School of Mathematics

An Application of the Universality Theorem for Tverberg Partitions

Imre Barany
Renyi Institute, Hungary and UCL, London
March 18, 2019

We show that, as a consequence of a remarkable new result of
Attila P\'or on universal Tverberg partitions, any large-enough set
$P$ of points in $\Re^d$ has a $(d+2)$-sized subset whose Radon point
has half-space depth at least $c_d \cdot |P|$, where $c_d \in (0, 1)$
depends only on $d$. We then give an application of this result to
computing weak $\eps$-nets by random sampling. Joint work with Nabil
Mustafa.

Minimal Sets and Properties of Feral Pseudoholomorphic Curves

Joel Fish
University of Massachusetts Boston
March 18, 2019

I will discuss some current joint work with Helmut Hofer, in which we define and establish properties of a new class of pseudoholomorhic curves (feral J-curves) to study certain divergence free flows in dimension three. In particular, we show that if H is a smooth, proper, Hamiltonian on R^4, then no non-empty regular energy level of H is minimal. That is, the flow of the associated Hamiltonian vector field has a trajectory which is not dense.

Localization and delocalization for interacting 1D quasiperiodic particles.

Ilya Kachkovskiy
Michigan State University
March 15, 2019
We consider a system of two interacting one-dimensional quasiperiodic particles as an operator on $\ell^2(\mathbb Z^2)$. The fact that particle frequencies are identical, implies a new effect compared to generic 2D potentials: the presence of large coupling localization depends on symmetries of the single-particle potential.

Local aspects of Venkatesh's thesis.

Yiannis Sakellaridis
Rutgers University
March 14, 2019

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; namely, there is a local transfer operator taking orbital measures for the Kuznetsov formula to test measures on the torus. The global comparison of trace formulas is then obtained as a Poisson summation formula for this transfer operator.

Halting problems for sandpiles and abelian networks

Lionel Levine
Cornell University; von Neumann Fellow
March 12, 2019

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 vertices of a finite directed graph, choose (however you like) a vertex with at least as many chips as out-neighbors, and send one chip from that vertex to each of its out-neighbors. Repeat, until there is no such vertex. 

Macroscopically minimal hypersurfaces

Hannah Alpert
Ohio State University
March 12, 2019

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 area at least that of
the hyperbolic surface of the same genus. We can prove a coarser
analogue of this statement, taking the appropriate notions of
macroscopic scalar curvature and macroscopic minimizing hypersurface
from Guth's 2010 proof of the systolic inequality for the

Near log-convexity of measured heat in (discrete) time and consequences

Mert Saglam
University of Washington
March 11, 2019

We answer a 1982 conjecture of Erd&‌#337;s 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 treatment, furthermore, through a refinement and strengthening of this inequality, we resolve two related open questions in complexity theory: the communication complexity of the $k$-Hamming distance is $\Omega(k \log k)$ and that consequently any property tester for k-linearity requires $\Omega(k \log k)$.

Geometry of 2-dimensional Riemannian disks and spheres.

Regina Rotman
University of Toronto; Member, School of Mathematics
March 11, 2019

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 of M, (joint with A. Nabutovsky, Y. Liokumovich).
This is a quantitative version of the well-known Lyusternik and Shnirelman theorem.