Equivariant and nonequivariant contact homology

Jo Nelson
Rice University
March 20, 2019

I will discuss joint work with Hutchings which constructs nonequivariant and a family floer equivariant version of contact homology. Both theories are generated by two copies of each Reeb orbit over Z and capture interesting torsion information. I will then explain how one can recover the original cylindrical theory proposed by Eliashberg-Givental-Hofer via our construction.

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
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.