Junior Analysis and Probability Seminar

The Junior Analysis and Probability Seminar hosts talks from early-career researchers working broadly in analysis and probability.

Organisers: Žan Bajuk ( ); Barnabás Gárgyán ( ); Hefin Lambley ( ); Grega Saksida ( ).

Time and room: Mondays, 15:00–16:00, weeks 2–10, in D1.07 (Zeeman), followed by discussions in Zeeman Common Room.

Schedule

13 October 2025: Laura Bradby (University of Warwick). A sharp local maximum principle for Ricci flow.
The maximum principle is a very useful tool in the study of Ricci flow, with many known applications. However, in the noncompact case, the maximum principle, as it is usually stated, fails. This means we need to switch to a local version of the result if we wish to work on noncompact manifolds. In this talk, we’ll discuss why the maximum principle is applicable to Ricci flow, as well as the effect that localisation has on both the statement and the proof. Finally, we’ll see a new sharp version of a known local maximum principle, and, time permitting, the idea of the proof.

20 October 2025: Rohan Shiatis (King's College London). The Integrable Snake Model.
On the integer lattice ℤ², a "pure snake configuration" is a permutation on the vertices containing no two-cycles and such that every vertex is mapped to either itself, the vertex to the right, the vertex above or the vertex below. As such, a pure snake configuration admits an interpretation as a collection of snaking, non-intersecting paths on ℤ². Pure snake configurations are a generalisation of lozenge tilings, which are in natural correspondence with paths that only travel right or up. We introduce a partition function on a finite version of this model and study the probabilistic properties of random pure snake configurations chosen according to their contribution to this partition function. Under a suitable weighting, the model is integrable in the sense that we have access to explicit formulas for its partition function and correlation function. We utilise the integrable and determinantal structure of this model in several applications through its various scaling limits, such as to prove a traffic representation of ASEP on the ring.

27 October 2025: Marc Truter (University of Warwick). K-stability: from differential to algebraic geometry.
Please note change of time to 13:00 and change of room to B1.12.
What is a Kähler-Einstein metric? What is Einstein doing here? How does this relate to Ricci curvature, and what on earth is a Fano, Calabi-Yau and general type variety? Most importantly, when does a complex manifold have a Kähler-Einstein metric? In 1997, Tian determined an analytic stability condition which answers this final question. Five years later, in 2002, Donaldson reformulated Tian's statements in purely algebraic terms. In this talk we will explore how this connection provides a bridge for answering questions in differential geometry using algebraic geometry.

3 November 2025: Ariadna León Quirós (University of Tübingen). Proof of the Willmore-type inequality for Riemannian manifolds with non-negative Ricci curvature via vector field method.
In 1977, D. C. Robinson developed a method for proving static vacuum Black Hole uniqueness in General Relativity. This method has recently been generalized to higher dimensions by C. Cederbaum, A. Cogo, B.Leandro, and J. Paolo dos Santos. It turns out that the same philosophy can also be used to prove geometric inequalities, such as the Willmore Inequality in Eucledian space by C. Cederbaum and A. Miehe. In my talk, I will show how to adjust this philosophy to prove the Agostiniani-Fogagnolo-Mazzieri Willmore-type inequality and other related inequalities for Riemannian manifolds with non-negative Ricci curvature. If time permits, I will explain the application of this method to Hamilton's pinching conjecture. This is joint work with C. Cederbaum.

10 November 2025: Isabella Gonçalves de Alvarenga (University of Warwick). Interface problems for variants of the contact process.
The contact process, introduced by Harris in 1974, is an interacting particle system modelling population dynamics. In this model, each site is either empty or occupied by an individual, which can reproduce by sending descendants to neighbouring empty sites. In this talk, we discuss two one-dimensional variants of the model: the multitype contact process and the contact-and-barrier process. We introduce the notion of an interface for these systems and present two main results: the tightness of the interface size and a central limit theorem for the position of the interface. Finally, we outline the proof strategy based on a new construction of those processes, called a patchwork construction.

17 November 2025: Riku Anttila (University of Jyväskylä). Heat kernels on metric spaces.
The heat kernel is, by definition, the fundamental solution of the heat equation. It can also be understood as the transition probability between densities of Brownian motion. In the context of more general metric spaces, heat kernels can be defined and analyzed using the theory of Dirichlet forms. In this talk, I will discuss some applications of heat kernels and their estimates to analysis on metric spaces. If time permits, I will also discuss some results in arxiv:2509.04155.

24 November 2025: Kinga Nagy (Osnabrück University). Large Clusters in Random Geometric Graphs.
Consider a geometrically constructed graph whose vertices are given by a random point set in Euclidean space, and an edge runs between two points whenever their distance is at most some fixed threshold. This construction, called the random geometric graph (RGG), is well-studied in both graph theoretical and geometric settings. In this talk, we discuss different classes of functionals related to the RGG, with a special focus on its large clusters.