Tue, 21 Jan 2025
16:00
C3

Quantum symmetries on Kirchberg algebras

Kan Kitamura
(Riken iThems)
Abstract

In subfactor theory, it has been observed that operator algebras often admit symmetries beyond mere groups, sometimes called quantum symmetries. Besides recent substantial progress on the classification programs of simple amenable C*-algebras and group actions on them, there has been increasing interest in their quantum symmetries. This talk is devoted to an attempt to ensure the existence of various quantum symmetries on simple amenable C*-algebras, at least in the purely infinite case, by providing a systematic way to produce them. As a technical ingredient, a simplicity criterion for certain Pimsner algebras is given.

Tue, 21 Jan 2025
15:30
L4

Deformations and lifts of Calabi-Yau varieties in characteristic p

Lukas Brantner
(Oxford)
Abstract

Derived algebraic geometry allows us to study formal moduli problems via their tangent Lie algebras. After briefly reviewing this general paradigm, I will explain how it sheds light on deformations of Calabi-Yau varieties. 
In joint work with Taelman, we prove a mixed characteristic analogue of the Bogomolov–Tian–Todorov theorem, which asserts that Calabi-Yau varieties in characteristic $0$ are unobstructed. Moreover, we show that ordinary Calabi–Yau varieties in characteristic $p$ admit canonical (and algebraisable) lifts to characteristic $0$, generalising results of Serre-Tate for abelian varieties and Deligne-Nygaard for K3 surfaces. 
If time permits, I will conclude by discussing some intriguing questions related to our canonical lifts.  
 

Tue, 21 Jan 2025
15:00
L6

Counting non-simple closed geodesics on random hyperbolic surfaces

Laura Monk
Abstract
The aim of this talk is to present new results related to the length spectrum of random hyperbolic surfaces. The Weil-Petersson model is a beautiful probabilistic model that was popularised by Mirzakhani to study random hyperbolic surfaces. In this continuous model, it is easy to argue that there exists a density function V_g(l) which "counts" how many closed geodesics of length l an average surface of genus g contains. In the case where we only count simple geodesics (with no self-intersections), Mirzakhani proved explicit formulas for this density, writing it as a polynomial function that can be interpreted in terms of volumes of moduli spaces. I will present joint work with Nalini Anantharaman where we obtain new explicit formulas for any fixed topology. Notably, I will present new coordinate systems on Teichmüller spaces in which the Weil-Petersson volume has a surprisingly simple expression.
 
Though purely geometric, those results were obtained in a project related to the spectral gap of the Laplacian. I will present applications of the techniques presented in this talk to this problem at the RMT seminar. Both talks will be disjoint and independent, with the intention that they can be viewed either separately or together.
Tue, 21 Jan 2025

14:00 - 15:00
L4

On inapproximability of hypergraph colourings and beyond

Standa Živný
(University of Oxford)
Abstract

I'll discuss how a certain notion of symmetry captures the computational complexity of approximating homomorphism problems between relational structures, also known as constraint satisfaction problems. I'll present recent results on inapproximability of conflict-free and linearly-ordered hypergraph colourings and solvability of systems of equations.

Tue, 21 Jan 2025

14:00 - 15:00
L6

Proof of the Deligne—Milnor conjecture

Dario Beraldo
(UCL)
Abstract

Let X --> S be a family of algebraic varieties parametrized by an infinitesimal disk S, possibly of mixed characteristic. The Bloch conductor conjecture expresses the difference of the Euler characteristics of the special and generic fibers in algebraic and arithmetic terms. I'll describe a proof of some new cases of this conjecture, including the case of isolated singularities. The latter was a conjecture of Deligne generalizing Milnor's formula on vanishing cycles. 

This is joint work with Massimo Pippi; our methods use derived and non-commutative algebraic geometry. 

Tue, 21 Jan 2025
13:00
L5

Celestial Holography and Self-Dual Einstein Gravity

David Skinner
Abstract

Celestial Holography posits the existence of a holographic description of gravitational theories in asymptotically flat space-times. To date, top-down constructions of such dualities involve a combination of twisted holography and twistor theory. The gravitational theory is the closed string B model living in a suitable twistor space, while the dual is a chiral 2d gauge theory living on a stack of D1 branes wrapping a twistor line. I’ll talk about a variant of these models that yields a theory of self-dual Einstein gravity (via the Plebanski equations) in four dimensions. This is based on work in progress with Roland Bittleston, Kevin Costello & Atul Sharma.

Mon, 20 Jan 2025
16:30
L4

Fluctuations around the mean-field limit for attractive Riesz interaction kernels in the moderate regime

Alexandra Holzinger
(Mathematical Institute)
Abstract

In this talk I will give a short introduction to moderately interacting particle systems and the general notion of fluctuations around the mean-field limit. We will see how a central limit theorem can be shown for moderately interacting particles on the whole space for certain types of interaction potentials. The interaction potential approximates singular attractive potentials of sub-Coulomb type and we can show that the fluctuations become asymptotically Gaussians. The methodology is inspired by the classical work of Oelschläger in the 1980s on fluctuations for the porous-medium equation. To allow for attractive potentials we use a new approach of quantitative mean-field convergence in probability in order to include aggregation effects. 

