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 23, 2023 at 14:30, Wachman 527

  Hecke algebras, I

  Martin W. Lorenz, Temple University
  
  "Hecke algebras" feature prominently in representation theory, knot theory, the Langlands program, and other areas. In three or four lectures, I will try to consolidate the various different versions of Hecke algebras. The first lecture will adopt the point of view taken in Shimura's "Introduction to the Arithmetic Theory of Automorphic Functions."
  
   

• Monday January 30, 2023 at 14:30, Wachman 527

  Hecke algebras, II

  Martin W. Lorenz, Temple University

  In the second talk of this series, I will focus on a specific example, first investigated by Iwahori: the Hecke algebra for the general linear group over a finite field and a Borel subgroup. Time permitting, I also will sketch some applications of this Hecke algebra to knot invariants.

   

• Monday February 13, 2023 at 14:30, Wachman 527

  Algebraic operations of string topology

  Ralph Kaufmann, Purdue University

  There are certain algebra structures which arise naturally in many areas of mathematics. We will discuss several of these and how they show up from the ideas of string topology which aims to provide operations on the loop space. This package includes a geometric background for Lie algebras, Poisson (or better odd Poisson = Gerstenhaber algebras). Newer results include the natural occurrence of bialgebra structures and so-called bibrackets. We will introduce these notions and relate them to geometric backgrounds. If time permits, we will show how these structures appear on Hochschild chains and cochain complexes.

   

• Monday February 20, 2023 at 14:30, Wachman 527

  Quiver moduli and effective global generation

  Svetlana Makarova, The University of Pennsylvania
  
  I will start with defining moduli problems in general and providing a modern understanding. The modern theory 
  ​"Beyond GIT", introduced by Alper and being developed by Alper, Halpern-Leistner, Heinloth and others, provides 
  a "coordinate-free" way of thinking about classification problems. Among giving a uniform philosophy, this allows 
  to treat problems that can't necessarily be described as global quotients. 
  
  I will then revisit and refine a classical result of King that moduli spaces of semistable representations of acyclic quivers 
  are projective using modern methods. I will define the stack of semistable quiver representations and use a recent 
  existence result to explain why it admits an adequate moduli space. Our methods allow us to improve the classical results: 
  I will define a determinantal line bundle on the stack which descends to a semiample line bundle on the moduli space and 
  provide effective bounds for global generation. For an acyclic quiver, we can observe that this line bundle is ample and thus 
  the adequate moduli space is projective over an arbitrary noetherian base. This talk is based on a preprint with Belmans, Damiolini, Franzen, Hoskins, Tajakka https://arxiv.org/abs/2210.00033
  
   

• Monday February 27, 2023 at 14:30, Wachman 527

  Does the Grothendieck-Teichmueller group act on algebraic numbers?

  Vasily Dolgushev, Temple University
  
  
  In their paper "Open problems in Grothendieck-Teichmueller theory", Pierre Lochak and Leila Schneps proposed a way to define an action of the Grothendieck-Teichmueller group, GT, on the set of algebraic numbers. There are many questions about this construction. In the first talk in this series, I will recall the group GT, Grothendieck's child's drawings and the action of GT on child's drawings. 
  
  
   

• Monday March 27, 2023 at 14:30, Wachman 527

  Does the Grothendieck-Teichmueller group act on algebraic numbers? Part 2

  Vasily Dolgushev, Temple University 

  I will recall the outer action of the absolute Galois group G_Q of rationals on the fundamental group of an algebraic curve. I will explain how this action gives us a homomorphism from G_Q to the Grothendieck-Teichmueller group and why Belyi's theorem implies that this homomorphism is injective. I hope to get to the open question formulated by Pierre Lochak and Leila Schneps in the title of this admittedly short series of talks.

   

• Monday April 10, 2023 at 14:30, Wachman 527

  Artin formalism for non-genuine Garrett-Rankin L-functions

  Kazim Buyukboduk, University College Dublin
  
  I will report joint work with D. Casazza and R. Sakamoto, where we formulate a conjecture (and prove it in many cases) on the factorization of a certain triple product p-adic L-function whose range of interpolation is empty. The relevant factorization statement reflects not only the Artin formalism for the underlying family of motives (which decompose as the sum of 2 motives of respective degrees 2 and 6) but also dwells on the interplay between various Gross--Zagier formulae for the relevant complex L-series, and the subtle relationship between the derivatives of complex L-series at their central critical point and p-adic L-functions. 

• Monday April 17, 2023 at 14:30, Wachman 527

  Finiteness theorems in arithmetic

  Jaclyn Lang, Temple University
  
  I will briefly recall the main finiteness theorems in algebraic number theory and use them to prove another one: any abelian extension of a number field that has exponent m and is unramified outside a finite set of primes is finite.  If time permits, I will briefly sketch how this is used to prove that the group of rational points on an elliptic curve is finitely generated.

