Algebra and Number Theory Seminars 2017/2018

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Title: Modular forms and quantum field theory

Speaker:  Marianne Leitner (DIAS)

Date:  Thursday 21st September March 2017

Time:  2pm

Location:  Agriculture. 1.01 (Seminar Room).


Abstract:

Quantum field theory is a treasure trove for mathematics which is exploited in small steps only.
The talk will discuss the well-established link between conformal field theory and modular forms,
and some recent work concerning mapping class groups for higher genus.

Title:  A Hopf algebra for Feynman diagrams and integrals

Speaker:  Ruth Britto (TCD)

Date:  Thursday 5th October 2017

Time:  2pm

Location:  Agriculture. 1.01 (Seminar Room).


Abstract:
I will present a combinatorial operation on one-loop Feynman graphs, described by cutting and pinching edges, that corresponds to the Hopf-algebraic coaction on the multiple polylogarithms that result from applying the Feynman rules. A generalization of this operation, expressed simply in terms of a pairing of integrands with contours of integration, can be applied directly to larger classes of functions, including hypergeometric functions.


Title:  Yang-Baxter equation, knots, cohomology: a golden triangle

Speaker:  Victoria Lebed (TCD)

Date:  Thursday 12th October 2017

Time:  2pm

Location:  Agriculture. 1.01 (Seminar Room).


Abstract:
In this talk I will explain how each of the three mathematical areas becomes a source of tools and inspirations for the remaining two. Among topics discussed will be state-sum knot invariants, deformations of solutions to the YBE, cohomology of different algebraic structures, and a braid-based graphical calculus for cohomology computations.

Title:  Algebraic constructions of semifields and maximum rank distance codes

Speaker:  John Sheekey (UCD)

Date:  Thursday 19th October 2017

Time:  2pm

Location:  Agriculture. 1.01 (Seminar Room).

Abstract: Rank-metric codes are codes consisting of matrices, with the distance between two matrices being the rank of their difference. Codes with maximum size for a fixed minimum distance are called Maximum Rank Distance (MRD) codes. These have received increased attention in recent years, in part due to their applications in Random Linear Network Coding.

(Finite) semifields are nonassociative division algebras over a field. Existence of non-trivial examples was established by Dickson in 1906. They have many connections with interesting objects in finite geometry, such as projective planes, spreads, flocks. The number of equivalence classes of semifields remains an open problem. By considering the maps defined by multiplication, there is a correspondence between semifields and MRD codes of a certain type.

In this talk we will review the known constructions for semifields and MRD codes, focusing in particular on those constructed using linearized polynomials and skew-polynomial rings. We will introduce a new family, which contains new examples of semifields and MRD codes, and incorporates previously distinct constructions into one family.


Title:    Decompositions into sums or products of two quadratic matrices

Speaker:    Clement de Sequins Pazzis (Versailles)

Date:        Thursday 26th October 2017

Time:        2pm

Location:    Agriculture. 1.01 (Seminar Room).

Abstract:

Let p and q be polynomials with degree 2 over a field F. A square matrix M in Mat_n(F) is called a (p,q)-sum (respectively, a (p,q)-product) whenever M=A+B (respectively, M=AB) for some matrices A, B in Mat_n(F) such that p(A)=q(B)=0.

Many special cases in the classification of (p,q)-sums and (p,q)-products have been studied in the past, reaching back to the 1960’s for the latter (see the works of Ballantine on products of idempotent matrices, and of Djokovic for products of two involutions), and the 1990's for the former (mostly by Hartwig, Putcha, Wang and Wu).

In a recent comprehensive article (see https://arxiv.org/abs/1703.01109), we have completed the classification of (p,q)-sums, and also the one of (p,q)-products under the additional condition p(0)q(0) is not equal to 0. In this lecture, we will give the main ideas behind those classifications. In particular, we will emphasize the role played by quaternion algebras.