Mon, 20 Jan 2025
15:30
L5

The Farrell--Jones Conjecture and automorphisms of relatively hyperbolic groups

Naomi Andrew
(Oxford University)
Abstract

The Farrell--Jones conjecture predicts that the algebraic K-theory of a group ring is isomorphic to a certain equivariant homology theory, and there are also versions for L-theory and Waldhausen's A-theory. In principle, this provides a way to calculate these K-groups, and has many applications. These include classifying manifolds admitting a given fundamental group and a positive resolution of the Borel conjecture.

I will discuss work with Yassine Guerch and Sam Hughes on the Farrell--Jones conjecture for extensions of relatively hyperbolic groups, as well as an application to their automorphism groups in the one-ended case. The methods are from geometric group theory: we go via the theory of JSJ decompositions to produce acylindrical actions on trees.

Mon, 20 Jan 2025
15:30
L3

Heat kernel for critical percolation clusters on the binary tree.

Prof Martin T Barlow
(University of British Columbia )
Abstract
Kesten defined the incipient infinite cluster (IIC) as the limit of large critical finite percolation clusters. We look at the (quenched) heat kernel on the IIC, and will see how it fluctuates due to the randomness of the cluster. 
 
This is a joint work with David Croydon and Takashi Kumagai. 
Mon, 20 Jan 2025
14:15
L5

Yang-Mills on an ALF-fibration

Jakob Stein
(UNICAMP)
Abstract

In this talk, we will make an explicit link between self-dual Yang-Mills instantons on the Taub-NUT space, and G2-instantons on the BGGG space, by displaying the latter space as a fibration by the former. In doing so, we will discuss analysis on non-compact manifolds, circle symmetries, and a new method of constructing solutions to quadratically singular ODE systems. This talk is based on joint work with Matt Turner: https://arxiv.org/pdf/2409.03886

Mon, 20 Jan 2025

13:00 - 14:00
L6

Symmetry Enhancement, SPT Absorption, and Duality in QED_3

Andrea Antinucci
Abstract

Abelian gauge theories in 2+1 dimensions are very interesting QFTs: they are strongly coupled and exhibit non-trivial dynamics. However, they are somewhat more tractable than non-Abelian theories in 3+1 dimensions. In this talk, I will first review the known properties of fermions in 2+1 dimensions and some conjectures about QED_3 with a single Dirac fermion. I will then present the recent proposal from [arXiv:2409.17913] regarding the phase diagram of QED_3 with two fermions. The findings reveal surprising (yet compelling) features: while semiclassical analysis would suggest two trivially gapped phases and a single phase transition, the actual dynamics indicate the presence of two distinct phase transitions separated by a "quantum phase." This intermediate phase exists over a finite range of parameters in the strong coupling regime and is not visible semiclassically. Moreover, these phase transitions are second-order and exhibit symmetry enhancement. The proposal is supported by several non-trivial checks and is consistent with results from numerical bootstrap, lattice simulations, and extrapolations from the large-Nf expansion.

Fri, 17 Jan 2025

11:00 - 12:00
L3

Do individuals matter? - From psychology, via wound healing and calcium signalling to ecology

Dr Ivo Siekmann
(School of Computer Science and Mathematics, Liverpool University)
Abstract
Should models in mathematical biology be based on detailed representations of individuals - biomolecules, cells, individual members of a population or agents in a social system? Or, alternatively, should individuals be described as identical members of a population, neglecting inter-individual differences? I will explore this question using recent examples from my own research.
 
In the beginning of my presentation I will ask you how you are feeling. Evaluating your answers, I will show how differences in personality can be represented in a model based on differential equations. I will then present an individual-based cell migration model based on the Ornstein-Uhlenbeck process that can help to design textured surfaces that enhance wound healing. In ecosystems, organisms that make decisions based on studying their environment such as fish might interact with populations that are unable of complex behaviour such as plankton. I will explain how piecewise-deterministic Markov (PDMP) models can be used for representing some populations as individuals and others as populations. PDMPs can also be used for modelling how interacting calcium channels generate calcium signals in cells. Finally, I will present a reaction-diffusion model of the photosynthetic activity of phytoplankton that explains how oxygen minimum zones emerge in the ocean.
Thu, 19 Dec 2024
16:00
L5

Geodesic cycles and Eisenstein classes for SL(2,Z)

Hohto Bekki
(MPIM Bonn)
Abstract

The geodesic cycles (resp. Eisenstein classes) for SL(2,Z) are special classes in the homology (resp. cohomology) of modular curve (for SL(2,Z)) defined by the closed geodesics (resp. Eisenstein series). It is known that the pairing between these geodesic cycles and Eisenstein classes gives the special values of partial zeta functions of real quadratic fields, and this has many applications. In this talk, I would like to report on some recent observations on the size of the homology subgroup generated by geodesic cycles and their applications. This is a joint work with Ryotaro Sakamoto.

Fri, 13 Dec 2024
12:00
L4

Asymptotic Higher Spin Symmetries in Gravity.

