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.

Title:    An introduction to K3 surfaces

Speaker:    Madeeha Khalid (UCD)

Date:    Thursday 24th January 2019

Time:    2pm

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

Abstract:
I will explain the notion of a K3 surface and describe some simple examples.  These are 4 dimensional spaces, similar to tori, which are of interest in mathematics and theoretical physics.  Time permitting, I will briefly describe some research directions in the theory of vector bundles on K3 surfaces.


Title:    Galois Theory and the cyclic quantum dilogarithm

Speaker:    Kevin Hutchinson (UCD)

Date:    Thursday 31st January 2019

Time:    2pm

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

Abstract:
A recent preprint by Calegari, Garoufalidis and Zagier gives an elegant explicit formula for a Chern class in K-theory in terms of the `cyclic quantum dilogarithm'. They use the formula to prove one direction of Nahm's conjecture on modularity of certain hypergeometric series. I will discuss some Galois-theoretic problems arising from their construction and show how the attempt to resolve them leads to intriguing identities -- some proved, some conjectural -- involving quantum dilogarithms.  

(This is work in progress. The questions and results I will discuss in the talk will involve only  basic Galois theory. K-theory will play no explicit role.)


Title:    Perfect and Pretty Good State Transfer on Graphs

Speaker:    Stephen Kirkland (Manitoba)

Date:    Thursday 7th February 2019

Time:    2pm

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

Abstract:
Transmitting a quantum state from one location to another is a key task within a quantum computer. This task can be realised through the use of a spin network, which can be modelled by an undirected graph G: vertices in G represent spins, and two edges are adjacent in G whenever the corresponding spins interact.

As the system evolves over time, the quality of the state transfer is measured by a function called the fidelity, which can be expressed in terms of the Hamiltonian for the system; that Hamiltonian is typically either the adjacency matrix of G or the Laplacian matrix of G. The fidelity at time t, f(t), is a number between 0 and 1, and if it happens that f(t_0)=1 for some t_0>0, then we say that there is perfect state transfer (PST). Similarly if f(t) can be made arbitrarily close to 1 via suitable choice of t, then there is pretty good state transfer (PGST).

In this talk we will give an introduction to perfect and pretty good state transfer on graphs, with a focus on identifying some families of graphs that possess PST/PGST, and other families that don't. If time permits, there will also be a discussion of the effect of errors on the fidelity. Tools from matrix analysis, spectral graph theory and number theory will each play a role.


Title:    Congruences of modular forms and special values of L-functions

Speaker:    Tadashi Ochiai (Osaka)

Date:    Thursday 14th February 2019

Time:    2pm

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

Abstract:
We discuss congruences between different modular forms. We start with a review of classical modular forms on the Poincare upper half plane and discuss when a congruence between modular forms modulo a certain prime number occurs within the space of modular forms of the same weight and level. We explain a classical result of Hida which proves that such congruences are detected by the special values of the adjoint L-function. If time permits, we will explain a joint work with Francesco Lemma on a generalization of Hida's result to Siegel modular forms.


Title:    On the theory of Euler systems

Speaker:    Ryotaro Sakamoto (Tokyo)

Date:    Thursday 21st February 2019

Time:    2pm

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

Abstract:
An Euler system is a collection of cohomology classes with certain norm interrelations. They were introduced by Kolyvagin in order to study Selmer groups (of elliptic curves). In this talk, I will first define the notion of an Euler system and a Selmer group associated to a p-adic Galois representation and I will give a simple example. I will then discuss Kolyvagin’s machinery in the language of Mazur and Rubin. If time permits, I will explain some recent developments on the general theory of Euler systems, and report on a joint work with David Burns and Takamichi Sano.

Title:       Dualities, trialities and graphs
Speaker: Klara Stokes (Tokyo)
Date:       Thursday 28th February 2019
Time:       2pm
Location: Room 1.25, O’Brien Centre for Science (North)

Abstract:
In projective geometry, a duality is a correlation sending points to lines and lines to points. A triality is a correlation of order three in a geometry of at least three types. This talk will be about some incidence geometries with interesting dualities and trialities and their related graphs.

 


Title:    Arithmetic of special L-values

Speaker:    Fabian Januszewski (Karlsruhe)

Date:    Thursday 7th March 2019

Time:    2pm

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

Abstract:
The study of special values of L-functions has a long history in number theory. While the Langlands Programme provides us with powerful tools for the study of analytic properties of L-functions, the arithmetic information we are seeking to extract from L-functions appears to be encoded in their special values. In this talk I will discuss classical examples alongside with recent results on rather general class of L-functions.


