Theoretical Statistics and Mathematics Unit, ISI Delhi

January 29, 2020,
3:30 PM at Seminar Room 2

Speaker:
Manish Mishra,
IISER, Pune

Title:
Regular Bernstein blocks

Abstract of Talk

I will recall some standard results of Bernstein in the representation theory of p-adic reductive groups. I will review the theory of Bernstein center and explain the Bernstein decomposition of the category of smooth representations of a p-adic group. I will then present an approach for understanding Bernstein blocks and state some of our results in that direction. This is a joint work with Jeffrey Adler.

February 4, 2020,
3:30 PM at Seminar Room 2

Speaker:
Pierre-Yves Bienvenu,
Camille Jordan Institute, Lyon, France

Title:
Additive bases in infinite abelian semigroups

Abstract of Talk

Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in a class of infinite abelian semigroups, which we term \textit{translatable} semigroups. These include all infinite abelian groups as well as the semigroup $\N$ of nonnegative integers. We show that, for every such semigroup $T$, the number of essential subsets of any additive basis is finite, and also that the number $E_T(h,k)$ of essential subsets of cardinality $k$ contained in an additive basis of order at most $ h$ can be bounded in terms of $h$ and $k$ alone. These theorems extend the reach of two important results, one due to Deschamps and Farhi and the other to Hegarty, bearing upon $\N$. Also, using invariant means, we address a classical problem, initiated by Erdös and Graham and extended by Nash
and Nathanson both in the case of $\N$, of estimating the maximal order $X_T(h,k)$ that a basis of co-cardinality $k$ contained in a given additive basis of order $h$ can have. Among other results, we prove that, whenever $T$ is a translatable semigroup, $X_T(h,k)$ is $O(h^{2k+1})$ for every integer $k \ge 1$. This result is new even in the case $k=1$ and $T$ is an infinite abelian group. Besides the maximal order $X_T(h,k)$, we also study the typical order $S_T(h,k)$.
Joint work with Benjamin Girard and Thai Hoang Lê.

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