Nicolas Cresto
(Perimeter Institute)
Abstract

 I will first give a short review of the concepts of Asymptotically Flat Spacetimes, IR triangle and Noether's theorems. I will then present what Asymptotic Higher Spin Symmetries are and how they were introduced as a candidate for an approximate symmetry of General Relativity and the S-matrix. Next, I'll move on to the recent developments of establishing these symmetries as Noether symmetries and describing how they are canonically and non-linearly realized on the asymptotic gravitational phase space. I will discuss how the introduction of dual equations of motion encapsulates the non-perturbativity of the analysis. Finally I'll emphasize the relation to twistor, especially with 2407.04028. Based on 2409.12178 and 2410.15219

Thu, 12 Dec 2024
14:00
(This talk is hosted by Rutherford Appleton Laboratory)

A Subspace-conjugate Gradient Method for Linear Matrix Equations

Davide Palitta
(Università di Bologna)
Abstract

 
The solution of multiterm linear matrix equations is still a very challenging task in numerical linear algebra.
If many different solid methods for equations with (at most) two terms exist in the literature, having a number of terms greater than two makes the numerical treatment of these equations much trickier. Very few options are available in the literature. In particular, to the best of our knowledge, no decomposition-based method for multiterm equations has never been proposed; only iterative procedures exist.
A non-complete list of contributions in this direction includes a greedy procedure designed by Sirkovi\'c and Kressner, projection methods tailored to the equation at hand, Riemannian optimization schemes, and matrix-oriented Krylov methods with low-rank truncations. The last class of solvers is probably one of the most commonly used ones. These schemes amount to adapting standard Krylov schemes for linear systems to matrix equations by leveraging the equivalence between matrix equations and their Kronecker form.
As long as no truncations are performed, this equivalence means that the algorithm itself is not exploiting the structure of the problem as it is not able to see that we are actually solving a matrix equation and not a linear system. The low-rank truncations we may perform in the matrix-equation framework can be viewed as a simple computational tool needed to make the solution process affordable in terms of storage allocation and not as an algorithmic advance.

 
By taking inspiration from the matrix-oriented cg method, our goal is to design a novel iterative scheme for the solution of multiterm matrix equations. We name this procedure the subspace-conjugate gradient method (Ss-cg) due to the peculiar orthogonality conditions imposed to compute some of the quantities involved in the scheme. As we will show in this talk, the main difference between Ss-cg and cg is the ability of the former to capitalize on the matrix equation structure of the underlying problem. In particular, we will make full use of the (low-rank) matrix format of the iterates to define appropriate ``step-sizes''. If these quantities correspond to scalars alpha_k and beta_k in cg, they will amount to small-dimensional matrices in our fresh approach.
This novel point of view leads to remarkable computational gains making Ss-cg a very competitive option for the solution of multi-term linear matrix equations.

 
This is a joint work with Martina Iannacito and Valeria Simoncini, both from the Department of Mathematics, University of Bologna.
 
Mon, 09 Dec 2024
15:30
L4

Unstable cohomology of SL(n,Z) and Hopf algebras

Peter Patzt
(University of Oklahoma)
Abstract

I want to give a survey about the rational cohomology of SL_n
Z. This includes recent developments of finding Hopf algebras in the
direct sum of all cohomology groups of SL_n Z for all n. I will give a
quick overview about Hopf algebras and what this structure implies for
the cohomology of SL_n Z.

Fri, 06 Dec 2024
16:00
L1

Fridays@4 – A start-up company? 10 things I wish I had known

Professor Peter Grindrod
(Mathematical Institute (University of Oxford))
Abstract

Are you thinking of launching your own start-up or considering joining an early-stage company? Navigating the entrepreneurial landscape can be both exciting and challenging. Join Pete for an interactive exploration of the unwritten rules and hidden insights that can make or break a start-up journey.

Drawing from personal experience, Pete's talk will offer practical wisdom for aspiring founders and team members, revealing the challenges and opportunities of building a new business from the ground up.

Whether you're an aspiring entrepreneur, a potential start-up team member, or simply curious about innovative businesses, you'll gain valuable perspectives on the realities of creating something from scratch.

This isn't a traditional lecture – it will be a lively conversation that invites participants to learn, share, and reflect on the world of start-ups. Come prepared to challenge your assumptions and discover practical insights that aren't found in standard business guides.
 

A Start-Up Company? Ten Things I Wish I Had Known


Speaker: Professor Pete Grindrod

Fri, 06 Dec 2024
15:00
L5

From single neurons to complex human networks using algebraic topology

Lida Kanari
(École Polytechnique Fédérale de Lausanne (EPFL))

Note: we would recommend to join the meeting using the Teams client for best user experience.

Abstract

Topological data analysis, and in particular persistent homology, has provided robust results for numerous applications, such as protein structure, cancer detection, and material science. In the field of neuroscience, the applications of TDA are abundant, ranging from the analysis of single cells to the analysis of neuronal networks. The topological representation of branching trees has been successfully used for a variety of classification and clustering problems of neurons and microglia, demonstrating a successful path of applications that go from the space of trees to the space of barcodes. In this talk, I will present some recent results on topological representation of brain cells, with a focus on neurons. I will also describe our solution for solving the inverse TDA problem on neurons: how can we efficiently go from persistence barcodes back to the space of neuronal trees and what can we learn in the process about these spaces. Finally, I will demonstrate how algebraic topology can be used to understand the links between single neurons and networks and start understanding the brain differences between species. The organizational principles that distinguish the human brain from other species have been a long-standing enigma in neuroscience. Human pyramidal cells form highly complex networks, demonstrated by the increased number and simplex dimension compared to mice. This is unexpected because human pyramidal cells are much sparser in the cortex. The number and size of neurons fail to account for this increased network complexity, suggesting that another morphological property is a key determinant of network connectivity. By comparing the topology of dendrites, I will show that human pyramidal cells have much higher perisomatic (basal and oblique) branching density. Therefore greater dendritic complexity, a defining attribute of human L2 and 3 neurons, may provide the human cortex with enhanced computational capacity and cognitive flexibility.