Title:    Modular forms and quantum field theory: new results


Speaker:    Marianne Leitner (DIAS)

Date:        Thursday 2nd November 2017

Time:        2pm

Location:     Agriculture. 1.01 (Seminar Room).


Abstract:
Quantum field theory is a treasure trove for mathematics which is exploited in small steps only. The talk will discuss the well-established link between conformal field theory and modular forms, and some recent work concerning mapping class groups for higher genus. It has deep links to the old work on the icosahedron. This is a continuation of my talk on September 21.


 

Title:        Stability properties of the colored Jones polynomial

Speaker:    Christine Lee (UT-Austin and MPIM)

Date:        Thursday 9th November 2017

Time:        2pm

Location:     Agriculture. 1.01 (Seminar Room).


Abstract:
The colored Jones polynomial is a sequence of Laurent polynomials J_N(K;q) for N greater than or equal to 2 assigned to a knot in a 3-sphere. The tail of the colored Jones polynomial is a power series whose first N-1 coefficients coincides with J_N(K;q) for all N > 2. Stability properties of the colored Jones polynomial is an interesting subject in quantum topology because of its connection to the volume of a hyperbolic knot, recurrence relation, and number-theoretic identities. In this talk I will discuss my work on the stability of the colored Jones polynomial and its categorification for all knots in the context of the work by Armond and Garoufalidis-Le.

 

Title:    Positive cones and gauges on algebras with involution

Speaker:    Thomas Unger (UCD)

Date:        Thursday 16th November 2017

Time:        2pm

Location:     Agriculture. 1.01 (Seminar Room).

Abstract:
Let (A,\sigma) be a central simple algebra with involution over a formally real field F. Orderings on F can be extended to (pre-)positive cones on (A,\sigma). Let X_{(A,\sigma)} denote the (spectral) space of positive cones on (A,\sigma). For example, X_{(M_n(\mathbb{R}), t)} has two elements, the PSD and NSD real matrices. In this talk I will explain how elements of X_{(A,\sigma)} give rise in a natural way to Tignol-Wadsworth gauges, which extend valuations on F to A. I will also discuss the interplay between positive cones and gauges. This is joint work with Vincent Astier.

 Title:    A mathematical framework for adversarial network communications

Speaker:    Alberto Ravagnani (UCD)

Date:        Thursday 23rd November 2017

Time:        2pm

Location:     Agriculture. 1.01 (Seminar Room).


Abstract:
In the context of Network Coding, one or multiple sources of information attempt to transmit messages to multiple receivers through a network of intermediate nodes. In order to maximize the network throughput, the nodes may be allowed to recombine the received packets before forwarding them towards the sinks.

In this talk, we present a general mathematical model for adversarial network transmissions, studying the scenario where one or multiple (possibly coordinated) adversaries can maliciously corrupt some of the transmitted messages, according to certain restrictions. For example, the adversaries may be constrained to operate on a vulnerable region of the network.

If noisy channels (traditionally studied in Information Theory) are described within a theory of "probability", adversarial channels are described within a theory of "possibility". Accordingly, in this talk we take a discrete combinatorial approach to define and study network adversaries.

We propose various combinatorial notions of capacity of an adversarial network, and present a general technique that allows to port upper bounds for the capacities of point-to-point channels to the networking context. We then briefly discuss the existence of capacity-achieving communication schemes for some adversarial scenarios of applied interest.

The new results in this talk are joint work with Frank R. Kschischang (University of Toronto).


Title:      Slope estimates of Artin-Schreier F-crystals


Speaker:    Stiofain Fordham (UCD)


Date:        Thursday 8th February 2018


Time:        2pm

Location:     Agriculture. 1.01 (Seminar Room).

Abstract:

Abstract: From a number theory perspective, the a priori complicated
structure of the cohomology of an abelian variety over a field of
positive characteristic can be reduced to a (semi-)-linear-algebraic
construction called a crystal and the action of Frobenius on the
crystal associated to the variety can be described fairly explicitly and
allows one to obtain information about the arithmetic properties
of the variety. I will describe an extension of results of Scholten-Zhu
in this direction, to arbitrary Artin-Schreier covers of the projective
line and their fibre products. This is joint work with S.E. Yılmaz.


