On Zimmer's conjecture

Sebastian Hurtado-Salazar
University of Chicago
April 6, 2017

The group $\mathrm{SL}_n(\mathbb Z)$ (when $n > 2$) is very rigid, for example, Margulis proved all its linear representations come from representations of $\mathrm{SL}_n(\mathbb R)$ and are as simple as one can imagine. Zimmer's conjecture states that certain "non-linear" representations ( group actions by diffeomorphisms on a closed manifold) come also from simple algebraic constructions.

Algebra and geometry of the scattering equations

Peter Goddard
Professor Emeritus, School of Natural Sciences
April 3, 2017
Four years ago, Cachazo, He and Yuan found a system of algebraic equations, now named the "scattering equations", that effectively encoded the kinematics of massless particles in such a way that the scattering amplitudes, the quantities of physical interest, in gauge theories and in gravity could be written as sums of rational functions over their solutions.

A time-space lower bound for a large class of learning problems

Ran Raz
Princeton University
April 3, 2017

We prove a general time-space lower bound that applies for a large class of learning problems and shows that for every problem in that class, any learning algorithm requires either a memory of quadratic size or an exponential number of samples. As a special case, this gives a new proof for the time-space lower bound for parity learning [R16].

Rigid holomorphic curves are generically super-rigid

Chris Wendl
Humboldt-Universität zu Berlin
March 31, 2017
I will explain the main ideas of a proof that for generic compatible almost complex structures in symplectic manifolds of dimension at least 6, closed embedded J-holomorphic curves of index 0 are always "super-rigid", implying that their multiple covers are never limits of sequences of curves with distinct images. This condition is especially interesting in Calabi-Yau 3-folds, where it follows that the Gromov-Witten invariants can be "localized" and computed in terms of Euler classes of obstruction bundles for a finite set of disjoint embedded curves.

Speculations about homological mirror symmetry for affine hypersurfaces

Denis Auroux
University of California, Berkeley; Member, School of Mathematics
March 31, 2017
The wrapped Fukaya category of an algebraic hypersurface $H$ in $(C*)^n$ is conjecturally related via homological mirror symmetry to the derived category of singularities of a toric Calabi-Yau manifold $X$, whose moment polytope is determined by the tropicalization of $H$. (The case of the pair of pants discussed in the first talk is a special case of this construction).