Fri, 06 Dec 2024

14:00 - 15:00
Mezzanine

End-of-term mathematical board games

Abstract

Would you like to meet some of your fellow students, and some graduate students and postdocs, in an informal and relaxed atmosphere, while building your communication skills? In this Friday@2 session, you'll be able to play a selection of board games, meet new people, and practise working together. What better way to spend the final Friday afternoon of term?! We'll play the games in the south Mezzanine area of the Andrew Wiles Building.

Fri, 06 Dec 2024

12:00 - 13:00
Quillen Room

Some Uniserial Specht Modules

Zain Kapadia
(Queen Mary University of London)
Abstract
The Representation Theory of the Symmetric Groups is a classical and rich area of combinatorial representation theory. Key objects of study include Specht modules, the irreducible ordinary representations, which can be reduced modulo p (for p prime). In general, these are no longer irreducible and finding their decomposition numbers and submodule structures are key questions in the area. We give sufficient and necessary conditions for a Specht module in characteristic 2, labelled by a hook partition to be a direct sum of uniserial summands.


 

Fri, 06 Dec 2024
12:00
L2

Combinatorial proof of a Non-Renormalization theorem

Paul-Hermann Balduf
(Oxford)
Abstract

In "Higher Operations in Perturbation Theory", Gaiotto, Kulp, and Wu discussed Feynman integrals that control certain deformations in quantum field theory. The corresponding integrands are differential forms in Schwinger parameters. Specifically, the integrand $\alpha$ is associated to a single topological direction of the theory.
I will show how the combinatorial properties of graph polynomials lead to a relatively simple, explicit formula for $\alpha$, that can be evaluated quickly with a computer. This is interesting for two reasons. Firstly, knowing the explicit formula leads to an elementary proof of the fact that $\alpha$ squares to zero, which asserts the absence of quantum corrections in topological field theories of two (or more) dimensions, known as Kontsevich's formality theorem. Secondly, the underlying constructions and proofs are not intrinsically limited to topological theories. In this sense, they serve as a particularly instructive example for simplifications that can occur in Feynman integrals with numerators.

Fri, 06 Dec 2024

11:00 - 12:00
L5

Spatial mechano-transcriptomics of mouse embryogenesis

Prof Adrien Hallou
(Dept of Physics University of Oxford)
Abstract

Advances in spatial profiling technologies are providing insights into how molecular programs are influenced by local signalling and environmental cues. However, cell fate specification and tissue patterning involve the interplay of biochemical and mechanical feedback. Here, we propose a new computational framework that enables the joint statistical analysis of transcriptional and mechanical signals in the context of spatial transcriptomics. To illustrate the application and utility of the approach, we use spatial transcriptomics data from the developing mouse embryo to infer the forces acting on individual cells, and use these results to identify mechanical, morphometric, and gene expression signatures that are predictive of tissue compartment boundaries. In addition, we use geoadditive structural equation modelling to identify gene modules that predict the mechanical behaviour of cells in an unbiased manner. This computational framework is easily generalized to other spatial profiling contexts, providing a generic scheme for exploring the interplay of biomolecular and mechanical cues in tissues.

Thu, 05 Dec 2024
17:00

Model-theoretic havens for extremal and additive combinatorics

Mervyn Tong
(Leeds University)
Abstract

Model-theoretic dividing lines have long been a source of tameness for various areas of mathematics, with combinatorics jumping on the bandwagon over the last decade or so. Szemerédi’s regularity lemma saw improvements in the realm of NIP, which were further refined in the subrealms of stability and distality. We show how relations satisfying the distal regularity lemma enjoy improved bounds for Zarankiewicz’s problem. We then pivot to arithmetic regularity lemmas as pioneered by Green, for which NIP and stability also imply improvements. Unsettled by the absence of distality in this picture, we discuss the role of distality in additive combinatorics, appealing to our result connecting distality with arithmetic tameness.

Thu, 05 Dec 2024

16:00 - 17:00
Virtual

Transportation market rate forecast using signature transform

Dr Xin Guo
Further Information
Abstract

Freight transportation marketplace rates are typically challenging to forecast accurately. In this talk, I will present a novel statistical technique based on signature transforms and  a predictive and adaptive model to forecast these marketplace rates. Our technique is based on two key elements of the signature transform: one being its universal nonlinearity property, which linearizes the feature space and hence translates the forecasting problem into linear regression, and the other being the signature kernel, which allows for comparing computationally efficiently similarities between time series data. Combined, it allows for efficient feature generation and precise identification of seasonality and regime switching in the forecasting process. 

