The thesis of Akshay Venkatesh obtains a ``Beyond Endoscopy''
proof of stable functorial transfer from tori to ${\rm SL}(2)$, by
means of the Kuznetsov formula. In this talk, I will show that
there is a local statement that underlies this work...
A decades-old application of the second variation formula proves
that if the scalar curvature of a closed 3--manifold is bounded
below by that of the product of the hyperbolic plane with the line,
then every 2--sided stable minimal surface has...
Will this procedure be finite or infinite? If finite, how long
can it last? Bjorner, Lovasz, and Shor asked these questions in
1991 about the following procedure, which goes by the name “abelian
sandpile”: Given a configuration of chips on the...
I will discuss some geometric inequalities that hold on
Riemannian 2-disks and 2-spheres.
For example, I will prove that on any Riemannian 2-sphere there M
exist at least three simple periodic geodesics of length at most
20d, where d is the diameter...
We answer a 1982 conjecture of Erdős and Simonovits
about the growth of number of $k$-walks in a graph, which
incidentally was studied earlier by Blakley and Dixon in 1966. We
prove this conjecture in a more general setup than the
earlier...
Abstract: Uryson k-width of a metric space X measures how close
X is to being k-dimensional. Several years ago Larry Guth proved
that if M is a closed n-dimensional manifold, and the volume of
each ball of radius 1 in M does not exceed a certain...
Abstract: For surfaces immersed into a compact Riemannian
manifold, we consider the curvature functional given by the $L^2$
integral of the second fundamental form. We discuss an an area
bound in terms of that functional, with application to the...
Abstract: We present new-curvature one-cycle sweepout estimates
in Riemannian geometry, both on surfaces and in higher dimension.
More precisely, we derive upper bounds on the length of
one-parameter families of one-cycles sweeping out essential...
Abstract : It is a joint work with G. Courtois, S. Gallot and
A.Sambusetti. We prove a compactness theorem for metric spaces with
anupper bound on the entropy and other conditions that will be
discussed.Several finiteness results will be drawn. It...
