Algebra and Number Theory Seminars 2018/2019

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Title: The length of commutative group algebras

Speaker: Olga Markova (Moscow State University)

Date: Thursday 13th September 2018

Time: 2pm

Location: Room 1.25, O’Brien Centre for Science (North)

Abstract: 
By the length of a finite system of generators for a finite-dimensional algebra over an arbitrary field we mean the least positive integer k such that the products of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. The length evaluation can be a difficult problem, for example, the length of the full matrix algebra is unknown (Paz’s Problem, 1984).

We examine some basic properties of the length function, including the length of a direct sum of algebras and the length of an algebra after the field extension, and relationship of this characteristic with other numerical invariants of the given algebra.

The main topic of this talk is the length of commutative algebras. We give an exact upper bound for the length of a commutative algebra in general and provide more specific bounds and exact length computation for commutative subalgebras in the matrix algebra and for group algebras of finite Abelian groups.

Title: The colored Jones polynomial for double twist knots

Speaker: Robert Osburn (UCD)

Date: Thursday 20th September 2018

Time: 2pm

Location: Room 1.25, O’Brien Centre for Science (North)


Abstract:
Over the past two decades, there has been substantial interest in the overlap between quantum knot invariants, q-series and modular forms. In this talk, we discuss one such instance, namely an explicit
q-hypergeometric expression for the colored Jones polynomial for double twist knots. As an application, we generalize a duality at roots of unity between the Kontsevich-Zagier series and the generating function for strongly unimodal sequences.

This is joint work with Jeremy Lovejoy (Paris 7 and Berkeley).

Title: p-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou

Speaker: Kazim Buyukboduk (UCD)

Date: Thursday 27th September 2018

Time: 2pm

Location: Room 1.25, O’Brien Centre for Science (North)


Abstract:
I will report joint work in progress with R. Pollack and S. Sasaki, where we prove a p-adic Gross-Zagier formula for critical slope p-adic L-functions. Besides the strategy for our proof, I will illustrate a number of applications. The first is the proof of a conjecture of Perrin-Riou, which predicts an explicit construction of the generator of the Mordell-Weil group in terms of p-adic L-values when the analytic rank is one. The second is towards a Birch and Swinnerton-Dyer formula when the analytic rank is one, yielding an improvement of the recent results of Jetchev-Skinner-Wan in this context (and simplifying their proof).

Title:    Integrability of algebraic foliations of the plane

Speaker:    Francisco Monserrat (Valencia)

Date:    Thursday 29th November 2018

Time:    2pm

Location:     Room 1.25, O’Brien Centre for Science (North)

Abstract:


A classical problem, proposed by H. Poincare, consists of determining whether a differential equation with polynomial coefficients, of first order and degree 1, is algebraically integrable or, equivalently, whether an algebraic foliation of the projective plane has a rational first integral. We will show some results concerning this problem that use tools of algebraic geometry. More specifically, we shall show that, assuming certain geometric conditions concerning the resolution of singularities of the foliation, there exists an algorithm that provides a solution to this problem. Also, we shall show a similar result that determines whether an algebraic plane foliation has a Darboux first integral of certain type



Title:    Some families of curves over finite fields for algebraic geometry codes and quantum codes

Speaker:    Gary McGuire (UCD)

Date:    Thursday 22nd November 2018

Time:    2pm

Location:     Room 1.25, O’Brien Centre for Science (North)

Abstract:
Good quantum error-correcting codes can be constructed from algebraic geometry codes that have particular properties. These algebraic geometry codes can be constructed from algebraic curves over finite fields that have particular properties. We shall describe some recent work that finds some families of curves with the desirable properties.

Joint work with Francisco Monserrat, Julio Moyano, Fernando Hernando, Robin Chapman.

 

 

Title:         Indivisibility and divisibility of class numbers of imaginary quadratic fields

 Speaker:  Olivia Beckwith (Bristol)

 Date:        Thursday 15th November 2018

 Time:        2pm

 Location:  Room 1.25, O’Brien Centre for Science (North)

 

Abstract:

For any prime p > 3, the strongest lower bounds for the number of imaginary quadratic fields with discriminant down to -X for which the class group has trivial (non-trivial) p-torsion are due to Kohnen and Ono (Soundararajan). I will discuss recent refinements of these classic results in which we consider the imaginary quadratic fields whose class number is indivisible (divisible) by p such that a given finite set of primes factor in a prescribed way. We prove a lower bound for the number of such fields with discriminant down to -X which is of the same order of magnitude as Kohnen and Ono's (Soundararajan's) results. For the indivisibility case, we rely on a result of Wiles establishing the existence of imaginary quadratic fields with trivial p-torsion in their class groups satisfying almost any given finite set of local conditions, and a result of Zagier which says that the Hurwitz class numbers are the Fourier coefficients of a mock modular form.

