The Heriot-Watt Analysis Seminar takes place on Wednesdays during term time. Unless otherwise specified, our venue for Semester 1 is in Room 1.20, Dugald Stewart Building, 3 Charles St, Edinburgh EH8 9AD. The building is located near the Maxwell Institute. All are welcome to attend the meetings.
If you have any questions or suggestions, please contact one of the organiser: Lyonell Boulton and Oana Pocovnicu.
Wednesday 14 October 2026: Room 1.20, Dugald Stewart Building
3:00 pm Alpár Meszaros (Durham University)
On Cross-Diffusion Systems with Independent Drifts and no Self-Diffusion: Mixing Dynamics
We present a global existence theory for weak solutions to a two-species cross-diffusion system, in one space dimension, in which the evolution of each species is governed by two mechanisms: a diffusion which acts only on the sum of the species with a logarithmic or fast diffusion type pressure law, and a drift term, which can differ between the two species. We will discuss two types of phenomena, regarding qualitative properties of solutions. First, we will show that if the initial densities are ‘totally mixed’, this property will propagate over time. Second, we will show how to remove this condition on the initial data, resulting in weak solutions which can be only partially mixed. All these solutions are in contrast with the segregated solutions, known previously in the literature. The talk will be based on joint works with Guy Parker (Lyon).
Wednesday 7 October 2026: Room 1.20, Dugald Stewart Building
3:00 pm Jonathan Eckhardt (University of Loughborough)
Wednesday 23 September 2026: Room 1.20, Dugald Stewart Building
3:00 pm Jacob Denson (University of Edinburgh)
A Characterization of Boundedness For Multipliers of Spherical Harmonic Expansions
Any function f: Sᵈ → ℂ on the d-dimensional sphere has a unique expansion of the form f = Σ fₖ, where fₖ is a spherical harmonic of degree k. For a function m: (0,∞) → ℂ, define a family of operators Tᴿ on Sᵈ by setting Tᴿ f = Σ f(k/R) fₖ. Such operators are called multipliers for spherical harmonic expansions, or zonal convolution operators. In this talk, we explore the boundedness of the operators {Tᴿ} on Lᵖ(Sᵈ). Our main result is that, for d ≥ 4 and a limited range of p, the operators {Tᴿ} are uniformly bounded on Lᵖ(Sᵈ) if and only if the Fourier multiplier operator on ℝᵈ with symbol m(|·|): ℝᵈ → ℂ is bounded on Lᵖ(ℝᵈ). This is the first transference principle of its kind for spectral multipliers on any compact manifold for p ≠ 2. Our methods are closely connected to endpoint local smoothing inequalities for the wave equation on Sᵈ.
Wednesday 17th June 2026
Maxwell Day
Venue: G.159 MacLaren Stuart Room, Old College University of Edinburgh, EH8 9YL
Programme:
2:00pm Irina Nenciu (University of Illiois Chicago)
Confinement and essential self-adjointness via weak anisotropic Hardy inequalities
We consider the problem of essential self-adjointness of the drift-diffusion operator $H=-\frac{1}{\rho}\nabla\cdot\rho D\nabla+V$ on domains $\Omega$ in $\mathbb R^d$. We give criteria showing how the behavior as $x\to\partial\Omega$ of the coefficients $\rho$, $D$ and the potential $V$ balances to ensure essential self-adjointness of $H$, which in turn is closely connected to confinement of quantum particles to $\Omega$. In the process, we will discuss an essential tool, which are new anisotropic Hardy inequalities. This is based on joint work with G. Nenciu (IMAR, Bucharest).
3:00pm Coffee Break
3:30pm Christof Sparber (University of Illiois Chicago)
Schrödinger equations with competing nonlinearities
We consider Schrödinger equations with competing nonlinearities in spatial dimensions up to three, for which global existence holds (i.e., no finite-time blow-up). A typical example is the case of the (focusing-defocusing) cubic-quintic NLS. We recall the notions of energy minimizing versus action minimizing ground states and show that, in general, the two must be considered as nonequivalent. The question of long-time behavior of solutions, in particular the problem of ground-state (in-)stability will be discussed using analytical results and numerical simulations. Joint work with R. Carles (U Rennes) and C. Klein (U. Bourgogne).
