Thu, 15 Oct 2020

16:00 - 17:00
Virtual

Inversion in Volvox: Forces and Fluctuations of Cell Sheet Folding

Pierre Haas
(University of Oxford)
Abstract

Tissue folding during animal development involves an intricate interplay
of cell shape changes, cell division, cell migration, cell
intercalation, and cell differentiation that obfuscates the underlying
mechanical principles. However, a simpler instance of tissue folding
arises in the green alga Volvox: its spherical embryos turn themselves
inside out at the close of their development. This inversion arises from
cell shape changes only.

In this talk, I will present a model of tissue folding in which these
cell shape changes appear as variations of the intrinsic stretches and
curvatures of an elastic shell. I will show how this model reproduces
Volvox inversion quantitatively, explains mechanically the arrest of
inversion observed in mutants, and reveals the spatio-temporal
regulation of different biological driving processes. I will close with
two examples illustrating the challenges of nonlinearity in tissue
folding: (i) constitutive nonlinearity leading to nonlocal elasticity in
the continuum limit of discrete cell sheet models; (ii) geometric
nonlinearity in large bending deformations of morphoelastic shells.
 

Tue, 08 Sep 2020

17:00 - 18:00

Joshua Bull - Can maths tell us how to win at Fantasy Football?

Joshua Bull
(University of Oxford)
Further Information

Fantasy Football is played by millions of people worldwide, and there are countless strategies that you can choose to try to beat your friends and win the game. But what’s the best way to play? Should you be patient and try to grind out a win, or are you better off taking some risks and going for glory? Should you pick players in brilliant form, or players with a great run of fixtures coming up? And what is this Fantasy Football thing anyway?

As with many of life’s deep questions, maths can help us shed some light on the answers. We’ll explore some classic mathematical problems which help us understand the world of Fantasy Football. We’ll apply some of the modelling techniques that mathematicians use in their research to the problem of finding better Fantasy Football management strategies. And - if we’re lucky - we’ll answer the big question: Can maths tell us how to win at Fantasy Football?

Joshua Bull is a Postdoctoral Research Associate in the Mathematical Institute in Oxford and the winner of the 2019-2020 Premier League Fantasy Football competition (from nearly 8 million entrants).

Watch live (no need to register):
https://twitter.com/OxUniMaths
https://www.facebook.com/OxfordMathematics/
https://livestream.com/oxuni/bull
Oxford Mathematics YouTube Channel

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

 

Thu, 06 Aug 2020

16:00 - 17:00
Virtual

Path signatures in topology, dynamics and data analysis

Vidit Nanda
(University of Oxford)
Abstract

The signature of a path in Euclidean space resides in the tensor algebra of that space; it is obtained by systematic iterated integration of the components of the given path against one another. This straightforward definition conceals a host of deep theoretical properties and impressive practical consequences. In this talk I will describe the homotopical origins of path signatures, their subsequent application to stochastic analysis, and how they facilitate efficient machine learning in topological data analysis. This last bit is joint work with Ilya Chevyrev and Harald Oberhauser.

Thu, 04 Jun 2020

16:00 - 17:00

Multi-agent reinforcement learning: a mean-field perspective

Renyuan Xu
(University of Oxford)
Abstract

Multi-agent reinforcement learning (MARL) has enjoyed substantial successes in many applications including the game of Go, online Ad bidding systems, realtime resource allocation, and autonomous driving. Despite the empirical success of MARL, general theories behind MARL algorithms are less developed due to the intractability of interactions, complex information structure, and the curse of dimensionality. Instead of directly analyzing the multi-agent games, mean-field theory provides a powerful approach to approximate the games under various notions of equilibria. Moreover, the analytical feasible framework of mean-field theory leads to learning algorithms with theoretical guarantees. In this talk, we will demonstrate how mean-field theory can contribute to the simultaneous-learning-and-decision-making problems with unknown rewards and dynamics. 

To approximate Nash equilibrium, we first formulate a generalized mean-field game (MFG) and establish the existence and uniqueness of the MFG solution. Next we show the lack of stability in naive combination of the Q-learning algorithm and the three-step fixed-point approach in classical MFGs. We then propose both value-based and policy-based algorithms with smoothing and stabilizing techniques, and establish their convergence and complexity results. The numerical performance shows superior computational efficiency. This is based on joint work with Xin Guo (UC Berkeley), Anran Hu (UC Berkeley), and Junzi Zhang (Stanford).

If time allows, we will also discuss learning algorithms for multi-agent collaborative games using mean-field control. The key idea is to establish the time consistent property, i.e., the dynamic programming principle (DPP) on the lifted probability measure space. We then propose a kernel-based Q-learning algorithm. The convergence and complexity results are carried out accordingly. This is based on joint work with Haotian Gu, Xin Guo, and Xiaoli Wei (UC Berkeley).

