Stanford University
Department of Mathematics

# Geometry Seminar Winter 2014

Organizers: Richard Bamler (rbamler@math.*) and Yi Wang (wangyi@math.*)

Time: Wednesday at 4 PM (NOTE THE CHANGE BACK!)

Location: 383N (TENTATIVE, NOTE THE CHANGE BACK!)

(*=stanford.edu)

## Next Seminar

10 March (Monday)

Speaker: Kazuo Akutagawa (Tokyo Institute of Technology)

Title: The Yamabe problem on edge-cone manifolds and Aubin's inequality

Abstract: This talk is based on a joint work with Gilles Carron (Univ. of Nantes) and Rafe Mazzeo (Stanford Univ.). We consider the Yamabe problem on a compact edge-cone manifold. For an almost Riemannian compact metric-measure $$n$$-space $$(X,\Omega,g)$$ with some mild additional conditions, the following refined Aubin's inequality holds: $Y(X,[g]) \leq Y_l (X, [g]) (\leq Y(S^n)),$ where $$Y(X,[g])$$ and $$Y_l(X,[g])$$ denote respectively the Yamabe constant and the local Yamabe constant of $$(X,\Omega, g)$$. When $$Y(X, [g]) < Y_l (X, [g])$$, we have proved that the Yamabe problem is solvable. But contrary to the smooth case, when $$Y(X, [g]) = Y_l (X, [g])$$, it is not generally solvable. Indeed, J. Viaclovsky gave such examples of orbifolds. Note that both compact edge-cone manifolds as well as compact orbifolds are almost Riemannian compact metric-measure spaces. In this talk, we consider a family of edge-cone Einstein metrics $$g_{\beta}$$ of cone angle $$2\pi\beta > 0$$ on the $$n$$-sphere $$S^n = (S^n, S^{n-2})$$. Then, we show some results on the Yamabe problem for $$(S^n, [g_\beta])$$, including the computation of $$Y(S^n, [g_\beta]) = Y_l (S^n, [g_\beta])$$ for $$0< \beta < 1$$. In particular, on $$(S^n, [g_\beta]) \; ( \beta \in \mathbb{N}, \beta \geq 2$$) the Yamabe problem is also not solvable.

12 March

(Note: special time 3:15pm - 4:15pm)

Room 380-D

there are 2 seminars on this day

Speaker: Frédéric Rochon (UQAM, Montreal)

Title: The moduli space of asymptotically cylindrical Calabi-Yau manifolds

Abstract: We show that the examples of asymptotically cylindrical Calabi-Yau manifolds recently obtained by Haskins-Hein-Nordstrom admit a full polyhomogeneous expansion at infinity. Making use of the b-calculus of Melrose, we then establish at Tian-Todorov result in that context, namely, we show that the deformation theory of such complex manifolds is unobstructed. Time permitting, we will also discuss how to define the Weil-Peterson metric in that context as well as some of its properties. This is a joint work with Ronan Conlon and Rafe Mazzeo.

12 March

(Note: special time 4:20pm - 5:20pm)

Room 383-N

there are 2 seminars on this day

Speaker: Pierre Albin (Urbana-Champaign)

Title: Hodge cohomology of stratified spaces

Abstract: The cohomology of any smooth closed manifold can be represented analytically as the de Rham group of closed forms modulo exact forms. If the manifold has a Riemannian metric, then in each cohomology class we can find a unique harmonic representative.
On singular spaces the situation is more complicated. If the singularities are geometrically controlled, in that the space is stratified,' then there is an analogous story as long as the cohomology and the metric are adapted to the singularities. These spaces arise naturally when studying smooth spaces or maps, for instance, as algebraic varieties, orbit spaces or moduli spaces.
The seminal work on these cohomologies is due to Goresky-MacPherson and Cheeger. I will report on joint work with Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza extending and refining these theories to general stratified spaces.

## Winter Quarter

8 January

Speaker: TBA

Title: TBA

Abstract: TBA

15 January

Speaker: Piotr Chrusciel (Vienna)

Title: The Trautman-Bondi mass on light-cones

Abstract: I will review the notion of Trautman-Bondi mass, and the characteristic Cauchy problem for the Einstein equations with data on a light-cone, and will present an elementary proof of positivity of the Trautman-Bondi mass in this setting.

22 January

Speaker: TBA

Title: TBA

Abstract: TBA

29 January

Speaker: Frank Pacard (Polytechnique)

Title: Solutions without any symmetry for some nonlinear problems in the plane

Abstract: I will present a construction of entire solutions which are defined in the plane and have finite energy, for some nonlinear problems arising from physics. In particular, this yields the existence of solutions which have no symmetry. The construction is inspired from the construction of compact constant mean curvature surfaces in Euclidean 3-space by N. Kapouleas.

5 February

Speaker: TBA

Title: TBA

Abstract: TBA

12 February

Speaker: TBA

Title: TBA

Abstract: TBA

19 February

Speaker: Ivan Sterling (St Mary's College of Maryland)

Title: Pseudospherical Surfaces of Low Differentiability

Abstract: We will bring together several new ideas in order to classify some $$C^1$$ surfaces $$f$$ in $$\mathbb{R}^3$$ with $$K = −1$$. We consider various special circumstances including:

• Surfaces which are not rank 2, but have $$f_x \neq 0$$ and $$f_y \neq 0$$.
• Surfaces which are not $$C^2$$, but for which $$f_{xy}$$ and $$f_{yx}$$ exist, are equal, and are continuous.
• Surfaces where $$\frac{f_x \times f_y}{\Vert f_x \times f_y\Vert}$$ is not continuous, but for which there exist some continuous $$N$$ perpendicular to $$f_x$$ and $$f_y$$.
Versions of Hilbert’s theorem and Minding’s theorem will be given.