Wednesday 27th May 2026
Presentations by the first cohort of Jack Carr interns
Venue: G.159 MacLaren Stuart Room, Old College University of Edinburgh, EH8 9YL
Programme:
3:00pm Filone Gilson Longmou Moffo (AIMS Rwanda)
H^1-Global wellposedness of the nonlinear Schrödinger equation with exponential non-linearity on compact surface
In this work, we establish a probabilistic global theory (existence for all times, uniqueness and continuity) in $H^1$ for the NLS equation with a Moser-Trudinger nonlinearity posed on compact surfaces. This equation is known to be the two dimensional counterpart to the classical energy-critical Schrödinger equations and there exists a trichotomy around the criticality of the equation based on the size of the total energy. In particular, for supercritical regimes (large energy), the equation is known to exhibit instabilities : the (uniform) continuity of the flow fails to hold. Also large data distributional non unique probabilistic solutions have been obtained in the work of Casteras and Monsaingeon but their setting does not handle the uniqueness issue for the $H^1$-data and therefore could not define a flow for this regularity. Our main focus here is to build a single probabilistic framework that provides both existence, uniqueness, and continuity with respect to the initial data in $H^1$. Our uniqueness and continuity are based on the so-called Yudowich argument , and the probabilistic estimates are derived through the IID limit procedure.
3:40pm Emma Nel (Stellenbosch University, South Africa)
Laguerre Spectral Collocation
Spectral collocation methods are powerful numerical techniques for solving differential equations, offering high accuracy through global polynomial approximation. However, classical spectral methods face significant challenges when applied to complicated geometries, large-scale systems, and problems posed on infinite or semi-infinite domains. This talk discusses several approaches aimed at extending the practicality and efficiency of spectral collocation techniques, including the treatment of complicated domains through domain decomposition, the development of sparse formulations, and stable Laguerre-based methods for problems on unbounded intervals. Particular attention is given to numerical stability and convergence in Laguerre spectral collocation, where mitigating overflow and underflow issues in differentiation matrices, as well as accelerating convergence through scaling and truncation strategies, is essential for making these methods competitive with alternative approaches.
4:10pm Coffee break (MIGS space, 5th floor, Bayes Centre)
Wednesday 6 May 2026: Room 1.17 Dugald Stewart Building, 3 Charles St, Edinburgh EH8 9AD
3:00pm Casper Oelen (Heriot-Watt University)
Non-Birkhoff periodic orbits in symmetric planar billiards
A mathematical billiard is a dynamical system describing the free motion of a point mass inside a domain, subject to elastic reflections at the boundary. Such systems are simple to describe yet exhibit a wide range of dynamical behaviours. A classical result of Poincaré and Birkhoff implies that every smooth, strictly convex planar billiard admits at least two periodic orbits of any rational rotation number. These are called Birkhoff orbits, and they admit a simple geometric description – they are homeomorphic to regular polygons. In this talk, we discuss the existence of non-Birkhoff periodic orbits in symmetric convex planar billiards – orbits that lack the ordering property characteristic of regular polygons. We present a quantitative criterion for the existence of such orbits with prescribed period, winding number, and spatio-temporal symmetry. Furthermore, we show that arbitrarily small analytic perturbations of the circular billiard admit non-Birkhoff periodic orbits of any rational rotation number. This is joint work with Bob Rink and Mattia Sensi.
Friday 20 March 2026
Maxwell Institute mini-symposium on analysis and PDEs
Venue: G1 Lecture Theatre, Paterson’s Land Building, Holyrood Rd, EH8 8AQ
Speakers:
2:20pm Iosif Polterovich (Université de Montréal)
Pólya’s eigenvalue conjecture, lattice points and Bessel zeros
The celebrated Pólya conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacians on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl’s asymptotics. The conjecture is known to hold for domains that tile the Euclidean space but remains largely open in general. In this talk, we will explain the motivation behind the conjecture and discuss some recent advances, particularly for balls and annuli. These developments, in turn, are linked to new results on lattice point counting and zeros of Bessel functions. The talk is based on joint work with Nikolay Filonov, Michael Levitin, and David Sher.
3:20pm Coffee Break
4:00pm Nikolay Tzvetkov (ENS Lyon)
Gauge transforms, random averaging operator ansatz and improved probabilistic well-posedness for the radial NLS on the three dimensional ball
We construct probabilistic strong solutions to the cubic nonlinear Schrödinger equation on the three-dimensional ball with radial initial data. These solutions lie in the supercritical regime with respect to the probabilistic scaling introduced by Deng-Nahmod-Yue. We achieve such a result thanks to the use of a gauge transform (without changing the equation) and refined modulation analysis with random averaging operators. This improves on a result by Bourgain-Bulut. This is a joint work with Burq, Camps and Chenmin Sun.