Title:    Hypergeometric differential equations and hypergeometric motives


Speaker:        Bartosz Naskrecki  Adam Mickiewicz University


Date:    Thursday 15th February 2018


Time:    2pm

Location:     Agriculture. 1.01 (Seminar Room)


Abstract:


In this talk we will discuss what are the so-called hypergeometric motives and how one can approach the problem of their explicit construction as Chow motives in explicitely given algebraic varieties. The class of hypergeometric motives corresponds to Picard-Fuchs equations of hypergeometric type and forms a rich family of pure motives with nice L-functions. Following recent work of Beukers-Cohen-Mellit we will show how to realise certain hypergeometric motives of weights 0 and 2 as submotives in elliptic and hyperelliptic surfaces. An application of this work is the computation of minimal polynomials of hypergeometric series with finite monodromy groups and proof of identities between certain hypergeometric finite sums, which mimics well-known identities for classical hypergeometric series. This is a part of the larger program conducted by Villegas et al. to study the hypergeometric differential equations (special cases of differential equations "coming from algebraic geometry") from the algebraic perspective.

Title:       Topological data analysis

Speaker: Graham Ellis (NUI Galway)

Date:       Thursday 22nd February 2018

Time:       2pm

Location:   Agriculture. 1.01 (Seminar Room)


Abstract:

I'll try to convince the audience that basic algebraic structures from topology -- vector spaces, abelian groups, rings, groupoids and maybe even multiple groupoids -- have a role to play in the computational analysis of large data sets. The data sets I have in mind include high-dimensional point cloud data, digital images, proteins, and so forth. This approach to data analysis necessitates the development of efficient algorithms in computational algebra. I'll discuss some algorithms in this context.


Title:    Cyclic models for finite vector spaces and applications

Speaker:    Alessandro Siciliano (University of Basilicata)

Date:    Thursday 1st March 2018

Time:        2pm

Location:     Agriculture. 1.01 (Seminar Room)

Abstract:


It is known that every dimensional projective space over a finite field admits a cyclic model. In the past years many authors used this model to construct relevant geometric objects in this geometry. In this talk the linear algebra of cyclic models for a vector space over a finite field will be discussed. Applications of these models to rank-metric codes and to finite geometries will be presented.


Title:  
 Number of solutions of algebraic equations over finite fields

Speaker:    Carlos Moreno (CUNY)

Date:    Monday 12th March 2018

Time:        12 pm

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

Abstract:

The Chevalley-Warning Theorem states that a (homogeneous) polynomial equation in n variables of degree d has solutions over any finite field if d < n. This result has had important applications in many branches of mathematics, most notably in logic (Ax-Kochen Theorem), in number theory (p-adic solutions) and Error Correcting Codes. In this survey talk, we will highlight some of the main tools and improvements and describe some problems that examine the validity of the result when n=d.

Title:    Spinor regular ternary quadratic forms

Speaker:    Anna Haensch (Duquesne University and MPIM)

Date:    Thursday 29th March 2018

Time:        2pm

Location:     Agriculture. 1.01 (Seminar Room)

Abstract:
It is well known that there is no direct integral analogue to Hasse’s local-global principle.  An integral lattice which satisfies a local-global principle (that is, a lattice that represents everything globally which is represented locally at every prime) is called regular.  Extending this notion of regularity introduced by Dickson in 1939, a positive definite ternary integral quadratic form is said to be spinor regular if it represents all the positive integers represented by its spinor genus (that is, all positive integers represented by any form in its spinor genus). Jagy conducted an extensive computer search for primitive ternary quadratic forms that are spinor regular, but not regular, resulting in a list of 29 such forms. In this talk, I will discuss recent work with A. G. Earnest in which we verify the completeness of this list.
Statistics Seminar