• Monday May 1, 2023 at 14:30, Wachman 527

  TBA

  Marco Zambon, KU Leuven

  TBA

   

• Monday September 11, 2023 at 13:30, Wachman 617

  Algebra Seminar. Organizational Meeting 

  This is an organizational meeting of the Algebra Seminar. We will discuss where (and how) to steer the "algebra boat" during this fall semester.

   

• Monday September 18, 2023 at 13:30, Wachman 527

  Algebra Seminar. An introduction to vexing Hecke algebras, I

  Jackie Lang, Temple University
  
  I will discuss work in progress with Robert Pollack and Preston Wake about counting congruences between "vexing" modular forms.
   

• Monday September 25, 2023 at 13:30, tba

  Algebra Seminar. An introduction to vexing Hecke algebras, II

  Jackie Lang, Temple University
  
  We will discuss the computation of the endomorphism algebra that was introduced last time.
  
   

• Monday October 2, 2023 at 13:30, Wachman 307

  Algebraic Neural Networks: Stability to Deformations

  Alejandro Parada-Mayorga, University of Pennsylvania
  
  
  Convolutional architectures play a central role on countless scenarios in machine learning, and the numerical evidence that proves the advantages of using them is overwhelming. Theoretical insights have provided solid explanations about why such architectures work well. These analysis apparently different in nature, have been performed considering signals defined on different domains and with different notions of convolution, but with remarkable similarities in the final results, posing then the question of whether there exists an explanation for this at a more structural level. In this talk we provide an affirmative answer to this question with a first principles analysis introducing algebraic neural networks (AlgNNs), which rely on algebraic signal processing and representation theory of algebras. In particular, we study the stability properties of algebraic neural networks showing that stability results for traditional CNNs, graph neural networks (GNNs), group neural networks, graphon neural networks, or any formal convolutional architecture, can be derived as particular cases of our results. This shows that stability is a universal property – at an algebraic level – of convolutional architectures, and this also explains why the remarkable similarities we find when analyzing stability for each particular type of architecture.
  
   

• Monday October 9, 2023 at 13:30, Wachman 527

  The Cartan-Brauer triangle

  Aniruddha Sudarshan, Temple University
  
  The Cartan-Brauer triangle is a commutative diagram involving certain Grothendieck groups. It is a useful tool in representation theory when passing between different base fields. The maps in the Cartan-Brauer triangle are traditionally denoted by c, d, and e, and the triangle is also commonly referred to as the "cde triangle". In this talk, I will define the Grothendieck groups in question and the aforementioned maps. Time permitting, I will also discuss some basic properties of the Cartan-Brauer triangle. 
  
   

• Monday October 16, 2023 at 13:30, Wachman 527

  The one-sided cycle shuffles in the symmetric group algebra

  Darij Grinberg, Drexel University

  Given a positive integer n, we define n elements t_1, t_2, ..., t_n in the group algebra of the symmetric group S_n by

  t_i = the sum of the cycles (i), (i, i+1), (i, i+1, i+2), ..., (i, i+1, ..., n)

  (where the cycle (i) is the identity permutation). Note that t_1 is the famous "top-to-random shuffle" element studied by many.

  These n elements t_1, t_2, ..., t_n do not commute. However, we show that they can be simultaneously triangularized in an appropriate basis of the group algebra (the "descent-destroying basis"). As a consequence, any rational linear combination of these n elements has rational eigenvalues. Various surprises emerge in describing these eigenvalues and their multiplicities; in particular, the Fibonacci numbers appear prominently.

  This talk will include an overview of other families (both well-known and exotic) of elements of these group algebras. A card-shuffling interpretation will be given and some tempting conjectures stated. This is joint work with Nadia Lafrenière.

• Monday October 23, 2023 at 13:30, Wachman 527

  Exploration of GT-shadows for the dihedral poset

  Vasily A. Dolgushev, Temple University
  
  Grothendieck-Teichmueller (GT) shadows form an infinite groupoid GTSh whose set of objects is the poset of certain finite index normal subgroups of the Artin braid group on 3 strands. After a brief reminder of GT-shadows, I will introduce the dihedral subposet of the poset of objects of GTSh. I will also talk about connected components of elements of the dihedral poset in the groupoid GTSh. My talk is based on a paper written this year by Ivan Bortnovskyi and Vadym Pashkovskyi.    
  
   

• Monday November 13, 2023 at 13:30, Wachman 527

  Quaternion algebras over local fields

  Aniruddha Sudarshan, Temple University

  TBA