Malignant gliomas are highly invasive brain tumors. Recent attention has focused on their capacity for network-driven invasion, whereby mitotic events can be followed by the migration of nuclei along long thin cellular protrusions, termed tumour microtubes (TM). Here I develop a mathematical model that describes this microtube-driven invasion of gliomas. I show that scaling limits lead to well known glioma models as special cases such as go-or-grow models, the PI model of Swanson, and the anisotropic model of Swan. I compute the invasion speed and I use the model to fit experiments of cancer resection and regrowth in the mouse brain. (Joint work with N. Loy, K.J. Painter, R. Thiessen, A. Shyntar).

In 2005, Bollobás, Janson and Riordan introduced and extensively studied a general model of inhomogeneous random graphs parametrised by graphons. In particular, they studied the emergence of a giant component in these inhomogeneous random graphs by relating them to a broad collection of inhomogeneous Galton-Watson branching processes. Fractional isomorphism of finite graphs is an important and well-studied concept at the intersection of graph theory and combinatorial optimisation. It has many different characterizations that involve a range of very different and seemingly unrelated properties of graphs. Recently, Grebík and Rocha developed a theory of fractional isomorphism for graphons. In our work, we characterise inhomogeneous random graphs that yield the same inhomogeneous Galton-Watson branching process (and hence have a similar component structure). This is joint work with Jan Hladký and Anna Margarethe Limbach.

The effect of malaria on the developing world is devastating. Each year there are more than 200 million cases and over 400,000 deaths, with children under the age of five the most vulnerable. Ambitious malaria elimination targets have been set by the World Health Organization for 2030. These involve the elimination of the disease in at least 35 countries. However, these malaria elimination targets rest precariously on being able to treat the disease appropriately; a difficult feat with the emergence and spread of antimalarial drug resistance, along with many other challenges. In this talk, I will introduce several statistical and mathematical models that can be used to monitor malaria transmission and to support malaria elimination. For example, I’ll present mechanistic models of disease transmission, statistical models that allow the emergence and spread of antimalarial drug resistance to be monitored, mechanistic models that capture the role of bioclimatic factors on the risk of malaria and optimal geospatial sampling schemes for future malaria surveillance. I will discuss how the results of these models have been used to inform public health policy and support ongoing malaria elimination efforts.

Ehrhart theory is the study of lattice polytopes, specifically aimed at understanding how many lattice points are inside dilates of a given lattice polytope, and the study has a wide range of connections ranging from coloring graphs to mirror symmetry and representation theory. Recently, we introduced new algebraic tools to understand this theory, and resolve some classical conjectures. I will explain the combinatorial underpinnings behind two of the key techniques: Parseval identities for semigroup algebras, and the character algebra of a semigroup.

We will give an overview of the recent attempts of building a structure theory for graphs centered around First-Order transductions: a notion of containment inspired by finite model theory. Particularly, we will speak about the notions of monadic dependence and monadic stability, their combinatorial characterizations, and the developments on the algorithmic front.

Latent space dynamics identification (LaSDI) is an interpretable data-driven framework that follows three distinct steps, i.e., compression, dynamics identification, and prediction. Compression allows high-dimensional data to be reduced so that they can be easily fit into an interpretable model. Dynamics identification lets you derive the interpretable model, usually some form of parameterized differential equations that fit the reduced latent space data. Then, in the prediction phase, the identified differential equations are solved in the reduced space for a new parameter point and its solution is projected back to the full space. The efficiency of the LaSDI framework comes from the fact that the solution process in the prediction phase does not involve any full order model size. For the identification, various approaches are possible, e.g., a fixed form as in dynamic mode decomposition and thermodynamics-based LaSDI, a regression form as in sparse identification of nonlinear dynamics (SINDy) and weak SINDy, and a physics-driven form as projection-based reduced order model. Various physics prob- lems were accurately accelerated by the family of LaSDIs, achieving a speed-up of 1000x, e.g., kinetic plasma simulations, pore collapse, and computational fluid problems.

A classical problem in combinatorial geometry, posed by Erdős in 1946, asks to determine the maximum number of unit segments in a set of $n$ points in the plane. Since then a great variety of extremal problems in finite planar point sets have been studied. Here, we look at such questions concerning triangles. Among others we answer the following question asked by Erdős and Purdy almost 50 years ago: Given $n$ points in the plane, how many triangles can be approximate congruent to equilateral triangles? For our proofs we use hypergraph Turán theory. This is joint work with Balogh and Dumitrescu.

This talk will first introduce combinatorics on permutations and patterns, presenting the basic notions and some fundamental results: the Marcus-Tardos theorem which bounds the density of matrices avoiding a given pattern, and the Guillemot-Marx algorithm for pattern detection using the notion now known as twin-width. I will then present a decomposition result: permutations avoiding a pattern factor into bounded products of separable permutations. This can be rephrased in terms of twin-width: permutation with bounded twin-width are build from a bounded product of permutations of twin-width 0. Comparable results on graph encodings follow from this factorisation. This is joint work with Édouard Bonnet, Romain Bourneuf, and Stéphan Thomassé.

There has been an increased use of scoring systems in clinical settings for the purpose of assessing risks in a convenient manner that provides important evidence for decision making. Machine learning-based methods may be useful for identifying important predictors and building models; however, their ‘black box’ nature limits their interpretability as well as clinical acceptability. This talk aims to introduce and demonstrate how interpretable machine learning can be used to create scoring systems for clinical decision making.

In general, random walks on fractal graphs are expected to exhibit anomalous behaviors, for example heat kernel is significantly different from that in the case of lattices. Alexander and Orbach in 1982 conjectured that random walks on critical percolation, a prominent example of fractal graphs, exhibit mean field behavior; for instance, its spectral dimension is 4/3. In this talk, I will talk about this conjecture for a canonical dependent percolation model.

Rödl and Ruciński established Ramsey’s theorem for random graphs. In particular, for fixed integers $r$, $\ell\geq 2$ they showed that $n^{-\frac{2}{\ell+1}}$ is a threshold for the Ramsey property that every $r$-colouring of the edges of the binomial random graph $G(n,p)$ yields a monochromatic copy of $K_\ell$. We investigate how this result extends to arbitrary colourings of $G(n,p)$ with an unbounded number of colours. In this situation Erdős and Rado showed that canonically coloured copies of $K_\ell$ can be ensured in the deterministic setting. We transfer the Erdős-Rado theorem to the random environment and show that for $\ell\geq 4$ both thresholds coincide. As a consequence the proof yields $K_{\ell+1}$-free graphs $G$ for which every edge colouring yields a canonically coloured $K_\ell$. This is joint work with Nina Kamčev.

In this talk we discuss the paper “Analysis of a detailed multi-stage model of stochastic gene expression using queueing theory and model reduction” by Muhan Ma, et.al., Mathematical Biosciences, 2024.

The study of sleep and circadian rhythms at scale requires novel technologies and approaches that are valid, cost effective and do not pose much of a burden to the participant. Here we will present our recent studies in which we have evaluated several classes of technologies and approaches including wearables, nearables, blood based biomarkers and combinations of data with mathematical models.