15:00
15:00
13:00
Topologically good cover from gradient descent
Note: we would recommend to join the meeting using the Teams client for best user experience.
Abstract
The cover of a dataset is a fundamental concept in computational geometry and topology. In TDA (topological data analysis), it is especially used in computing persistent homology and data visualisation using Mapper. However only rudimentary methods have been used to compute a cover. In this talk, we formulate the cover computation problem as a general optimisation problem with a well-defined loss function, and use gradient descent to solve it. The resulting algorithm, ShapeDiscover, substantially improves quality of topological inference and data visualisation. We also show some preliminary applications in scRNA-seq transcriptomics and the topology of grid cells in the rats' brain. This is a joint work with Luis Scoccola and Heather Harrington.
Block decompositions for p-adic groups
Abstract
I will give an overview on some aspects of the theory, with particular emphasis on decomposition of categories of smooth representations. I will also discuss passing from complex representations to other coefficient rings.
Mathematical modeling of some aspects of Age-related Macular Degeneration (AMD)
Abstract
Our visual perception of the world heavily relies on sophisticated and delicate biological mechanisms, and any disruption to these mechanisms negatively impacts our lives. Age-related macular degeneration (AMD) affects the central field of vision and has become increasingly common in our society, thereby generating a surge of academic and clinical interest. I will present some recent developments in the mathematical modeling of the retinal pigment epithelium (RPE) in the retina in the context of AMD; the RPE cell layer supports photoreceptor survival by providing nutrients and participating in the visual cycle and “cellular maintenance". Our objectives include modeling the aging and degeneration of the RPE with a mechanistic approach, as well as predicting the progression of atrophic lesions in the epithelial tissue. This is a joint work with the research team of Prof. M. Paques at Hôpital National des Quinze-Vingts.
17:00
Globally valued fields, adelic curves and Siu inequality
Abstract
In this talk, I will introduce the frameworks of globally valued fields (Ben Yaacov-Hrushovski) and adelic curves (Chen-Moriwaki). Both of these frameworks aim at understanding the arithmetic of fields sharing common features with global fields. A lot of examples fit in this scope (e.g. global fields, finitely generated extension of the prime fields, fields of meromorphic functions) and we will try to describe some of them.
Although globally valued fields and adelic curves came from different motivations and might seem quite different, they are related (and even essentially equivalent). This relation opens the door for new methods in the study of global arithmetic. As an application, we will sketch the proof of an arithmetic analogue of Siu inequality in algebraic geometry (a fundamental tool to detect the existence of global sections of line bundles in birational geometry). This is a joint work with Michał Szachniewicz.
16:00
Refined conjectures of ‘Birch—Swinnerton-Dyer type’ and the theory of Euler systems
Abstract
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.
14:00
Solving sparse linear systems using quantum computing algorithms
Abstract
The currently available quantum computers fall into the NISQ (Noisy Intermediate Scale Quantum) regime. These enable variational algorithms with a relatively small number of free parameters. We are now entering the FTQC (Fault Tolerant Quantum Computer) regime where gate fidelities are high enough that error-correction schemes are effective. The UK Quantum Missions include the target for a FTQC device that can perform a million operations by 2028, and a trillion operations by 2035.
This talk will present the outcomes from assessments of two quantum linear equation solvers for FTQCs– the Harrow–Hassidim–Lloyd (HHL) and the Quantum Singular Value Transform (QSVT) algorithms. These have used sample matrices from a Computational Fluid Dynamics (CFD) testcase. The quantum solvers have also been embedded with an outer non-linear solver to judge their impact on convergence. The analysis uses circuit emulation and is used to judge the FTQC requirements to deliver quantum utility.
13:30
Seiberg-Witten theory
Abstract
Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.
12:00
A modeling perspective on retinal degeneration
Abstract
This talk introduces an ongoing research project focused on building mechanistic models to study retinal degeneration, with a particular emphasis on the geometric aspects of the disease progression.
As we develop a computational model for retinal degeneration, we will explore how cellular materials behave and how wound-healing mechanisms influence disease progression. Finally, we’ll detail the numerical methods used to simulate these processes and explain how we work with medical data.
