- Wednesday, March 4, 2020, 3:15pm, Physics 119, Number Theory Seminar
*Arithmetic loci of étale rank $1$ local systems*

Helene Esnault (Freie Universitat Berline and IAS)- I’ll give our definition of them, in analogy to Simpson's bialgebraicity notion over the complex numbers, explain what special properties they have, and mention some corollaries (notably hard Lefschetz in rank $1$ in positive characteristic). Joint with Moritz Kerz

- Wednesday, March 18, 2020, 3:15pm, Physics 119, Number Theory Seminar
*TBD*

Margaret Bilu (Courant Institute of Mathematical Sciences (NYU), Mathematics)- TBD

- Wednesday, April 8, 2020, 3:15pm, Physics 119, Number Theory Seminar
*The Semiring of Formal Differences*

Keith Pardue- For rings, there is a natural bijection between (two-sided) ideals and congruences, but for semirings (where we do not require additive inverses), this correspondence breaks. Many basic notions concerning ideals in rings thus split into two different notions for semirings, one for ideals and the other for congruences. Currently there is interest in extensions of scheme theory to commutative semirings, motivated especially by ideas concerning the "field of one element" in arithmetic geometry. To do so, we must decide if we would rather talk about prime ideals or about prime congruences; if the latter then we must decide what we mean by a prime congruence. Of the several definitions of a prime congruence put forward, the most promising is due to Joó and Mincheva. In this talk, we will see that the congruences in a semiring \(R\) are themselves special ideals in a different semiring \(R_{fd}\), the semiring of formal differences for \(R\). Then Joó and Mincheva's prime congruences are precisely those congruences that are prime ideals in \(R_{fd}\). We will also examine Joó and Mincheva's natural arithmetic of congruences in comparison with classical arithmetic of ideals. The natural numbers \(\mathbb{N}\) is already a rich example for this story. We will end with an alternative multiplication law for congruences on \(\mathbb{N}\) that better extends the arithmetic of ideals in \(\mathbb{Z}\) than Joó and Mincheva's. In particular, this alternative multiplication law distributes over sum of congruences and admits unique factorization of congruences into prime congruences.

- Wednesday, April 15, 2020, 12:00pm, Physics 119, Number Theory Seminar
*TBA*

Jordan Ellenberg (University of Wisconsin-Madison)- TBA

- Wednesday, April 15, 2020, 3:15pm, Physics 119, Number Theory Seminar
*TBA*

Lennart Gehrmann (University of Duisburg-Essen/McGill University) - Wednesday, April 22, 2020, 3:15pm, 119 Physics, Number Theory Seminar
*TBA*

Wei Zhang (MIT)

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