7:00pm Dinner
Thursday 26 February 2026
2026 Carr Memorial Lecture
Venue: ICMS Lecture Theatre (Potterrow)
4:00pm Moustapha Fall (AIMS Senegal)
Equilibrium state of some surfaces and their patterns
Wednesday 25 February 2026: 50 George Square Room G.06
3:00 pm Moustapha Fall (AIMS Senegal)
Uniqueness and non-degeneracy properties of the Generalized Benjamin-Ono equation
Solitary waves of the Benjamin-Ono equation are completely described by nontrivial solutions to the equation (-\Delta)^{\frac{1}{2}} u+u=u^2 on $\mathbb{\R}$. The uniqueness of such solutions, up to translations, was established in a classical work of Amick and Toland (Acta Math., 1991). In this talk, we present a recent result, obtained in collaboration with Tobias Weth, which extends the uniqueness result of Amick and Toland to the more general equation (-\Delta)^{s} u+u=u^2 on $\mathbb{\R}$, for the optimal range of the parameter s.
Wednesday 21 January 2026: Lecture Theatre G.03, 50 George Square.
3:00 pm Georgios Papamikos (University of Essex)
Lie symmetry reductions of the p-Laplace equation and invariant solutions
I will present the construction of the complete symmetry group of Lie point transformations for the p-Laplace equation for all real p, including p=∞ (Aronsson equation), and classify the corresponding symmetry generators. This classification leads to a classification of symmetry reductions to ODEs. I will also present exact invariant solutions (p-harmonic functions) which we construct as solutions of the obtained reduced ODEs. I will also explain how to derive a known duality of the p-Laplace equation directly from Noether's theorem. More general symmetries of contact type, and their use, will also be presented. This is joint work with Dr. M. Akman.
Wednesday 14 January 2026: EM G.83
Analysis and Probability Colloquium
2:00 pm Anke Wiese (Heriot-Watt University)
A Chen-Strichartz series expansion for Lévy models
For deterministic differential equations and for continuous stochastic differential equations, the Chen-Strichartz series expansion, an expansion of the solution in terms of commutators of vector fields and iterated integrals, is well-known to play a central role in understanding analytic and geometric properties of the solution to the equations and is key in the development of numerical integration schemes that preserve qualitative properties of the solution.
For stochastic differential equations that are driven by Lévy processes, the flow map, which describes the evolution of the solution, involves additional operators arising from jumps. We will derive a Chen-Strichartz series representation, depending only on commutators of vector fields and iterated integrals, and we will provide an explicit expression for the components in this series, generalising previous results for deterministic and continuous stochastic differential equations. We will illustrate the new Chen-Strichartz series representation in the application to a multi-dimensional stochastic volatility model.
Wednesday 12 November 2025
Maxwell Analysis Mini-symposium
Venue: Old College (G.159 - MacLaren Stuart Room)
Speakers:
1:30pm David Beltran (Universitat de València)
A fractal local smoothing conjecture for the wave equation
The local smoothing phenomenon for the wave equation consists in obtaining a regularity gain over the known fixed-time L^p estimates for the half-wave propagator if one takes an L^p-integration in time over the interval [1,2]. This phenomenon was first observed by Sogge, who conjectured the sharp regularity gain depending on the Lebesgue exponent. Obtaining the conjectured bounds has become a central problem in Harmonic Analysis. It continues open in dimensions 3 and higher, but there has been lots of partial progress in recent years. In this talk we present a fractal version of that conjecture, in which the integration is now taken over an arbitrary subset E of [1,2]. This new family of conjectured inequalities “interpolates” between the fixed-time estimates and the standard local smoothing estimates. The conjectured regularity gain for each Lebesgue exponent depends on a quantity involving the Assouad spectrum of and the Legendre transform. We provide positive evidence for this conjecture, verifying it for radial functions and any Lebesgue exponent, and for arbitrary functions in the off-diagonal Strichartz regime. This is joint work with Joris Roos, Alex Rutar and Andreas Seeger.
