||||||
| Boyer, Steve | Univ. Quebec | boyer@math.uqam.ca | |||||||||||
| Calegari, Danny | Cal Tech | Groups acting on the plane | We discuss some examples of groups of homeomorphisms of the plane, and suggest some tentative connections with topology, analysis and algebra. | dannyc@its.caltech.edu | |||||||||
| Cochran, Tim | Rice | Filtering Knots and Knot Concordance Classes by Gropes | We consider the monoid K/G(n) of all knots modulo the relation that two are equivalent if they cobound in S3 an embedded symmetric grope of height n. We also consider the group C/F(n) of all concordance classes of knots modulo the relation that two are equivalent if they cobound in S3 x [0,1] an embedded symmetric grope of height n. There is a natural surjection from the former to the latter. We show that these "filtrations" are non-trivial for each n>2. Part of this work is joint with Peter Teichner. Further results are joint with Tahee Kim. The strongest theorem is that for any knot J whose Alexander polynomial has degree at least 4, the preimage of J under the map K/G(n+1)--K/G(n)---C/F(n) is an infinite set, even when restricting to knots with the same Seifert matrix and genus as J. The techniques range from hands-on Kirby calculus, to homological algebra over noncommutative rings, to von Neumann rho invariants of Cheeger-Gromov. In this talk we highlight the techniques. | cochran@math.rice.edu | |||||||||
| Coffey, James | Melbourne | J.Coffey@ms.unimelb.edu.au | |||||||||||
| Collin, Olivier | UQAM | collin@math.uqam.ca | |||||||||||
| Cooper, Daryl | UCSB | cooper@math.ucsb.edu | |||||||||||
| Culler, Marc | UIC | culler@math.uic.edu | |||||||||||
| Dunfield, Nathan | Cal Tech | What is a random 3-manifold, what does it look like, and why should you care? | After warming up by discussing random surfaces, I'll discuss various notions of random 3-manifolds. One natural notion is picking among all triangulations with some fixed number of simplices, but since this seems very difficult to understand I'll focus on "random Heegaard splittings". In this context, I'll explain how to calculate, say the probability that a genus-2 3-manifold has a cover with covering group A5. In the large Heegaard genus limit such probabilities depend only on basic properties of the covering group in question. Finally, I'll discuss experimental evidence about the percentage of tunnel-number 1 3-manifolds that fiber over the circle. | nathand@math.harvard.edu | |||||||||
| Etnyre, John | Penn | Knot invariants via contact geometry | I will describe a method of associating a Legendrian T2 to a topological knot. By studying this T2 one can define invariants of the knot. In particular the contact homology of the T2 seems to be an interesting new invariant of the knot. Lenny Ng has combinatorially studied this new invariant, and shown connections with the Alexander polynomial. In this talk I will concentrate on the more geometric aspects of the invariant and ongoing work of Ekholm, Sullivan and myself aimed at better understanding the relation between Ng's combinatorics and the actual contact homology of the T2. | etnyre@math.upenn.edu | |||||||||
| Eudave-Munoz, Mario | UNAM | mario@matem.unam.mx | |||||||||||
| Frohman, Charles | Iowa | frohman@math.uiowa.edu | |||||||||||
| Hambleton, Ian | McMaster | Permutations, isotropy and smooth cyclic group actions on definite 4-manifolds (math.GT/0307297) | Joint work with Mihail Tanase. We use the equivariant Yang-Mills moduli space to investigate the relation between the singular set, isotropy representations at fixed points, and permutation modules realized by the induced action on homology for smooth group actions on certain 4-manifolds. | ian@math.mcmaster.ca | |||||||||
| Harvey, Shelley | MIT | Invariants of 3-manifolds from Non-commutative Algebra | We will discuss some non-commutative generalizations of the Alexander polynomial of a 3-manifold and some of their applications to 3- and 4-manifolds. In particular, we will define two sequences of integral invariants rn and dn and discuss their behavior. We show that they give computable algebraic obstructions to a 3-manifolds fibering over S1, a 3-manifold being a Seifert fibered space and a 4-manifold of the form X x S1 admitting a symplectic structure. We will also present the new result that, for 3-manifolds, the dn are non-increasing while the rn are non-decreasing (as a function of n). | sharvey@math.mit.edu | |||||||||
| Kerckhoff, Steven | Stanford | spk@geom.Stanford.EDU | |||||||||||
| Kirby, Rob | Berkeley | Singular symplectic 2-forms on 4-manifolds | It is known that an arbitray smooth 4-manifold with b2+ > 0 has a closed 2-form which is vanishes on a set Z of circles and is symplectic off the circles. In joint work with David Gay, we explicitly construct such singular 2-forms. | kirby@Math.Berkeley.EDU | |||||||||
| Klaff, Benjamin | UIC | klaff@math.uic.edu | |||||||||||
| Klodginski, Elizabeth | Michigan | eklodgin@umich.edu | |||||||||||
