Past

Come to this session to learn how to get started in consultancy from Dawn Gordon at Oxford University Innovation (OUI). After an introduction to what consultancy is, we'll explore case studies of consultancy work performed by mathematicians and statisticians within the university. This session will also include practical advice on how you can explore consultancy opportunities alongside your research work, from finding potential clients to the support that OUI can offer.

In this talk, I will present the notion of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto R. Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). I will explain how the fibered barcode is a particular instance of projected barcodes and how the ISM and the SCD provide lower bounds for the convolution distance. 

Furthermore, in the case where the persistence module considered is the sublevel-sets persistence modules of a function f : X -> R^n, we will explain how, under mild conditions, the projected barcode of this module by a linear map u : R^n \to R is the collection of sublevel-sets barcodes of the composition uf . In particular, it can be computed using software dedicated to one-parameter persistence modules. This is joint work with Nicolas Berkouk.

Habegger showed that a subvariety of a fibre power of the Legendre family of elliptic curves is special if and only if it contains a Zariski-dense set of special points. I'll discuss joint work with Gal Binyamini, Harry Schmidt, and Margaret Thomas in which we use pfaffian methods to obtain an effective uniform version of Manin-Mumford for products of CM elliptic curves. Using this we then prove an effective version of Habegger's result.

I will explain a new construction of an Euler system for the symmetric square of an eigenform and its connection with L-values. The construction makes use of some simple Eisenstein cohomology classes for Sp(4) or, equivalently, SO(3,2). This is an example of a larger class of similarly constructed Euler systems.  This is a report on joint work with Marco Sangiovanni Vincentelli.

Given a non-orientable Lagrangian surface L in a symplectic 4-manifold, how far
can the cohomology class of the symplectic form be deformed before there is no
longer a Lagrangian isotopic to L? I will properly introduce this and a
related question, both of which are less interesting for orientable
Lagrangians due to topological conditions. The majority of this talk will be
an exposition on Evans' 2020 work in which he solves this deformation
question for a particular Klein bottle. The proof employs the heavy machinery
of symplectic field theory and more classical pseudoholomorphic
curve theory to severely constrain the topology and intersection properties of
the limits of certain pseudoholomorphic curves under a process called
neck-stretching. The treatment of SFT-related material will be light and focus
mainly on how one can use the compactness theorem to prove interesting things.

Data-driven reduced-order modeling aims at constructing models describing the underlying dynamics of unknown systems from measurements. This has become an increasingly preeminent discipline in the last few years. It is an essential tool in situations when explicit models in the form of state space formulations are not available, yet abundant input/output data are, motivating the need for data-driven modeling. Depending on the underlying physics, dynamical systems can inherit differential structures leading to specific physical interpretations. In this work, we concentrate on systems that are described by differential equations and possess linear dynamics. Extensions to more complicated, nonlinear dynamics are also possible and will be briefly covered here if time permits.

The methods developed in our study use rational approximation based on Loewner matrices. Starting with the approach by Antoulas and Anderson in '86, and moving forward to the one by Mayo and Antoulas in '07, the Loewner framework (LF) has become an established methodology in the model reduction and reduced-order modeling community. It is a data-driven approach in the sense that what is needed to compute the reduced models is solely data, i.e., samples of the system's transfer function. As opposed to conventional intrusive methods that require an actual large-scale model to reduce (described by many differential equations), the LF only needs measurements in compressed format. In the former category of approaches, we mention balanced truncation (BT), arguably one of the most prevalent model reduction methods. Introduced in the early 80s, this method constructs reduced-order models (ROMs) by using balancing and truncating steps (with respect to classical system theory concepts such as controllability and observability). We show that BT can be reinterpreted as a data-driven approach, by using again the Loewner matrix as a central ingredient. By making use of quadrature approximations of certain system theoretical quantities (infinite Gramian matrices), a novel method called QuadBT (quadrature-based BT) is introduced by G., Gugercin, and Beattie in '22. We show parallels with the LF and, if time permits, certain recent extensions of QuadBT. Finally, all theoretical considerations are validated on various numerical test cases.