2:30pm Oana Pocovnicu (Heriot-Watt University)
Invariant Gibbs dynamics for fractional wave equations in negative Sobolev spaces
In this talk, we consider a fractional nonlinear wave equation with a general power-type nonlinearity (FNLW) on the two-dimensional torus. Our main goal is to construct invariant global-in-time Gibbs dynamics for FNLW. We first construct the Gibbs measure associated with this equation. By introducing a suitable renormalisation, we then prove almost sure local well-posedness with respect to Gibbsian initial data. Finally, we extend solutions globally in time by applying Bourgain's invariant measure argument. We also consider the case of initial data consisting of the randomisation of a given pair of functions of negative regularities. We show that, in this case, probabilistic well-posedness fails unless we impose that the given pair has additional Fourier-Lebesgue regularity.
3:30pm Coffee Break
4:00pm Shaoming Guo (Chern Institute of Mathematics)
Wednesday 5 November 2025: room G.06, 50 George Square.
3:00 pm Guopeng Li (Beijin Institute of Technology)
Probabilistic well-posedness of dispersive PDEs beyond variance blowup
In this talk, we use the fourth moment theorem to investigate a possible extension of probabilistic well-posedness theory of nonlinear dispersive PDEs with random initial data beyond variance blowup. As a model equation, we study the Benjamin-Bona-Mahony equation (BBM) with Gaussian random initial data. By introducing a suitable vanishing multiplicative renormalization constant on the initial data, we show that solutions to BBM with the renormalized Gaussian random initial data beyond variance blowup converge in law to a solution to the stochastic BBM forced by the derivative of a spatial white noise. Based on the joint work with Jiawei Li & Tadahiro Oh (Edinburgh), and Nikolay Tzvetkov (ENS Lyon).
Wednesday 1 October 2025: room G.06, 50 George Square.
3:00 pm Katie Gittins (Durham University)
Nodal counts for the Robin problem on Lipschitz domains
We consider the Courant-sharp eigenvalues of the Laplacian (with boundary conditions) on Euclidean domains. That is, the eigenvalues that have a corresponding eigenfunction which achieves the maximum number of nodal domains given by Courant's theorem. We will first give an overview of previous results for the Courant-sharp Dirichlet, Neumann, and Robin eigenvalues of the Laplacian. In particular, Pleijel's theorem and upper bounds for the number of Courant-sharp eigenvalues. We will then present recent joint work with Asma Hassannezhad, Corentin Léna, and David Sher which extends previous results in various directions.
Wednesday 3 September 2025: room G.06, 50 George Square.
3:00 pm Alexandra Melnig (Alexandru Ioan Cuza University)
Controllability for coupled parabolic systems
In this talk we focus on internally distributed controls acting on part of the system, while the equations are coupled through zero-order terms in star-like or tree-like configurations. Under appropriate assumptions on the coupling functions, we prove local exact controllability to stationary solutions. The key point is establishing Carleman estimates for the adjoint to the linearized system, with suitable observation operators.
Wednesday 9 April 2025: Appleton Tower room 2.04
3:00 pm Maryna Kachanovska (ENSTA)
Limiting absorption principle for a degenerate problem modelling a plasma resonance
In this talk, we study the two-dimensional Maxwell equations in cold plasmas. Various degeneracies of the cold plasma dielectric tensor lead to the appearance of plasma resonances, namely, solutions with low regularity. We focus on one such case, describing the so-called hybrid resonances. In this situation, the Maxwell equations reduce to a second-order PDE with a degenerate coefficient in the principal part of the operator. We prove that the viscosity limit of the problem yields solutions with very peculiar singularities. The key difficulty is then to formulate a well-posed problem satisfied by this solution, with the main issue being uniqueness. We show how to equip this problem with a radiation-type condition, which ensures the uniqueness of the solutions. Towards the end of the talk, we discuss some difficulties that arise in the numerical modelling of the problem.
Wednesday 26 March 2025
Maxwell Analysis Mini-symposium
Venue: Usha Kasera Lecture Theatre (Old College)
Speakers:
1:00pm Pierre Germain (Imperial)
Asymptotic stability of solitary waves in nonlinear dispersive equations
I will present some recent developments on the asymptotic stability of solitary waves in nonlinear dispersive equations, focusing on the 1D case, in particular nonlinear Klein-Gordon and nonlinear Schrodinger equations. New ideas involving the distorted Fourier transform and nonlinear resonances have led to important progress. I will rely on joint work with Fabio Pusateri and Charles Collot.
