Previous semesters: Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.

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Date | Speaker | Title |
---|---|---|

Jan. 23, 1:00 p.m. | Organizational meeting | |

Jan. 30 | Jonathan Bloom (Columbia) | A link surgery spectral sequence in monopole Floer homology |

Feb. 6 | John Etnyre (Georgia Tech) | Fibered knots and the Bennequin bound |

Feb. 13 | Jeremy Kahn (Stony Brook) | Random Ideal Triangulations and the Weil-Petersson Ehrenpreis conjecture (PDF) |

Feb. 20, 1:10 PM |
Mikhail Gromov (Courant / IHÉS) (Joint with SGGT seminar.) |
TBA |

Feb. 20, 2:30 PM | Dieter Kotschick (LMU Munich / IAS) | Ordering manifolds by maps of non-zero degree |

Feb. 27 | Joseph Maher (Oklahoma State) | Random Heegaard splittings |

Mar. 6 | Jesse Johnson (Yale) | Common stabilizations for Heegaard splittings |

Mar. 6 | Alex Kontorovich (Brown) | Apollonian circle packings and horospherical flows on hyperbolic 3-manifolds |

Mar. 13 | Tom Church (Chicago) | Groups of mapping classes that cannot be realized by diffeomorphisms (PDF) |

Mar. 27 | Alexander Fel'shtyn (University of Szczecin / Boise State) | How to categorify dynamical zeta functions |

Apr. 3 | Sergio Fenley (FSU / Princeton) | Existence and rigidity of
pseudo-Anosov flows transverse to R-covered foliations |

April 10 | Keiko Kawamuro (Rice / IAS) | Braids and open book decompositions |

Apr. 17 | Andrew Putman (MIT) | The Picard group of the moduli space of curves with level structures |

Apr. 24 | Guoliang Yu (Vanderbilt) | Geometric complexity and topological rigidity |

May 1 | No seminar (NY Joint Symplectic Geometry Seminar) | |

May 8 | Peter Teichner (Berkeley) | Whitney towers in 4-manifolds |

## Abstracts.

#### January 30, 2009.

Jonathan Bloom, Columbia University, “A link surgery spectral sequence in monopole Floer homology”.Abstract: In Heegaard Floer homology, the surgery exact sequence of a knot has a natural generalization to links, which takes the form of a spectral sequence and leads to an important connection with Khovanov homology (see "Branched-double covers" by Ozsvath and Szabo). In this talk, I will derive such a link surgeries spectral sequence for monopole Floer homology. In some ways, the proof is more direct and geometric in this context. The main idea involves stretching 2-handle cobordisms in a controlled manner using polytopes of metrics. These polytopes take the form of permutohedra and, more generally, graph-associahedra. No background in monopole Floer homology (or polytopes) required.

#### February 6, 2009.

John Etnyre, Georgia Tech, “Fibered knots and the Bennequin bound”.

Abstract: In the early 80's Bennequin proved that the
self-linking number of a transverse knot in the standard contact
structure on S^{3} was bounded above by minus the Euler
characteristic of any Seifert surface for the knot. Eliashberg
later proved the same bound in any tight contact manifold. It
has been know for quit some time now that this bound is not
optimal for many knot types. It turns out there is a elegant
interaction between the optimality of the Bennequin inequality
for fibered knots and Giroux's work on the relation between open
books and contact structures. In this talk I will explain this
interaction and give a precise characterization of when the
Bennequin bound is optimal for fibered knots. I will also
discuss several interesting corollaries, including applications
to braid theory and a proof that “transverse knots
classify contact structures”.

#### February 13, 2009.

Jeremy Kahn, SUNY Stony Brook, “Random Ideal Triangulations and the Weil-Petersson Ehrenpreis conjecture” (PDF).

Abstract: The Ehrenpreis conjecture states that given any two compact hyperbolic Riemann surfaces, or any two non-compact finite area hyperbolic Riemann surfaces, there are finite covers of the two surfaces that are arbitrarily close in the Teichmuller metric.

We prove the same statement for the normalized Weil-Petersson metric, in the case where the two surfaces are non-compact. In the course of doing so we construct “Random ideal triangulations” of the covers, where fairly accurate estimations of the proportion of each immersed triangle can be made.

#### February 20, 2009.

Dieter Kotschick, Ludwig Maxmilian University of Munich and IAS, “Ordering manifolds by maps of non-zero degree”.

Abstract: The existence of a map of non-zero degree defines an interesting transitive relation, called the domination relation, between homotopy types of closed oriented manifolds. In dimension two the relation coincides with the ordering given by the genus. We study this relation in higher dimensions, with special emphasis on the case where the domain is a non-trivial product and the target has a large universal covering in a suitable sense, e.g., the target could be non-positively curved. In many such situations we prove that there are no maps of non-zero degree. When the target is an irreducible locally symmetric space of non-compact type this was conjectured by Gromov. (Joint work with C. Loeh.)

#### February 27, 2009.

Joseph Maher, Oklahoma State, “Random Heegaard splittings”.

Abstract: A random Heegaard splitting is a 3-manifold obtained
by using a random walk of length n on the mapping class group as
the gluing map between two handlebodies. We show that the joint
distribution of random walks of length *n* and their
inverses is asymptotically independent, and converges to the
product of the harmonic and reflected harmonic measures defined
by the random walk. We use this to show that the translation
length on the curve complex of a random walk grows linearly in
the length of the walk, and similarly, that distance in the
curve complex between the disc sets of a random Heegaard
splitting grows linearly in *n*. In particular, this
implies that a random Heegaard splitting is hyperbolic with
asymptotic probability one.