Title:    Birational RSK correspondence and Whittaker functions

Speaker:    Neil O’Connell (UCD)

Date:    Thursday 5th April 2018

Time:        2pm

Location:     Agriculture. 1.01 (Seminar Room)

Abstract:
The Robinson-Schensted-Knuth (RSK) correspondence is a combinatorial bijection which plays an important role in the theory of Young tableaux and symmetric functions, particularly in understanding combinatorial aspects of Schur polynomials (Cauchy-Littlewood identity, Littlewood-Richardson rule, etc.). I will give some background on this and then explain how a birational version of the RSK correspondence provides a similar `combinatorial’ framework for the study of GL(n,R)-Whittaker functions.  These functions arise in the context of automorphic forms associated with GL(n,R), and reduce to the classical Whittaker functions in the case n=2.

Title:    Combinatorics on words for Markoff numbers


Speaker:    Laurent Vuillon (University Savoie Mont Blanc)


Date:    Thursday 12th April 2018


Time:    2pm


Location:     Agriculture. 1.01 (Seminar Room)

Abstract:
Markoff numbers are fascinating integers related to number theory, hyperbolic geometry, continued fractions and Christoffel words. Many great mathematicians have worked on these numbers and the 100 years famous uniqueness conjecture by Frobenius is still unsolved. In this talk, we state a new formula to compute the Markoff numbers using iterated palindromic closure and the Thue-Morse substitution. The main theorem shows that for each Markoff number m, there exists a word v in {a, b}∗ such that m − 2 is equal to the length of the iterated palindromic closure of the iterated antipalindromic closure of the word av. This construction gives a new recursive construction of the Markoff numbers by the lengths of the words involved in the palindromic closure.


Title:    Commuting graph of the algebra of matrices

Speaker:    Damjana Kokol Bukovsek (Ljubljana)

Date:    Thursday 19th April 2018

Time:        2pm

Location:     Agriculture. 1.01 (Seminar Room)

Abstract:
Let A be a groupoid, that is, a nonempty set equipped with an inner operation, written as a product ab. The essence of commutativity relation on A is captured in its commuting graph \Gamma(A). By definition, this is a simple graph whose vertices are all non central elements of A and where two distinct vertices a and b are connected if they commute in A. One of the basic properties of a graph is its diameter and connectedness. This question turned out to be surprisingly hard for the commuting graphs.
We will consider the commuting graph of M_n(F), the algebra of n x n matrices over a field F. If F is algebraically closed and n is greater than or equal to 3, then the diameter of the commuting graph \Gamma(M_n(F)) is 4. The diameter of a connected commuting graph of M_n(F) is bounded above by 6. The commuting graph of M_n(\mathbb{Q}) is disconnected for any n greater than or equal to 2, but for each prime greater than or equal to 7 there exist two similar matrices A and B in M_{2p}(\mathbb{Q}) at maximal possible distance 6. If F is a finite field, then \Gamma(M_n(F)) is a finite graph and it has the following properties: if n is greater than or equal to 4 and even, then its diameter is 4. If n is prime, then it is disconnected. If n is odd and is neither a prime nor a square of a prime, then its diameter is at most 5.


Title:                Modular forms from partitions

Speaker:         Chris Jennings-Shaffer (Cologne)

Date:               Thursday 26th April 2018

Time:               2pm

Location:         Agriculture. 1.01 (Seminar Room).

Abstract:
We give a brief introduction to integer partitions and how they produce modular objects. We begin with q-series representations of generating functions and examples of modular forms, including the celebrated Rogers-Ramanujan identities.  Moving beyond modular forms, we also explain how integer partition type functions arise as mock modular forms. Going even further, we give an example of a higher depth mock modular form coming from a certain partition function, as established in joint work with Bringmann and Mahlburg. No prior knowledge of modular forms, partitions, or q-series will be assumed.