2:00pm Coffee Break
2:30pm Laure Saint-Raymond (IHES)
The effect of viscosity on internal waves
In presence of topography, forcing 2D inviscid internal waves leads to the formation of singular geometric patterns, called ``attractors", which accumulate most of the energy. Our goal is to understand how this phenomenlogy is modified by a small viscosity.
Wednesday 12 March 2025: Appleton Tower room 2.04
3:00 pm Lukas Schimmer (Loughborough)
Spectral inequalities for Jacobi operators as conjectured by Hundertmark and Simon
In 2002, Hundertmark and Simon proved Lieb–Thirring inequalities for Jacobi operators. They conjectured that their bounds could be improved by replacing a term (which depends on the off-diagonal parts of the operator) by its positive part. In this talk, I will present a proof of their conjecture. Subsequently I will relate (sharp) Lieb–Thirring inequalities for Jacobi operators to (sharp) Lieb–Thirring inequalities for Schrödinger operators on the real line. This talk is partly based on joint work with A. Laptev and M. Loss.
Wednesday 19 February 2025 (note unusual date and venue): Appleton Tower room 2.05
3:00 pm Daniel Eceizabarrena (BCAM)
Multifractality in the evolution of vortex filaments. Abstract
Wednesday 12 February 2025: Appleton Tower room 2.04
3:00 pm Jonathan Fraser (St Andrews)
L^2 restriction estimates from the Fourier spectrum
The Stein--Tomas restriction theorem is an important result in Fourier restriction theory. It gives a range of q for which Lq→L2 restriction estimates hold for a given measure, in terms of the Fourier and Frostman dimensions of the measure. I will discuss recent work where we improve this theorem using the Fourier spectrum; a family of dimensions that interpolate between the Fourier and Sobolev dimensions for measures. This is joint work with Marc Carnovale and Ana de Orellana.
Wednesday 29 January 2025: Appleton Tower room 2.04
3:00 pm Antonio Arnal (Belfast) - POSTPONED!
Energy decay of solutions of the wave equation with unbounded damping at infinity
We study the long-time behaviour of the solutions of the wave equation with damping a unbounded at infinity and defined in an open (possibly unbounded) set Ω ⊂ R^n . Using multiplier methods, Ikehata and Takeda (2017) proved polynomial decay rates for unbounded continuous damping when n ≥ 3. Our results significantly extend their findings by assuming minimal regularity conditions on a and placing no restriction on the space dimension n. In addition, we apply different methods that rely upon some recent advances in the theory of operator semigroups on Hilbert spaces and upon the spectral analysis of the resolvent of the generator G of the equation. This presentation is based on joint work with Borbala Gerhat, Julien Royer and Petr Siegl.
Wednesday 15 January 2025: Appleton Tower room 2.11 - Notice the Double Bill
3:00 pm Duvan Henao (Universidad de O'Higgins)
Swelling-induced debonding of polymer gels
We present a thin-film asymptotic analysis of the debonding of a polymer hydrogel from a rigid substrate when exposed to solvent [SIAM J. Appl. Math. 84 (2024); J. Elast. 153 (2023); J. Elast. 141 (2020)]. When a confined simplified setting is considered, the higher integrability of Jacobians leads to a rigorous existence theorem for the polyconvex functional that results from adding the Flory-Huggins entropic mixing energy to the energy of the elastic distortion of polymer chains. The study of the energy release rate leads to a concise and explicit formula, whose range of validity is assessed by means of a particular finite element scheme. A numerical study also shows that the simplified formula also serves as an upper bound for the energy release rate in the unconfined three-dimensional case. This is work in collaboration with C. Calderer, C. Garavito, S. Lyu, L. Tapia, M. Sanchez, R. Siegel, and S. Song.
4:30 pm Ewelina Zatorska (Warwick) - CANCELLED DUE TO TRAVEL DISRUPTION!
One-dimensional compressible Euler equations with non-local effects
This talk will be devoted to a one-dimensional model of collective motion. The considered system consists of compressible Euler equations with nonlocal interaction term playing the same role as the pressure term in fluids’ equations. I will present some of our recent results concerning existence of strong and weak solutions, long-time asymptotic of solutions, and singular limits through relative entropy method based on the two-velocity formulation.
