# Colloquium Series

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**Speaker:** Professor Maria Victoria Velasco Collado (University of Granada)

**Title:** An introduction to evolution algebras**Time:** Friday, 13 October 2017 15:00**Venue:** Science South S3.56, UCD O'Brien Centre

**ABSTRACT:** Highly abstract mathematical tools, dating from Gregor Mendel (1822-1824) himself, have been used to study laws of genetic inheritance. Many different non-associative algebraic structures, generally known as 'genetic algebras', and represented here by evolution algebras, have attracted the interest of geneticists and also become independently interesting, with applications and connections to other fields of mathematics.

**Speaker:** Professor Kelly Cline (Carroll College, Helena, Montana, USA)

**Title:** Teaching Undergraduate Mathematics with Clickers and Classroom Voting**Time:** Monday, 13th November 2017 15:00**Venue:** Science Hub H2.38 UCD O'Brien Centre

**ABSTRACT:** Classroom voting with clickers is a powerful way to create a highly interactive lesson and to engage students in discussions about mathematics. This talk will report on what we’ve learned while conducting several studies of classroom voting in mathematics. How do we organize voting to maximize student engagement and learning? How do we teach all the necessary topics, given the amount of time that classroom voting requires? Research indicates that creating student discussions is a key to how classroom voting impacts student learning. What types of questions produce memorable discussions? What are the best ways to guide student discussions after a vote? What insights can we gain by studying how students vote on different questions? Finally, we’ll introduce our free web-based library containing over 2,000 clicker questions designed for classroom voting in mathematics.

**Speaker:** Professor Jon Keating , FRS (University of Bristol)

**Title:** Primes and Polynomials in Short Intervals**Time:** Wednesday, 21st February 2018, 15:00—16.00**Venue:** Science North N1.25 UCD O'Brien Centre

**ABSTRACT:** I will discuss a classical problem in Number Theory concerning the distribution of primes in short intervals and explain how an analogue of this problem involving polynomials can be solved by evaluating certain matrix integrals. I will also explain a generalisation to other arithmetic questions with a similar flavour.

**Speaker:** Professor John D. Gibbon (Imperial College London)

**Title:** Turbulence and the 3D Navier-Stokes equations

**Time:** Wednesday, 4th April 2018, 16:00—17.00

**Venue:** Science South S1.67

**Abstract:**

The regularity of solutions of the 3D incompressible Navier-Stokes equations remains one of the great open problems in modern applied mathematics. I will begin my talk by giving a very short summary of Kolmogorov’s 1941 theory of turbulence before surveying what we know about the Navier-Stokes equations themselves, and where our knowledge breaks down. In the last part of the talk I will present some new Navier-Stokes weak solution results that address the issue of whether Richardson’s cascade theory has a finite limit.

Coffee and tea will be served in the School of Mathematicss and Statistics Common Room at 3.15pm

**Speaker:** FRANCIS BROWN (University of Oxford)

**Title:** A new class of modular forms

**Time:** Tuesday, 23rd October 2018, 15:00—16.00

**Venue:** Science North N1.25

**Abstrct:**

**Speaker:** Professor CARLOS MATHEUS (CNRS and Ecole Polytechnique, University Paris Saclay)

**Title**: "Counting periodic trajectories in Sinai billiards"

**Time:** Monday, 17th December 2018, 11:00 am

**Venue:** Science North N1.25

**Abstract:**

The study of the dynamics of billiards was initially motivated by some fundamental questions in Statistical Mechanics (such as Boltzmann's ergodic hypothesis, etc.). From the mathematical point of view, they are extremely useful when testing new ideas in Dynamical Systems and Ergodic Theory.

In the first part of the talk, we review three classes of (planar) billiards in order to illustrate how the curvature of obstacles affects the behaviour of trajectories in the corresponding billiards. In the second part of the talk, we discuss the recent works by Baladi, Demers, Lima and myself allowing to count periodic trajectories in the so-called Sinai billiards.

**Speaker: **Professor STEPHEN KIRKLAND (University Manitoba, Canada)

**Title**: "Kemeny's Constant for Markov Chains"

**Time: **Friday, 25th January 2019, 13:00 - 13.50

**Venue: **Science North N1.25

**Abstract: **Markov chains are a much-studied class of stochastic processes, and it is well-known that if the transition matrix A associated with a Markov chain possesses a certain property (called primitivity), then the long-term behaviour of the Markov chain is described by a particular eigenvector of A, known as the stationary distribution vector. Rather less well-known is Kemeny’s constant for a Markov chain, which can be interpreted in terms of the expected number of time steps taken to arrive at a randomly chosen state, starting from initial state i. In particular, if Kemeny’s constant is small, then we can think of the Markov chain as possessing good mixing properties.

