# Seminars & Events for Algebraic Topology Seminar

##### Loop space homology, string homology, and closed geodesics

The homology of free loop space of a manifold enjoys additional structure first identified by Chas and Sullivan. The string multiplication has been studied by Ralph Cohen and John Jones and together with J.~Yan, they have introduced a spectral sequence converging to string homology that is related to the Serre spectral sequence for the free loop space. Using this tool, and the work of Felix, Halperin, Lemaire and Thomas, Jones and I establish some conditions on manifolds that guarantee the existence of infinitely many closed geodesics on the manifold in any Riemannian metric.

##### Finiteness properties for the fundamental groups of complex algebraic varieties

We describe some relations obtained in joint work with S.Papadima and A. Suciu between finiteness properties of fundamental groups and resonance and characteristic varieties.

##### On the topological complexity of 2-torsion lens spaces

The topological complexity of a topological space is the minimum number of rules required to specify how to move between any two points of the space. A ``rule'' must satisfy the requirement that the path varies continuously with the choice of end points. We use connective complex K-theory to obtain new lower bounds for the topological complexity of 2-torsion lens spaces. We follow a program set up by Jesus Gonzalez, and answer a question posed by him.

##### Act globally, compute locally: group actions, fixed points, and localization

Localization is a topological technique that allows us to make global equivariant computations in terms of local data at the fixed points. For example, we may compute a global integral by summing integrals at each of the fixed points. Or, if we know that the global integral is zero, we conclude that the sum of the local integrals is zero. This often turns topological questions into combinatorial ones and vice versa. I will give an overview of how this technique arises in symplectic geometry, and some details about some recent projects in this area.

##### Topological Witt groups and Reality

This is joint work with Karoubi. Given an algebraic variety over the real numbers, we compare its Witt groups with the topological Witt groups of the underlying manifold with involution using Atiyah's Real K-theory, Brumfiel's Theorem and a conjecture of Bruce Williams.

##### The Persistent Homology Group

The persistence of a function f: X -> R is a collection of measurements, one for each open interval of the real line. We call each measurement a persistent homology group. If f is stratifiable, then its persistence can be visualized by something called the persistence diagram. The persistent homology group is special because it is stable to perturbations of the function f. Through the lens of intersection theory, the persistent homology group is simply the image of a cap product. This interpretation leads to a definition of the persistent homology group for Whitney stratifiable maps g: X -> M to manifolds M.

##### Combinatorial covers and cohomological vanishing

I will describe some joint work with Alex Suciu and Sergey Yuzvinsky. We construct a combinatorial framework for proving cohomological vanishing results on certain classes of spaces, by means of a Mayer-Vietoris-type spectral sequence and certain Cohen-Macaulayness hypotheses. The spaces include complex hyperplane complements, their De Concini-Procesi compactifications, and configuration spaces of points in tori. In particular, we generalize classical vanishing results due to Kohno, Esnault, Schechtman and Vieweg, and more recent work of Davis, Januszkiewicz, Leary and Okun.

##### Deformation and Extension of Fibrations of Spheres by Great Circles

In a 1983 paper with Frank Warner, we proved that the space of all great circle fibrations of the 3-sphere S3 deformation retracts to the subspace of Hopf fibrations, and so has the homotopy type of a pair of disjoint two-spheres. Since that time, no generalization of this result to higher dimensions has been found, and so we narrow our sights here and show that in an infinitesimal sense explained below, the space of all smooth oriented great circle fibrations of the 2n+1 sphere S2n+1 deformation retracts to its subspace of Hopf fibrations. The tools gathered to prove this also serve to show that every germ of a smooth great circle fibration of S2n+1 extends to such a fibration of all of S2n+1, a result previously known only for S3 . Joint work with Patricia Cahn (UPenn) and Haggai Nuchi (Univ. of Toronto)

##### Geometry and Topology: A conference in honor of Martin Bendersky's seventieth birthday and in commemoration of our friend and colleague Sam Gitler

For more information, please see the conference web page at: http://web.math.princeton.edu/conference/Bendersky-Gitler/

##### Geometry and Topology: A conference in honor of Martin Bendersky's seventieth birthday and in commemoration of our friend and colleague Sam Gitler