Wednesday 27 November 2024
Maxwell Analysis Mini-symposium
Venue: Sydney Smith Lecture Theatre (Medical School, Teviot)
Speakers:
1:30pm François Genoud (EPFL)
Finite time blow-up for the critical nonlinear Schrödinger equation on a star graph
The talk will start with a brief history of finite time blow-up solutions for critical nonlinear Schrödinger equations (NLS) in various contexts. Once the stage is set, the construction of a finite time blow-up solution for the critical NLS on a star graph with delta interaction will be presented. The simplest configuration of a graph with two branches corresponds to the NLS on the line with a "delta potential" at the origin. The general case involves a one-dimensional Laplace operator on the graph with Robin boundary conditions at the vertex. The blow-up analysis relies on the resolution of the nonlinear Cauchy problem within the domain of the corresponding linear operator.
This is joint work with Stefan Le Coz and Julien Royer.
2:30pm Kaj Nyström (Upsala University)
Caloric Measure and Parabolic Uniform Rectifiability
The heat equation stands as a cornerstone in mathematics, physics, and various applied fields, describing the distribution of heat (or diffusion of particles) over time. A fundamental question associated with this equation is the Dirichlet problem, which seeks solutions given boundary conditions---a challenge that links the behavior of the heat equation to the geometry of underlying spaces. In recent collaborative work with S. Bortz, S. Hofmann, and J-M. Martell, we address two longstanding conjectures on the $L^p$ Dirichlet problem for the heat equation, caloric measure, and parabolic uniform rectifiability. First, we establish that, for parabolic Lipschitz graphs, the solvability of the $L^p$ Dirichlet problem is equivalent to parabolic uniform rectifiability. Second, we show that, in the broader setting of parabolic Ahlfors-David regular boundaries, the solvability of the $L^p$ Dirichlet problem necessitates parabolic uniform rectifiability. This talk will outline these results, highlighting the intricate interplay between the heat equation and geometric regularity.
3:30pm Coffee Break
4:00pm Máté Weirdl (The University of Memphis)
Basecamp for convergence of ergodic averages along subsequences
Wednesday 13 November 2024
3:00 PM David Seifert (Newcastle)
Rates of decay for operator semigroups and damped waves
Semigroup theory has long played a central role in the study of damped waves and other linear evolution equations. One of the most influential results of recent times has been a theorem due to Borichev and Tomilov (Math. Annalen, 2010), which yields optimal polynomial rates of decay for classical semigroup orbits provided the resolvent satisfies a corresponding polynomial estimate along the imaginary axis. In this talk I shall present an extension of the Borichev-Tomilov theorem beyond the purely polynomial case to a much larger class of resolvent bounds. This result is optimal in several ways. I shall also give examples illustrating how the abstract theory can be used to obtain sharp rates of energy decay in wave equations subject to different types of damping.
Wednesday 6 November 2024
3:00 PM Federico Dipasquale (Scuola Superiore Meridionale)
Two-dimensional ferronematics: Asymptotics for critical points. Abstract
28-30 October 2024
14-18 October 2024
Wednesday 2 October 2024
3:00 PM Edoardo Tolotti (Universita di Pavia)
Stability of the Von Kármán regime for thin plates under Neumann boundary conditions
In this talk we analyze the stability of the Von Kármán model for thin plates subject to pure Neumann conditions and to dead loads, with no restriction on their direction. We show a stability alternative, which extends previous results by Lecumberry and Müller in the Dirichlet case. Because of the rotation invariance of the problem, their notions of stability have to be modified and combined with the concept of optimal rotations due to Maor and Mora. Finally, we prove that the Von Kármán model is not compatible with some specific types of forces. Thus, for such, only the Kirchhoff model applies.
Wednesday 25 September 2024
No Seminar: MACS - Lyell Workshop
8-13 September 2024
Wednesday 4 September 2024
3:00 PM Matthias Täufer (Hagen) [Room 4.18, 40 George Square]
Quantitative unique continuation and applications
Unique continuation is a rigidity property which many solutions of PDEs have: If they vanish on a subdomain they cannot vanish identically. Quantitative variants of unique continuation establish a relation between the norm of functions and the norm of their restriction to a subdomain and are an exciting field connecting harmonic analysis with spectral theory and applications to engineering and Mathematical Physics.
