Thu, 20 Oct 2022

12:00 - 13:00
L1

### OCIAM TBC

Prof Halim Kusumaatmaja
(Durham University)

The join button will be published 30 minutes before the seminar starts (login required).

Abstract

Prof Halim Kusumaatmaja's group is interested in theoretical and computational soft matter and biophysics. Their work is interdisciplinary, lying at the interface between Physics, Chemistry, Engineering and Biology. They are also part of the Durham Centre for Soft Matter, the Biophysical Sciences Institute, and the SOFI CDT. See here for more detailed descriptions.

Thu, 05 May 2022

16:00 - 17:00
L5

### Gaussian distribution of squarefree and B-free numbers in short intervals

Alexander Mangerel
(Durham University)
Abstract
(Joint with O. Gorodetsky and B. Rodgers) It is a classical quest in analytic number theory to understand the fine-scale distribution of arithmetic sequences such as the primes. For a given length scale h, the number of elements of a "nice" sequence in a uniformly randomly selected interval $(x,x+h], 1 \leq x \leq X$, might be expected to follow the statistics of a normally distributed random variable (in suitable ranges of $1 \leq h \leq X$).  Following the work of Montgomery and Soundararajan, this is known to be true for the primes, but only if we assume several deep and long-standing conjectures among which the Riemann Hypothesis. In fact, previously such distributional results had not been proven for any (non-trivial) sequence of number-theoretic interest, unconditionally.

As a model for the primes, in this talk I will address such statistical questions for the sequence of squarefree numbers, i.e., numbers not divisible by the square of any prime, among other related sifted'' sequences called B-free numbers. I hope to further motivate and explain our main result that shows, unconditionally, that short interval counts of squarefree numbers do satisfy Gaussian statistics, answering several questions of R.R. Hall.

Fri, 04 Mar 2022

15:00 - 16:00
L6

### Open questions on protein topology in its natural environment.

Christopher Prior
(Durham University)
Abstract

Small angle x-ray scattering is one of the most flexible and readily available experimental methods for obtaining information on the structure of proteins in solution. In the advent of powerful predictive methods such as the alphaFold and rossettaFold algorithms, this information has become increasingly in demand, owing to the need to characterise the more flexible and varying components of proteins which resist characterisation by these and more standard experimental techniques. To deal with structures about little of which is known a parsimonious method of representing the tertiary fold of a protein backbone as a discrete curve has been developed. It represents the fundamental local Ramachandran constraints through a pair of parameters and is able to generate millions of potentially realistic protein geometries in a short space of time. The data obtained from these methods provides a treasure trove of information on the potential range of topological structures available to proteins, which is much more constrained that that available to self-avoiding walks, but still far more complex than currently understood from existing data. I will introduce this method and its considerations then attempt to pose some questions I think topological data analysis might help answer. Along the way I will ask why roadies might also help give us some insight….

Mon, 10 Jun 2019
15:45
L6

### Unitary group integrals, surfaces, and mapping class groups

Michael Magee
(Durham University)
Abstract

For any word w in a free group of rank r>0, and any compact group G, w induces a `word map' from G^r to G by substitutions of elements of G for the letters of w. We may also choose the r elements of G independently with respect to Haar measure on G, and then apply the word map. This gives a random element of G whose distribution depends on w. An interesting observation is that this distribution doesn't change if we change w by an automorphism of the free group. It is a wide open question whether the measures induced by w on compact groups determine w up to automorphisms.
My talk will be mostly about the case G = U(n), the n by n complex unitary matrices. The technical tool we use is a precise formula for the moments of the distribution induced by w on U(n). In the formula, there is a surprising appearance of concepts from infinite group theory, more specifically, Euler characteristics of mapping class groups of surfaces. I'll explain how our formula allows us to make progress on the question described above.
This is joint work with Doron Puder (Tel Aviv).

Mon, 25 Feb 2019
15:45
L6

### Twisted Blanchfield pairings and Casson-Gordon invariants

Anthony Conway
(Durham University)
Abstract

In the late seventies, Casson and Gordon developed several knot invariants that obstruct a knot from being slice, i.e. from bounding a disc in the 4-ball. In this talk, we use twisted Blanchfield pairings to define twisted generalisations of the Levine-Tristram signature function, and describe their relation to the Casson-Gordon invariants. If time permits, we will present some obstructions to algebraic knots being slice. This is joint work with Maciej Borodzik and Wojciech Politarczyk.