An algorithm based on our technique has been deployed by Amazon trucking operations, with far superior forecast accuracy and better interpretability versus commercially available industry models, even during the COVID-19 pandemic and the Ukraine conflict. Furthermore, our technique is in production in Amazon and has been adopted for Amazon finance planning,  with an estimated annualized saving of $50MM in the transportation sector alone. 

Thu, 05 Dec 2024
16:00
L4

Mean Field Games in a Stackelberg problem with an informed major player

Dr Philippe Bergault
(Université Paris Dauphine-PSL)
Further Information

Please join us for refreshments outside the lecture room from 15:30.

Abstract

We investigate a stochastic differential game in which a major player has a private information (the knowledge of a random variable), which she discloses through her control to a population of small players playing in a Nash Mean Field Game equilibrium. The major player’s cost depends on the distribution of the population, while the cost of the population depends on the random variable known by the major player. We show that the game has a relaxed solution and that the optimal control of the major player is approximatively optimal in games with a large but finite number of small players. Joint work with Pierre Cardaliaguet and Catherine Rainer.

Thu, 05 Dec 2024
16:00
Lecture Room 3

Zeros of polynomials with restricted coefficients: a problem of Littlewood

Benjamin Bedert
(University of Oxford)
Abstract

The study of polynomials whose coefficients lie in a given set $S$ (the most notable examples being $S=\{0,1\}$ or $\{-1,1\}$) has a long history leading to many interesting results and open problems. We begin with a brief general overview of this topic and then focus on the following old problem of Littlewood. Let $A$ be a set of positive integers, let $f_A(x)=\sum_{n\in A}\cos(nx)$ and define $Z(f_A)$ to be the number of zeros of $f_A$ in $[0,2\pi]$. The problem is to estimate the quantity $Z(N)$ which is defined to be the minimum of $Z(f_A)$ over all sets $A$ of size $N$. We discuss recent progress showing that $Z(N)\geqslant (\log \log N)^{1-o(1)}$ which provides an exponential improvement over the previous lower bound. 

A closely related question due to Borwein, Erd\'elyi and Littmann asks about the minimum number of zeros of a cosine polynomial with $\pm 1$-coefficients. Until recently it was unknown whether this even tends to infinity with the degree $N$. We also discuss work confirming this conjecture.

 

Thu, 05 Dec 2024

16:00 - 17:00
L1

The Art of Cancer Modelling

Prof. Mark Chaplain
(University of St. Andrews)
Further Information

Mark Chaplain is the Gregory Chair of Applied Mathematics at the University of St. Andrews. 

Here's a little about his research from the St. Andrews website:

Research areas

Cancer is one of the major causes of death in the world, particularly the developed world, with around 11 million people diagnosed and around 9 million people dying each year. The World Health Organisation (WHO) predicts that current trends show the number rising to 11.5 million in 2030. There are few individuals who have not been touched either directly or indirectly by cancer. While treatment for cancer is continually improving, alternative approaches can offer even greater insight into the complexity of the disease and its treatment. Biomedical scientists and clinicians are recognising the need to integrate data across a range of spatial and temporal scales (from genes through cells to tissues) in order to fully understand cancer. 

My main area of research is in what may be called "mathematical oncology" i.e. formulating and analysing  mathematical models of cancer growth and treatment. I have been involved in developing a variety of novel mathematical models for all the main phases of solid tumour growth, namely: avascular solid tumour growth, the immune response to cancer, tumour-induced angiogenesis, vascular tumour growth, invasion and metastasis. 

The main modelling techniques involved are the use and analysis of nonlinear partial and ordinary differential equations, the use of hybrid continuum-discrete models and the development of multiscale models and techniques. 

Much of my current work is focussed on what may be described as a "systems approach" to modelling cancer growth through the development of quantitative and predictive mathematical models. Over the past 5 years or so, I have also helped develop models of chemotherapy treatment of cancer, focussing on cell-cycle dependent drugs, and also radiotherapy treatment. One of the new areas of research I have started recently is in modelling intracellular signalling pathways (gene regulation networks) using partial differential equation models. 

The long-term goal is to build a "virtual cancer" made up of different but connected mathematical models at the different biological scales (from genes to tissue to organ). The development of quantitative, predictive models (based on sound biological evidence and underpinned and parameterised by biological data) has the potential to have a positive impact on patients suffering from diseases such as cancer through improved clinical treatment.

Further details of my current research can be found at the Mathematical Biology Research Group web page.

Abstract

In this talk we will provide an overview of a number of mathematical models of cancer growth and development - gene regulatory networks, the immune response to cancer, avascular solid tumour growth, tumour-induced angiogenesis, cancer invasion and metastasis. In the talk we will also discuss (the art of) mathematical modelling itself giving illustrations and analogies from works of art. 

