In the definition of the skein lasagna module of a $4$-manifold $X$, it is essential that the input TQFT be fully functorial for link cobordisms in $S^3 \times [0, 1]$. I will describe an approach to resolve existing sign ambiguities in Kronheimer and Mrowka's spectral sequence from Khovanov homology to singular instanton link homology. The goal is to obtain a theory that is fully functorial for link cobordisms in $S^3 \times [0,1]$, and where the $E_2$ page carries a canonical isomorphism to Khovanov-Rozansky $\mathfrak{gl}_2$ link homology. Possible applications include non-vanishing theorems for $4$-manifold Khovanov skein lasagna modules à la Ren-Willis.

Low-rank approximation of parameter-dependent matrices A(t) is an important task in the computational sciences, with applications in areas such as dynamical systems and the compression of series of images. In this talk, we introduce AdaCUR, an efficient randomised algorithm for computing low-rank approximations of parameter-dependent matrices using the CUR decomposition. The key idea of our approach is the ability to reuse column and row indices for nearby parameter values, improving efficiency. The resulting algorithm is rank-adaptive, provides error control, and has complexity that compares favourably with existing methods. This is joint work with Yuji Nakatsukasa.

Solving large-scale continuous-time algebraic Riccati equations is a significant challenge in various control theory applications. 

This work demonstrates that when the matrix coefficients of the equation are quasiseparable, the solution also exhibits numerical quasiseparability. This property enables us to develop two efficient Riccati solvers. The first solver is applicable to the general quasiseparable case, while the second is tailored to the particular case of banded coefficients. Numerical experiments confirm the effectiveness of the proposed algorithms on both synthetic examples and case studies from the control of partial differential equations and agent-based models. 

The rheological (deformation and flow) properties of biological tissues  are important in processes such as embryo development, wound healing and 
tumour invasion. Indeed, processes such as these spontaneously generate  stresses within living tissue via active process at the single cell level. 
Tissues are also continually subject to external stresses and deformations  from surrounding tissues and organs. The success of numerous physiological 
functions relies on the ability of cells to withstand stress under some conditions, yet to flow collectively under others. Biological tissue is 
furthermore inherently viscoelastic, with a slow time-dependent mechanics.  Despite this rich phenomenology, the mechanisms that govern the 
transmission of stress within biological tissue, and its response to bulk deformation, remain poorly understood to date.

This talk will describe three recent research projects in modelling the rheology of biological tissue. The first predicts a strain-induced 
stiffening transition in a sheared tissue [1]. The second elucidates the interplay of external deformations applied to a tissue as a whole with 
internal active stresses that arise locally at the cellular level, and shows how this interplay leads to a host of fascinating rheological 
phenomena such as yielding, shear thinning, and continuous or discontinuous shear thickening [2]. The third concerns the formulation of 
a continuum constitutive model that captures several of these linear and nonlinear rheological phenomena [3].

[1] J. Huang, J. O. Cochran, S. M. Fielding, M. C. Marchetti and D. Bi, 
Physical Review Letters 128 (2022) 178001

[2] M. J. Hertaeg, S. M. Fielding and D. Bi, Physical Review X 14 (2024) 
011017.

[3] S. M. Fielding, J. O. Cochran, J. Huang, D. Bi, M. C. Marchetti, 
Physical Review E (Letter) 108 (2023) L042602.

In the 1970s, Serre conjectured that any continuous, irreducible and odd mod p representation of the absolute Galois group G_Q is modular. Serre furthermore conjectured that there should be an explicit minimal weight "k" such that the Galois representation is modular of this weight, and that this weight only depends on the restriction of the Galois representation to the inertial subgroup I_p. This is often called the weight part of Serre's conjecture. Both the weight part, and the modularity part, of the Serre's conjecture are nowadays known to be true. In this talk, I want to explain how to rephrase the conjecture in representation theoretic terms (for k >= 2), so that the weight k is replaced by a certain (mod p) irreducible representation of GL_2(F_p), and how upon rephrasing the conjecture one can realize it as a statement about local-global compatibility with the mod p local Langlands correspondence.

Alesker's theory of generalized valuations unifies smooth measures and constructible functions on real analytic manifolds, extending classical operations on measures. Therefore, operations on generalized valuations can be used to define integral transforms that unify both classical Radon transforms and their topological analogues based on the Euler characteristic, which have been successfully used in shape analysis. However, this unification is proven under rather restrictive assumptions in Alesker's original paper, leaving key aspects conjectural. In this talk, I will present a recent result obtained with A. Bernig that significantly closes this gap by proving that the two approaches indeed coincide on constructible functions under mild transversality assumptions. Our proof relies on a comparison between these operations and operations on characteristic cycles.

Explicit examples of the AdS/CFT correspondence where both bulk and boundary theories are tractable are hard to come by, but the minimal tension string on AdS_3 x S^3 x T^4  is one notable example. In this paper, we discuss how one can construct sigma models on twistor space, with a particular focus on applying these techniques to the aforementioned string theory. We derive novel incidence relations, which allow us to understand to what extent the minimal tension string encodes information about the bulk. We identify vertex operators in terms of bulk twistor variables and a map from twistor space to spacetime is presented. We also demonstrate the presence of a partially broken global supersymmetry algebra in the minimal tension string and we argue that this implies that there exists an N=2 formulation of the theory. The implications of this are studied and we demonstrate the presence of an additional constraint on physical states. This is based on work with Ron Reid-edwards https://arxiv.org/abs/2411.08836.

I will present some ongoing work on solving parametric linear systems arising from the application of the finite elements method on elliptic partial differential trial equations. The focus of the talk will be on leveraging randomised numerical linear algebra to solve these equations in high-dimensional parameter spaces with special emphasis on the multi-query context where optimal sampling is not practical. In this context I will discuss some ideas on choosing a suitable low-dimensional approximation of the solution, as well as reducing the variance of the sketched systems. This research aims at exploring the potential of randomisation as a probabilistic framework for model order reduction, with potential applications to online simulations, uncertainty quantification and inverse problems, via the research grant EPSRC EP/V028618/1

Bio: Nick Polydorides is a professor in computational engineering at the University of Edinburgh and has interests in randomised numerical linear algebra, inverse problems and edge computing. Previously, he was a faculty at the Cyprus Institute, and a postdoctoral fellow at MIT’s lab for Information and Decision Systems. He has a PhD in Electrical Engineering from the University of Manchester.  

Scaling arguments give a natural guess at the regularity condition on the noise in a stochastic PDE for a local solution theory to be possible, using the machinery of regularity structures or paracontrolled distributions. This guess of ``subcriticality'' is often, but not always, correct. In cases when it is not, a the blowup of the variance of certain nonlinear functionals of the noise necessitates a different, multiplicative renormalisation. This led to a general prediction and the first results in the case of the KPZ equation in [Hairer '24]. We discuss recent developments towards confirming this prediction. Based on joint works with Fabio Toninelli and Yueh-Sheng Hsu.

TBC

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.

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.
 

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 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.

TBA

TBA

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.

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.