Algebra Seminar

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.

• Monday January 22, 2024 at 13:30, Wachman 527

  Algebra Seminar. Organizational Meeting

  This is the organizational meeting of the Algebra Seminar. 

   

• Monday January 29, 2024 at 13:30, Wachman 527

  The Herbrand—Ribet Theorem

• Monday February 5, 2024 at 13:30, Wachman 527

  Toward a local Langlands correspondence in families

  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. 

• Monday February 12, 2024 at 13:30, Wachman 527

  Galois theory for infinite algebraic extensions

  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.  
  
  
   

• Monday February 19, 2024 at 13:30, Wachman 527

  Introduction to topological groups

  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.
  
   

• Monday February 26, 2024 at 09:30, Wachman Hall 108

  Zeroes of Period Polynomials of Cusp Forms

  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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• Monday March 11, 2024 at 13:30, Wachman 307

  Number fields with small discriminants

  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.

   

• Monday March 18, 2024 at 13:30, Wachman 527

  Introduction to profinite groups

  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.
  
  
   

• Monday April 8, 2024 at 13:30, Wachman 527

  Non-open subgroups of finite index

  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. 

   

• Monday April 15, 2024 at 13:30, Wachman 527

  Orbit problems and the mod p properties of Markoff numbers

  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.

   

• Monday August 26, 2024 at 13:30, Wachman 617

  Algebra Seminar: organizational meeting 

   

• Monday September 9, 2024 at 13:30, Wachman 617

  Galois theory for infinite algebraic extensions

  Vasily Dolgushev, Temple University

  We will show that the Galois group of an infinite (algebraic) Galois extension is naturally a profinite group and give several examples. We will formulate the main theorem of the Galois theory for infinite algebraic extensions (the theorem is due to W. Krull) and show that, in this set-up, Galois groups have non-closed subgroups.  

   

• Monday September 16, 2024 at 13:30, Wachman 617

  Local Fields, Inertia Groups, And Frobenius Elements

  Xiaoyu Huang, Temple University

  We will begin by discussing local fields and valuations, followed by an exploration of inertia groups and Frobenius elements. Next, we will cover unramified extensions and then connect these concepts to the global field setting.

   

• Monday September 23, 2024 at 13:30, Wachman 617

  The Chebotarev Density Theorem

  Aniruddha Sudarshan, Temple University

  Dirichlet's prime number theorem states that there exists infinitely many primes in a given arithmetic progression. Chebotarev's theorem is a vast generalization of this classical result. We state Chebotarev's result and talk about a few of its applications.  We will mainly focus on the density result of Frobenius elements in the absolute Galois group of a number field. 

   

• Monday September 30, 2024 at 13:20, Wachman 617

  Cyclotomic Characters and Compatible Systems of p-adic Galois Representations

  Stephen Liu, Temple University

  The p-adic cyclotomic character is an important example of one-dimensional Galois representations. We will introduce the construction and some properties of cyclotomic characters, such as ramification and certain compatibility across different primes. Motivated by these properties, we will try to define compatible systems of  p-adic Galois representations. 

   

• Monday October 7, 2024 at 13:20, Wachman 617

  Galois representations on Elliptic curves

  Erik Wallace, Temple University

  This talk will give a brief introduction to Galois Representations on Elliptic curves, and a survey of some of the main results. We will draw particular attention to connections with previous talks, such as on the inertia group, on the Chebotarev density theorem. Specific examples will be included, as well as some sage code.

   

• Monday October 14, 2024 at 13:20, Wachman 617

  Complex Galois representations

  Chathumini Kondasinghe, Temple University

  In this talk we will show that every complex Galois representation factors through the Galois group of a finite extension of Q. Then we will use this result to show that every Galois representation over C can be seen as a compatible system of Galois representations. 

   

• Monday October 21, 2024 at 13:20, Wachman 617

  Class Field Theory and 1-Dimensional Galois Representations

  Sean O'Donnell, Temple University

  This talk will give a brief summary of some results of class field theory and their relevance to the study of 1-dimensional Galois representations of local and global fields. Results covered will include the existence of the local and global Artin homomorphisms, the Kronecker-Weber Theorem, and a brief overview of the connection between local and global class field theory.

   

• Thursday October 31, 2024 at 15:30, Wachman 617

  An invitation to the BSD conjecture

  Ashay Burungale, University of Texas Austin
  
  The talk plans to present an introduction to the BSD conjecture predicting a mysterious link between rational points on an elliptic curve defined over rational numbers, and analytic properties of the associated Hasse-Weil L-function. Some recent progress will be discussed.