Thu, 07 Mar 2019

14:00 - 15:00
L4

### Flexible computational abstractions for complex preconditioners

Dr Lawrence Mitchell
(Durham University)
Abstract

Small block overlapping, and non-overlapping, Schwarz methods are theoretically highly attractive as multilevel smoothers for a wide variety of problems that are not amenable to point relaxation methods.  Examples include monolithic Vanka smoothers for Stokes, overlapping vertex-patch decompositions for $H(\text{div})$ and  $H(\text{curl})$ problems, along with nearly incompressible elasticity, and augmented Lagrangian schemes.

While it is possible to manually program these different schemes,  their use in general purpose libraries has been held back by a lack   of generic, composable interfaces. We present a new approach to the   specification and development such additive Schwarz methods in PETSc  that cleanly separates the topological space decomposition from the  discretisation and assembly of the equations. Our preconditioner is  flexible enough to support overlapping and non-overlapping additive  Schwarz methods, and can be used to formulate line, and plane smoothers, Vanka iterations, amongst others. I will illustrate these new features with some examples utilising the Firedrake finite element library, in particular how the design of an approriate computational interface enables these schemes to be used as building blocks inside block preconditioners.

This is joint work with Patrick Farrell and Florian Wechsung (Oxford), and Matt Knepley (Buffalo).

Thu, 07 Jun 2018
12:00
L5

### On singular limits for the Vlasov-Poisson system

Mikaela Iacobelli
(Durham University)
Abstract

The Vlasov-Poisson system is a kinetic equation that models collisionless plasma. A plasma has a characteristic scale called the Debye length, which is typically much shorter than the scale of observation. In this case the plasma is called ‘quasineutral’. This motivates studying the limit in which the ratio between the Debye length and the observation scale tends to zero. Under this scaling, the formal limit of the Vlasov-Poisson system is the Kinetic Isothermal Euler system. The Vlasov-Poisson system itself can formally be derived as the limit of a system of ODEs describing the dynamics of a system of N interacting particles, as the number of particles approaches infinity. The rigorous justification of this mean field limit remains a fundamental open problem. In this talk we present the rigorous justification of the quasineutral limit for very small but rough perturbations of analytic initial data for the Vlasov-Poisson equation in dimensions 1, 2, and 3. Also, we discuss a recent result in which we derive the Kinetic Isothermal Euler system from a regularised particle model. Our approach uses a combined mean field and quasineutral limit.

Mon, 20 Nov 2017
15:45
L6

### Stable diffeomorphism of 4-manifolds

Mark Powell
(Durham University)
Abstract

I will talk about the diffeomorphism classification of 4-manifolds up to
connected sums with the complex projective plane, and how the resulting
equivalence class of a manifold can be detected by algebraic topological
invariants of the manifold.  I may also discuss related results when one
takes connected sums with another favourite 4-manifold, S^2 x S^2, instead.

Tue, 27 Feb 2018
14:15
L4

### The regular representations of GL_N over finite local principal ideal rings

Alexander Stasinski
(Durham University)
Abstract

Let $F$ be a non-Archimedean local field with ring of integers $\mathcal O$ and maximal ideal $\mathfrak p$. T. Shintani and G. Hill independently introduced a large class of smooth representations of $GL_N(\mathcal O)$, called regular representations. Roughly speaking they correspond to elements in the Lie algebra $M_N(\mathcal O)$ which are regular mod $\mathfrak p$ (i.e, having centraliser of dimension $N$). The study of regular representations of $GL_N(\mathcal O)$ goes back to Shintani in the 1960s, and independently and later, Hill, who both constructed the regular representations with even conductor, but left the much harder case of odd conductor open. In recent simultaneous and independent work, Krakovski, Onn and Singla gave a construction of the regular representations of $GL_N(\mathcal O)$ when the residue characteristic of $\mathcal O$ is not $2$.

In this talk I will present a complete construction of all the regular representations of $GL_N(\mathcal O)$. The approach is analogous to, and motivated by, the construction of supercuspidal representations of $GL_N(F)$ due to Bushnell and Kutzko. This is joint work with Shaun Stevens.

Fri, 19 May 2017
14:15
C3

### Modelling a glacial cycle using three equations

Pippa Whitehouse
(Durham University)
Abstract

In my research I model three components of the Earth system: the ice sheets, the ocean, and the solid Earth. In the first half of this talk I will describe the traditional approach that is used to model the impact of ice sheet growth and decay on global sea-level change and solid Earth deformation. I will then go on to explain how collaboration across the fields of glaciology, geodynamics and seismology is providing exciting new insight into feedbacks between ice dynamics and solid Earth deformation.

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