Our series of departmental colloquia will be of interest to all academic staff, postdoctoral researchers and PhD students. Contact Karl-Mikael Perfekt for further information.

### 23 March 2021

16:00 (via Zoom Meeting ID: 615 8447 3398

**Speaker: **Christiane Tretter, University of Bern

**Title: **Challenges in non-selfadjoint spectral problems

**Abstract:** In this talk different techniques to address the challenges arising in spectral problems for non-selfadjoint linear operators will be presented. The methods and results will be illustrated by several applications from mathematical physics.

### 15 October 2020

11:00 (via Microsoft Teams, joining instructions will be circulated in due course)

**Speaker:** Björn Schmalfuss, Friedrich Schiller Universität Jena

**Title: **Random Dynamical Systems - an Overview

**Abstract:** The theory of random dynamical systems generalizes the theory of (deterministic) dynamical systems if the system is influenced by an (ergodic) noise. These systems are generated for instance by stochastic differential equations driven by a Brownian motion, or more generally by other kinds of noise, like a fractional Brownian motion. In this talk, we discuss some problems of generation of such a system. In addition, we introduce some special objects from this theory like random attractors, random invariant manifolds, and random fixed points. Finally, we will discuss some applications.

### 26 November 2019

13:00 - 14:00 (Slingo Lecture Theatre, JJ Thomson Building)

**Speaker:** Gordon Blower (Lancaster University)

**Title:** Algebraic Approaches to Integrable Operators in Random Matrix Theory

**Abstract:** In the context of random matrix theory, many of the fundamental ensembles are described by integrable operators, which are defined on intervals on the real line, or more generally cuts on a hyperelliptic Riemann surface. In this largely expository talk, I discuss how these operators can be understood algebraically in terms of commutative and noncommutative differentials. While these results are implicit in papers of Cuntz and Quillen from the 1990s, their relevance has not been fully realized in random matrix theory.

### 15 February 2019

16:00 - 17:00 (M314)

**Speaker:** Nicholas Young (Newcastle University)

**Title:** Newton-Girard and Waring-Lagrange theorems for two non-commuting variables

**Abstract:** In 1629 Albert Girard gave formulae for the power sums of several commuting variables in terms of the elementary symmetric functions; his result was subsequently often attributed to Newton. Over a century later Waring proved that an arbitrary symmetric polynomial in finitely many commuting variables can be expressed as a polynomial in the elementary symmetric functions of those variables.

In 1939 Margarete Wolf studied the analagous questions for non-commuting variables. She showed that there is no finite algebraic basis for the algebra of symmetric functions in d > 1 non-commuting variables, so there is no finite set of 'elementary symmetric functions' in the non-commutative case.

Nevertheless, Jim Agler, John McCarthy and I have recently proved analogues of Girard's and Waring's theorems for symmetric functions in two non-commuting variables. We find three free polynomials f, g, h in two non-commuting indeterminates x, y such that every symmetric polynomial in x and y can be written as a polynomial in f, g, h, and 1/g. In particular, power sums can be written explicitly in terms of f, g, and h.

### 9 November 2018

16:00 - 17:00 (M113)

**Speaker: **Carola-Bibiane Schönlieb (Cambridge)

**Title:** Deep and shallow learning approaches for regularised inversion in imaging

**Abstract:** In this talk we discuss the idea of data-driven regularisers for inverse imaging problems, investigating two parametrisation: total variation type regularisers and deep neural networks. This talk is based on joint works with J. C. De Los Reyes, L. Calatroni, C. Chung, T. Valkonen, S. Lunz and O. Oektem.

Carola-Bibiane works in image processing and PDE. She won the Whitehead prize in 2016 "for her spectacular contributions to the mathematics of image analysis", and the Philip Leverhulme Prize in 2017.

### 3 March 2017

13:00 - 14:00 (M113)

**Speaker:** Daniel Lawson (University of Bristol)

**Theme:** Genetics for Mathematicians

**Title:** The mathematics behind fine-scale personal ancestry inference, and what it can tell us

**Abstract: **Personal Genomics is a booming industry and allows people to go on a discovery process for their own history. The methods behind it allow for a discovery process for whole a population and can inform fields such as history[1], archaeology[2] and sociology. In this talk we discuss the underlying genetics models and how they are related to (but more complex than) the well-studied problem of clustering a graph.

We will compare two recent advances that allow extremely accurate personal ancestry inference. The first, recently released by Ancestry DNA [3] uses hundreds of thousands of samples with known location and uses graph heuristics to achieve geographical localisation. The second, developed by ourselves [4], uses careful algorithms on fewer samples to achieve similar clustering. This is based on the "Stochastic Block Model" which is a generative description of a graph. A number of approaches to inference are available, including Markov-Chain Monte Carlo and algorithmic approaches such as maximising modularity. We will discuss their relative merits, as well as extensions to the model to allow nodes to be mixtures of the blocks, called the mixed membership model. This has great importance in genetics because it describes admixture, i.e. what proportion of a genome is from different regions.

Scaling these models, and the improvements in accuracy that scale provides, is invaluable in genetics, and we will describe some of the very exciting consequences of having this methodology available, which range from the practical (personal genomics) to the bizarre (what we can learn about 1700s social class from the amorous congress of Captain Cook's crew in the Society Islands).

[1] Leslie et al 2015, Nature 519:309-314

[2] Pagani & Lawson et al 2016 Nature 538:238-242

[3] Han et al 2017 Nat Comms 8:14238

[4] Lawson et al 2012, PLoS Genet. 8:e1002453

### 21 February 2017

13:00 - 14:00 (RH Theatre, JJ Thomson Building)

**Speaker: **Lara Alcock (Loughborough)

**Title: **Tilting the Classroom: Engaging Students in Large Lectures

**Abstract:** There is much discussion currently about flipping the classroom or otherwise making dramatic adjustments to teaching. But for most lecturers, especially those with large classes, this is not practical. My view is that lectures are not inherently bad, and that there are numerous ways to make them more engaging without dramatic changes.

This talk will be about 18 approaches that I use - these work well together, but each can be implemented independently so they can be tried out according to personal taste. There will be lots of examples and some light-touch discussion of how this approach relates to evidence from psychological research on learning.