In this talk, I will present several quantitative unique continuation estimates for functions in spectral subspaces of Schrödinger operators and two applications thereof: (1) Control cost estimates for the heat equation with explicit estimates on the cost of controllability which allow to study controllability in the so-called homogenization regime, and (2) Anderson localization for random Schrödinger operators.
Wednesday 17 January 2024
4:30 PM Asma Hassannezhad (Bristol)
Steklov eigenvalues of negatively curved manifolds
The geometry and topology of negatively curved manifolds are subtly reflected in a geometric bound for the Laplace eigenvalues, a connection that has been explored since the 1980s. Among these results, we can mention the celebrated result of Schone in 1982 on the Laplace spectral gap on closed negatively curved manifolds of dimension at least three. Building upon these foundational studies in the Laplace case, we study the Steklov eigenvalues of pinched negatively curved manifolds with totally geodesic boundaries. The Steklov eigenvalues are associated with a first-order elliptic pseudodifferential operator known as the Dirichlet-to- Neumann operator. We discuss a counterpart of Schone's result for the Steklov problem on negatively curved manifolds of dimension at least three. This talk is based on joint work with Ara Basmajian, Jade Brisson, and Antoine Métras.
Wednesday 7 February 2024
4:30 PM Angeliki Menegaki (Imperial College London)
L2-stability for the 4-waves kinetic equation around the Rayleigh-Jeans equilibrium
We consider the four-waves spatial homogeneous kinetic equation arising in weak wave turbulence theory. In this talk I will present some new results on the existence and long-time behaviour of solutions around the Rayleigh-Jeans thermodynamic equilibrium. In particular, introducing a cut-off on the frequencies, I will present an L^2 stability of mild solutions for initial data close to Rayleigh-Jeans. If time permits, I will discuss an ongoing work on the same questions for radial solutions, without the cut-off on the frequencies. The last part is joint work with Miguel Escobedo (UPV/EHU).
Wednesday 14 February 2024
4:30 PM Cleopatra Christoforou (Cyprus)
An exposition on hyperbolic balance laws with application to an Euler-type flocking model
Mathematical models introduced to capture the emergent behavior of self-organized systems have brought new mathematical challenges and most studies on flocking models investigate so far smooth solutions. In this talk, we will first have an overview of the theory of entropy weak solutions to hyperbolic balance laws and then describe an Euler-type flocking problem with an all-to-all interaction kernel. We will discuss how the theory of balance laws is further developed to establish the global existence of entropy weak solutions to Euler-type flocking systems with arbitrary initial data and also capture unconditional time-asymptotic flocking behavior for weak solutions.
Thursday 22 February 2024
11 AM Beatrice Costeri (Pavia)
A microlocal approach to the stochastic nonlinear Dirac equation
We present a novel framework for the study of a wide class of nonlinear Fermionic stochastic partial differential equations of Dirac type, which is inspired by the functional approach to the λ Φ^3 model. The main merit is that, by realizing random spinor fields within a suitable algebra of functional-valued Dirac distributions, we are able to use specific techniques proper of microlocal analysis. These allow us to deal with renormalization using an Epstein-Glaser perspective, hence without resorting to any specific regularization scheme. As a concrete example we shall use this method to discuss the stochastic Thirring model in two Euclidean dimensions and we shall comment on its applicability to a larger class of Fermionic SPDEs. Based on joint work with A. Bonicelli, C. Dappiaggi and P. Rinaldi -- ArXiv: 2309.16376.
Friday 23 February 2024, 1:20pm - 5:20pm
Microlocal analysis & PDEs: advances and perspectives
Venue: ICMS Lecture Theatre
Speakers:
1:20 PM Matteo Capoferri (Heriot-Watt)
“Pre-seminar”-style talk: Microlocal analysis and QFT: what’s it all about
1:45 PM Gabriel Schmid (Genova)
The Quantization of Maxwell Theory in the Cauchy Radiation Gauge: Hodge Decomposition and Hadamard States
2:20 PM Claudia Garetto (Queen Mary)
Higher order hyperbolic equations with multiplicities
3:10 PM Coffee Break
3:40 PM Claudio Dappiaggi (Pavia)
Interplay between Stochastic Partial Differential Equations and microlocal analysis
4:30 PM Linhan Li (Edinburgh)
A Green function characterization of uniform rectifiability of any codimension
The support of the London Mathematical Society (Research Grant - Scheme 9) is gratefully acknowledged.Wednesday 28 February 2024
4:30 PM Heiko Gimperlein (Innsbruck)
Regularity theory for fractional Laplacians: a geometric approach and applications
We consider the sharp boundary regularity of solutions to the Dirichlet problem for the fractional Laplacian on a smoothly bounded domain in Euclidean space. The fractional Laplacian is defined via the extension method, as a Dirichlet-to-Neumann operator for a degenerate elliptic problem in a half-space of one higher dimension. We use techniques from geometric microlocal analysis to analyse the regularity of solutions, with particular emphasis on asymptotic expansions and Hölder continuous data. Detailed and sharp results about this problem have been obtained by Gerd Grubb. Our complementary approach extends to polygonal domains and to the Calderon (inverse) problem.