Ongoing research in collaboration with the group of M. Paques (Paris Eye Imaging - Quinze Vingts National Ophthalmology Hospital and Vision Institute).
Reducing acquisition time and radiation damage: data-driven subsampling for spectromicroscopy
Abstract
Spectro-microscopy is an experimental technique with great potential to science challenges such as the observation of changes over time in energy materials or environmental samples and investigations of the chemical state in biological samples. However, its application is often limited by factors like long acquisition times and radiation damage. We present two measurement strategies that significantly reduce experiment times and applied radiation doses. These strategies involve acquiring only a small subset of all possible measurements and then completing the full data matrix from the sampled measurements. The methods are data-driven, utilizing spectral and spatial importance subsampling distributions to select the most informative measurements. Specifically, we use data-driven leverage scores and adaptive randomized pivoting techniques. We explore raster importance sampling combined with the LoopASD completion algorithm, as well as CUR-based sampling where the CUR approximation also serves as the completion method. Additionally, we propose ideas to make the CUR-based approach adaptive. As a result, capturing as little as 4–6% of the measurements is sufficient to recover the same information as a conventional full scan.
Constitutive Modeling of the Microstructure of Arterial Walls Including Collagen Cross-Linking
Extended Bio
Gerhard A. Holzapfel is a world-leading figure in biomechanics, currently serving as Professor and Head of the Institute of Biomechanics at Graz University of Technology (TUG), Austria. He also holds appointments as Adjunct Professor at the Norwegian University of Science and Technology (NTNU) in Trondheim and Visiting Professor at the University of Glasgow. From 2004 to 2013, he was Professor of Biomechanics at the Royal Institute of Technology (KTH) in Stockholm.
Following a PhD in Mechanical Engineering from Graz, Professor Holzapfel was awarded an Erwin Schrödinger Scholarship, enabling him to conduct research at Stanford University. He achieved his Habilitation at TU Vienna in 1996 and was the recipient of Austria’s prestigious START Award in 1997. Over subsequent decades, he has led pioneering work in computational biomechanics, including as Head of the Computational Biomechanics research group at TUG (1998–2004).
Professor Holzapfel has received numerous accolades, including the Erwin Schrödinger Prize of the Austrian Academy of Sciences (2011), listings among “The World’s Most Influential Scientific Minds” (Thomson Reuters, 2014), the William Prager Medal and Warner T. Koiter Medal (2021), an honorary doctorate from École des Mines de Saint-Étienne (2024), and election to the U.S. National Academy of Engineering (2025). In 2024, he was awarded a prestigious Synergy Grant from the European Research Council (ERC).
His research spans experimental and computational biomechanics and mechanobiology, with a particular focus on soft biological tissues and the cardiovascular system in both health and disease. His expertise includes nonlinear continuum mechanics, constitutive modelling, growth and remodeling, imaging and image-based modeling, and the mechanics of therapeutic interventions such as angioplasty and stenting.
Professor Holzapfel is the author of the widely adopted graduate textbook Nonlinear Solid Mechanics (Wiley), has co-edited seven additional books, and contributed chapters to over 30 volumes. He has published more than 300 peer-reviewed journal articles. He is also the co-founder and co-editor of the journal Biomechanics and Modeling in Mechanobiology (Springer). His work has been funded by numerous national and international agencies, including the Austrian Science Fund, NIH, the European Commission, and industry collaborators.
Abstract
Nowadays, the 3D ultrastructure of a fibrous tissue can be reconstructed in order to visualize the complex nanoscale arrangement of collagen fibrils including neighboring proteoglycans even in the stretched loaded state [1]. In particular, experimental data of collagen fibers in human artery layers have shown that the f ibers are not symmetrically dispersed [2]. In addition, it is known that collagen f ibers are cross-linked and the density of cross-links in arterial tissues has a stiffening effect on the associated mechanical response. A first attempt to characterize this effect on the elastic response is presented and the influence of the cross-link density on the mechanical behavior in uniaxial tension is shown [3]. A recently developed extension of the model that accounts for dispersed fibers connected by randomly distributed cross-links is outlined [4]. A simple shear test focusing on the sign of the normal stress perpendicular to the shear planes (Poynting effect) is analyzed. In [5] it was experimentally observed that, in contrast to rubber, semi-flexible biopolymer gels show a tendency to approach the top and bottom faces under simple shear. This so-called negative Poynting effect and its connection with the cross-links as well as the fiber and crosslink dispersion is also examined.
