Past

It is known that there exists certain randomness in the values of the Moebius function. It is widely believed that this randomness predicts significant cancellations in the summation of the Moebius function times any 'reasonable' sequence. This rather vague principle is known as an instance of the 'Moebius randomness principle'. Sarnak made this principle precise by identifying the notion 'reasonable' as deterministic. More precisely, Sarnak's Moebius Disjointness Conjecture predicts the disjointness of the Moebius function from any arithmetic functions realized in any topological dynamical systems of zero topological entropy. In this talk, I will firstly introduce some background and progress on this conjecture. Secondly, I will talk about some of my work on this. Thirdly, I will talk some related problems to this conjecture.

Given a 4-dimensional Einstein orbifold that cannot be desingularized by smooth Einstein metrics, we investigate the existence of an ancient solution to the Ricci flow coming out of such a singular space. In this talk, we will focus on singularities modeled on a cone over $\mathbb{R}P^3$ that are desingularized by gluing Eguchi-Hanson metrics to get a first approximation of the flow. We show that a parabolic version of the corresponding obstructed gluing problem has a  smooth solution: the bubbles are shown to grow exponentially in time, a phenomenon that is intimately connected to the instability of such orbifolds. Joint work with Tristan Ozuch.

Accurately predicting turbulent fluid mechanics remains a significant challenge in engineering and applied science. Reynolds-Averaged Navier–Stokes (RANS) simulations and Large-Eddy Simulation (LES) are generally accurate, though non-Boussinesq turbulence and/or unresolved multiphysical phenomena can preclude predictive accuracy in certain regimes. In turbulent combustion, flame–turbulence interactions lead to inverse-cascade energy transfer, which violates the assumptions of many RANS and LES closures. We survey the regime dependence of these effects using a series of high-resolution Direct Numerical Simulations (DNS) of turbulent jet flames, from which an intermediate regime of heat-release effects, associated with the hypothesis of an “active cascade,” is apparent, with severe implications for physics- and data-driven closure models. We apply adjoint-based data assimilation method to augment the RANS and LES equations using trusted (though not necessarily high-fidelity) data. This uses a Python-native flow solver that leverages differentiable-programming techniques, automatic construction of adjoint equations, and solver-in-the-loop optimization. Applications to canonical turbulence, shock-dominated flows, aerodynamics, and flow control are presented, and opportunities for reacting flow modeling are discussed.

There has been recent advances in understanding the microscopic origin of the Bekenstein-Hawking entropy of supersymmetric ant de Sitter (AdS) black holes using holography and localization applied to the dual quantum field theory. In this talk, after a brief overview of the general picture, I will propose a BPS partition function -- based on gluing elementary objects called gravitational blocks -- for known AdS black holes with arbitrary rotation and generic magnetic and electric charges. I will then show that the attractor equations and the Bekenstein-Hawking entropy can be obtained from an extremization principle.

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.

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.

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.

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.

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.

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.

TBC

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.