One-relator groups, non-positive immersions and coherence

Henry Wilton
Cambridge University
March 26, 2019

Abstract: There seems to be an analogy between the classes of fundamental groups of compact 3-manifolds and of one-relator groups.  (Indeed, many 3-manifold groups are also one-relator groups.) For instance, Dehn’s Lemma for 3-manifolds (proved by Papakyriakopoulos) can be seen as analogous to Magnus’ Freiheitssatz for one-relator groups.  But the analogy is still very incomplete, and since there are deep results on each side that have no analogue on the other, there is a strong incentive to flesh it out.

On the Approximation Resistance of Balanced Linear Threshold Functions

Aaron Potechin
University of Chicago
March 25, 2019
Constraint satisfaction problems (CSPs) are a central topic of study in computer science. A fundamental question about CSPs is as follows. Given a CSP where each constraint has the form of some predicate P and almost all of the constraints can be satisfied, is there a randomized polynomial time algorithm which is guaranteed to do significantly better in expectation than a random assignment? If so, then we say that the predicate P is approximable. If not, then we say that P is approximation resistant.

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.

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.