Schedule

Welcome

Mark Roelands  (Universiteit Leiden)

Title:  Hilbert's metric isometries on simplices of arbitrary dimension

Abstract:  Hilbert's metric was first introduced by Hilbert, as the name suggests, in the 19th century in order to study metrics on subsets for which straight lines are geodesics. This is related to Hilbert's 4th problem. By the work of G. Birkhoff, these metric spaces can be represented on cones where the distance is determined by the partial ordering. Other than providing some historical background of Hilbert's metric, a sketch of the proof characterising the isometries on simplices will be presented which uses the cone representation.

Coffee Break

Sven van Golden  (University of Birmingham)

Title:  Infinitely generated self-affine sets and restricted digit sets for signed Lüroth expansions 

Abstract:  In recent years much work has been done towards finding the fractal dimensions of the limit sets of finite affine iterated function systems, also known as self-affine sets. Of particular interest are conditions under which the Hausdorff dimension of such sets equals the affinity dimension, a value introduced by Falconer in 1988.

In this joint work with C. Kalle, S. Kombrink and T. Samuel we extend this notion of affinity dimension to self-affine sets generated by countably infinite iterated function systems where each affine map has diagonal linear parts. We introduce a family of both finitely and infinitely generated self-affine sets in the plane that arise from the restricted digit sets for signed Lüroth expansions. Moreover, we study the vertical fibres of these sets to find conditions under which their Hausdorff dimensions equal the affinity dimension.

Jonny Imbierski  (Universiteit Leiden)

Title: Hausdorff Dimension of Besicovitch-Eggleston Sets for Non-autonomous GLS Maps 

Abstract: We introduce the family of Non-autonomous Generalised Lüroth Series (NGLS) maps. Each NGLS map is a number system described by a pair (T,w), where T is a collection of GLS maps on the unit interval and w gives the order of maps of T to be applied at each time-step. Given a fixed NGLS map (T,w), all but countably many points in the unit interval have a unique (T,w)-expansion. This allows us to study the level sets of points that have each digit in their (T,w)-expansion appear with a specified frequency, which are known as the Besicovitch-Eggleston sets for (T,w). We determine when each Besicovitch-Eggleston set for (T,w) is non-empty and then derive a formula for its Hausdorff dimension. This is joint work with Charlene Kalle (Leiden).

Lunch

Bram Mesland  (Universiteit Leiden)

Title: The Friedrichs angle and alternating projections in Hilbert C*-modules 

Abstract:  In his foundational work on rings of operators, John von Neumann proved that given two projections P,Q on a Hilbert space, the sequence (PQ)^n converges in the *-strong topology to the projection onto the intersection of the ranges of P and Q. The finer convergence properties of the sequence (PQ)^n are detected by a numerical invariant called the Friedrichs angle between P and Q. The Friedrichs angle has found various applications in approximation theory. If we allow for continuous families of projections, various technicalities present itself. The natural setting for this problem is that of Hilbert C*-modules, a notion from the theory of operator algebras. In this context, von Neumann’s convergence result does not hold for an arbitrary pair of projections, and the Friedrichs angle cannot be straightforwardly defined. In this talk I will show that a satisfactory definition of the Friedrichs angle can be obtained in this generality as well as discuss (counter)examples and interpretations in (noncommutative) geometry. 

Cold Refreshments & Collaboration

Wine Reception - Basement of Watson Building (R15 on the Campus map)

Sander Hille  (Universiteit Leiden)

Title: On best approximation in Fortet-Mourier norm of a measure by a finite sum of Dirac measures

Abstract: Measure-valued solutions to partial differential equations provide a functional analytic framework to compare continuum models with interacting particle systems as description for particular phenomena, or as numerical method for simulation. The particle system corresponds to a (weighted) sum of Dirac measures, while the continuum model yields a density with respect to Lebesgue measure. The question then arises to compute the distance between the two. We shall discuss recently obtained expressions for the Fortet-Mourier norm between such measures and the problem of best approximation in this norm by means of a fixed maximal number of point measures. The formulas obtained for the norm show that this problem is related to the so-called Fermat-Weber problem in logistics, which is – as yet – only partially solved.