A growing plant is a fascinating system involving multiple fields. Biologically, it is a multi-cellular system controlled by bio-chemical networks. Physically, it is an example of an "active solid" whose element (cells) are active, performing mechanical work to drive the evolving geometry. Computationally, it is a distributed system, processing a multitude of local inputs into a coordinated developmental response. In this talk I will discuss how plants, a living information-processing organism, uses physical laws and biological mechanisms to alter its own shape, and negotiate its environment. Here I will focus on two examples reflecting the computational and mechanical aspects: (i) probing temporal integration in gravitropic responses reveals plants sum and subtract signals, (ii) the interplay between active growth-driven processes and passive mechanics.

A common pastime of geometric group theorists is to try and derive algebraic information about a group from the geometric properties of its Cayley graphs. One of the most classical demonstrations of this can be seen in the work of Maschke (1896) in characterising those finite groups with planar Cayley graphs. Since then, much work has been done on this topic. In this talk, I will attempt to survey some results in this area, and show that the class group with planar Cayley graphs is QI-rigid.

Mathematics has always been part of the fabric of culture. References to mathematics in literature go back at least as far as Aristophanes, and encompass everyone from Dostoevsky to Oscar Wilde. In this talk I’ll explore some of the ways that literature has engaged with mathematical ideas, from the 17^th and 18^th century obsession with the cycloid (the “Helen of Geometry”) to the 19^th century love of the fourth dimension.

Spectral properties of Schrödinger operators are studied a lot in mathematical physics. They can give the description of trapped fermionic particles. This presentation will focus on the non-interacting case. I will explain why it is relevant to estimate L^p bounds of orthonormal families of eigenfuntions at the semiclassical regime and then, give the main ideas of the proof.

Lie groupoids are closely connected to pseudo-differential calculus. On a vector bundle considered as a `commutative Lie groupoid' (i.e. as a family of commutative Lie groups), they can be treated using the Fourier transform. In this talk, we explore the extension of this idea to the noncommutative space by employing the tubular neighborhood construction and subsequently adopting a global approach through the introduction of deformation to the normal cone (groupoid). By utilizing this groupoid, we can construct the analytic index of pseudo-differential operators without relying on pseudo-differential calculus.

Furthermore, through the canonical construction of the space of functions with Schwartz decay, pseudo-differential operators on a manifold can be represented as an integral associated with smooth functions on the deformation to the normal cone. This perspective provides a geometric characterization that allows for the direct proof of fundamental properties of pseudo-differential operators.

COURSE_PROPOSAL (12)_2.pdf

The space of unordered tuples. The notion of differentiability and the theory of metric Sobolev in the context of multi-valued functions. Multivalued maximum principle and Holder regularity. Estimates on the Hausdorff dimension of the singular set of Dir-minimizing functions. If time permits: mass minimizing currents and their link with Dir-minimizers. 

The study of the fundamental group of an affine manifold has a long history that goes back to Hilbert’s 18th problem. It was asked if the fundamental group of a compact Euclidian affine manifold has a subgroup of a finite index such that every element of this subgroup is translation. The motivation was the study of the symmetry groups of crys- talline structures which are of fundamental importance in the science of crystallography. A natural way to generalize the classical problem is to broaden the class of allowed mo- tions and consider groups of affine transformations. In 1964, L. Auslander in his paper ”The structure of complete locally affine manifolds” stated the following conjecture, now known as the Auslander conjecture: The fundamental group of a compact complete locally flat affine manifold is virtually solvable.

In 1977, in his famous paper ”On fundamental groups of complete affinely flat manifolds”, J.Milnor asked if a free group can be the fundamental group of complete affine flat mani- fold.
The purpose of the talk is to recall the old and to talk about new results, methods and conjectures which are important in the light of these questions .

The talk is aimed at a wide audience and all notions will be explained 1

The Stokes–Onsager–Stefan–Maxwell (SOSM) equations model the flow of concentrated mixtures of distinct chemical species in a common thermodynamic phase. We derive a novel variational formulation of these nonlinear equations in which the species mass fluxes are treated as unknowns. This new formulation leads to a large class of high-order finite element schemes with desirable linear-algebraic properties. The schemes are provably convergent when applied to a linearization of the SOSM problem.

