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).