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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.
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."
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.
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.
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
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.
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.
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.
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.
Marco Zambon, KU Leuven
TBA
This is an organizational meeting of the Algebra Seminar. We will discuss where (and how) to steer the "algebra boat" during this fall semester.
Jackie Lang, Temple University
I will discuss work in progress with Robert Pollack and Preston Wake about counting congruences between "vexing" modular forms.
Jackie Lang, Temple University
We will discuss the computation of the endomorphism algebra that was introduced last time.
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.
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.
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