Prevalence of ρ-irregularity and related properties
 Galeati, L Gubinelli, M Annales de l Institut Henri Poincaré Probabilités et Statistiques volume 60 issue 4 (01 Nov 2024)
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, 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.

Eighty-six billion and counting: do we know the number of neurons in the human brain?
 Goriely, A Brain : a journal of neurology volume 148 issue 3 689-691 (Mar 2025)
Thu, 27 Feb 2025

16:00 - 17:00
Lecture Room 4

The wild Brauer-Manin obstruction

Margherita Pagano
(Imperial College London)
Abstract

A way to study rational points on a variety is by looking at their image in the p-adic points. Some natural questions that arise are the following: is there any obstruction to weak approximation on the variety? Which primes might be involved in it? I will explain how primes of good reduction can play a role in the Brauer-Manin obstruction to weak approximation, with particular emphasis on the case of K3 surfaces.

The entanglement membrane in 2d CFT: reflected entropy, RG flow, and
information velocity
 Jiang, H Mezei, M Virrueta, J (25 Nov 2024) http://arxiv.org/abs/2411.16542v1

Mirzakhani Society

Hello all, we are the Mirzakhani Society, a group for all the women and non-binary mathematicians at Oxford. We run lots and lots of relaxed events from pizza nights to career conferences. Our main aim is to create a warm and welcoming environment for women* to come together and chat about life, maths and anything in-between. 

 

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