Past Fridays@4

18 June 2021
16:00
Helena Webb
Abstract

How aware should we be of letting AI make decisions on prison sentences? Or what is our responsibility in ensuring that mathematics does not predict another global stock crash?

In this talk, Helena will outline how we can view ethics and responsibility as central to processes of innovation and describe her experiences applying this perspective to teaching in the Department of Computer Science. There will be a chance to open up discussion about how this same approach can be applied in other Departments here in Oxford.

Helena is an interdisciplinary researcher working in the Department of Computer Science. She works on projects that involve examining the social impacts of computer-based innovations and identifying the ways in which these innovations can better meet societal needs and empower users. Helena is very passionate about the need to embed ethics and responsibility into processes of learning and research in order to foster technologies for the social good.

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11 June 2021
16:00
Jaclyn Lang and Jan Sbierski
Abstract

Jaclyn Lang
Explicit Class Field Theory
Class field theory was a major achievement in number theory
about a century ago that presaged many deep connections in mathematics
that today are known as the Langlands Program.  Class field theory
associates to each number field an special extension field, called the
Hilbert class field, whose ring of integers satisfies unique
factorization, mimicking the arithmetic in the usual integers.  While
the existence of this field is always guaranteed, it is a difficult
problem to find explicit generators for the Hilbert class field in
general.  The theory of complex multiplication of elliptic curves is
essentially the only setting where there is an explicit version of class
field theory.  We will briefly introduce class field theory, highlight
what is known in the theory of complex multiplication, and end with an
example for the field given by a fifth root of 19.  There will be many
examples!

 

Jan Sbierski
The strength of singularities in general relativity
One of the many curious features of Einstein’s theory of general relativity is that the theory predicts its own breakdown at so-called gravitational singularities. The gravitational field in general relativity is modelled by a Lorentzian manifold — and thus a gravitational singularity is signalled by the geometry of the Lorentzian manifold becoming singular. In this talk I will first review the classical definition of a gravitational singularity along with a classification of their strengths. I will conclude with outlining newly developed techniques which capture the singularity at the level of the connection of Lorentzian manifolds.

 

 

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28 May 2021
16:00
Clemens Koppensteiner and David Gómez-Castro
Abstract

Clemens Koppensteiner
Categorifying Heisenberg algebras

Categorification replaces set-theoretic structures with category-theoretic analogues. We discuss what this means and why it is useful. We then discuss recent work on categorifying Heisenberg algebras and their Fock space representations. In particular this gives a satisfying answer to an observation about equivariant K-theory made by Ian Grojnowski in 1996.

 

Aggregation-Diffusion Equations
David Gómez-Castro

The aim of this talk is to discuss an evolution problem modelling particles systems exhibiting aggregation and diffusion phenomena, and we will focus mostly on the so-called Aggregation-Diffusion Equation: ∂ρ ∂t = ∇ · (ρ ∇(U′ (ρ) + V + W ∗ ρ)) (ADE)

First, we will discuss the modelling. The famous case U′ (ρ) = log ρ and W = 0 is the famous Heat Equation. In the classical literature, the term U′(ρ) is typically deduced from Darcy’s law and models an internal energy of the system. We will show through particle systems how the term V models a confinement energy and W ∗ ρ an aggregation energy. The complete model covers many famous examples from different disciplines: Porous Media, Fokker-Plank, Keller-Segel and others. After this modelling, we discuss the mathematical treatment of (ADE). As in the case of the Heat Equation, the diffusion cases where W = V = 0 are typically studied in the Lebesgue and Sobolev spaces. However, as in the Keller-Segel problem, a Dirac measures may appear in finite time. We present the Wasserstein distance between measures, which is a natural framework for these equations, connecting with the theory of Optimal Transport. In fact, when U, V and W are convex, (ADE) can be studied as the gradient-flow of a free-energy functional (i.e. curves minimising this energy) in this Wasserstein distance, applying Calculus of Variations techniques. We will discuss the minimisation problem associated to F, with an interest to the existence of Dirac measures. Finally, we will present new results showing that indeed, in some cases besides Keller-Segel, states with a Delta can be achieved through solutions of the evolution problem

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14 May 2021
16:00
Candy Bowtell and Luci Basualdo Bonatto
Abstract

In this session we will host a Q&A with current researchers who have recently gone through successful applications as well as more senior staff who have been on interview panels and hiring committees for postdoctoral positions in mathematics. The session will be a chance to get varied perspectives on the application process and find out about the different types of academic positions to apply for.