Title:    Yang-Baxter algebras and Gromov-Witten invariants of Grassmannians

Speaker:    Christian Korff (Glasgow)

Date:    Thursday 28th March 2019

Time:    2pm

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

Abstract:
Solutions of the Yang-Baxter equation (often also known as the braid relation) play an important role in the representation theory of quantum groups and vice versa. Using a simple combinatorial model we show how such solutions to the Yang-Baxter equation arise when counting exact matchings or dimer configurations on the honeycomb lattice. With each such solution one can associate an associative algebra, called the Yang-Baxter algebra, which contains a commutative subalgebra. For the dimer model the latter turns out to be isomorphic to the quantum cohomology of Grassmannians and we state an independent geometric construction of the Yang-Baxter algebra.





Title:    Jordan triple product homomorphisms on Hermitian and triangular matrices

Speaker:    Damiana Kokol Bukovsek (Ljubljana

Date:    Thursday 4th April 2019

Time:    2pm

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

Abstract:
A map Φ : M_n(F) →M_m(F) is a Jordan triple product (J.T.P.) homomorphism whenever Φ(ABA) = Φ(A)Φ(B)Φ(A)  for all A, B ∈ M_n(F). We study J.T.P. homomorphisms on Hermitian matrices H_n(C) and upper triangular matrices T_n(F). We characterize J.T.P. homomorphisms of real or complex Hermitian matrices for the cases n=1, m=1, and n=m=2.

We characterize J.T.P. homomorphisms of real or complex upper triangular matrices for the cases n=1, and m=1. In some cases we consider only continuous maps and the implications of omitting the assumption of continuity.

Title:           Explicit Reciprocity Laws

Speaker:     Denis Benois (Bordeaux)

Date:          Thursday 11th April 2019

Time:           2pm

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

Abstract:


The search for generalizations of the quadratic reciprocity law was a major theme in 19th century mathematics.  Class Field Theory reduces this problem to an explicit computation of the Hilbert symbol for local fields. This problem was completely solved in the late 70's.
The pioneering work of Bloch and Kato on special values of L-functions (1990) showed that classical explicit reciprocity laws can be considered as particular cases of general reciprocity laws for p-adic representations of Galois groups. This picture remains mainly conjectural, but in all known cases reciprocity laws relate arithmetic invariants of p-adic representations to special values of associated L-functions.
In this talk, I will give an overview of this large theory.


Title:    A characterization of annular domains by quadrature identities II

Speaker:    Stephen Gardiner (UCD)

Date:    Tuesday 9th April 2019

Time:    4pm

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

Abstract:
We will show that annular domains may be characterized as quadrature domains for harmonic functions with respect to a uniformly distributed measure on a sphere. This verifies an old conjecture of Armitage and Goldstein. (Joint work with Tomas Sjödin).

Title:         Third homology of perfect central extensions of groups

Speaker:  Behrooz Mirzaii (Universidade de São Paulo & UCD)

Date:        Thursday 18th April 2019 

Time:        2pm

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

 

Abstract: In this talk I will discuss the homology exact sequence associated to a perfect central extension of groups A >--> G -->> Q. We will especially concentrate on the third homology of this extension. To study these groups, we will need certain tools from algebraic topology, such as Serre fibration, Eilenberg-MacLane spaces and Whitehead's quadratic functor. Finally when BG^+, the plus construction of the classifying space of G, is an H-space, we will study the kernel of the homomorphism H_3(G,\mathbb{Z}) --> H_3(Q, \mathbb{Z}).

Title:           Morphisms of complex Hadamard matrices

Speaker:     Ronan Egan (Galway)

Date:           Thursday 25th April 2019

Time:           2pm

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

 

Abstract: Let M be a nxn matrix with complex entries of norm less than or equal to 1. A well-known theorem of Hadamard bounds the magnitude of the determinant of M as a function of its dimension. Hadamard proved that this bound is met precisely when all entries of M have norm exactly 1, and the rows are pairwise orthogonal under the Hermitian inner product. Thus M is said to be a complex Hadamard matrix if this bound is attained, or equivalently, if MM* = nI where M* denotes the conjugate transpose of M. Butson Hadamard matrices are complex Hadamard matrices with entries restricted to the set of kth roots of unity for some positive integer k. In this talk I will introduce the concept of a morphism from the set of Butson matrices over kth roots of unity to the set of Butson matrices over rth rootsof unity. As concrete examples of such morphisms, I will describe tensor-product-like maps which reduce the order of the roots of unity appearing in a Butson matrix at the cost of increasing the dimension. This work generalises previous constructions of Turyn and Compton-Craigen-de Launey of real Hadamard matrices from certain complex Hadamard matrices with entries in the 4th and 6th roots of unity respectively. This is a report on recent joint workwith Padraig Ó Catháin and Eric Swartz.