For more information, please see the conference web page at: http://web.math.princeton.edu/conference/Bendersky-Gitler/

##### Geometry and Topology: A conference in honor of Martin Bendersky's seventieth birthday and in commemoration of our friend and colleague Sam Gitler

For more information, please see the conference web page at: http://web.math.princeton.edu/conference/Bendersky-Gitler/

##### Outer space and the combinatorics of character varieties

For a discrete group $\pi$ and a reductive group $G$, the character variety $\mathcal{X}(\pi, G)$ is the moduli space of semi-stable representations of $\pi$ into $G.$ Many moduli spaces appear in the guise of character varieties, for example for a manifold $M$ with fundamental group $\pi_1(M) = \pi$, $\mathcal{X}(\pi, G)$ is the space of flat topological $G-$bundles on $M$, and if $M$ is equipped with a K\"ahler structure, $\mathcal{X}(\pi, G)$ is the space of $G-$Higgs bundles. We will discuss several related combinatorial and geometric features of the character variety $\mathcal{X}(F_g, SL_2(C))$, where $F_g$ is the free group on $g$ generators and $G = SL_2(C)$ is the special linear group of $2\times2$ matrices.

##### Motivic Adams spectral sequence

To any field $k$, there is a motivic stable homotopy category of schemes over $k$. In this setting, one can construct a motivic Adams spectral sequence (MASS) which converges to something related to the homotopy groups of spheres. This talk will introduce the motivic stable homotopy category and show how the MASS over the complex numbers relates to the classical Adams spectral sequence. I will then discuss joint work with Paul Ostvaer on the MASS over finite fields.

##### Intersections of Quadrics, old and new II

**THIS IS A SPECIAL ALGEBRAIC TOPOLOGY SEMINAR. PLEASE NOTE DIFFERENT DAY (TUESDAY). **\noindent Consider $F:\mathbf{R}^n\rightarrow\mathbf{R}^k$ given by $k$ quadratic forms and the varieties $V=F^{-1}(0)$ and $Z=V\cap S^{n-1}$. In 1984-86 I described the topology of the generic $Z$ when $k=2$ and the quadratic forms are simultaneously diagonalizable. In 2007 Sam Gitler showed me a construction called the \textit{polyhedral product functor} (due to him, Tony Bahri, Martin Bendersky, and Fred Cohen) that includes $Z$ as a special case and this old work gained new life: Sam and I obtained in 2008 new results about the case $k>2$ and after that many old and new problems have been solved and the work continues intensely until today.

##### Cohomology of spaces with torus action and toral rank conjecture

In this talk we discuss various approaches to the long-stated toral rank conjecture. This conjecture gives lower bound on the cohomology rank of a finite dimensional topological space X with (almost) free torus (S^1)^m action. The question turns out to be closely related to the famous algebraic Buchsbaum-Eisenbud-Horrocks conjecture. We formulated graded (in fact filtered) version of toral rank conjecture and demonstrate how Lerray-Serre spectral sequences allows to derive some lower bounds.

##### The Cohomology of G-spaces (G compact group)

**Please note special time. **

##### The total surgery obstruction

The 1960's Browder-Novikov-Sullivan-Wall high-dimensional surgery theory for deciding if an n-dimensional Poincare duality space X is homotopy equivalent to an n-dimensional topological manifold has two obstructions. There is a primary topological K-theory obstruction to the existence of a topological bundle structure on the Spivak spherical fibration \nu{X \subset S^{n+k}}. There is also a secondary algebraic L-theory surgery obstruction in the Wall group L_n(Z[\pi_1(X)]) of quadratic forms over the fundamental group ring Z[\pi_1(X)], which depends on the resolution of the primary obstruction. In 1979 (in Princeton) I united these two obstructions in a single "total surgery obstruction" s(X) \in S_n(X).

##### Moduli space actions and cyclic operads

**Please note different location. **I will describe a combinatorial dg model for the homology of the open moduli spaces of punctured Riemann spheres. This model acts (in the operadic sense) on chain complexes computing eg string topology, cyclic cohomology of a Frobenius algebra, or equivariant cohomology of an S^1 space. We thus find homology operations old and new, as well as a homotopy invariant description of the chain level structures which produce these operations.