References
[1]A. Pukaluk et al.: An ultrastructural 3D reconstruction method for observing the arrangement of collagen fibrils and proteoglycans in the human aortic wall under mechanical load. Acta Biomaterialia, 141:300-314, 2022.
[2] G.A. Holzapfel et al.: Modelling non-symmetric collagen fibre dispersion in arterial walls. Journal of the Royal Society Interface, 12:20150188, 2015.
[3] G.A. Holzapfel and R.W. Ogden: An arterial constitutive model accounting for collagen content and cross-linking. Journal of the Mechanics and Physics of Solids, 136:103682, 2020.
[4] S. Teichtmeister and G.A. Holzapfel: A constitutive model for fibrous tissues with cross-linked collagen fibers including dispersion – with an analysis of the Poynting effect. Journal of the Mechanics and Physics of Solids, 164:104911, 2022.
[5] P.A. Janmey et al.: Negative normal stress in semiflexible biopolymer gels. Nature Materials, 6:48–51, 2007.
Relativistically invariant wave equations in the realist theory
Abstract
17:00
Responsible modelling and the ethics of mathematics for decision support - Erica Thompson
Mathematical models are used to inform decisions across many sectors including climate change, finance, and epidemics. But models are not perfect representations of the real world – they are partial, uncertain and often biased. What, then, does responsible modelling look like? And how can we apply this ethical framework to new AI modelling methods?
Erica Thompson is Associate Professor of Modelling for Decision Making at UCL’s Department of Science, Technology, Engineering and Public Policy (STEaPP), and the author of 'Escape From Model Land' (2022).
Please email @email to register to attend in person.
The lecture will be broadcast on the Oxford Mathematics YouTube Channel on Wednesday 25 June at 5-6pm and any time after (no need to register for the online version).
The Oxford Mathematics Public Lectures are generously supported by XTX Markets.
16:00
Even the Loch Ness monster deserves a curve graph
Abstract
16:00
The Fourier coefficients of the holomorphic multiplicative chaos
Abstract
In this talk, we consider the coefficients of the Fourier series obtained by exponentiating a logarithmically correlated holomorphic function on the open unit disc, whose Taylor coefficients are independent complex Gaussian variables, the variance of the coefficient of degree k being theta/k where theta > 0 is an inverse temperature parameter. In joint articles with Paquette, Simm and Vu, we show a randomized version of the central limit theorem in the subcritical phase theta < 1, the random variance being related to the Gaussian multiplicative chaos on the unit circle. We also deduce, from results on the holomorphic multiplicative chaos, other results on the coefficients of the characteristic polynomial of the Circular Beta Ensemble, where the parameter beta is equal to 2/theta. In particular, we show that the central coefficient of the characteristic polynomial of the Circular Unitary Ensembles tends to zero in probability, answering a question asked in an article by Diaconis and Gamburd.
16:00
Dual properties for abelian group actions
Abstract
A landmark result in the study of locally compact, abelian groups is the Pontryagin duality. In simple terms, it says that for a given locally compact, abelian group G, one can uniquely associate another locally compact, abelian group called the Pontryagin dual of G. In the realm of C*-algebras, whenever such an abelian group G acts on a C*-algebra A, there is a canonical action of the dual group of G on the crossed product of A by G. In particular, it is natural to ask to what extent one can relate properties of the given G-action to those of the dual action.
In this talk, I will first introduce a property for actions of locally compact abelian groups called the abelian Rokhlin property and then state a duality type result for this property. While the abelian Rokhlin property is in general weaker than the known Rokhlin property, these two properties coincide in the case of the acting group being the real numbers. Using the duality result mentioned above, I will give new examples of continuous actions of the real numbers which satisfy the Rokhlin property. Part of this talk is based on joint work with Johannes Christensen and Gábor Szabó.
