University College London

In the 1980s, Mazur and Tate proposed refinements of the Birch–Swinnerton-Dyer conjecture that also capture congruences between twists of Hasse–Weil L-series by Dirichlet characters. In this talk, I will report on new results towards these refined conjectures, obtained in joint work with Matthew Honnor. I will also outline how the results fit into a more general approach to refined conjectures on special values of L-series via an enhanced theory of Euler systems. This final part will touch upon joint work with David Burns.

Given a Hamiltonian circle action on a symplectic manifold, Fukaya and Teleman tell us that we can relate the equivariant Fukaya category to the Fukaya category of a symplectic reduction.  Yanki Lekili and I have some conjectures that extend this story - in certain special examples - to singular values of the moment map. I'll also explain the mirror symmetry picture that we use to support our conjectures, and how we interpret our claims in Teleman's framework of `topological group actions' on categories.

Topically, my goal is to provide a fun and instructive introduction to graph cumulants: a hierarchical set of subgraph statistics that extend the classical cumulants (mean, (co)variance, skew, kurtosis, etc) to relational data.  

Intuitively, graph cumulants quantify the propensity (if positive) or aversion (if negative) for the appearance of any particular subgraph in a larger network.  

Concretely, they are derived from the “bare” subgraph densities via a Möbius inversion over the poset of edge partitions.  

Practically, they offer a systematic way to measure similarity between graph distributions, with a notable increase in statistical power compared to subgraph densities.  

Algebraically, they share the defining properties of cumulants, providing clever shortcuts for certain computations.  

Generally, their definition extends naturally to networks with additional features, such as edge weights, directed edges, and node attributes.  

Finally, I will discuss how this entire procedure of “cumulantification” suggests a promising framework for a motif-centric statistical analysis of general structured data, including temporal and higher-order networks, leaving ample room for exploration. 

Recent years have witnessed a surge of interest in deep learning methods to tackle inverse problems arising in various domains such as medical imaging, remote sensing, and the arts and humanities. This talk offers an overview of recent advances in the foundations and applications of deep learning for inverse problems, with a focus on model-based deep learning methods. Concretely, this talk will overview our work relating to theoretical advances in the area of mode-based learning, including learning guarantees; algorithmic advances in model-based learning; and, finally it will showcase a portfolio of emerging signal & image processing challenges that benefit from model based learning, including image separation / deconvolution challenges arising in the arts and humanities.

Bio:

Miguel Rodrigues is a Professor of Information Theory and Processing at University College London; he leads the Information, Inference and Machine Learning Lab at UCL, and he has also been the founder and director of the master programme in Integrated Machine Learning Systems at UCL. He has also been the UCL Turing University Lead and a Turing Fellow with the Alan Turing Institute — the UK National Institute of Data Science and Artificial Intelligence.

He held various appointments with various institutions worldwide including Cambridge University, Princeton University, Duke University, and the University of Porto, Portugal. He obtained the undergraduate degree in Electrical and Computer Engineering from the Faculty of Engineering of the University of Porto, Portugal and the PhD degree in Electronic and Electrical Engineering from University College London.

Dr. Rodrigues's research lies in the general areas of information theory, information processing, and machine learning. His most relevant contributions have ranged from the information-theoretic analysis and design of communications systems, information-theoretic security, information-theoretic analysis and design of sensing systems, and the information-theoretic foundations of machine learning.

He serves or has served as Editor of IEEE BITS, Editor of the IEEE Transactions on Information Theory, and Lead Guest Editor of various Special Issues of the IEEE Journal on Selected Topics in Signal Processing, Information and Inference, and Foundations and Trends in Signal Processing.

Dr. Rodrigues has been the recipient of various prizes and awards including the Prize for Merit from the University of Porto, the Prize Engenheiro Cristian Spratley, the Prize Engenheiro Antonio de Almeida, fellowships from the Portuguese Foundation for Science and Technology, and fellowships from the Foundation Calouste Gulbenkian. Dr. Rodrigues research on information-theoretic security has also attracted the IEEE Communications and Information Theory Societies Joint Paper Award 2011.  

He has also been elevated to Fellow of the Institute of Electronics and Electrical Engineers (IEEE) for his contributions to the ‘multi-modal data processing and reliable and secure communications.’

It is a long-standing open problem to generalize sheaf-counting invariants of complex projective three-folds to symplectic manifolds of real dimension six. One approach to this problem involves counting  J-holomorphic curves  C, for a generic almost complex structure J, with weights depending on J. Various existing symplectic invariants (Gromov-Witten, Gopakumar-Vafa, Bai-Swaminathan) can be expressed as such weighted counts. In this talk, based on joint work with Thomas Walpuski, I will discuss a new construction of weights associated with curves and a closely related problem about the structure of the space of Cauchy-Riemann operators on  C.

I will discuss how some ideas from Geometric Langlands can be used to obtain new results in birational geometry and on the topology of algebraic varieties.

Following the 2012 breakthrough in deep learning for classification and visions problems, the last decade has seen tremendous raise of interest in machine learning in a wider mathematical research community from foundational research through field specific analysis to applications. 

As data is at the core of any inverse problem, it was a natural direction for the field to investigate how machine learning could aid various aspects of inversion yielding numerous approaches from somewhat ad-hoc but very effective like learned unrolled methods to provably convergent learned regularisers with everything in between. In this talk I will review some on these developments through a lens of the research of our group.   

This talk will introduce Khovanov and Knot Floer Homology as tools for studying knots. I will then cover some applications to problems in knot theory including distinguishing embedded surfaces and how they can be used in the context of ribbon concordances. No prior knowledge of either will be necessary and lots of pictures are included.

The feedback loop between simulations and observations is the driving force behind almost all discoveries in astronomy. However, as technological innovations allow us to create ever more complex simulations and make ever more detailed observations, it becomes increasingly difficult to combine the two: since we cannot do controlled experiments, we need to simulate whatever we can observe. This requires efficient simulation pipelines, including (general-relativistic-)(magneto-)hydrodynamics, particle physics, chemistry, and radiation transport. In this talk, we explore the challenges associated with these modelling efforts and discuss how adopting data-driven surrogate modelling and proper control over model uncertainties, promises to unlock a gold mine of future discoveries. For instance, the application to stellar wind simulations can teach us about the origin of chemistry in our Universe and the building blocks for life, while supernova simulations can reveal exotic states of matter and elucidate the formation black holes.