Coffee Break

Onno van Gaans  (Universiteit Leiden)

Title: Survival in delayed stochastic population models

Abstract:  We consider a class of population models given by a scalar stochastic delay differential equation with a fixed delay in the birth term. If the coefficient of the noise is roughly proportional to the population size, then it can be shown that the population size will be bounded away from zero with probability 1, provided the birth term does not tend to zero near zero. In our analysis, we investigate the logarithm of the population size and aim to show that it is bounded below in probability. This is joint work with Mark van den Bosch and Sjoerd Verduyn Lunel.

It may be more realistic to suppose that the random fluctuations are relatively larger in small populations, instead of proportional. For equations with such a noise coefficient and without delay, very recent results of Julius Busse show that extinction is possible with a positive probability. 

Devi-Fitri Ferdania  (University of Birmingham)

Title: Exploring the Morphology of Mitochondria in Heart Cancer Cells through Fractal Analysis in Fluorescence Imaging

Abstract: Fractal analysis is a powerful mathematical tool used extensively in texture analysis to quantify and characterize complex patterns and structures within images. It operates on the principle that many natural and synthetic textures exhibit self-similarity across different scales, meaning patterns at smaller scales resemble those at larger scales. Two pivotal measurements in fractal analysis, fractal dimension and lacunarity, play crucial roles in assessing the complexity and irregularity of structures and quantifying the distribution of gaps within the objects, respectively. In this research, we delve into the application of fractal analysis techniques to examine fluorescence image data of mitochondria in heart cancer cells subjected to various treatments. These treatments include the chemotherapy drug Paclitaxel, mitochondria division inhibitor (MDivi-1), hypoxia-induced chemical (CoCl2), and tumour-growth drug (MYLS22). By employing fractal dimension and lacunarity as analytical tools, our research endeavours to unveil distinct patterns and characteristics associated with different treatments, aiming to enhance our understanding of the corresponding mitochondrial dynamics.

Lunch

Lauren Thomas-Seale  (University of Birmingham)

Title: Geometry, Topology and Interfaces with Mechanical Engineering

Abstract: The application of topological methodologies in engineering, are sometimes a very loose reflection of the mathematical definition. Within mechanical engineering the academic and industrial interpretation, and applications, differ further still. Interviews with engineering organisations, revealed that topology is almost exclusively interpreted as ‘Topology Optimisation’ in the context of advanced design and manufacturing. From which, there is a lot of confusion and interchange between the use of the words topology and geometry. As such, there exists an opportunity to translate the mathematical definition of topology, and subsequent implementation, into engineering. This presentation will explore the current interpretation of topology in mechanical engineering, which largely involves applications of Topology Optimisation, and explore what opportunities exist, should a more robust definition be adopted. 

Mabel Lizzi Rajendran  (University of Birmingham)

Title: Nonlocal operators in cancer models

Abstract: Cancer growth and spread are complex processes involving nonlocal phenomena such as memory and interactions with the surrounding environment. Including these phenomena in the mathematical model results in partial differential equation (PDE) with nonlocal operators, which pose interesting challenges in demonstrating their well-posedness and in numerical simulation.

In the first model, we capture the subdiffusive behaviour observed during the initial stages of cancer growth. By modifying Fick’s law of diffusion and employing continuum mixture theory, we obtain a PDE with a nonlocal time operator (fractional time derivative). 

The second model captures cell-cell adhesion behaviour mediated by cell-adhesion molecule (CAM) bindings, which plays a key role in cancer migration. Through multiscale modelling, it is demonstrated that including this characteristic results in a system of a PDE with nonlocal operator in space coupled with a novel nonlinear integral equation with two nonlocal operators.

Sebastian Gilbert  (University of Birmingham)

Title: Topological and Spatial Utensils for Neighbourhood Analysis in Multiplex Images

Abstract: Evidence shows that, throughout their life, a person will develop tumours which are identified and eradicated by normal immunological mechanisms, without notice or consequence by the individual. However, in many cases when ‘normal’ immunological responses fail, we observe the detrimental results as cancer. Multiplex immuno-fluorescence imaging identifies the proximity of many different immune cell types within the tumour microenvironment. Designing mathematical and computational methods to interpret the spatially-resoled ex-vivo dataset (specifically in colorectal cancer) can lead to crucial ways in understanding and predicting disease progression. We introduce tools which combine topological and spatial intuition to interpret communities (a.k.a neighbourhoods) within these multi-layered networks. Combining these techniques with prior knowledge of immunological mechanisms, we present intuitive and predictive biomarkers which go beyond previous prognostic tools such as grading, multi-satellite stability/instability, and Immunoscore®.

