Fri, 02 May 2025
13:00
L5

An algebraic derivation of Morse Complexes for poset-graded chain complexes

Ka Man Yim
(Cardiff University)

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

Abstract

The Morse-Conley complex is a central object in information compression in topological data analysis, as well as the application of homological algebra to analysing dynamical systems. Given a poset-graded chain complex, its Morse-Conley complex is the optimal chain-homotopic reduction of the initial complex that respects the poset grading.  In this work, we give a purely algebraic derivation of the Morse-Conley complex using homological perturbation theory. Unlike Forman’s discrete Morse theory for cellular complexes, our algebraic formulation does not require the computation of acyclic partial matchings of cells.  We show how this algebraic perspective also yields efficient algorithms for computing the Conley complex.  This talk features joint work with Álvaro Torras Casas and Ulrich Pennig in "Computing Connection Matrices of Conley Complexes via Algebraic Morse Theory" (arXiv:2503.09301). 
 

Tue, 25 Feb 2025
16:00
C3

Equivariant higher Dixmier-Douady theory for UHF-algebras

Valerio Bianchi
(Cardiff University)
Abstract

A classical result of Dixmier and Douady enables us to classify locally trivial bundles of C*-algebras with compact operators as fibres via methods in homotopy theory. Dadarlat and Pennig have shown that this generalises to the much larger family of bundles of stabilised strongly self-absorbing C*-algebras, which are classified by the first group of the cohomology theory associated to the units of complex topological K-theory. Building on work of Evans and Pennig I consider Z/pZ-equivariant C*-algebra bundles over Z/pZ-spaces. The fibres of these bundles are infinite tensor products of the endomorphism algebra of a Z/pZ-representation. In joint work with Pennig, we show that the theory refines completely to this equivariant setting. In particular, we prove a full classification of the C*-algebra bundles via equivariant stable homotopy theory.

Thu, 30 May 2024
12:00
L5

Description of highly symmetric RCD-spaces

Diego Corro
(Cardiff University)
Abstract
RCD-spaces arise naturally from optimal transport theory by the work of Otto-Villanni-Sturm. Moreover, these spaces have a very rich (local) analysis, and several properties of Riemannian manifolds hold for these spaces. But so far the global underlying topological structure of RCD-spaces is not fully understood. 
 
In this talk we consider RCD-spaces with a lot of symmetry, that is a large Lie group acting on it by measure preserving isometries, and fully describe the underlying topological structure. We prove this by taking ideas from optimal transport to construct a canonical space transverse to the orbit. Moreover, I also present a systematic method of constructing such RCD-spaces with high symmetry.
 
This is joint work with Jesús Núñez-Zimbrón and Jaime Santos-Rodríguez.
Fri, 10 Nov 2023

15:00 - 16:00
L5

Topological Data Analysis (TDA) for Geographical Information Science (GIS)

Padraig Corcoran
(Cardiff University)
Further Information

Dr Padraig Corcoran is a Senior Lecturer and the Director of Research in the School of Computer Science and Informatics (COMSC) at Cardiff University.

Dr Corcoran has much experience and expertise in the fields of graph theory and applied topology. He is particularly interested in applications to the domains of geographical information science and robotics.

Abstract

Topological data analysis (TDA) is an emerging field of research, which considers the application of topology to data analysis. Recently, these methods have been successfully applied to research problems in the field of geographical information science (GIS). This includes the problems of Point of Interest (PoI), street network and weather analysis. In this talk I will describe how TDA can be used to provide solutions to these problems plus how these solutions compare to those traditionally used by GIS practitioners. I will also describe some of the challenges of performing interdisciplinary research when applying TDA methods to different types of data.

Mon, 14 Feb 2022

16:30 - 17:30
L3

Stability from rigidity via umbilicity

Julian Scheuer
(Cardiff University)
Abstract

The soap bubble theorem says that a closed, embedded surface of the Euclidean space with constant mean curvature must be a round sphere. Especially in real-life problems it is of importance whether and to what extent this phenomenon is stable, i.e. when a surface with almost constant mean curvature is close to a sphere. This problem has been receiving lots of attention until today, with satisfactory recent solutions due to Magnanini/Poggesi and Ciraolo/Vezzoni.
The purpose of this talk is to discuss further problems of this type and to provide two approaches to their solutions. The first one is a new general approach based on stability of the so-called "Nabelpunktsatz". The second one is of variational nature and employs the theory of curvature flows. 

