A Brief Tour of Proof Complexity: Lower Bounds and Open Problems

Toniann Pitassi
University of Toronto; Visiting Professor, School of Mathematics
March 19, 2019

I will give a tour of some of the key concepts and ideas in proof complexity. First, I will define all standard propositional proof systems using the sequent calculus which gives rise to a clean characterization of proofs as computationally limited two-player games. I will also define algebraic and semi-algebraic systems (SOS, IPS, Polynomial Calculus). 

Multiplicity One Conjecture in Min-max theory

Xin Zhou
University of California, Santa Barbara; Member, School of Mathematics
March 19, 2019

I will present a proof with some substantial details of the Multiplicity One Conjecture in Min-max theory, raised by Marques and Neves. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are all two-sided and have multiplicity one. 

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

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.