Meets on Wednesdays from 1 - 2pm in DRL 3C6

Organizer: Rafal Komendarczyk (rako AT math.upenn.edu)

Click for the Official Calendar of the seminar

The Geometry-Topology Graduate Student Seminar has been meeting for more than twenty years. All graduate students (at any level) with an interest in geometry and/or topology are strongly encouraged to attend. In this seminar, graduate students give all the talks. Interested faculty attend, and make suggestions of various papers suitable for reading and presentation by the graduate students.

Suggested introductory papers during the organizational meeting in the Fall'08:

• "The total curvature of a knotted, simple closed curve in 3-space is greater than 4 Pi" by John Milnor, 1950

  Milnor's proof of this appeared in the Annals of Math in 1950, when he was 19 years old. Additional references are do Carmo's "Differential Geometry of Curves and Surfaces", pages 390-404, and Chern's article, "Curves and Surfaces in Euclidean Space," in MAA Studies in Global Geometry and Analysis, Vol. 4, pages 16 - 56.

• "Mappings of the 3-sphere to the 2-sphere" by Heinz Hopf, 1931 (in German)

  This is one of the most important papers ever in topology, and from it sprang the study of homotopy groups of spheres.

• "On the volume elements on a manifold" by Jurgen Moser, 1965

  Any two volume elements on a closed manifold which have the same total volume are equivalent via a diffeomorphism. One talk for this paper, which is probably accessible to an ambitious student currently taking Math 600.

• "The Sakharov-Zeldovich minimization problem" from the book by Arnold and Khesin "Topological hydrodynamics", 1998

  Description of the Sakharov-Zeldovich magnetic energy minimization for the neutron star. More formal approach is also presented in "Zeldovich's neutron star and the prediction of magnetic froth". From MathSci reviews : Consider a ball of perfectly conducting incompressible medium with close to zero viscosity. It is shown that the magnetic energy along the orbit of the rotationally symmetric magnetic field $(-y,x,0)$ (Zeldovich's initial condition) under the group of volume-preserving $C^\infty$-diffeomorphism is arbitrarily close to zero.

• "Geometry of higher helicities", by Boris Khesin, 2003

  Survey various results related to Arnold's theorem on the asymptotic Hopf invariant on three-dimensional manifolds and recent work on linking of a vector field with a foliation, the asymptotic crossing number, short path systems, and relations with the Calabi invariant.

• "On the existence of contact forms" by W. Thurston and H. Winkelnkempler

  This is an introductory paper to contact geometry. The authors give a short proof of the known result that $M^3$ admits a contact form, i.e. a smooth 1-form $\omega$ for which $\omega\wedge d\omega\neq 0$. Their proof uses elementary constructions of forms, with a theorem on the structure of $M^3$ due to J. W. Alexander [Proc. Nat. Acad. Sci. U.S.A. 9 (1923), 93--95]. That theorem, as they remark, also implies immediately the existence of a codimension-1 foliation on $M$.

• "Contact structures on 3-manifolds are deformations of foliations", by John Etnyre, 2006:

  The author show that contact structures can be obtained (up to isotopy) as perturbations of foliations.

Suggested introductory topics:

1. The H-cobordism thm & and the Poincaré conjecture

   Milnor's book, Lectures on the H-Cobordism Theorem.

2. Morse Theory & Bott Periodicity

   Milnor's book Morse Theory.

3. Fenchel Nilsen coordinates for Teichmuller space, proof of bijection to R^{6g-6}

   book by Ratcliffe: Foundations of Hyperbolic Manifolds.

4. Bohm-Wilking space form paper

   Only for advanced students. "Manifolds with positive curvature operators are space forms." Ann. of Math. (2) 167 (2008) Using Ricci-flow techniques they prove, under technical assumptions, the Ricci-flow converges to a space form.

5. Division algebras & the Hopf invariant 1 problem

   Adams' paper On the Nonexistence of Elements of Hopf Invariant One. Bredon's book. Atiyah's K-Theory book.

6. Infinitely many closed geodesics (Gromoll-Meyer) (nondegenerate case)

   The paper by Gromoll and Meyer Periodic geodesics on compact riemannian manifolds. J. Differential Geometry 3 1969 493--510. Paper by Wolfgang Ziller on index of iterated geodesics. Existence of closed geodesics on positively curved manifolds. J. Differential Geom. 18 (1983), no. 2, 221--252

7. Milnor's Growth of groups paper

   Milnor's paper Growth of finitely generated solvable groups. J. Differential Geometry 2 1968 447--449.

8. On Hopf Algebras:

   Milnor paper On the structure of Hopf algebras. Ann. of Math. (2) 81, 1965 211--264.

9. Milnor's exotic sphere paper

   Kervaire Milnor: Groups of homotopy spheres. I. Ann. of Math. (2) 77 1963 504--537. Differentiable structures on spheres. Amer. J. Math. 81 1959 962--972.

10. Higher dimensional Gauss-Bonnet thm

    In the notes http://www.johno.dk/papers/fiberbundles.pdf

11. Quater pinched sphere theorem (fits well with 4))

    Brendle, Schoen Classification of manifolds with weakly $1/4$-pinched curvatures. Acta Math. 200 (2008), no. 1, 1--13.