Thu, 18 Jan 2024

12:00 - 13:00
L3

Coupling rheology and segregation in granular flows

Nico Gray
(University of Manchester)
Further Information

Professor Nico Gray is based in the Department of Mathematics at the University of Manchester. 

This is from his personal website:

My research interests lie in understanding and modelling the flow of granular materials, in small scale experiments, industrial processes and geophysical flows.

[Mixing in a rotating drum][Flow past a rearward facing pyramid]

Current research is aimed at understanding fundamental processes such as the flow past obstacles, shock waves, dead-zones, fluid-solid phase transitions, particle size segregation and pattern formation. A novel and important feature of all my work is the close interplay of theory, numerical computation and experiment to investigate these nonlinear systems. I currently have three active experiments which are housed in two laboratories at the Manchester Centre for Nonlinear Dynamics. You can click on the videos and pictures as well as the adjacent toolbar to find out more about specific problems that I am interested in.

Abstract

During the last fifteen years, there has been a paradigm shift in the continuum modelling of granular materials; most notably with the development of rheological models, such as the μ(I)-rheology (where μ is the friction and I is the inertial number), but also with significant advances in theories for particle segregation. This talk details theoretical and numerical frameworks (based on OpenFOAM®) which unify these disconnected endeavours. Coupling the segregation with the flow, and vice versa, is not only vital for a complete theory of granular materials, but is also beneficial for developing numerical methods to handle evolving free surfaces. This general approach is based on the partially regularized incompressible μ(I)-rheology, which is coupled to a theory for gravity/shear-driven segregation (Gray & Ancey, J. Fluid Mech., vol. 678, 2011, pp. 353–588). These advection–diffusion–segregation equations describe the evolving concentrations of the constituents, which then couple back to the variable viscosity in the incompressible Navier–Stokes equations. A novel feature of this approach is that any number of differently sized phases may be included, which may have disparate frictional properties. The model is used to simulate the complex particle-size segregation patterns that form in a partially filled triangular rotating drum. There are many other applications of the theory to industrial granular flows, which are the second most common material used after fluids. The same processes also occur in geophysical flows, such as snow avalanches, debris flows and dense pyroclastic flows. Depth-averaged models, that go beyond the μ(I)-rheology, will also be derived to capture spontaneous self-channelization and levee formation, as well as complex segregation-induced flow fingering effects, which enhance the run-out distance of these hazardous flows.

 

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Tue, 27 Feb 2024

14:00 - 15:00
L5

Modular Reduction of Nilpotent Orbits

Jay Taylor
(University of Manchester)
Abstract

Suppose 𝐺𝕜 is a connected reductive algebraic 𝕜-group where 𝕜 is an algebraically closed field. If 𝑉𝕜 is a 𝐺𝕜-module then, using geometric invariant theory, Kempf has defined the nullcone 𝒩(𝑉𝕜) of 𝑉𝕜. For the Lie algebra 𝔤𝕜 = Lie(𝐺𝕜), viewed as a 𝐺𝕜-module via the adjoint action, we have 𝒩(𝔤𝕜) is precisely the set of nilpotent elements.

We may assume that our group 𝐺𝕜 = 𝐺 × 𝕜 is obtained by base-change from a suitable ℤ-form 𝐺. Suppose 𝑉 is 𝔤 = Lie(G) or its dual 𝔤* = Hom(𝔤, ℤ) which are both modules for 𝐺, that are free of finite rank as ℤ-modules. Then 𝑉 ⨂ 𝕜, as a module for 𝐺𝕜, is 𝔤𝕜 or 𝔤𝕜* respectively.

It is known that each 𝐺 -orbit 𝒪 ⊆ 𝒩(𝑉) contains a representative ξ ∈ 𝑉 in the ℤ-form. Reducing ξ one gets an element ξ𝕜 ∈ 𝑉𝕜 for any algebraically closed 𝕜. In this talk, we will explain two ways in which we might want ξ to have “good reduction” and how one can find elements with these properties. We will also discuss the relationship to Lusztig’s special orbits.

This is on-going joint work with Adam Thomas (Warwick).

Thu, 01 Jun 2023
17:00
L4

Cancelled: An effective mixed André-Oort result

Gareth Jones
(University of Manchester)
Abstract

Habegger showed that a subvariety of a fibre power of the Legendre family of elliptic curves is special if and only if it contains a Zariski-dense set of special points. I'll discuss joint work with Gal Binyamini, Harry Schmidt, and Margaret Thomas in which we use pfaffian methods to obtain an effective uniform version of Manin-Mumford for products of CM elliptic curves. Using this we then prove an effective version of Habegger's result.

Tue, 06 Jun 2023

14:00 - 15:00
C5

Simplicity of Nekrashevych algebras of contracting self-similar groups

Nora Szakacs
(University of Manchester)
Abstract

A self-similar group is a group $G$ acting on a regular, infinite rooted tree by automorphisms in such a way that the self-similarity of the tree is reflected in the group. The most common examples are generated by the states of a finite automaton. Many famous groups, like Grigorchuk's 2-group of intermediate growth, are of this form. Nekrashevych associated $C^*$-algebras and algebras with coefficients in a field to self-similar groups. In the case $G$ is trivial, the algebra is the classical Leavitt algebra, a famous finitely presented simple algebra. Nekrashevych showed that the algebra associated to the Grigorchuk group is not simple in characteristic 2, but Clark, Exel, Pardo, Sims, and Starling showed its Nekrashevych algebra is simple over all other fields. Nekrashevych then showed that the algebra associated to the Grigorchuk-Erschler group is not simple over any field (the first such example). The Grigorchuk and Grigorchuk-Erschler groups are contracting self-similar groups. This important class of self-similar groups includes Gupta-Sidki p-groups and many iterated monodromy groups like the Basilica group. Nekrashevych proved algebras associated to contacting groups are finitely presented.

