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Current contacts: Vasily Dolgushev, Ed Letzter or Martin Lorenz.
The Seminar usually takes place on Mondays at 1:30 PM in Room 617 on the sixth floor of Wachman Hall. Click on title for abstract.
This is the organizational meeting of the Algebra Seminar.
Aniruddha Sudarshan, Temple University
This talk will give an overview of the main ingredients for Ribet's proof of the converse to Herbrand theorem. We will start with the statement of Herbrand's theorem, followed by reformulations of its converse using class field theory and Galois cohomology. Then, we venture into the world of congruences between modular forms, and Galois representations to prove a crucial lemma from which the converse to Herbrand's theorem follows.
Gilbert Moss, University of Maine
Let $G$ be a connected reductive algebraic group, such as $GL_n$, and let $F$ be a nonarchimedean local field, such as the p-adic numbers $\mathbb{Q}_p$. The local Langlands program describes a connection, which has been established in many cases, between irreducible smooth representations of $G(F)$ and Langlands parameters, which are described in terms of the absolute Galois group of $F$. The local Langlands correspondence "in families" is concerned with an aspect of the local Langlands program that seeks to upgrade this connection beyond irreducible representations to a smoothly varying morphism between natural moduli spaces of $G(F)$ representations and Langlands parameters. We will describe a precise conjecture in this direction and summarize past work establishing the conjecture for $GL_n(F)$, as well as ongoing work toward establishing it for classical groups.
Vasily Dolgushev, Temple University
This is an overview of the series of talks on Galois theory for infinite algebraic extensions. I will introduce the set-up and formulate the main theorem of the Galois theory for infinite algebraic extensions (the theorem is due to W. Krull). I will formulate the Nikolov-Segal theorem on finitely generated profinite groups and talk about examples of non-open subgroups of finite index in the absolute Galois group of rational numbers. If time permits, I will also formulate the Shafarevich conjecture.
Chathumini Kondasinghe, Temple University
This is a brief introduction to topological groups. I will define topological groups, give several examples and prove selected statements. This is a part of the series on talks on Galois theory for infinite algebraic extensions.
Wissam Raji, American University of Beruit
We consider the period polynomials $r_f (z)$ associated with cusp forms $f$ of weight $k$ on all of $SL_2(\mathbb{Z})$, which are generating functions for the critical L-values of the modular L-function associated to f. In 2014, El-Guindy and Raji proved that if $f$ is an eigenform, then $r_f (z)$ satisfies a “Riemann hypothesis” in the sense that all its zeros lie on the natural boundary of its functional equation. We show that this phenomenon is not restricted to eigenforms, and we provide large natural infinite families of cusp forms whose period polynomials almost always satisfy the Riemann hypothesis. For example, we show that for weights $k ≥ 120$, linear combinations of eigenforms with positive coefficients always have unimodular period polynomials.
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Frauke Bleher, University of Iowa
The discriminant d_F of a number field F is a basic invariant of F. The smaller d_F is relative to [F:Q], the more elements there are in the ring of integers O_F of F that have a given bounded size. This is relevant, for example, to cryptography using elements of O_F. In 2007, two cryptographers (Peikert and Rosen) asked whether one could give an explicit construction of an infinite family of number fields F having d_F^{1/[F:Q]} bounded by a constant times [F:Q]^d for some d < 1. By an explicit construction we mean an algorithm requiring time bounded by a polynomial in log([F:Q]) for producing a set of polynomials whose roots generate F. In this talk I will describe work with Ted Chinburg showing how this can be done for any d > 0. The proof uses the group theory of profinite 2-groups as well as recent results in analytic number theory.
Sean O'Donnell, Temple University
We will start the talk with a review of limits of functors and present selected examples for categories of groups, topological spaces and topological groups. We will also discuss natural transformations and upgrade the limit assignment to a functor. Motivated by Galois Theory, we will present several properties of limits from downward directed posets and their co-initial sub-posets. We will define the profinite completion of a group G as the topological group. If time permits, we will conclude the talk with a practical description of the profinite completion of the ring of integers.
Aniruddha Sudarshan, Temple University
In this talk, we will show the existence of a non-open subgroup of finite index of the absolute Galois group of the rationals. If time permits, we will also talk about the Nikolov-Segal theorem. Among other things, this theorem implies that every finite index subgroup of a topologically finitely generated profinite group G is open in G.
William Chen, Rutgers University
Markoff numbers are positive integers which encode how resistant certain irrational numbers are to being approximated by rationals. In 1913, Frobenius asked for a description of all congruence conditions satisfied by Markoff numbers modulo primes p. In 1991 and 2016, Baragar, Bourgain, Gamburd, and Sarnak conjectured a refinement of Frobenius’s question, which amounts to showing that the Markoff equation x^2 + y^2 + z^2 - xyz = 0 satisfies “strong approximation”; that is to say: they conjecture that its integral points surject onto its mod p points for every prime p. In this talk we will show how to prove this conjecture for all but finitely many primes p, thus reducing the conjecture to a finite computation. A key step is to understand this problem in the context of describing the orbits of certain group actions. Primarily, we will consider the action of the mapping class group of a topological surface S on (a) the set of G-covers of S, where G is a finite group, and (b) on the character variety of local systems on S. Questions of this type have been related to many classical problems, from proving that the moduli space of curves of a given genus is connected, to Grothendieck’s ambitious plan to understand the structure of the absolute Galois group of the rationals by studying its action on "dessins d’enfant". We will explain some of this history and why such problems can be surprisingly difficult.
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