| Kobayashi, Tsuyoshi | Nara | On the growth rate of tunnel number of knots | This is a joint work with Yo'av Rieck. In this
talk we study the asymptotic behavior of the tunnel number of
knots in
closed 3-manifolds under repeated connected sum operation. For
convenience
we denote the connected sum of n copies of K, by nK. Let t(K)
be the
tunnel number of a knot K. It is well known that for any pair of
knots
K1, K2 the following inequality holds:
Theorem. Let K be a knot in a closed, orientable manifold M with g(E(K)) > g(M). Let n be the bridge number of K with respect to a Heegaard surface of genus t(K) (=g(E(K)) - 1). Then K has growth at most [(n-1)/n]. In particular, for sufficiently large n equality does not hold in inequality t(n K) ≤ n t(K) + (n-1). Note that the assumption of the theorem trivially holds for any knot in S3. I will also give some explanations on topics related to this subject. |
tsuyoshi@cc.nara-wu.ac.jp | |||||||||
| Lackenby, Marc | Oxford | Heegaard genus and covering spaces | The Heegaard gradient conjecture proposes that a hyperbolic 3-manifold has a sequence of covering spaces with sublinear growth of Heegaard genus if and only if the manifold is virtually fibred. The relevance of this conjecture is that it is one half of a programme for proving the virtually Haken conjecture. In my talk, I will outline recent progress on the Heegaard gradient conjecture. I will explain why a negatively curved 3-manifold is virtually fibred if it has a sequence of finite regular covering spaces whose Heegaard genus grows more slowly than the fourth root of the degree. | lackenby@maths.ox.ac.uk | |||||||||
| Long, Darren | UCSB | long@math.ucsb.edu | |||||||||||
| Menasco, William | SUNY Buffalo | Applications of the Markov Theorem Without Stabilization | This work is joint with Joan
Birman. The Markov Theorem Without Stabilization (MTWS)
establishes the
existence of a finite set of moves which allow one to pass from an
arbitrary closed braid representative X+ of an
arbitrary oriented
knot or link type in oriented 3-space to an arbitrary
representative
X- of minimum braid index, through a sequence of closed
braids
whose braid index is non-increasing. Using the MTWS, we will give several constructions for producing two transversal knots that have the same topological knot type in the 3-sphere, have the same transversal invariants in the standard contact structure on the 3-sphere, but are not transversally isotopic. |
avoid 16th & 17th | menasco@tait.math.buffalo.edu | ||||||||
| Rolfsen, Dale | UBC | Untwisting Heegaard diagrams in 3-space | In joint work with David Gillman, I will outline a proof of a theorem that contradicts a remark made by Haken in 1969. Suppose V is a genus g handlebody embedded in 3-space and J is a family of g curves on the boundary of V, which describe a Heegaard splitting of a homology 3-sphere, then there is a self-homeomorphism of V so that the image of each curve of J bounds a surface in the complement of the interior of V. These surfaces will not be disjoint unless the manifold is a homotopy sphere. | rolfsen@math.ubc.ca | |||||||||
| Ruberman, Daniel | Brandeis | Rohlin's invariant and Gauge theory | I will describe my joint work with Nikolai Saveliev on a topic in 3.5 dimensional topology: the relation between Rohlin's invariant of 3-manifolds and 4-dimensional gauge theory. For 4-manifolds with the Z[Z]-homology of S1× S3, there is a Casson-type invariant defined via gauge theory, and also a Rohlin invariant. I will discuss the natural conjecture that these are the same, modulo 2, as in the 3-dimensional case. This conjecture has some interesting implications about the homology cobordism group and other classical problems. I will outline the proof for the case when the manifold fibers over the circle with finite order monodromy. Similar invariants exist for 3 and 4-dimensional homology tori, and the conjecture holds in these cases. | ruberman@brandeis.edu | Rubinstein, J. Hyam | Melbourne | Structures on ideal triangulations of 3-manifolds | This is joint work with Ensil Kang. For an orientable irreducible atoroidal 3-manifold with incompressible tori boundary, Lackenby constructed taut ideal triangulations and asked if these admit angle structures. We give a necessary and sufficient condition for when there is such an angle structure. The obstruction is the existence of certain types of generalized almost normal surfaces. (This corrects an earlier version of this work - thanks to Cooper and Lackenby for helpful comments). Other results include a boundary operator from the space of spun normal surfaces to homology classes in the peripheral tori and existence theory for spun normal surfaces | rubin@ms.unimelb.edu.au | ||||
| Scharlemann, Martin | UCSB | mgscharl@math.ucsb.edu | |||||||||||