In this talk, we discuss a result of the speaker and Benjamin Steinberg characterizing simplicity of Nekrashevych algebras of contracting groups. In particular, we give an algorithm for deciding simplicity given an automaton generating the group. We apply our results to several families of contracting groups like GGS groups and Sunic's generalizations of Grigorchuk's group associated to polynomials over finite fields.

Mon, 06 Mar 2023
15:30
L4

Homeomorphisms of surfaces: a new approach

Richard Webb
(University of Manchester)
Abstract

Despite their straightforward definition, the homeomorphism groups of surfaces are far from straightforward. Basic algebraic and dynamical problems are wide open for these groups, which is a far cry from the closely related and much better understood mapping class groups of surfaces. With Jonathan Bowden and Sebastian Hensel, we introduced the fine curve graph as a tool to study homeomorphism groups. Like its mapping class group counterpart, it is Gromov hyperbolic, and can shed light on algebraic properties such as scl, via geometric group theoretic techniques. This brings us to the enticing question of how much of Thurston's theory (e.g. Nielsen--Thurston classification, invariant foliations, etc.) for mapping class groups carries over to the homeomorphism groups. We will describe new phenomena which are not encountered in the mapping class group setting, and meet some new connections with topological dynamics, which is joint work with Bowden, Hensel, Kathryn Mann and Emmanuel Militon. I will survey what's known, describe some of the new and interesting problems that arise with this theory, and give an idea of what's next.

 

Tue, 21 Feb 2023
14:00
L6

A Prolog-assisted search for simple Lie algebras

David Stewart
(University of Manchester)
Abstract

(jt work with David Cushing and George Stagg)

Prolog is a rather unusual programming language that was developed by Alain Colmerauer 50 years ago in one of the buildings on the way to the CIRM in Luminy. It is a declarative language that operates on a paradigm of first-order logic -- as distinct from imperative languages like C, GAP and Magma. Prolog operates by loading in a list of axioms as input, and then responds at the command line to queries that ask the language to achieve particular goals, given those axioms. It gained some notoriety through IBM’s implementation of ‘Watson’, which was a system designed to play the game show Jeopardy. Through a very efficiently implemented constraint logic programming module, it is also the worlds fastest sudoku solver. However, it has had barely any serious employment by pure mathematicians. So the aim of this talk is to advertise Prolog through an extended example: my co-authors and I used it to search for new simple Lie algebras over the field GF(2) and were able to classify a certain flavour of absolutely simple Lie algebra in dimensions 15 and 31, discovering a dozen or so new examples. With some further examples in dimension 63, we then extrapolated two previously undocumented infinite families of simple Lie algebras.

Thu, 23 Feb 2023
16:00
L4

Upper bounds for moments of the Riemann zeta-function

Hung Bui
(University of Manchester)
Abstract

Assuming the Riemann Hypothesis, Soundararajan established almost sharp upper bounds for all positive moments of the Riemann zeta-function. This result was later improved by Harper, who proved upper bounds of the right order of magnitude. I will describe some of the ideas in their proofs, and then discuss recent joint work with Alexandra Florea, where we consider negative moments of the Riemann zeta-function. For example, we can obtain asymptotic formulas for negative moments when the shift in the zeta function is large enough, confirming a conjecture of Gonek.  We also obtain an upper bound for the average of the generalised Mobius function.

Tue, 08 Nov 2022
14:00
L6

Generalising Vogan's conjecture across Schur-Weyl duality

Kieran Calvert
(University of Manchester)
Abstract

We outline Dirac cohomology for Lie algebras and Vogan’s conjecture. We then cover some basic material on Schur-Weyl duality and Arakawa-Suzuki functors. Finishing with current efforts and results on generalising Vogan’s conjecture to a Schur-Weyl duality setting. This would relate the centre of a Lie algebra with the centre of the relevant tantaliser algebra. We finish by considering a unitary module X and giving a bound on the action of the tantalizer algebra.

Thu, 17 Feb 2022

16:00 - 17:00
L5

Values of the Ramanujan tau-function

Vandita Patel
(University of Manchester)
Abstract

The infamous Ramanujan tau-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan tau-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves. This is joint work with Mike Bennett (UBC), Adela Gherga (Warwick) and Samir Siksek (Warwick).

Fri, 11 Mar 2022

14:00 - 15:00
L6

An example of the Lyndon-Hochschild-Serre spectral sequence

Anja Meyer
(University of Manchester)
Abstract

Spectral sequences are computational tools to find the (co-)homology of mathematical objects and are used across various fields. In this talk I will focus on the LHS spectral sequence, which we associate to an extension of groups to compute group cohomology. The first part of the talk will serve as introduction to both group cohomology and general spectral sequences, where I hope to provide and intuition and some reduced formalism. As main example, and core of this talk, we will look at the LHS spectral sequence associated to the group extension $(\mathbb{Z}/3\mathbb{Z})^3 \rightarrow S \rightarrow \mathbb{Z}/3\mathbb{Z}$, where $S$ is a Sylow-3-subgroup of $SL_2(\mathbb{Z}/9\mathbb{Z})$. In particular I will present arguments that all differentials on the $E^2$ page vanish.

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