Virtual

Contagion models on networks can be used to describe the spread of information, rumours, opinions, and (more topically) diseases through a population. In the simplest contagion models, each node represents an individual that can be in one of a number of states (e.g. Susceptible, Infected, or Recovered), and the states of the nodes evolve according to specified rules. Even with simple Markovian models of transmission and recovery, it can be difficult to compute the dynamics of contagion on large networks: running simulations can be slow, and the system of master equations is typically too large to be tractable.

 One approach to approximating contagion dynamics is to assume that each node state is independent of the neighbouring node states; this leads to a system of ODEs for the node state probabilities (the “first-order approximation”) that always overestimates the speed of infection spread. This approach can be made more sophisticated by introducing pair approximations or higher-order moment closures, but this dramatically increases the size of the system and slows computations. In this talk, I will present some alternative node-based approximations for contagion dynamics. The first of these is exact on trees but will always underestimate the speed of infection spread on a network with loops. I will show how this can be combined with the classic first-order node-based approximation to obtain a node-based approximation that has similar accuracy to the pair approximation, but which is considerably faster to solve.

Crystal Structure Prediction aims to reveal the properties that stable crystalline arrangements of a molecule have without stepping foot in a laboratory, consequently speeding up the discovery of new functional materials. Since it involves producing large datasets that themselves have little structure, an appropriate classification of crystals could add structure to these datasets and further streamline the process. We focus on geometric invariants, in particular introducing the density fingerprint of a crystal. After exploring its computations via Brillouin zones, we go on to show how it is invariant under isometries, stable under perturbations and complete at least for an open and dense space of crystal structures.

Giancarlo Rota once wrote of matroids that "It is as if one were to
condense all trends of present day mathematics onto a single
structure, a feat that anyone would a priori deem impossible, were it
not for the fact that matroids do exist" (Indiscrete Thoughts, 1997).
This makes matroid theory a natural hub through which ideas flow from
one field of mathematics to the next. At the end of our three-day
workshop, participants will understand the most common objects and
constructions in matroid theory to the depth suitable for exploring
many of these interesting connections. We will also pick up some
highly practical matroid tools for working through problems in
persistent homology, (optimal) cycle representatives, and other
objects of interest in TDA.

Day 3, Lecture 1

Circuits in persistent homology

Day 3, Lecture 2

Exercise: write your own persistent homology algorithm!
 

Giancarlo Rota once wrote of matroids that "It is as if one were to
condense all trends of present day mathematics onto a single
structure, a feat that anyone would a priori deem impossible, were it
not for the fact that matroids do exist" (Indiscrete Thoughts, 1997).
This makes matroid theory a natural hub through which ideas flow from
one field of mathematics to the next. At the end of our three-day
workshop, participants will understand the most common objects and
constructions in matroid theory to the depth suitable for exploring
many of these interesting connections. We will also pick up some
highly practical matroid tools for working through problems in
persistent homology, (optimal) cycle representatives, and other
objects of interest in TDA.

Day 2, Lecture 1, 10-10.45am

Matroid representations, continued

Day 2, Lecture 2, 11-11.45am

Matroids in homological algebra

Giancarlo Rota once wrote of matroids that "It is as if one were to
condense all trends of present day mathematics onto a single
structure, a feat that anyone would a priori deem impossible, were it
not for the fact that matroids do exist" (Indiscrete Thoughts, 1997).
This makes matroid theory a natural hub through which ideas flow from
one field of mathematics to the next. At the end of our three-day
workshop, participants will understand the most common objects and
constructions in matroid theory to the depth suitable for exploring
many of these interesting connections. We will also pick up some
highly practical matroid tools for working through problems in
persistent homology, (optimal) cycle representatives, and other
objects of interest in TDA.

Condensed outline

Day 1, Lecture 1, 10-10.45am

Definitions There are many definitions of matroids. Here's how to organize them.
Examples We work with matroids every day. Here are a few you have seen.
Important properties What's so great about a matroid?

Day 1, Lecture 2, 11-11.45am

The essential operations: deletion, contraction, and dualization
Working with matroids: matrix representations

That neural networks may be pruned to high sparsities and retain high accuracy is well established. Recent research efforts focus on pruning immediately after initialization so as to allow the computational savings afforded by sparsity to extend to the training process. In this work, we introduce a new `DCT plus Sparse' layer architecture, which maintains information propagation and trainability even with as little as 0.01% trainable kernel parameters remaining. We show that standard training of networks built with these layers, and pruned at initialization, achieves state-of-the-art accuracy for extreme sparsities on a variety of benchmark network architectures and datasets. Moreover, these results are achieved using only simple heuristics to determine the locations of the trainable parameters in the network, and thus without having to initially store or compute with the full, unpruned network, as is required by competing prune-at-initialization algorithms. Switching from standard sparse layers to DCT plus Sparse layers does not increase the storage footprint of a network and incurs only a small additional computational overhead.

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A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Topology optimisation finds the optimal material distribution of a fluid or solid in a domain, subject to PDE, volume, and box constraints. The optimisation problem is normally nonconvex and can support multiple local minima. In recent work [1], the authors developed an algorithm for systematically discovering multiple minima of two-dimensional problems through a combination of barrier methods, active-set strategies, and deflation. The bottleneck of the algorithm is solving the Newton systems that arise. In this talk, we will present preconditioning methods for these linear systems as they occur in the topology optimization of Stokes flow. The strategies involve a mix of block preconditioning and specialized multigrid relaxation schemes that reduce the computational work required and allow the application of the algorithm to three-dimensional problems.

[1] “Computing multiple solutions of topology optimization problems”, I. P. A. Papadopoulos, P. E. Farrell, T. M. Surowiec, 2020, https://arxiv.org/abs/2004.11797

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

We combine the notions of graded Clifford algebras and Drinfeld algebras. This gives us a framework to study algebras with a PBW property and underlying vector space $\mathbb{C}[G] \# Cl(V) \otimes S(U) $ for $G$-modules $U$ and $V$. The class of graded Clifford-Drinfeld algebras contains the Hecke-Clifford algebras defined by Nazarov, Khongsap-Wang. We give a new example of a GCD algebra which plays a role in an Arakawa-Suzuki duality involving the Clifford algebra.

A wide variety of fixed-point iterative methods for the solution of nonlinear operator equations in Hilbert spaces exists. In many cases, such schemes can be interpreted as iterative local linearisation methods, which can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative methods. It will be shown that for strongly monotone operators this unified iteration scheme satisfies an energy contraction property. Consequently, the generated sequence converges to a solution of the original problem.

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A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Suspensions are composed of mixtures of particles and fluid and are
ubiquitous in industrial processes (e.g. waste disposal, concrete,
drilling muds, metalworking chip transport, and food processing) and in
natural phenomena (e.g. flows of slurries, debris, and lava). The
present talk focusses on the rheology of concentrated suspensions of
non-colloidal particles. It addresses the classical shear viscosity of
suspensions but also non-Newtonian behaviour such as normal-stress
differences and shear-induced migration. The rheology of dense
suspensions can be tackled via a diversity of approaches that are
introduced. In particular, the rheometry of suspensions can be
undertaken at an imposed volume fraction but also at imposed values of
particle normal stress, which is particularly well suited to yield
examination of the rheology close to the jamming transition. The
influences of particle roughness and shape are discussed.