 

 

Thu, 05 Dec 2024
16:00
C3

C*-diagonals in the C*-algebras of non-principal twisted groupoids

Anna Duwenig
(KU Leuven)
Abstract

The reduced twisted C*-algebra A of an étale groupoid G has a canonical abelian subalgebra D: functions on G's unit space. When G has no non-trivial abelian subgroupoids (i.e., G is principal), then D is in fact maximal abelian. Remarkable work by Kumjian shows that the tuple (A,D) allows us to reconstruct the underlying groupoid G and its twist uniquely; this uses that D is not only masa but even what is called a C*-diagonal. In this talk, I show that twisted C*-algebras of non-principal groupoids can also have such C*-diagonal subalgebras, arising from non-trivial abelian subgroupoids, and I will discuss the reconstructed principal twisted groupoid of Kumjian for such pairs of algebras.

Thu, 05 Dec 2024

14:00 - 15:00
Lecture Room 3

Solving (algebraic problems from) PDEs; a personal perspective

Andy Wathen
(Oxford University)
Abstract

We are now able to solve many partial differential equation problems that were well beyond reach when I started in academia. Some of this success is due to computer hardware but much is due to algorithmic advances. 

I will give a personal perspective of the development of computational methodology in this area over my career thus far. 

Thu, 05 Dec 2024
13:00
N3.12

Resurgence

Clément Virally
Abstract

Perturbation theory is one of the main tools in the modern physicist's toolbox to solve problems. Indeed, it can often the only approach we have to computing any quantity of interest in a physical theory. However, perturbative contributions can actually grow as we increase the order. Thus, many perturbative series in physics are asymptotic, with 0 radius of convergence. In this talk, I will describe resurgence, which gives us a way of treat such series, by adding non-perturbative effects in a systematic manner.

 

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.

Thu, 05 Dec 2024

12:00 - 12:30
Lecture Room 6

Who needs a residual when an approximation will do?

Nathaniel Pritchard
(University of Oxford)
Abstract

The widespread need to solve large-scale linear systems has sparked a growing interest in randomized techniques. One such class of techniques is known as iterative random sketching methods (e.g., Randomized Block Kaczmarz and Randomized Block Coordinate Descent). These methods "sketch" the linear system to generate iterative, easy-to-compute updates to a solution. By working with sketches, these methods can often enable more efficient memory operations, potentially leading to faster performance for large-scale problems. Unfortunately, tracking the progress of these methods still requires computing the full residual of the linear system, an operation that undermines the benefits of the solvers. In practice, this cost is mitigated by occasionally computing the full residual, typically after an epoch. However, this approach sacrifices real-time progress tracking, resulting in wasted computations. In this talk, we use statistical techniques to develop a progress estimation procedure that provides inexpensive, accurate real-time progress estimates at the cost of a small amount of uncertainty that we effectively control.

Thu, 05 Dec 2024

12:00 - 13:00
L3

Chaotic flows in polymer solutions: what’s new?

Prof. Rich Kerswell
(University of Cambridge)
Further Information

Rich Kerswell is a professor in the Department of Applied Mathematics and Theoretical Physics (DAMTP) at the University of Cambridge. His research focuses on fluid dynamics, particularly in the transition to turbulence, geophysical fluid flows, and nonlinear dynamics. Kerswell is known for studying how simple fluid systems can exhibit complex, chaotic behavior and has contributed to understanding turbulence's onset and sustainment in various contexts, including pipes and planetary atmospheres. His work integrates mathematical modeling, theoretical analysis, and computational simulations to explore instabilities and the fundamental mechanisms governing fluid behavior in nature and industry.

Abstract

It is well known that adding even small amounts of  long chain polymers (e.g. few parts per million) to Newtonian solvents can drastically change the flow behaviour by introducing elasticity. In particular,  two decades ago, experiments in curved geometries  demonstrated  that polymer flows can be  chaotic even at vanishingly small Reynolds numbers. The situation in `straight’ flows  such as pressure-driven flow down a channel is less clear  and hence an area of current focus. I will discuss recent progress.

Thu, 05 Dec 2024

11:00 - 12:00
C1

Local-Global Principles and Fields Elementarily Characterised by Their Absolute Galois Groups

Benedikt Stock
(University of Oxford)
Abstract

Jochen Koenigsmann’s Habilitation introduced a classification of fields elementarily characterised by their absolute Galois groups, including two conjecturally empty families. The emptiness of one of these families would follow from a Galois cohomological conjecture concerning radically closed fields formulated by Koenigsmann. A promising approach to resolving this conjecture involves the use of local-global principles in Galois cohomology. This talk examines the conceptual foundations of this method, highlights its relevance to Koenigsmann’s classification, and evaluates existing local-global principles with regard to their applicability to this conjecture.

Wed, 04 Dec 2024
16:00
L6

Tambara-Yamagami Fusion Categories

Adrià Marín-Salvador
(University of Oxford)
Abstract

In this talk, I will introduce fusion categories as categorical versions of finite rings. We will discuss some examples which may already be familiar, like the category of representations of a finite group and the category of vector spaces graded over a finite group. Then, we will define Tambara-Yamagami categories, which are a certain type of fusion categories which have one simple object which is non-invertible. I will provide the classification results of Tambara and Yamagami on these categories and give some small examples. Time permitting, I will discuss current work in progress on how to generalize Tambara-Yamagami fusion categories to locally compact groups. 

This talk will not assume familiarity with category theory further than the definition of a category and a functor.

