## Geometry and Topology Seminar

This seminar discusses geometry and topology in a broad sense including (but not exclusively) Riemannian geometry, low dimensional topology and geometry, and geometric group theory.

### Loop products and resonance for loop homology on spheres

by Nancy Hingston | The College of New Jersey,
Time: 16:10  (Monday, Jan. 17, 2011)
Location: HU1018, 215 Huron St
Abstract:
The critical points of the length function on the free loop space $L(M)$ of a compact Riemannian manifold $M$ are the closed geodesics on $M$. Filtration by the length function gives a link between the geometry of closed geodesics and the algebraic structure given by the Chas-Sullivan product on the homology of $L(M)$. Geometry reveals the existence of a related product on the cohomology of $L(M)$. For manifolds such as spheres and projective spaces for which there is a metric with all geodesics closed, the resulting homology and cohomology rings are nontrivial, and closely linked to the geometry. If $x$ is a homology class on $L(M)$, the "minimax" critical level $cr(x)$ is a critical value. Let $M$ be a sphere, and fix a metric and a coefficient field. We prove a resonance result for homology (and cohomology) on $L(M)$. In particular the limit as $deg(x)$ goes to infinity of $cr(x)/deg(x)$ exists.

Mark Goresky and Hans-Bert Rademacher are collaborators. I will not assume any knowledge of the Chas-Sullivan or other loop products.

#### Dates in this series

· Monday, Mar. 08, 2010: The fundamental group of random 2-complexes (Matthew Kahle)
· Monday, Mar. 15, 2010: Three Dimensional Manifolds All of Whose Geodesics Are Closed (John Olsen)
· Monday, Mar. 22, 2010: Curves on surfaces with large topology (Hugo Parlier)
· Friday, Mar. 26, 2010: New invariants in singularity theory (Rustam Sadykov)
· Friday, Mar. 26, 2010: The hidden smoothness of the space of stable maps (Jonathan Wise)
· Monday, Apr. 05, 2010: Moduli spaces of conics in curves (Chris Judge)
· Monday, Sep. 20, 2010: Total Q-curvature for CR manifolds (Kengo Hirachi)
· Monday, Sep. 27, 2010: Group completions of moduli spaces (Rustam Sadykov)
· Tuesday, Oct. 12, 2010: Pontryagin invariants and integral formulas for Milnor's triple linking number (Herman Gluck)
· Monday, Oct. 18, 2010: Positively curved polar manifolds (Fuquan Fang)
· Monday, Nov. 01, 2010: Rational formality and special holonomy (Manuel Amann)
· Monday, Nov. 08, 2010: Isoperimetric Inequalities -- methods and applications (Emanuel Milman)
· Monday, Nov. 15, 2010: From the sandwich to the waist (Alfredo Hubard)
· Monday, Nov. 29, 2010: TBA (Keith Burns)
· Monday, Nov. 29, 2010: Conjugate points and non-injectivity of the exponential map (Keith Burns)
· Monday, Jan. 17, 2011: Loop products and resonance for loop homology on spheres (Nancy Hingston)
· Monday, Jan. 31, 2011: A Variational Characterization of the Catenoid (Jacob Bernstein)
· Monday, Apr. 18, 2011: On the structure of conformally covariant powers of the Laplacian (Andreas Juhl)
· Monday, May. 09, 2011: Causality, Low conjecture and globally hyperbolic spacetimes (Vladimir Chernov)
· Monday, Sep. 26, 2011: Mapping degrees of self-maps of simply-connected manifolds (Manuel Amann)
· Monday, Oct. 17, 2011: Formality of Positive Quaternion Kähler manifolds (Vitali Kapovitch)
· Friday, Oct. 21, 2011: Kahler-Einstein metrics singular along a divisor (Yanir Rubinstein)
· Monday, Oct. 24, 2011: Diastolic inequality on a 2-sphere (Yevgeny Liokumovich)
· Monday, Nov. 14, 2011: TBA (Ian Hambleton)
· Monday, Nov. 14, 2011: Coarse geometry and group actions (Ian Hambleton)
· Monday, Nov. 21, 2011: Fundamental groups of manifolds with lower Ricci curvature bounds (Vitali Kapovitch)
· Monday, Nov. 28, 2011: Fundamental groups of manifolds with lower Ricci curvature bounds II (Vitali Kapovitch)
· Tuesday, Nov. 29, 2011: Asymptotic geometry, finite generation of fundamental groups, and harmonic functions (Bruce Kleiner)
· Friday, Dec. 09, 2011: Topological Complexity and H-Spaces (Gregory Lupton)
· Tuesday, Jan. 24, 2012: A new construction of compact $G_2$ manifolds by glueing families of Eguchi-Hanson spaces (Spiro Karigiannis)
· Thursday, Jan. 26, 2012: Coarse Differentiation and the rank of Teichmüller space (Kasra Rafi)
· Thursday, Feb. 16, 2012: Manifolds and spaces with Ricci curvature bounds (Toby Colding)
· Monday, Feb. 27, 2012: Homological and homotopical filling functions (Robert Young)
· Monday, Mar. 12, 2012: Algorithmic decidability of Thurston equivalence and geometrization of branched coverings (Michael Yampolsky)
· Monday, Mar. 26, 2012: On some problems in the geometry of surfaces in normed spaces (Dmitri Burago)
· Monday, Apr. 02, 2012: CANCELLED -- Scattering rigidity with trapped geodesics; the two dimensional case (Chris Croke)
· Monday, Sep. 10, 2012: Realisation and dismantlability (Piotr Przytycki)
· Friday, Oct. 05, 2012: The metric completion of Outer Space (Yael Algom Kfir)
· Monday, Oct. 15, 2012: Complexes which require large filling distortion in Euclidean space (Dominic Dotterrer)
· Tuesday, Nov. 06, 2012: A theorem of existence for Asymptotically Conical Calabi-Yau manifolds (Ronan Conlon)
· Monday, Nov. 19, 2012: Ricci Solitons of Cohomogeneity One (McKenzie Wang)
· Monday, Nov. 26, 2012: Complete affine 3-manifolds, Lorentzian and hyperbolic geometry (Bill Goldman)
· Monday, Dec. 03, 2012: Hyperbolic Group Boundaries and Cannon-Thurston Maps (Owen Baker)