Algebraic fibring is the group-theoretic analogue of fibration over the circle for manifolds. Generalising the work of Agol on hyperbolic 3-manifolds, Kielak showed that many groups virtually fibre. In this talk we will discuss the geometry of groups which fibre, with some fun applications to Poincare duality groups - groups whose homology and cohomology invariants satisfy a Poincare-Lefschetz type duality, like those of manifolds - as well as to exotic subgroups of Gromov hyperbolic groups. No prior knowledge of these topics will be assumed.

Disclaimer: This talk will contain many manifolds.

Kaplansky's Zerodivisor Conjecture predicts that the group algebra kG is a domain, where k is a field and G is a torsion-free group. Though the general sentiment is that the conjecture is false, it still remains wide open after more than 70 years. In this talk we will survey known positive results surrounding the Zerodivisor Conjecture, with a focus on the technique of embedding group algebras into division rings. We will also present some new results in this direction, which are joint with Pablo Sánchez Peralta.

Speaker: Ximena Laura Fernandez
Title: Let it Be(tti): Topological Fingerprints for Audio Identification

Abstract: Ever wondered how music recognition apps like Shazam work or why they sometimes fail? Can Algebraic Topology improve current audio identification algorithms? In this talk, I will discuss recent collaborative work with Spotify, where we extract low-dimensional homological features from audio signals for efficient song identification despite continuous obfuscations. Our approach significantly improves accuracy and reliability in matching audio content under topological distortions, including pitch and tempo shifts, compared to Shazam.

Talk based on the work: https://arxiv.org/pdf/2309.03516.pdf
 

Speaker: Brett Kolesnik
Title: Coxeter Tournaments

Abstract: We will present ongoing joint work with three Oxford PhD students: Matthew Buckland (Stats), Rivka Mitchell (Math/Stats) and Tomasz Przybyłowski (Math). We met last year as part of the course SC9 Probability on Graphs and Lattices. Connections with geometry (the permutahedron and generalizations), combinatorics (tournaments and signed graphs), statistics (paired comparisons and sampling) and probability (coupling and rapid mixing) will be discussed.

A previously known function-theoretic characterisation of compactness for a weighted composition operator on BMOA is improved. Moreover, the same function-theoretic condition also characterises weak compactness and complete continuity. In order to close the circle of implications, the operator-theoretic property of fixing a copy of c0 comes in useful. 

Artificial Intelligence (AI) will strongly determine our future prosperity and well-being. Due to its generic nature, AI will have an impact on all sciences and business sectors, our private lives and society as a whole. AI is pre-eminently a multidisciplinary technology that connects scientists from a wide variety of research areas, from behavioural science and ethics to mathematics and computer science.

Without downplaying the importance of that variety, it is apparent that mathematics can and should play an active role. All the more so as, alongside the successes of AI, also critical voices are increasingly heard. As Robert Dijkgraaf (former director of the Princeton Institute of Advanced Studies) observed in May 2019: ”Artificial intelligence is in its adolescent phase, characterised by trial and error, self-aggrandisement, credulity and lack of systematic understanding.” Mathematics can contribute to the much-needed systematic understanding of AI, for example, greatly improving reliability and robustness of AI algorithms, understanding the operation and sensitivity of networks, reducing the need for abundant data sets, or incorporating physical properties into neural networks needed for super-fast and accurate simulations in the context of digital twinning.

Mathematicians absolutely recognize the potential of artificial intelligence, machine learning and (deep) neural networks for future developments in science, technology and industry. At the same time, a sound mathematical treatment is essential for all aspects of artificial intelligence, including imaging, speech recognition, analysis of texts or autonomous driving, implying it is essential to involve mathematicians in all these areas. In this presentation, we highlight the role of mathematics as a key enabling technology within the emerging field of scientific machine learning. Or, as I always say: ”Real intelligence is needed to make artificial intelligence work.”

We study the distribution of eigenvalues of kernel random matrices where each element is the empirical covariance between the feature map evaluations of a random fully-connected neural network. We show that, under mild assumptions on the non-linear activation function, namely Lipschitz continuity and measurability, the limiting spectral distribution can be written as successive free multiplicative convolutions between the Marchenko-Pastur law and a nonrandom measure specific to the neural network. The latter has no known analytical expression but can be simulated empirically, separately from the random matrices of interest.

Complex dynamics underlying industrial manufacturing depend in part on multiphase multiphysics, in which fluids and materials interact across orders of magnitude variations in time and space. In this talk, we will discuss the development and application of a host of numerical methods for these problems, including Level Set Methods, Voronoi Implicit Interface Methods, implicit adaptive representations, and multiphase discontinuous Galerkin Methods.  Applications for industrial problems will include modeling how foams evolve, how electro-fluid jetting devices work, and the physics and dynamics of rotary bell spray painting across the automotive industry.

In this talk, we discuss two-dimensional Riemann problems in the framework of potential flow
equation and isentropic Euler system. We first review recent results on the existence, regularity and properties of
global self-similar solutions involving transonic shocks for several 2D Riemann problems in the
framework of potential flow equation. Examples include regular shock reflection, Prandtl reflection, and four-shocks
Riemann problem. The approach is to reduce the problem to a free boundary problem for a nonlinear elliptic equation
in self-similar coordinates. A well-known open problem is to extend these results to a compressible Euler system,
i.e. to understand the effects of vorticity. We show that for the isentropic Euler system, solutions have
low regularity, specifically velocity and density do not belong to the Sobolev space $H^1$ in self-similar coordinates.  
We further discuss the well-posedness of the transport equation for vorticity in the resulting low regularity setting.

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