15:30
Bordism categories and orientations of moduli spaces
Abstract
15:00
Perturbing circle invariant solution to LMCF in the Eguchi-Hanson space to solutions in the Kummer K3 surface
Abstract
In this talk I will describe a possible strategy to obtain new solution to LMCF in the Kummer K3 surface by a fixed point argument. The key idea is that the regions where curvature concentrates in the Kummer K3 surface are modeled on the Eguchi-Hanson space.
15:00
Proper versus trivial actions on Lp-spaces
Abstract
Property (T) (respectively aTmenability) is equivalent to admitting only a trivial action (respectively, a proper action) on a median space, and is also equivalent to admitting only a trivial action (respectively, a proper action) on a Hilbert space (so some L2). For p>2 I will investigate an analogous equivalent characterisation.
A new lower bound for the Ramsey numbers $R(3,k)$
Abstract
In this talk I will discuss a new lower bound for the off-diagonal Ramsey numbers $R(3,k)$. For this, we develop a version of the triangle-free process that is significantly easier to analyse than the original process. We then 'seed' this process with a carefully chosen graph and show that it results in a denser graph that is still sufficiently pseudo-random to have small independence number.
This is joint work with Marcelo Campos, Matthew Jenssen and Marcus Michelen.
14:00
A geometric approach to Nichols algebras and their approximations
Abstract
Nichols algebras, also known as small shuffle algebras, are a family of graded bialgebras including the symmetric algebras, the exterior algebras, the positive parts of quantized enveloping algebras, and, conjecturally, Fomin-Kirillov algebras. As the case of Fomin-Kirillov algebra shows, it can be very
difficult to determine the maximum degree of a minimal generating set of relations of a Nichols algebra.
Building upon Kapranov and Schechtman’s equivalence between the category of perverse sheaves on Sym(C) and the category of graded connected bialgebras, we describe the geometric counterpart of the maximum degree of a generating set of relations of a graded connected bialgebra, and we show how this specialises to the case o Nichols algebras.
The talk is based on joint work with Francesco Esposito and Lleonard Rubio y Degrassi.
13:00
Finite-temperature quantum topological order in three dimensions
Abstract
We identify a three-dimensional system that exhibits long-range entanglement at sufficiently small but nonzero temperature--it therefore constitutes a quantum topological order at finite temperature. The model of interest is known as the fermionic toric code, a variant of the usual 3D toric code, which admits emergent fermionic point-like excitations. The fermionic toric code, importantly, possesses an anomalous 2-form symmetry, associated with the space-like Wilson loops of the fermionic excitations. We argue that it is this symmetry that imbues low-temperature thermal states with a novel topological order and long-range entanglement. Based on the current classification of three-dimensional topological orders, we expect that the low-temperature thermal states of the fermionic toric code belong to an equilibrium phase of matter that only exists at nonzero temperatures. We conjecture that further examples of topological orders at nonzero temperatures are given by discrete gauge theories with anomalous 2-form symmetries. Our work therefore opens the door to studying quantum topological order at nonzero temperature in physically realistic dimensions.
12:30
On the Limits of PAC Learning Opinion Dynamics
Abstract
Agents in social networks with threshold-based dynamics change opinions when influenced by sufficiently many peers. Existing literature typically assumes that the network structure and dynamics are fully known, which is often unrealistic. In this work, we ask how to learn a network structure from samples of the agents' synchronous opinion updates. Firstly, if the opinion dynamics follow a threshold rule where a fixed number of influencers prevent opinion change (e.g., unanimity and quasi-unanimity), we give an efficient PAC learning algorithm provided that the number of influencers per agent is bounded. Secondly, under standard computational complexity assumptions, we prove that if the opinion of agents follows the majority of their influencers, then there is no efficient PAC learning algorithm. We propose a polynomial-time heuristic that successfully learns consistent networks in over 97% of our simulations on random graphs, with no failures for some specified conditions on the numbers of agents and opinion diffusion examples.
12:00