• Monday November 4, 2024 at 13:20, Wachman 617

  Automorphic functions and harmonic analysis on groups

  Ross Griebenow, Temple University

  We introduce the theory of automorphic functions for a discrete cofi- nite group Γ ≤ PSL2(R), and the spectral theory of the Laplace operator on H. Combining these ideas, we develop the “Selberg trace formula” which relates the spectral decomposition of square-integrable functions on Γ\H with the geometry of Γ\H. The trace formula allows us to under- stand the properties of a certain zeta function which is analogous to the Riemann zeta-function, and characterize the number of closed geodesics on Γ\H with bounded length.

• Friday November 8, 2024 , University of Pennsylvania

  Zeta functions and symplectic duality

  Yiannis Sakellaridis, Johns Hopkins University
  
  Abstract:  The Riemann zeta function was introduced by Euler, but carries Riemann's name because he was the one who extended it to a meromorphic function on the entire complex plane, and discovered its importance for the distribution of primes. It admits a vast class of generalizations, called L-functions, but, as in Riemann's case, one usually cannot prove anything about them without relying on seemingly unrelated integral representations. 
  
  In joint work with David Ben-Zvi and Akshay Venkatesh, we elucidate the origin of such integral representations, showing that they are manifestations of a duality between nice Hamiltonian spaces for a pair $(G,\check G)$ of ``Langlands dual'' groups. Over the geometric cousins of number fields -- algebraic curves and Riemann surfaces -- such dualities had been anticipated and constructed in many cases by Gaiotto and others, motivated by mathematical physics. 
  
  The first talk will be a gentle and example-oriented introduction to problems in the ``relative'' Langlands program, introducing automorphic L-functions, and various ways of generalizing Riemann's integral representation. We will also talk about the idea of quantization, and why it might be an appropriate framework for studying such constructions. In the second talk, I will introduce a conjectural duality between nice (``hyperspherical'') Hamiltonian spaces, and how it gives rise to a hierarchy of conjectures, both function- and sheaf-theoretic, refining the Langlands correspondence.

• Friday November 8, 2024 , University of Pennsylvania

  Mac Lane valuations and algebraic geometry

  Andrew Obus, CUNY Baruch
  
  Abstract:  Almost 90 years ago, Mac Lane discovered the technique of "inductive valuations", which allows one to write down valuations on a rational function field over a discretely valued field in a particularly explicit way.  The first talk will be a hands-on introduction to the theory, requiring no background beyond the definition of a discrete valuation (which we will recall).  At the end of the talk, we will fast forward 80 years or so and discuss the relationship between Mac Lane valuations and models of the projective line.

   

  In the second talk give various examples of how the relationship between Mac Lane valuations and models of the projective line can be used to resolve singularities and find regular models of arithmetic surfaces.  In particular, we will overview how Mac Lane valuations can be used to give explicit minimal regular normal crossings models of superelliptic curves (joint with Padmavathi Srinivasan), and also how they can used to understand “stabilization indices” of curves with potentially multiplicative reduction (joint with Daniele Turchetti).

   

• Monday November 11, 2024 at 13:30, Wachman 617

  Sum of Two Squares and Four Squares: A Proof by Modular Forms

  Anthony Pasles, Temple University

  We solve the Sum of Two Squares and Sum of Four Squares Problems using modular forms. In particular, we develop explicit formulas for the number of representations of a positive integer as both a sum of two squares of integers and as a sum of four squares of integers. We introduce the Jacobi theta series (theta), and remark that understanding the Fourier expansions of powers of this function will provide solutions to the problems in question. Then we verify theta satisfies certain functional equations in order to illustrate that theta squared is a modular form of weight 1 and level Gamma (a specific congruence subgroup). We show this space of modular forms has dimension 1, and then use an Eisenstein series living in the same space to determine the Fourier coefficients of theta squared and solve the sum of two squares problem. We employ a similar method with modular forms of weight 2 and level Gamma to solve the sum of four squares problem.

   

• Monday November 18, 2024 at 13:30, Wachman 617

  Introduction to Atkin-Lehner Theory

  Violet Nguyen, Temple University

  In the 1930s, E. Hecke introduced linear operators to study the space of modular forms of level N, which are now fundamental to the theory. He showed that there is a basis of normalized eigenforms with respect to the commuting operators T_n for n coprime to N, and each eigenspace is one-dimensional. An extension of these results would be done by A.O.L. Atkin and J. Lehner in 1970 for the congruence subgroups Gamma_0(N) and all positive integers n. This talk gives a detailed overview of Atkin-Lehner theory for Gamma_1(N), starting from our knowledge of the Hecke operators T_n on modular forms of level Gamma(1) and coprime n.