The panel members will be Candy Bowtell, Luci Basualdo Bonatto, Mohit Dalwadi, Ben Fehrman and Frances Kirwan. 

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12 March 2021
16:00
Elena Gal and Alexandre Bovet
Abstract

Speaker: Elena Gal (4pm)

Title: Associativity and Geometry

Abstract: An operation # that satisfies a#(b#c)=(a#b)#c is called "associative". Associativity is "common" - if we are asked to give an example of operation we are more likely to come up with one that has this property. However if we dig a bit deeper we encounter in geometry, topology and modern physics many operations that are not associative "on the nose" but rather up to an equivalence. We will talk about how to describe and work with this higher associativity notion.

Speaker: Alexandre Bovet (4:30pm)

Title: Investigating disinformation in social media with network science

Abstract:
While disinformation and propaganda have existed since ancient times, their importance and influence in the age of
social media is still not clear.  We investigate the spread of disinformation and traditional misinformation in Twitter in the context of the 2016 and 2020 US presidential elections. We analyse the information diffusion networks by reconstructing the retweet networks corresponding to each type of news and the top news spreaders of each network are identified. Our investigation provides new insights into the dynamics of news diffusion in Twitter, namely our results suggests that disinformation is governed by a different diffusion mechanism than traditional centre and left-leaning news. Centre and left leaning traditional news diffusion is driven by a small number of influential users, mainly journalists, and follow a diffusion cascade in a network with heterogeneous degree distribution which is typical of diffusion in social networks, while the diffusion of disinformation seems to not be controlled by a small set of users but rather to take place in tightly connected clusters of users that do not influence the rest of Twitter activity. We also investigate how the situation evolved between 2016 and 2020 and how the top news spreaders from the different news categories have driven the polarization of the Twitter ideological landscape during this time.

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5 March 2021
16:00
William Durham
Abstract

In this session, William Durham from the Bank of England will give a presentation about working as a mathematician in the BoE, and will give advice on interviewing for non-academic jobs. He has previously provided mock interviews in our department for jobs aimed at mathematicians with PhDs, and is happy to conduct some mock interviews (remotely, of course) for individuals as well.

Please email Helen McGregor (helen.mcgregor@maths.ox.ac.uk) by Monday 22 February if you might be interested in having a mock interview with William Durham on 5 March.
 

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12 February 2021
16:00
Ben Fehrman and Markus Upmeier
Abstract

In this session, Ben Fehrman and Markus Upmeier will give their thoughts on how to deliver a good talk for a conference or a seminar and tips for what to do and what to avoid. There will be a particular emphasis on how to give a good talk online. 

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5 February 2021
16:00
Katherine Staden and Pierre Haas
Abstract

Speaker: Katherine Staden
Introduced by: Frances Kirwan
Title: Inducibility in graphs
Abstract: What is the maximum number of induced copies of a fixed graph H inside any graph on n vertices? Here, induced means that both edges and non-edges have to be correct. This basic question turns out to be surprisingly difficult, and it is not even known for all 4-vertex graphs H. I will survey the area and discuss some key results, ideas and techniques -- combinatorial, analytical and computer-assisted.

Speaker: Pierre Haas
Introduced by: Alain Goriely
Title: Shape-Shifting Droplets
Abstract: Experiments show that small oil droplets in aqueous surfactant solution flatten, upon slow cooling, into a host of polygonal shapes with straight edges and sharp corners. I will begin by showing how plane (and rather plain) geometry explains the sequence of these polygonal shapes. I will go on to show that geometric considerations of that ilk cannot however explain the three-dimensional polyhedral shapes that the initially spherical droplets evolve through while flattening. I will conclude by showing that the experimental data agree with the predictions of a model based on a partial phase transition of the oil near the droplet edges.

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