Wed, 17 Jun 2020
10:00
Virtual

TBA

Jonathan Fruchter
(University of Oxford)
Wed, 10 Jun 2020
10:00
Virtual

TBA

Mehdi Yazdi
(University of Oxford)
Fri, 12 Jun 2020

15:00 - 16:00
Virtual

Contagion Maps for Manifold Learning

Barbara Mahler
(University of Oxford)
Abstract

Contagion maps are a family of maps that map nodes of a network to points in a high-dimensional space, based on the activations times in a threshold contagion on the network. A point cloud that is the image of such a map reflects both the structure underlying the network and the spreading behaviour of the contagion on it. Intuitively, such a point cloud exhibits features of the network's underlying structure if the contagion spreads along that structure, an observation which suggests contagion maps as a viable manifold-learning technique. We test contagion maps as a manifold-learning tool on several different data sets, and compare its performance to that of Isomap, one of the most well-known manifold-learning algorithms. We find that, under certain conditions, contagion maps are able to reliably detect underlying manifold structure in noisy data, when Isomap is prone to noise-induced error. This consolidates contagion maps as a technique for manifold learning. 

Fri, 29 May 2020

15:00 - 16:00
Virtual

Persistent Homology with Random Graph Laplacians

Tadas Temcinas
(University of Oxford)
Abstract


Eigenvalue-eigenvector pairs of combinatorial graph Laplacians are extensively used in graph theory and network analysis. It is well known that the spectrum of the Laplacian L of a given graph G encodes aspects of the geometry of G  - the multiplicity of the eigenvalue 0 counts the number of connected components while the second smallest eigenvalue (called the Fiedler eigenvalue) quantifies the well-connectedness of G . In network analysis, one uses Laplacian eigenvectors associated with small eigenvalues to perform spectral clustering. In graph signal processing, graph Fourier transforms are defined in terms of an orthonormal eigenbasis of L. Eigenvectors of L also play a central role in graph neural networks.

Motivated by this we study eigenvalue-eigenvector pairs of Laplacians of random graphs and their potential use in TDA. I will present simulation results on what persistent homology barcodes of Bernoulli random graphs G(n, p) look like when we use Laplacian eigenvectors as filter functions. Also, I will discuss the conjectures made from the simulations as well as the challenges that arise when trying to prove them. This is work in progress.
 

Wed, 13 May 2020

17:00 - 18:00

Renaud Lambiotte - Smartphones vs COVID-19

Renaud Lambiotte
(University of Oxford)
Further Information

For several weeks news media has been full of how contact tracing Smartphone apps could help fight COVID-19. However, mobile phones can do more than just trace - they are vital tools in the measurement, prediction and control of the virus.

Looking at recent epidemics as well as COVID-19, Renaud will discuss the different types of data that researchers have been collecting, demonstrating their pros and cons as well as taking a wider view of where mobile data can help us understand the impact of lockdowns on social behaviour and improve our ways of calibrating and updating our epidemiological models. And he will discuss the issue that underpins all this and which is vital for widespread take-up from the Public: privacy and data protection.

Renaud Lambiotte is Associate Professor of Networks and Nonlinear Systems in Oxford.

Watch live:
https://twitter.com/OxUniMaths
https://www.facebook.com/OxfordMathematics/
https://livestream.com/oxuni/lambiotte

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

Thu, 07 May 2020

12:00 - 13:00
Virtual

Vectorial problems: sharp Lipschitz bounds and borderline regularity

Cristiana De FIlippis
(University of Oxford)
Abstract

Non-uniformly elliptic functionals are variational integrals like
\[
(1) \qquad \qquad W^{1,1}_{loc}(\Omega,\mathbb{R}^{N})\ni w\mapsto \int_{\Omega} \left[F(x,Dw)-f\cdot w\right] \, \textrm{d}x,
\]
characterized by quite a wild behavior of the ellipticity ratio associated to their integrand $F(x,z)$, in the sense that the quantity
$$
\sup_{\substack{x\in B \\ B\Subset \Omega \ \small{\mbox{open ball}}}}\mathcal R(z, B):=\sup_{\substack{x\in B \\ B\Subset \Omega \ \small{\mbox{open ball}}}} \frac{\mbox{highest eigenvalue of}\ \partial_{z}^{2} F(x,z)}{\mbox{lowest eigenvalue of}\  \partial_{z}^{2} F(x,z)} $$
may blow up as $|z|\to \infty$. 
We analyze the interaction between the space-depending coefficient of the integrand and the forcing term $f$ and derive optimal Lipschitz criteria for minimizers of (1). We catch the main model cases appearing in the literature, such as functionals with unbalanced power growth or with fast exponential growth such as
$$
w \mapsto \int_{\Omega} \gamma_1(x)\left[\exp(\exp(\dots \exp(\gamma_2(x)|Dw|^{p(x)})\ldots))-f\cdot w \right]\, \textrm{d}x
$$
or
$$
w\mapsto \int_{\Omega}\left[|Dw|^{p(x)}+a(x)|Dw|^{q(x)}-f\cdot w\right] \, \textrm{d}x.
$$
Finally, we find new borderline regularity results also in the uniformly elliptic case, i.e. when
$$\mathcal{R}(z,B)\sim \mbox{const}\quad \mbox{for all balls} \ \ B\Subset \Omega.$$

The talk is based on:
C. De Filippis, G. Mingione, Lipschitz bounds and non-autonomous functionals. $\textit{Preprint}$ (2020).

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