Activities This Week

Chernoff’s inequality is one of the most basic inequalities in probability theory. It states that the sum of independent random variables is close to its mean with very high probability. The `Expander-Graph Chernoff inequality’, proven by Ajtai, Komlos, and Szemeredi in 1987, is a powerful extension of Chernoff’s inequality, asserting that Chernoff’s inequality holds even when sampling the random variables using paths in an expander graph instead of sampling them independently.

In this talk, we will discuss further extensions and generalizations of Chernoff’s inequality coming from high dimensional expanders. High dimensional expanders are sparse hypergraphs exhibiting pseudo-random properties. These are more sophisticated structures than the expander graphs of [AKS], but in return the generalization of Chernoff’s bound obtained from them applies in a far more general setup.

We will see why Chernoff-type bounds hold for high dimensional expanders, and how this inequality arises naturally in problems in high dimensional geometry and computational complexity.

This is based on joint work with Max Hopkins. The talk will not assume prior background in complexity theory and is aimed at a broad audience.

בתיאוריה של קשיחות של גרפים, מסתכלים על מבנים שמורכבים מנקודות וממוטות שמחברים ביניהן, ושואלים מתי מבנה כזה הוא קשיח, כלומר לא יכול להתעוות או ״להתקמט״ בלי לשבור את המוטות. בהרצאה אציג את המושג של גרף קשיח, נדבר על תכונות של גרפים כאלה ונדון בשאלה כיצד ניתן לקבוע אם גרף נתון הוא קשיח או לא.

TBA

Surface holonomy and the Wess–Zumino phase are fundamental in string theory and Chern–Simons theory, but giving a fully analytic description of their nonabelian versions has been a longstanding challenge. In this talk, I will explain how Yekutieli’s theory of nonabelian multiplicative integration on surfaces provides such a framework. The starting point is a smooth 2-connection (α,β) on a Lie crossed module. I will describe how one constructs multiplicative integrals associated to this data, and then show that these integrals satisfy the axioms of a transport 2-functor in the sense of Schreiber and Waldorf, providing an explicit model of nonabelian surface holonomy. I will conclude by discussing the resulting three-dimensional Stokes theorem and its relation to the Wess–Zumino phase, including the abelian case as a special instance.

We count the number of minimally ramified quartic dihedral exten- sions of the rational numbers by discriminant unconditionally, and by the product of ramified primes assuming the Generalized Riemann Hypothe- sis.