 




Title:    Subspace Designs in Coding Theory

Speaker:    Eimear Byrne (UCD)

Date:    Thursday 8th November 2018

Time:    2pm

Location:     Room 1.25, O’Brien Centre for Science (North)

Abstract:
The notion of a subspace design has appeared in the literature since the 1970s and a few constructions were given in the following decades. There was a resurgence of interest in such designs in recent years, as some subspace designs are optimal as constant weight subspace codes, which have been shown to have applications to error correction in network coding. It is known that non-trivial subspace designs with sufficiently large parameters exist over any finite field and there are now several papers showing the existence of such objects for various parameter sets. In most cases these constructions rely on a prescribed automorphism group.  In this talk, we'll show a connection between rank metric codes and subspace designs, essentially giving a q-analogue of the Assmus-Mattson theorem. We'll give a brief overview of the topic and outline some open problems.

Title:    An algebraic toolbox in Iwasawa theory

Speaker:    Antonio Lei (Laval)

Date:    Thursday 1st November 2018

Time:    2pm

Location:     Room 1.25, O’Brien Centre for Science (North)

Abstract:


In Iwasawa Theory, we study asymptotic behaviours of arithmetic objects over infinite towers of field extensions. For example, let E be an elliptic curve, that is, a curve which is defined by a cubic equation and is equipped with a group structure. It is an interesting question in Number Theory to ask how many points on E are defined over the rational numbers. We may replace rational numbers by a bigger field, and ask how many points on E are defined over this bigger field. We may continue this process on replacing this bigger field by a series of bigger fields and keep asking the same question. It turns out that it is possible to construct a series of field extensions systematically so that the number of points on E defined over these fields do not grow very fast. In this talk, we will explore some of the algebraic tools Iwasawa theorists use to study such asymptotic behaviours.



Title:    The projective characters of metacyclic p-groups

Speaker:    Conor Finnegan (UCD)

Date:    Thursday 25th October 2018

Time:    2pm

Location:     Room 1.25, O’Brien Centre for Science (North)

Abstract:


I will give a general introduction to projective representation theory, assuming a basic prior knowledge of ordinary representation theory. I will then present a strategy for finding the projective character tables of metacyclic p-groups of positive type, using the previously understood abelian case as an example. I will discuss the main results in the application of this strategy to the non-abelian case, and will describe the difficulties which arise when trying to apply the same methods to metacyclic p-groups of negative type.

Title:    Properties of rank metric codes and analogies with the Hamming metric

Speaker:    Alessandra Neri (Zurich)

Date:    Thursday 18th October 2018

Time:    2pm

Location:     Room 1.25, O’Brien Centre for Science (North)

Abstract:
Rank metric codes have been introduced in 1978, but only in the last decade they gained a lot of interest due to their application to network coding. These codes are linear subspaces of n x m matrices over a finite field F_q, but they can be also seen as vectors of length n over an extension field F_q^m. Codes that are optimal in this metric are called Maximum Rank Distance (MRD) codes. The first and most studied construction of rank metric codes was proposed in the seminal works of Delsarte (1978), Gabidulin (1985) and Roth (1991). These codes are known as generalized Gabidulin codes, and they represent the analogue of generalized Reed-Solomon codes for the rank metric.

 In this talk we give an overview on the analogies between MRD codes and their counterpart in the classical Hamming metric, i.e. the so-called Maximum Distance Separable (MDS) codes. This will be done focusing in particular on the generator matrix of these two families of codes. In particular, we also examine the structure of generalized Gabidulin codes, that represent the analogue of Generalized Reed-Solomon codes in the rank metric. The study of these codes and their encoders leads to matrices with a deep structure in the context of finite fields and finite geometry.

Title:    Integer completely positive matrices of order two.

Speaker:    
Helena Šmigoc (UCD)

Date:    
Thursday 11th October 2018

Time:    
2pm

Location:     
Room 1.25, O’Brien Centre for Science (North)


Abstract:
We will show that every 2 x 2 integer matrix A that is both positive definite and entrywise nonnegative can be written in the form A=VV^T, where the entries of V are nonnegative integers.

This is joint work with Thomas J. Laffey.


Title:    Hyperbolicity of quadratic forms over function fields of quadrics

Speaker:    James O’Shea (DCU)

Date:    Thursday 4th October 2018

Time:    2pm

Location:     Room 1.25, O’Brien Centre for Science (North)


Abstract:
The study of the behaviour of quadratic forms under scalar extension to function fields of quadrics is an important theme in quadratic form theory. In particular, a natural problem in this regard is to seek to determine which forms defined over the base field become trivial under such a scalar extension. We will discuss some results in this regard and their connection to the study of prominent classes of quadratic forms, such as round forms and multiples of Pfister forms.