Tue, 10 Nov 2020
12:00
Virtual

Conformal Field Theory through Subfactors and K-theory

Dai Evans
(Cardiff University)
Abstract

Subfactors and K-theory are useful mechanisms for understanding modular tensor categories and conformal field theories. As part of this programme, one issue to try and construct or reconstruct a conformal field theory as the representation theory of a conformal net of algebras, or as a vertex operator algebra from a given abstractly presented modular tensor category. Orbifold models play an important role and orbifolds of Tambara-Yamagami systems are relevant to understanding the double of the Haagerup as a conformal field theory. This is joint work with Andreas Aaserud, Terry Gannon and Ulrich Pennig.

Tue, 03 Mar 2020

15:30 - 16:30
L4

Skein-triangulated representations of generalized braid categories

Timothy Logvinenko
(Cardiff University)
Abstract

The ordinary braid group ${\mathrm Br}_n$ is a well-known algebraic structure which encodes configurations of $n$ non-touching strands (“braids”) up to continious transformations (“isotopies”). A classical result of Khovanov and Thomas states that there is a natural categorical action of ${\mathrm Br}_n$ on the derived category of the cotangent bundle of the variety of complete flags in ${\mathbb C}^n$. 

In this talk, I will introduce a new structure: the category ${\mathrm GBr}_n$ of generalised braids. These are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. In the context of triangulated categories, it is natural to impose certain relations which result in the notion of a skein-triangulated representation of ${\mathrm GBr}_n$. A decade-old conjecture states that there is a skein-triangulated action of ${\mathrm GBr}_n$ on the cotangent bundles of the varieties of full and partial flags in ${\mathbb C}^n$. We prove this conjecture for $n = 3$. We also show that, in fact, any categorical action of ${\mathrm Br}_n$ can be lifted to a categorical action of ${\mathrm GBr}_n$, generalising a result of Ed Segal. This is a joint work with Rina Anno and Lorenzo De Biase.

Mon, 24 Feb 2020

16:00 - 17:00
L4

$\Gamma$- convergence and homogenisation for a class of degenerate functionals

Federica Dragoni
(Cardiff University)
Abstract

I will present a $\Gamma$-convergence for degenerate integral functionals related to homogenisation problems  in the Heisenberg group. In our  case, both the rescaling and the notion of invariance or periodicity are chosen in a way motivated by the geometry of the Heisenberg group. Without using special geometric features, these functionals would be neither coercive nor periodic, so classic results do not apply.  All the results apply to the more general case of Carnot groups. Joint with Nicolas Dirr, Paola Mannucci and Claudio Marchi.

Fri, 01 Nov 2019

15:00 - 16:00
N3.12

The Persistence Mayer-Vietoris spectral sequence

Alvaro Torras Casas
(Cardiff University)
Abstract

In this talk, linear algebra for persistence modules will be introduced, together with a generalization of persistent homology. This theory permits us to handle the Mayer-Vietoris spectral sequence for persistence modules, and solve any extension problems that might arise. The result of this approach is a distributive algorithm for computing persistent homology. That is, one can break down the underlying data into different covering subsets, compute the persistent homology for each cover, and join everything together. This approach has the added advantage that one can recover extra geometrical information related to the barcodes. This addresses the common complaint that persistent homology barcodes are 'too blind' to the geometry of the data.

Thu, 30 May 2019

16:00 - 17:30
L3

Likely instabilities in stochastic hyperelastic solids

Angela Mihai
(Cardiff University)
Abstract

Likely instabilities in stochastic hyperelastic solids

L. Angela Mihai

School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK

E-mail: @email.uk

 

Nonlinear elasticity has been an active topic of fundamental and applied research for several decades. However, despite numerous developments and considerable attention it has received, there are important issues that remain unresolved, and many aspects still elude us. In particular, the quantification of uncertainties in material parameters and responses resulting from incomplete information remain largely unexplored. Nowadays, it is becoming increasingly apparent that deterministic approaches, which are based on average data values, can greatly underestimate, or overestimate, mechanical properties of many materials. Thus, stochastic representations, accounting for data dispersion, are needed to improve assessment and predictions. In this talk, I will consider stochastic hyperelastic material models described by a strain-energy density where the parameters are characterised by probability distributions. These models, which are constructed through a Bayesian identification procedure, rely on the maximum entropy principle and enable the propagation of uncertainties from input data to output quantities of interest. Similar modelling approaches can be developed for other mechanical systems. To demonstrate the effect of probabilistic model parameters on large strain elastic responses, specific case studies include the classic problem of the Rivlin cube, the radial oscillatory motion of cylindrical and spherical shells, and the cavitation and finite amplitude oscillations of spheres.

Subscribe to Cardiff University