Quantum computing with noninteracting particles

Alex Arkhipov
Massachusetts Institute of Technology
February 9, 2015
We introduce an abstract model of computation corresponding to an experiment in which identical, non-interacting bosons are sent through a non-adaptive linear circuit before being measured. We show that despite the very limited nature of the model, an exact classical simulation would imply a collapse of the polynomial hierarchy. Moreover, under plausible conjectures, a "noisy" approximate simulation would do the same.

Twisted matrix factorizations and loop groups

Daniel Freed
University of Texas, Austin; Member, School of Mathematics and Natural Sciences
February 9, 2015
The data of a compact Lie group $G$ and a degree 4 cohomology class on its classifying space leads to invariants in low-dimensional topology as well as important representations of the infinite dimensional group of loops in $G$. Previous work with Mike Hopkins and Constantin Teleman brought the twisted equivariant topological K-theory of $G$ into the game, but only on the level of equivalence classes. After reviewing these ideas I will describe ongoing work with Teleman which gives a geometric construction of the representation categories.

How to round subspaces: a new spectral clustering algorithm

Ali Kemal Sinop
Simons Institute for the Theory of Computing, Berkeley
February 10, 2015
Given a $k$-dimensional linear subspace, consider the problem of approximating it (with respect to the spectral norm) in terms of another subspace spanned by the indicators of a $k$-partition of the coordinates. This is known as the spectral clustering problem, first introduced by [Kannan, Kumar]. It is a generalization of the standard $k$-means problem, which corresponds to the Frobenius norm. In this talk, I will present a new spectral clustering algorithm.

Extending the Prym map

Samuel Grushevsky
Stony Brook University
February 10, 2015
The Torelli map associates to a genus g curve its Jacobian - a $g$-dimensional principally polarized abelian variety. It turns out, by the works of Mumford and Namikawa in the 1970s (resp. Alexeev and Brunyate in 2010s), that the Torelli map extends to a morphism from the Deligne-Mumford moduli of stable curves to the Voronoi (resp. perfect cone) toroidal compactification of the moduli of abelian varieties. The Prym map associates to an etale double cover of a genus g curve its Prym - a principally polarized $(g-1)$-dimensional abelian variety.

Modern Cosmology and the Origin of the Universe

Matias Zaldarriaga
Institute for Advanced Study
February 11, 2015
The last decades have seen great advances
in our understanding of the history of our
universe. I will summarize our current
knowledge, describe some of the puzzles
that still remain and speculate about future
developments in cosmology.
Matias Zaldarriaga has made many influential
and creative contributions to our understanding of the
early universe, particle astrophysics, and cosmology
As a probe of fundamental physics. Much of his work
centers on understanding the clues about the earliest

Symplectic homology via Gromov-Witten theory

Luis Diogo
Columbia University
February 13, 2015
Symplectic homology is a very useful tool in symplectic topology, but it can be hard to compute explicitly. We will describe a procedure for computing symplectic homology using counts of pseudo-holomorphic spheres. These counts can sometimes be performed using Gromov-Witten theory. This method is applicable to a class of manifolds that are obtained by removing, from a closed symplectic manifold, a symplectic hypersurface of codimension 2. This is joint work with Samuel Lisi.

The log-concavity conjecture and the tropical Laplacian

June Huh
Princeton University; Veblen Fellow, School of Mathematics
February 17, 2015
The log-concavity conjecture predicts that the coefficients of the chromatic (characteristic) polynomial of a matroid form a log-concave sequence. The known proof for realizable matroids uses algebraic geometry in an essential way, and the conjecture is open in its full generality. I will give a survey of known results and introduce a stronger conjecture that a certain Laplacian matrix associated to a matroid has exactly one negative eigenvalue.