Magnant and Martin conjectured that every $d$-regular graph on $n$ vertices can be covered by $n/(d+1)$ vertex-disjoint paths. Gruslys and Letzter verified this conjecture in the dense case, even for cycles rather than paths. We prove the analogous result for directed graphs and oriented graphs, that is, for all $\alpha>0$, there exists $n_0=n_0(\alpha)$ such that every $d$-regular digraph on $n$ vertices with $d \ge \alpha n $ can be covered by at most $n/(d+1)$ vertex-disjoint cycles. Moreover if $G$ is an oriented graph, then $n/(2d+1)$ cycles suffice. This also establishes Jackson's long standing conjecture for large $n$ that every $d$-regular oriented graph on $n$ vertices with $n\leq 4d+1$ is Hamiltonian.
This is joint work with Viresh Patel and  Mehmet Akif Yıldız.

TBC

Let K be an algebraically closed field. Given three elements a Lie algebra over K, we say that these elements form an sl_2-triple if they generate a subalgebra which is a homomorphic image of sl_2(K). In characteristic 0, the Jacobson-Morozov theorem provides a bijection between the orbits of nilpotent elements of the Lie algebra and the orbits of sl_2-triples. In this talk I will discuss the progress made in extending this result to fields of characteristic p, and discuss results for both the classical and exceptional Lie algebras. 

We describe a network model for the progression of Alzheimer's disease based on the underlying relationship to toxic proteins. From human patient data we construct a network of a typical brain, and simulate the concentration and build-up of toxic proteins, as well as the clearance, using reaction--diffusion equations. Our results suggest clearance plays an important role in delaying the onset of Alzheimer's disease, and provide a theoretical framework for the growing body of clinical results.

The usual approach to study 2d CFT relies on the Virasoro algebra and its representation theory. Moving away from the criticality, this infinite dimensional symmetry is lost so it is useful to have a look at 2d CFTs from the point of view of more general framework of quantum integrability. Every 2d conformal field theory has a natural infinite dimensional family of commuting higher spin conserved quantities that can be constructed out of Virasoro generators. Perhaps surprisingly two different sets of Bethe ansatz equations are known that diagonalise these. The first one is of Gaudin/Calogero type and was discovered by Bazhanov–Lukyanov–Zamolodchikov in the context of ODE/IM correspondence. The second set is a very natural generalisation of the Bethe ansatz for the Heisenberg XXX spin chain and was found more recently by Litvinov. I will discuss these constructions as well as their relation to W-algebras and the affine Yangian.

We build a model of a financial market where a large number of firms determine their dynamic emission strategies under climate transition risk in the presence of both environmentally concerned and neutral investors. The firms aim to achieve a trade-off between financial and environmental performance, while interacting through the stochastic discount factor, determined in equilibrium by the investors' allocations. We formalize the problem in the setting of mean-field games and prove the existence and uniqueness of a Nash equilibrium for firms. We then present a convergent numerical algorithm for computing this equilibrium and illustrate the impact of climate transition risk and the presence of environmentally concerned investors on the market decarbonization dynamics and share prices. We show that uncertainty about future climate risks and policies leads to higher overall emissions and higher spreads between share prices of green and brown companies. This effect is partially reversed in the presence of environmentally concerned investors, whose impact on the cost of capital spurs companies to reduce emissions. However, if future climate policies are uncertain, even a large fraction of environmentally concerned investors is unable to bring down the emission curve: clear and predictable climate policies are an essential ingredient to allow green investors to decarbonize the economy.

Joint work with Pierre Lavigne

PhD_course_Roux.pdf

We will start from the description of a particle system modelling a finite size network of interacting neurons described by their voltage. After a quick description of the non-rigorous and rigorous mean-field limit results, we will do a detailed analytical study of the associated Fokker-Planck equation, which will be the occasion to introduce in context powerful general methods like the reduction to a free boundary Stefan-like problem, the relative entropy methods, the study of finite time blowup and the numerical and theoretical exploration of periodic solutions for the delayed version of the model. I will then present some variants and related models, like nonlinear kinetic Fokker-Planck equations and continuous systems of Fokker-Planck equations coupled by convolution.