26 February

Speaker: TBA

Title: TBA

Abstract: TBA

5 March

Speaker: David Sher (Michigan)

Title: Inverse spectral problems for the Dirichlet-to-Neumann map

Abstract: The Dirichlet-to-Neumann operator on a compact Riemannian manifold with boundary is the map which takes the boundary value of any harmonic function to the boundary value of its normal derivative. It appears in many physical settings, such as electric impedance tomography, and has been extensively studied over the last thirty years. The inverse spectral problem for the Dirichlet-to-Neumann map is the following: given knowledge of the spectrum of the map, as an operator acting on functions on the boundary, what can be said about the geometry of the manifold? In this talk, I will first give an introduction to the subject (requiring no specialized knowledge) and then briefly discuss two recent specific results. The first, joint with I. Polterovich (Montreal), uses heat equation techniques to show that any compact three-dimensional Euclidean domain with connected boundary which has the same Dirichlet-to-Neumann spectrum as a ball must in fact be a ball. The second, joint with A. Girouard (U. Laval), L. Parnovski (UCL), and I. Polterovich, shows how to determine the set of lengths of the boundary components of any compact surface from the Dirichlet-to-Neumann spectrum.

10 March (Monday)

Speaker: Kazuo Akutagawa (Tokyo Institute of Technology)

Title: The Yamabe problem on edge-cone manifolds and Aubin's inequality

Abstract: This talk is based on a joint work with Gilles Carron (Univ. of Nantes) and Rafe Mazzeo (Stanford Univ.). We consider the Yamabe problem on a compact edge-cone manifold. For an almost Riemannian compact metric-measure $$n$$-space $$(X,\Omega,g)$$ with some mild additional conditions, the following refined Aubin's inequality holds: $Y(X,[g]) \leq Y_l (X, [g]) (\leq Y(S^n)),$ where $$Y(X,[g])$$ and $$Y_l(X,[g])$$ denote respectively the Yamabe constant and the local Yamabe constant of $$(X,\Omega, g)$$. When $$Y(X, [g]) < Y_l (X, [g])$$, we have proved that the Yamabe problem is solvable. But contrary to the smooth case, when $$Y(X, [g]) = Y_l (X, [g])$$, it is not generally solvable. Indeed, J. Viaclovsky gave such examples of orbifolds. Note that both compact edge-cone manifolds as well as compact orbifolds are almost Riemannian compact metric-measure spaces. In this talk, we consider a family of edge-cone Einstein metrics $$g_{\beta}$$ of cone angle $$2\pi\beta > 0$$ on the $$n$$-sphere $$S^n = (S^n, S^{n-2})$$. Then, we show some results on the Yamabe problem for $$(S^n, [g_\beta])$$, including the computation of $$Y(S^n, [g_\beta]) = Y_l (S^n, [g_\beta])$$ for $$0< \beta < 1$$. In particular, on $$(S^n, [g_\beta]) \; ( \beta \in \mathbb{N}, \beta \geq 2$$) the Yamabe problem is also not solvable.

12 March

(Note: special time 3:15pm - 4:15pm)

Room 380-D

there are 2 seminars on this day

Speaker: Frédéric Rochon (UQAM, Montreal)

Title: The moduli space of asymptotically cylindrical Calabi-Yau manifolds

Abstract: We show that the examples of asymptotically cylindrical Calabi-Yau manifolds recently obtained by Haskins-Hein-Nordstrom admit a full polyhomogeneous expansion at infinity. Making use of the b-calculus of Melrose, we then establish at Tian-Todorov result in that context, namely, we show that the deformation theory of such complex manifolds is unobstructed. Time permitting, we will also discuss how to define the Weil-Peterson metric in that context as well as some of its properties. This is a joint work with Ronan Conlon and Rafe Mazzeo.

12 March

(Note: special time 4:20pm - 5:20pm)

Room 383-N

there are 2 seminars on this day

Speaker: Pierre Albin (Urbana-Champaign)

Title: Hodge cohomology of stratified spaces

Abstract: The cohomology of any smooth closed manifold can be represented analytically as the de Rham group of closed forms modulo exact forms. If the manifold has a Riemannian metric, then in each cohomology class we can find a unique harmonic representative.
On singular spaces the situation is more complicated. If the singularities are geometrically controlled, in that the space is stratified,' then there is an analogous story as long as the cohomology and the metric are adapted to the singularities. These spaces arise naturally when studying smooth spaces or maps, for instance, as algebraic varieties, orbit spaces or moduli spaces.
The seminal work on these cohomologies is due to Goresky-MacPherson and Cheeger. I will report on joint work with Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza extending and refining these theories to general stratified spaces.

19 March

Speaker: Sergey Cherkis (Arizona)

Title: TBA

Abstract: TBA

## Past Quarters

For the Fall 2013 Schedule go here

For the Spring 2013 Schedule go here

For the Winter 2013 Schedule go here

For the Fall 2012 Schedule go here

For the Spring 2012 Schedule go here

For the Winter 2012 Schedule go here

For the Fall 2011 Schedule go here

For the Spring 2011 Schedule go here

For the Winter 2011 Schedule go here

For the Fall 2010 Schedule go here

For the Spring 2010 Schedule go here

For the Winter 2010 Schedule go here

For the Fall 2009 Schedule go here