Wed, 04 Dec 2024
11:00
L4

Effective Mass of the Polaron and the Landau-Pekar-Spohn Conjecture

Chiranjib Mukherjee
(University of Münster)
Abstract

According to a conjecture by Landau-Pekar (1948) and by Spohn (1986), the effective mass of the Fröhlich Polaron should diverge in the strong coupling limit like a quartic power of the coupling constant. In a recent joint with R. Bazaes, M. Sellke and S.R.S. Varadhan, we prove this conjecture.

Tue, 03 Dec 2024
16:00
C3

The space of traces of certain discrete groups

Raz Slutsky
(University of Oxford)
Abstract

A trace on a group is a positive-definite conjugation-invariant function on it. These traces correspond to tracial states on the group's maximal  C*-algebra. In the past couple of decades, the study of traces has led to exciting connections to the rigidity, stability, and dynamics of groups. In this talk, I will explain these connections and focus on the topological structure of the space of traces of some groups. We will see the different behaviours of these spaces for free groups vs. higher-rank lattices, and how our strategy for the free group can be used to answer a question of Musat and Rørdam regarding free products of matrix algebras. This is based on joint works with Arie Levit, Joav Orovitz, and Itamar Vigdorovich.

Tue, 03 Dec 2024
16:00
L6

Large deviations of Selberg’s CLT: upper and lower bounds

Emma Bailey
(University of Bristol)
Abstract

Selberg’s CLT informs us that the logarithm of the Riemann zeta function evaluated on the critical line behaves as a complex Gaussian. It is natural, therefore, to study how far this Gaussianity persists. This talk will present conditional and unconditional results on atypically large values, and concerns work joint with Louis-Pierre Arguin and Asher Roberts.

Tue, 03 Dec 2024
15:00
L6

Short loxodromics in graph products

Alice Kerr
(University of Bristol)
Abstract
Let G be a finitely generated group, with finite generating set S. Suppose G contains elements with some property that we’re interested in.  Can we find elements with this property uniformly quickly in G? That is, does S^n contain an element with this property for a bounded n?
 
We will discuss this question for graph products, where the elements we are looking for are ones with nice hyperbolic properties, such as loxodromic and Morse elements. We will also talk about consequences for the growth of these groups. This is joint work with Elia Fioravanti.

 
Tue, 03 Dec 2024
14:00
L5

Gecia Bravo-Hermsdorff: What is the variance (and skew, kurtosis, etc) of a network? Graph cumulants for network analysis

Gecia Bravo-Hermsdorff
(University College London)
Abstract

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. 

Tue, 03 Dec 2024

14:00 - 15:00
L4

A Zarankiewicz problem in tripartite graphs

Freddie Illingworth
(University College London)
Abstract

In 1975, Bollobás, Erdős, and Szemerédi asked the following Zarankiewicz-type problem. What is the smallest $\tau$ such that an $n \times n \times n$ tripartite graph with minimum degree $n + \tau$ must contain $K_{t, t, t}$? They further conjectured that $\tau = O(n^{1/2})$ when $t = 2$.

I will discuss our proof that $\tau = O(n^{1 - 1/t})$ (confirming their conjecture) and an infinite family of extremal examples. The bound $O(n^{1 - 1/t})$ is best possible whenever the Kővári-Sós-Turán bound $\operatorname{ex}(n, K_{t, t}) = O(n^{2 - 1/t})$ is (which is widely-conjectured to be the case).

This is joint work with Francesco Di Braccio (LSE).

Tue, 03 Dec 2024
14:00
L6

Hyperbolic intersection arrangements

Samuel Lewis
(University of Oxford)
Abstract

Consider a connected graph and choose a subset of its vertices. From this simple setup, Iyama and Wemyss define a collection of real hyperplanes known as an intersection arrangement, going on to classify all tilings of the affine plane that arise in this way. These "local" generalisations of Coxeter combinatorics also admit a nice wall-crossing structure via Dynkin involutions and longest Weyl elements. In this talk I give an analogous classification in the hyperbolic setting using the data of an "overextended" ADE diagram with three distinguished vertices. I then discuss ongoing work applying intersection arrangements to parametrise notions of stability conditions for preprojective algebras.

Tue, 03 Dec 2024
13:00
L2

Quantized axial charge of lattice fermions and the chiral anomaly

Arkya Chatterje
(MIT )
Abstract

Realizing chiral global symmetries on a finite lattice is a long-standing challenge in lattice gauge theory, with potential implications for non-perturbative regularization of the Standard Model. One of the simplest examples of such a symmetry is the axial U(1) symmetry of the 1+1d massless Dirac fermion field theory: it acts by equal and opposite phase rotations on the left- and right-moving Weyl components of the Dirac field. This field theory also has a vector U(1) symmetry which acts identically on left- and right-movers. The two U(1) symmetries exhibit a mixed anomaly, known as the chiral anomaly. In this talk, we will discuss how both symmetries are realized as ordinary U(1) symmetries of an "ultra-local" lattice Hamiltonian, on a finite-dimensional Hilbert space. Intriguingly, the anomaly of the Abelian U(1) symmetries in the infrared (IR) field theory is matched on the lattice by a non-Abelian Lie algebra. The lattice symmetry forces the low-energy phase to be gapless, closely paralleling the effects of the anomaly in the field theory.