| Schleimer, Saul | UIC | Heegaard splittings, the curve complex, and hyperbolic geometry | Fix H a Heegaard splitting surface of the closed orientable three-manifold M. Let V and W be the handlebodies which are the complement of H. We have natural subsets C(V) and C(W) of the curve complex of H, C(H). These are the curves of H which bound disks in V and W, respectively. Hempel defines the distance of H to be the length of the shortest path between C(V) and C(W) inside of C(H). I'll discuss what is known about this quantity and how it relates to other features of M such as incompressible surfaces, triangulations, mapping class groups, etc. I'll also discuss what is conjectured -- including the possibility of predicting the hyperbolic structure of M when the distance is sufficiently large. | saul@math.uic.edu | |||||||||
| Schultens, Jennifer | UC Davis | Some topology and algebra of graph manifolds | Graph manifolds are a fairly well understood class of 3-manifolds. They are modelled on a (1-dimensional) graph and a set of (2-dimensional) surfaces. This facilitates many computations. The structure of Heegaard splittings of graph manifolds can be completely described. In particular, each such Heegaard splitting can be obtained via a finite set of standard constructions. It follows that the Heegaard genus can be computed for a given graph manifold or certain classes of graph manifolds. The rank of the fundamental group of a 3-manifold, i.e., the smallest possible number of generators for the fundamental group of a 3-manifold is an algebraic analogue of the Heegaard genus. Interestingly, the rank of the fundamental group of a 3-manifold can be smaller than the Heegaard genus. This was first shown by Boileau-Collins-Zieschang. Richard Weidmann and I show that the discrepancy can in fact be arbitrarily large. | jcs@mathcs.emory.edu | |||||||||
| Shalen, Peter | UIC | Knots with only two strict essential surfaces | In this talk I will discuss recent joint work with Marc
Culler. The general results concern a knot K in a closed
orientable
3-manifold Σ with cyclic fundamental group. An essential
surface F in
the exterior M(K) is said to be strict if it is not a fiber
in a
fibration over S1 and is not the common associated
∂I-bundle of two twisted I-bundles whose union is
M(K).
Suppose that F1 and F2 are connected strict
essential
surfaces in M(K), and that every connected strict essential
surface in M is
isotopic to either F1 or F2. For simplicity
assume
that the Euler characteristic χi of Fi
is
strictly negative for i=1,2. Then ∂Fi ≠ ∅
for
i=1,2. For i=1,2, let si=pi/qi
and
mi denote, respectively, the boundary slope (with
respect to any
given framing) and the number of boundary components of
Fi.
Assume that q2 ≠ 0. Let Δ =
|p1q2 - p2q1| denote
the
geometric intersection number of the slopes. Then Δ ≠ 0
and
I will discuss the significance of this result and indicate some features of the proof, which combines character variety techniques with combinatorial estimates. |
shalen@math.uic.edu | |||||||||
| Teichner, Peter | UCSD | Knots, von Neumann Signatures, and Grope Cobordism | We explain new developments in classical knot theory in 3 and 4 dimensions, i.e. we study knots in 3-space, up to isotopy as well as up to concordance. In dimension 3 we give a geometric interpretation of the Kontsevich integral (joint with Jim Conant), and in dimension 4 we introduce new concordance invariants using von Neumann signatures (joint with Tim Cochran and Kent Orr). The common geometric feature is the notion of a grope cobordism and the main result (joint with Tim) is that these filtrations of the space of knots are nontrivial at every level. | arriving Sunday | pteichne@ucsd.edu | ||||||||
| Thompson, Abigail | UC Davis | thompson@math.ucdavis.edu | |||||||||||
| Tillman, Stephan | UQAM | tillman@math.uqam.ca | |||||||||||
| Walsh, Genevieve | UC Davis | Virtually Haken fillings of two-bridge knot complements | For any two-bridge knot complement, there is a finite cover that is the complement of a link of great circles in $S^3. We show that all great circle link complements are fibered, which implies that all two-bridge knot complements are virtually fibered. We also show that for many two-bridge knots, this cover contains a closed incompressible surface. Infinitely many fillings of the two-bridge knot complement lift to fillings of great circle link complement where the incompressibility of this surface is preserved. Using this, we show that infinitely many fillings of an infinite class of two-bridge knot complements are virtually Haken. | gwalsh@math.utexas.edu | |||||||||
| Wu, Ying-Qing | Iowa | wu@math.uiowa.edu | |||||||||||
| Zhang, Xingru | SUNY Buffalo | Traces of representations and topology of 3-manifolds | We introduce and study trace fields and invariant trace fields of SL(2,C)-representations of 3-manifold groups. We give conditions on such fields which imply that the underlying 3-manifold is virtually Haken or has virtually positive or infinite first Betti number. We define A-polynomial rows for link manifolds and apply them to construct large hyperbolic link manifolds with non-integral traces. We give a relation between the canonical Culler-Shalen norm and the invariant trace field of a hyperbolic knot manifold. | xinzhang@math.buffalo.edu |