Cold Refreshments

Ian Morris  (Queen Marry University of London)

Title:  Exceptional projections of self-affine sets 

Abstract:  It is widely believed that if a self-affine set is defined by an affine iterated function system which satisfies appropriate separation conditions and whose linear parts are in general position in GL(d,R), then its Hausdorff dimension equals a value predicted by Falconer in 1988. It is also expected that such self-affine sets should have no exceptional projections in the sense of Marstrand's theorem: if the set is orthogonally projected onto any linear subspace, then the Hausdorff dimension of its projected image should always equal either the dimension of the original set or the rank of the projection, whichever is smaller. Both of these statements have recently been proved in dimensions up to 3. I will discuss some work in progress with Çağrı Sert in which we show that if the linear parts are instead confined to a smaller linear algebraic group then an abundance of exceptional projections can arise.  

Cold Refreshments & Collaboration

Pub Visit  –  Wildcat Tap

Address:  1381-1383, Pershore Rd, Stirchley, Birmingham B30 2JR
Station:  Bournville (direct connection from University station) 

Option 1:  Taking the 18:01 train from University station. Meeting 17:45 in the entrance area of the Old Gym.
Option 2:  Weather permitting, a group of us might walk to the venue along the canal (approx. 45 minute walk). Meeting 17:25 in the entrance area of the Old Gym.

Conference Dinner  –  Patiala Indian Restaurant Birmingham 

Address:  1260 Pershore Rd, Stirchley, Birmingham B30 2XU
Station:  Bournville (direct connection from University station) 

Susana Gutierrez  (University of Birmingham)

Title: Self-similar solutions of the Landau-Lifshitz-Gilbert equation and related problems 

Abstract: The Landau-Lifshitz-Gilbert equation (LLG) is a continuum model describing the dynamics for the spin in ferromagnetic materials. In the first part of this talk we describe the properties and dynamical behaviour of self-similar solutions of this model in one dimension.  Time permitting, and motivated by the properties of these solutions, we consider the Cauchy problem for the LLG-equation and provide a global well-posedness result provided that the BMO norm of the initial data is small.  

Coffee Break

Dimitris Gerontogiannis  (Universiteit Leiden)

Title: The logarithmic Dirichlet Laplacian and dynamics on Ahlfors regular spaces

Abstract: The Laplace-Beltrami operator is a fundamental tool in the study of compact Riemannian manifolds. This talk is about the logarithmic analogue of this operator on Ahlfors regular spaces. These are metric-measure spaces that might lack any differential or algebraic structure. Examples are compact Riemannian manifolds, several fractals, self-similar Smale spaces and limit sets of hyperbolic isometry groups. Further, this operator is intrinsically defined, its spectral properties are analogous to those of elliptic pseudo-differential operators on manifolds and exhibits compatibility with non-isometric actions in the sense of noncommutative geometry. If time allows, I will describe how this can be used to study stable/unstable foliations. This is joint work with Bram Mesland (Leiden).

Lunch

David Hume  (University of Birmingham)

Title:  Conformal dimension via coarse geometry

Abstract:  Determining the quasi-symmetry type of a metric space is a key problem in geometric function theory. This problem has a natural coarse geometric analogue: determining the quasi-symmetry type of a metric space is essentially equivalent to determining the coarse geometric "quasi-isometry" type of a suitably defined cone over that space.

One of the most intensely studied quasi-symmetry invariants is conformal dimension, introduced by Pansu in the 80's as a tool for the coarse geometric classification of classical rank one symmetric spaces and their lattices. While highly useful, the conformal dimension is, even in the world of dimension theory, notoriously difficult to calculate. Indeed, it is not known for the Sierpiński carpet.

In this talk I will present new methods of obtaining lower bounds on the conformal dimension of a metric space using the coarse geometric structure of its cone. This is joint work with John Mackay and Romain Tessera.

Nataliya Balabanova  (University of Birmingham)

Title: Adaptive dynamics for Prisoner's Dilemma

Abstract: Adaptive dynamics is a method of determining the best way to alter the strategy of mutants in the population in order to maximise the profit. This mathematical apparatus can be applied in game theory to the repeated donation game, otherwise known as the prisoner's dilemma. In this talk, we are going to cover the construction of the payoff function for the prisoner's dilemma,  its symmetries, the corresponding adaptive dynamics and some of their properties.

Cold Refreshments & Collaboration