Thursday, February 15, 2018
02/15/2018 - 4:00pm to 5:00pm
The problem of error-free information transmission in presence of noise naturally leads to a graph parameter, called Shannon capacity. Despite sixty years of research, very little is known about this parameter. The reason is that only two upper bounds for Shannon capacity are known. In this talk, we review these bounds, and bring to light a new upper bound by Blasiak. We show that it is sometimes stronger than both of the existing bounds and that it is multiplicative. Joint workwith Chris Cox.
02/15/2018 - 4:15pm to 5:15pm
Deciding whether a given algebraic variety is rational, or birational to projective space, is an age-old and challenging problem in algebraic geometry. The rationality problem for rationally connected varieties has seen incredible advances in the last several years,thanks to a degeneration method for the Chow group of 0-cycles initiatedby Voisin, developed by Colliot-Thélène and Pirutka, and recently refined by Schreieder. After summarizing some of these advances, I will speak about joint work with Christian Boehning and Alena Pirutka on the rationality problem for two types of Fano fourfolds lying on the boundary of where different techniques are required: hypersurfaces of bidegree (2,3) in P2 x P3 and complete intersection of type (2,3) in P6. The first haveindex 1 and Picard rank 2, and we prove that the very general such hypersurface is not stably rational by exploiting conic bundle and cubic surface bundle structures. The second have index 2 and Picard rank 1, and are more challenging.
02/15/2018 - 6:00pm to 7:00pm
What to do if the measurements that you took were corrupted by a malicious spy? We will see how the natural geometric approachto the problem leads to a geometry where lines are crooked, and triangles are square.