Algebra & Combinatorics Seminar:   2020–21

Abstract. In classical Iwasawa theory, one studies a relationship called the Iwasawa main conjecture, between an analytic object (the p-adic L-function) and an algebraic object (the Selmer group). This relationship involves codimension one cycles of an Iwasawa algebra. The topic of higher codimension Iwasawa theory seeks to generalize this relationship. We will describe a result in this topic using codimension two cycles, involving an elliptic curve with supersingular reduction. This is joint work with Antonio Lei.

Abstract. Let $G$ be the group $SL_2$ over a finite extension $F$ of $\mathbb{Q}_p$, $p$ odd. I will discuss certain distributions on $G(F)$, belonging to what is called its Bernstein center (I will explain what this and many other terms in this abstract mean), supported in a certain explicit subset of $G(F)$ arising from the work of A. Moy and G. Prasad. The assertion is that these distributions form a subring of the Bernstein center, and that convolution with these distributions has very agreeable properties with respect to orbital integrals. These are 'depth $r$ versions' of results proved for general reductive groups by J.-F. Dat, R. Bezrukavnikov, A. Braverman and D. Kazhdan.

Abstract. Cohomology theories are one of the most important algebraic invariants of topological spaces and this has inspired the definition of several different cohomology theories in algebraic geometry. In this talk, we focus on algebraic K-theory, which is one such classical cohomological invariant of algebraic varieties. After motivating and introducing this notion, we discuss several fundamental properties of algebraic K-theory of varieties with algebraic group actions. Well-known examples of varieties with group actions include toric varieties and flag varieties.

Abstract. I will discuss totally positive/non-negative matrices and kernels, including Polya frequency (PF) functions and sequences. This includes examples, history, and basic results on total positivity, variation diminution, sign non-reversal, and generating functions of PF sequences (with some proofs). I will end with applications of total positivity to old and new phenomena involving Schur polynomials.

Abstract. A traditional way of assessing the size of a subset X of the integers is to use some version of density. An alternative approach, independently rediscovered by many authors, is to look at the closure of X in the profinite completion of the integers. This for example gives a quick, intuitive solution to questions like: what is the probability that an integer is square-free? Moreover, in many cases, one finds that the density of X can be recovered as the Haar measure of the closure of X. I will discuss some things that one can learn from this approach in the more general setting of rings of integers in global fields. This is joint work with Luca Demangos.

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