Cohomology for computer science

Alex Lubotzky
Hebrew University of Jerusalem
November 17, 2015
We will start with presenting the basic notions of (co)homomology of simplical complexes (which requires only basic linear algebra over the field of order 2) and then we will indicate its relevance for several topics in computer science and combinatorics: 1) Property testing. 2) Quantum error correcting codes (where triangulation of some 4-dim hyperbolic manifolds lead to such codes with unexpected parameters) 3) High dimensional expanders. 4) Random simplical complexes.

Cohomology for computer science II

Alex Lubotzky
Hebrew University of Jerusalem
November 17, 2015

We will start with presenting the basic notions of (co)homomology of simplical complexes (which requires only basic linear algebra over the field of order 2) and then we will indicate its relevance for several topics in computer science and combinatorics: 1) Property testing. 2) Quantum error correcting codes (where triangulation of some 4-dim hyperbolic manifolds lead to such codes with unexpected parameters) 3) High dimensional expanders. 4) Random simplical complexes.

Geometric deformations of orthogonal and symplectic Galois representations

Jeremy Booher
Stanford University
November 19, 2015
For a representation of the absolute Galois group of the rationals over a finite field of characteristic $p$, we would like to know if there exists a lift to characteristic zero with nice properties. In particular, we would like it to be geometric in the sense of the Fontaine-Mazur conjecture: ramified at finitely many primes and potentially semistable at $p$. For two-dimensional representations, Ramakrishna proved that under technical assumptions, odd representations admit geometric lifts.

Advances on Ramsey numbers

Jacob Fox
Stanford University
November 23, 2015
Ramsey theory refers to a large body of deep results in mathematics whose underlying philosophy is captured succinctly by the statement that "Every very large system contains a large well-organized subsystem." Ramsey numbers capture how very large the system should be in order for this to be true. Despite much attention, Ramsey numbers are generally not well understood. This talk will discuss some major problems and recent advances in this area.

Fun with finite covers of 3-manifolds: connections between topology, geometry, and arithmetic

Nathan Dunfield
University of Illinois, Urbana-Champaign
November 23, 2015
Following the revolutionary work of Thurston and Perelman, the topology of 3-manifolds is deeply intertwined with their geometry. In particular, hyperbolic geometry, the non-Euclidean geometry of constant negative curvature, plays a central role. In turn, hyperbolic geometry opens the door to applying tools from number theory, specifically automorphic forms, to what might seem like purely topological questions.

General systems of linear forms: equidistribution and true complexity

Pooya Hatami
Member, School of Mathematics
November 24, 2015
Higher-order Fourier analysis is a powerful tool that can be used to analyse the densities of linear systems (such as arithmetic progressions) in subsets of Abelian groups. We are interested in the group $\mathbb{F}_p^n$, for fixed $p$ and large $n$, where it is known that analysing these averages reduces to understanding the joint distribution of a family of sufficiently pseudorandom (formally, high rank) nonclassical polynomials applied to the corresponding system of linear forms.

Hausdorff dimension of Kleinian group uniformization of Riemann surface and conformal rigidity

Yong Hou
Princeton University; Visitor, School of Mathematics
November 24, 2015

For this talk I'll discuss uniformization of Riemann surfaces via Kleinian groups. In particular question of conformability by Hasudorff dimension spectrum. I'll discuss and pose some questions which also in particular will imply a conjecture due to Bers.

Lower bounds on the size of semidefinite programming relaxations

David Steurer
Cornell University
November 30, 2015
We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on $n$-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than $2^{n^{\delta}}$, for some constant $\delta > 0$. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes.