Geometric Analysis and Topology Seminar

Fall 2014

The seminar's usual time is Friday at 11:00am in 517 Warren Weaver Hall (Directions). Special times and dates are marked in red. Click on the title of a talk for the abstract (if available).

Upcoming seminars:

Oct 3
Karim Adiprasito
(IHES/Hebrew University)
Some geometric aspects of the theory of convex polytopes 517 WWH
Oct 10
Robert Young
Filling multiples of embedded curves 517 WWH
Oct 24
Adam Levine
Satellite operators and piecewise-linear concordance 517 WWH
Oct 31
Alex Lubotzky
(Hebrew University/NYU)
4-dimensional arithmetic hyperbolic manifolds and quantum error correcting codes. 517 WWH
Dec 5
Andrew Ranicki
(University of Edinburgh)
The total surgery obstruction 102 WWH
Feb 23
Mark Sapir
Feb 6
Raanan Schul
(Stony Brook)

Organizers: Sylvain Cappell, Jeff Cheeger, Robert Haslhofer, Bruce Kleiner, and Robert Young.

Previous semesters:


Some geometric aspects of the theory of convex polytopes, Karim Adiprasito.  After centuries of study, we are still mystified by many aspects of convex polytopes, be it in relation to algebraic geometry or to linear programming. I will survey some rather surprising relations between the simplex algorithm and CAT(1) geometry, curvature obstructions and faces of polytopes, and finally Betti numbers of projective toric varieties, finiteness theorems and approximation of smooth convex bodies.
Filling multiples of embedded curves, Robert Young.  Filling a curve with an oriented surface can sometimes be "cheaper by the dozen". For example, L. C. Young constructed a smooth curve drawn on a projective plane in R^n which is only about 1.3 times as hard to fill twice as it is to fill once and asked whether this ratio can be bounded below. We will use methods from geometric measure theory to answer this question and pose some open questions about systolic inequalities for surfaces embedded in R^n.
Satellite operators and piecewise-linear concordance, Adam Levine.  Every knot in the 3-sphere bounds a piecewise-linear (PL) disk in the 4-ball,  but Akbulut showed in 1990 that the same is not true for knots in the boundary of an  arbitrary contractible 4-manifold. We strengthen this result by showing that there exists  a knot K in a homology sphere Y (which is the boundary of a contractible 4-manifold)  such that K does not bound a PL disk in any homology 4-ball bounded by Y. The proof  relies on using bordered Heegaard Floer homology to show that the action of a certain  satellite operator on the knot concordance group is not surjective.
4-dimensional arithmetic hyperbolic manifolds and quantum error correcting codes, Alex Lubotzky.  A family of quantum error correcting codes (QECC) is constructed out of congruence quotients of the 4- dimensional hyperbolic space. Using methods of systolic geometry over Z/2Z, we evaluate the parameters of these codes and disprove a conjecture of Ze'mor who predicted that such homological QECC do not exist. All notions will be defined and explained. A joint work with Larry Guth.

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