In this talk, we will give a short overview of Kemeny’s constant, and discuss some results dealing with the problem of minimising Kemeny’s constant over transition matrices that are subject various constraints. We will also describe some results showing that Kemeny’s constant can exhibit surprising behaviour when the transition matrix is perturbed. Throughout, the techniques used rely on matrix theory and graph theory.

**Speaker:** DARRYL HOLM (Imperial College London)

**Title: ** Stochastic transport in Geophysical Fluid Dynamics

**Time:** Monday, 4th February 2019, 13:00 - 13.50

**Venue:** Science North N1.25

**Abstract:** We discuss a new approach for deriving stochastic fluid equations which describe the slow large-scale characteristics of GFD without having to resolve the small fast scales accurately via very costly high-resolution direct numerical simulations. Instead, we discuss parametrising the small fast scales by using a new approach based on the concept of stochastic transport, rather than stochastic diffusion.

**Speaker:** VAKHTANG PUTKARADZE (University of Alberta, Canada)

**Title:** Integrability and Chaos in Figure Skating

**Time:** Monday, 4th March 2019, 13:00 - 13.50

**Venue:** Science North N1.25

**Abstract:** We derive and analyze a three dimensional model of a figure skater. We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate's direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. For a static (non-articulated) skater, we show that the system is integrable if and only if the projection of the center of mass on skate's direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia's axes. The integrability is proved by showing the existence of two new constants of motion linear in momenta, providing a new and highly nontrivial example of an integrable non-holonomic mechanical system. We also consider the case when the projection of the center of mass on skate's direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior, by studying the divergence of nearby trajectories. We also demonstrate the intricate behavior during the transition from the integrable to chaotic case. Our model shows many features of real-life skating, especially figure skating, and we conjecture that real-life skaters may intuitively use the discovered mechanical properties of the system for the control of the performance on ice.

**Speaker:** Professor POLONA OBLAK (University of Ljubljana, Slovenia)**Title: ** On the multiplicities of eigenvalues of symmetric matrices whose pattern is constrained by a graph **Time: ** 15:00 - 15.50**Date:** Tuesday, 16th April 2019**Venue: ** Science North N1.25**Abstract:** The Inverse Eigenvalue Problem for a graph (IEPG) is a problem of determining all possible lists of real numbers that can occur as the lists of eigenvalues of symmetric matrices, whose pattern is constrained by a given graph G. This question is, for general graphs, difficult to answer. In this talk we discuss related questions and introduce some parameters that help us to better understand the Inverse Eigenvalue Problem for a graph. Those include multiplicity lists of eigenvalues, and the number of distinct eigenvalues. We will present some recent results and open problems.

**Speaker:** Professor DAVID KRIBS (University of Guelph, Canada)**Title:** Quantum Information: A brief (and somewhat self-indulgent) mathematical introduction**Time:** 15:00 - 15.50**Date:** Tuesday, 14th May 2019**Venue:** Science North N1.25**Abstract:** Quantum information is an umbrella term that has evolved to describe the collection of theoretical and experimental efforts over the past two decades focused on the development of quantum technologies. Advances in quantum information have also included the discovery of interesting new mathematics, and connected the subject with several well-developed areas of mathematics along the way, leading to new opportunities for applied mathematics as well. In this talk, I will review some central aspects of the basic mathematical foundation for quantum information. I'll then touch on as many different areas in the subject that I can fit into the rest of the talk, (possibly) including (very) brief introductions to quantum error correction, quantum algorithms, and quantum entanglement theory.

**Title:** The Caratheodory Conjecture and Beyond**Speaker:** Professor BRENDAN GUILFOYLE (Institute of Technology Tralee, Ireland)**Date:** Thursday, 6th June 2019, 15:00 - 15.50**Venue:** Science North N1.25**Abstract:**

In this talk we will discuss the Caratheodory Conjecture about the number of umbilic points on a smooth convex surface in Euclidean 3-space and its proof, originally proposed by the speaker and Wilhelm Klingenberg in 2008. Lying at the intersection of geometry, analysis and topology, the various parts of the proof are now appearing in publication. In this talk an outline will be given, together with recent counter-examples in nearly Euclidean 3-spaces. Finally, the resolution of a conjecture of Toponogov using the same techniques will be explored.

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