Junior Analysis and Probability Seminar

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

Organisers: Dimitrios Andreakis ( ); Žan Bajuk ( ); Barnabás Gárgyán ( ).

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

Schedule

Term 1

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.

8 December 2025: Sebastian Woodward (University of Oxford). Quantitative Stability Estimates for almost minimising maps of the Dirichlet Energy.
A natural question in the study of many variational problems is the quantitative stability of minimisers, i.e. the question of whether, and at what rate, the distance to the set of minimisers can be controlled in terms of the energy defect. In this talk, I will present joint work with Professor Melanie Rupflin, which establishes quantitative stability estimates for degree one maps of the Dirichlet Energy from either the torus or hyperbolic surfaces into the sphere. While the classical question of quantitative stability fails due to the non-existence of degree one minimisers from the given surface, the existence of a “singular” minimiser consisting of a “bubble” and “base map” can be shown. In this work, we establish a quantitative stability estimate for this set of “singular” minimisers through a carefully constructed combined gradient flow that evolves both domain and map.

Term 2

19 January 2026: Oliver Brown (University of Bristol). Anyons on many-particle quantum graphs.
It is known [1] that quantum statistics on planar graphs are richer than in ℝ² or ℝ³. We present the first general construction of self-adjoint Hamiltonians for anyons on metric graphs, resolving an open problem in the theory of quantum statistics on graphs [2]. Adapting existing many-particle quantum graph frameworks with two-body contact interactions [3] we construct the Hamiltonian as a self-adjoint operator via the Friedrichs extension. The Hamiltonian is parameterised by representations of the graph braid group to incorporate non-Abelian exchange statistics and Aharonov-Bohm phases via a systematic choice of branch cuts. We analyse eigenstate regularity by adapting elliptic regularity results [4], and compute numerical eigenstates for several simple graphs as the fractional statistics parameter varies. The results indicate that eigenfunction regularity is dependent on particle statistics and interactions.

[1] Harrison, J.M., Keating, J.P., Robbins, J.M., Sawicki, A.: n-particle quantum statistics on graphs. Communications in Mathematical Physics 330(3), 1293– 1326 (2014)https://doi.org/10.1007/s00220-014-2091-0
[2] Balachandran, A.P., Ercolessi, E.: Statistics on networks. International Journal of Modern Physics A; (United States) 7:19 (1992)https://doi.org/10.1142/S0217751X9200209X
[3] Bolte, J., Kerner, J.: Quantum graphs with two-particle contact interactions. Journal of Physics A: Mathematical and Theoretical 46(4), 045207 (2013)https://doi.org/10.1088/1751-8113/46/4/045207
[4] Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (2011).https://books.google.co.uk/books?id=lJTtujsZSgkC

26 January 2026: Leo Tyrpak (University of Oxford). Fluctuations in population models with non-local interactions..
We analyse the fluctuations of the (substantially generalised) Bolker-Pacala-Dieckmann-Law model as introduced in (Eth+24). These fluctuations are shown to converge to the mild solution of a linear SPDE. As a byproduct we obtain a Wright Malecot-type Isolation by Distance formula by deploying the approach ofFor22.

2 February 2026: Andrey Chernyshev (University of Warwick). Normalization flow.
In the talk we will consider a new approach to the normal form theory for a system of ODEs in a neighbourhood of an equilibrium point. Traditional procedures carry out the normalization step by step: the nonresonant terms in the Taylor expansion of the vector field are eliminated first in degree 2, then (by another change of variables) in degree 3, and so on. We propose a different strategy. Consider the infinite-dimensional space of all vector fields with a singular point (an equilibrium) at the origin. In this space we will construct a flow (generated by a certain differential equation) with the following properties. Shifts along the trajectories of this flow correspond to changes of variables. The flow moves in the direction of the subspace of normal forms. Thus the normalization procedure becomes continuous. The formal aspect of the theory (as in the traditional approach) causes no difficulties. The analytic aspect and the issues of convergence of the series are, as usual, nontrivial.
9 February 2026: Gabriele Cassese (University of Oxford).
16 February 2026: Dain Kim (Massachusetts Institute of Technology). Ricci flow on ALF manifolds.
Asymptotically Locally Flat (ALF) Ricci-flat metrics are expected to model certain long-time singularities in four-dimensional Ricci flow, so understanding their stability is essential. In this talk, I will discuss that conformally Kähler, non-hyperkähler Ricci-flat ALF metrics are dynamically unstable under Ricci flow. Our work establishes three key tools in this setting: a Fredholm theory for the Laplacian on ALF metrics, the preservation of the ALF structure along the Ricci flow, and an extension of Perelman’s λ-functional to ALF metrics. This is joint work with Tristan Ozuch.
23 February 2026: Andrew Roberts (University of Leeds).
2 March 2026: June Lo (King's College London).
9 March 2026: Tirumala Venkata Chakradhar (University of Bristol).
16 March 2026: Sidney Stanbury (University of Cambridge).