# 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.

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