#### March 6, 2009.

Jesse Johnson, Yale University, “Common stabilizations for Heegaard splittings”.

Abstract: A Heegaard surface is a surface embedded in a 3-manifold so that it cuts the 3-manifold into simple pieces called handlebodies. Given a Heegaard surface, there is a construction called stabilization that creates a new Heegaard surface of higher genus. It has long been known that any two Heegaard surfaces for the same manifold can be stabilized some number of times to produce isotopic surfaces, but it is not well understood how many stabilizations are, in general, necessary. I will describe a family of examples that require a relatively large number of stabilizations, showing that the problem is more complex than previously conjectured.

Alex Konotorovich, Brown University, “Apollonian circle packings and horospherical flows on hyperbolic 3-manifolds”.

Abstract: We prove an asymptotic formula for the number of circles in an Apollonian packing of bounded curvature. Using the affine linear sieve, we give sharp upper bounds for the number of circles of prime curvature, and the number of ”twin prime“ tangent circles. The main ingredient of our proof is the equidistribution of long horospherical flows in the unit tangent bundle of an infinite volume hyperbolic 3-manifold, under the assumption that the Hausdorff dimension of its limit set exceeds one. This is joint work with Hee Oh.

#### March 27, 2009.

Alexander Fel'shtyn, University of Szczecin / Boise State, “How to categorify dynamical zeta functions”.

Abstract: I will discuss Weil type dynamical zeta functions. These zeta functions count periodic points of dynamical system homologically or in the presence of fundamental group of a manifold and give rise to the Reidemeister torsion.

In the talk a programm of a categorification of Weil type dynamical zeta functions is proposed. The first indication of a possibility of such categorification is a formula of Milnor's that relates Weil zeta function of a monodromy map to the Alexander polynomial of a algebraic knot and Ozsváth-Szabó categorifications of the Alexander polynomial and Turaev torsion function.

I will show that the Weil zeta function of a symplectomorphism
of surface is a graded Euler characteristic in PFH homology or a
graded, regularised Euler characteristic of mapping tori in
HF^{+}, ECH or SWF homology.

It is interesting to generalise this result to higher dimensions.

There is also a strong indication that PFH and HF^{+} homology
give a categorification of the Nielsen periodic point theory
and corresponding minimal zeta function.

#### April 3, 2009.

Sergio Fenley, Florida State University /
Princeton University,
“Existence and rigidity of pseudo-Anosov flows transverse to
**R**-covered foliations”.

Abstract: Pseudo-Anosov flows are extremely common in three
manifolds and they are very useful. How many pseudo-Anosov flows
are there in a manifold up to topological conjugacy? We analyse
this question in the context of flows transverse to a given
foliation *F*. We prove that if *F* is **R**-covered (leaf space in the
universal cover is the real numbers) then there are at most two
pseudo-Anosov flows transverse to *F*. In addition if there are
two, then the manifold is hyperbolic and the the foliation *F*
blows down to a foliation topologically conjugate to the stable
foliation of a particular type of an Anosov flow. The results
use the topological theory of pseudo-Anosov flows, the universal
circle for foliations and the geometric theory of **R**-covered
foliations. We also discuss the existence of transverse
pseudo-Anosov flows in this setting.

#### April 10, 2009.

Keiko Kawamuro, Rice University / IAS, “Braids and open book decompositions”.

Abstract: We construct an immersed surface for a null-homologous
braid in an annulus open book decomposition, which is hinted at
the so called Bennequin surface for a braid in **R**^{3}. By
resolving the singularities of the immersed surface, we obtain
an embedded Seifert surface for the braid. We find a
self-linking number formula associated to the surface and see
that it is an extension of the Bennequin's self-linking formula
for a braid in **R**^{3}. We also prove that our
self-linking formula is invariant (changes by 2) under a
positive (negative) braid stabilization which preserves
(changes) the transverse knot class. (Joint work with Elena
Pavelescu.)

#### April 17, 2009.

Andrew Putman, MIT, “The Picard group of the moduli space of curves with level structures”.

Abstract: The moduli spaces of curves with level structures are the finite covers of the moduli spaces of curves associated with the linear congruence subgroups of the mapping class group. We will describe a sequence of results about the low-dimensional cohomology groups of these linear congruence subgroups which yield a complete description of the complex line bundles on the corresponding moduli spaces.

#### April 24, 2009.

Guoliang Yu, Vanderbilt, “Geometric complexity and topological rigidity”.

Abstract: In this talk, I will introduce a notion of geometric complexity and discuss its applications to geometric group theory and rigidity of manifolds. In particular, I will show how to prove various versions of the Borel conjecture under certain finiteness conditions on the geometric complexity. This is joint work with Erik Guentner and Romain Tessera.

# Other relevant information.

## Other area seminars.

- Columbia Symplectic Geometry/Gauge Theory Seminar
- CUNY Geometry and Topology Seminar
- CUNY Complex Analysis & Dynamics Seminar
- CUNY Magnus Seminar
- Princeton Topology Seminar.
- All Columbia Math Dept Seminars

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