Mon, 02 Dec 2024
16:30
L4

Introducing various notions of distances between space-times

Anna Sakovich
(University of Uppsala)
Abstract

I will introduce the class of causally-null-compactifiable spacetimes that can be canonically converted into compact timed-metric spaces using the cosmological time function of Andersson-Galloway-Howard and the null distance of Sormani-Vega. This class of space-times includes future developments of compact initial data sets and regions exhausting asymptotically flat space-times. I will discuss various intrinsic notions of distance between such space-times and show that some of them are definite in the sense that they are equal to zero if and only if there is a time-oriented Lorentzian isometry between the space-times. These definite distances allow us to define notions of convergence of space-times to limit space-times that are not necessarily smoothThis is joint work with Christina Sormani.

Mon, 02 Dec 2024
16:00
C3

TBC

Leo Gitin
(University of Oxford)
Abstract

TBC

Mon, 02 Dec 2024
15:30
L5

Building surfaces from equilateral triangles

Lasse Rempe
(Manchester University)
Abstract
In this talk, we consider the following question. Suppose that we glue a (finite or infinite) collection of closed equilateral triangles together in such a way that we obtain an orientable surface. The resulting surface is a Riemann surface; that is, it has a natural conformal structure (a way of measuring angles in tangent space). We ask which Riemann surfaces are *equilaterally triangulable*; i.e., can arise in this fashion.

The answer in the compact case is given by a famous classical theorem of Belyi, which states that a compact surface is equilaterally triangulable if and only if it is defined over a number field. These *Belyi surfaces* - and their associated “dessins d’enfants” - have found applications across many fields of mathematics, including mathematical physics.

In joint work with Chris Bishop, we give a complete answer of the same question for the case of infinitely many triangles (i.e., for non-compact Riemann surfaces). The talk should be accessible to a general mathematical audience, including postgraduate students.


 

Mon, 02 Dec 2024
15:30
L3

Chasing regularization by noise of 3D Navier-Stokes equations

Dr Antonio Agresti
(Delft University of Technology )
Abstract

Global well-posedness of 3D Navier-Stokes equations (NSEs) is one of the biggest open problems in modern mathematics. A long-standing conjecture in stochastic fluid dynamics suggests that physically motivated noise can prevent (potential) blow-up of solutions of the 3D NSEs. This phenomenon is often referred to as `regularization by noise'. In this talk, I will review recent developments on the topic and discuss the solution to this problem in the case of the 3D NSEs with small hyperviscosity, for which the global well-posedness in the deterministic setting remains as open as for the 3D NSEs. An extension of our techniques to the case without hyperviscosity poses new challenges at the intersection of harmonic and stochastic analysis, which, if time permits, will be discussed at the end of the talk.

Mon, 02 Dec 2024
14:15
L4

Open Gromov-Witten invariants and Mirror symmetry

Kai Hugtenburg
(Lancaster)
Abstract

This talk reports on two projects. The first work (in progress), joint  with Amanda Hirschi, constructs (genus 0) open Gromov-Witten invariants for any Lagrangian submanifold using a global Kuranishi chart construction. As an application we show open Gromov-Witten invariants are invariant under Lagrangian cobordisms. I will then describe how open Gromov-Witten invariants fit into mirror symmetry, which brings me to the second project: obtaining open Gromov-Witten invariants from the Fukaya category.

Mon, 02 Dec 2024

14:00 - 15:00
Lecture Room 3

Enhancing Accuracy in Deep Learning using Marchenko-Pastur Distribution

Leonid Beryland
(Penn State University)
Abstract

We begin with a short overview of Random Matrix Theory (RMT), focusing on the Marchenko-Pastur (MP) spectral approach. 

Next, we present recent analytical and numerical results on accelerating the training of Deep Neural Networks (DNNs) via MP-based pruning ([1]). Furthermore, we show that combining this pruning with L2 regularization allows one to drastically decrease randomness in the weight layers and, hence, simplify the loss landscape. Moreover, we show that the DNN’s weights become deterministic at any local minima of the loss function. 
 

Finally, we discuss our most recent results (in progress) on the generalization of the MP law to the input-output Jacobian matrix of the DNN. Here, our focus is on the existence of fixed points. The numerical examples are done for several types of DNNs: fully connected, CNNs and ViTs. These works are done jointly with PSU PhD students M. Kiyashko, Y. Shmalo, L. Zelong and with E. Afanasiev and V. Slavin (Kharkiv, Ukraine). 

 

[1] Berlyand, Leonid, et al. "Enhancing accuracy in deep learning using random matrix theory." Journal of Machine Learning. (2024).

Mon, 02 Dec 2024
13:30
C4

Extended TQFT, gauge theory, and Measurement Based Quantum Computation

Gabriel Wong
Abstract

Measurement-Based Quantum Computation (MBQC) is a model of quantum computation driven by measurements instead of unitary gates.   In 2D it is capable of supporting universal quantum computations.   Interestingly, while all measurements are local, the computational output involves non local observables.   We will use the simpler case of 1D MBQC to illustrate how these features can be captured by ideas from gauge theory and extended TQFT. We will also explain  MBQC from the perspective of the extended Hilbert space construction in gauge theories, in which the entanglement edge modes play the role of the logical qubit.