Wednesday 6 March 2024
4:30 PM Francesco Ferraresso (Sassari)
Spectral analysis of dissipative Maxwell systems
Dissipative Maxwell systems find frequent application in the modelling of electromag- netic wave propagation through conductive media. In this scenario, the medium absorbs part of the EM energy of the wave, resulting in loss of energy. From a mathematical point of view, conductivity makes the underlying Maxwell operator non-selfadjoint. I will discuss a few recent results regarding the essential spectrum and the spectral approximation of dissipative Maxwell systems in unbounded domains of the three dimen- sional Euclidean space. Under certain assumptions on the behaviour of the coefficients at infinity, the essential spectrum decomposes in two parts, one of which is non-empty even in bounded domains. Moreover, the eigenvalues of finite multiplicity of the system can be computed exactly by means of the domain truncation method.
We will then see how these results generalise to two situations of relevance in applica- tions: 1) Drude-Lorentz metamaterials; 2) semi-transparent Faraday layers.
Based on joint work with S. Bogli (Durham), M. Marletta (Cardiff), and C. Tretter (Bern).
Friday 15 March 2024
Maxwell Analysis Mini-Symposium
Venue: ICMS Lecture Theatre
Speakers:
Grigori Rozenblum (Chalmers)
Eugene Shargorodsky (King's College London)
Susanna Terracini (Torino)
Wednesday 20 March 2024
2:30 PM Alexander Mielke (Berlin)
Asymptotic self-similar behavior in reaction-diffusion systems on R^d
Wednesday 10 April 2024
4:30 PM Matteo Novaga (Pisa)
Some mathematical models of charged liquid drops
I will give an overview of recent mathematical developments in the study of equilibrium configurations of liquid drops in the presence of repulsive Coulombic forces. These problems share a common feature that the equilibrium shape of a charged drop is determined by an interplay of the cohesive action of surface tension and the repulsive effect of long-range forces that favor fragmentation. In the talk, I will focus on two classical models: Gamow's liquid drop model of an atomic nucleus and Rayleigh's model of a perfectly conducting liquid drop. Surprisingly, despite a very similar physical background these two models exhibit drastically different mathematical properties. I will discuss the basic questions of existence vs. non-existence, as well as some qualitative properties of global energy minimizers in these models, and present the current state of the art.
Wednesday 24 April 2024
4:00 PM Gui-Qiang Chen (Oxford) [40 George Square, LT B]
Entropy Analysis and Singularities of Solutions for Hyperbolic Conservation Laws
In this talk, we present some reflections and recent developments in solving longstanding open problems related to the singularities of entropy solutions for nonlinear hyperbolic conservation laws and related nonlinear partial differential equations through entropy analysis and associated methods. These problems especially include establishing the minimal entropy conditions for well-posedness, understanding the phenomena of cavitation/decavitation and concentration/deconcentration, and the rigorous analysis for entropy solutions via the theory of divergence-measure fields, among others. Further related topics, perspectives, and open problems will also be addressed.
Wednesday 1 May 2024
4:00 PM Matthias Langer (Strathclyde)
Discrete coagulation-fragmentation equations
In many situations in nature and industrial processes clusters of particles can combine into larger clusters or fragment into smaller ones. The evolution of the cluster size distribution can often be described by an integro-differential equation (when the size of the clusters is arbitrary) or an infinite system of differential equations (when the cluster size assumes only discrete values). I shall discuss how operator semigroup theory and evolution families can be applied to study the discrete version of the coagulation-fragmentation equation. In particular, positivity and analyticity play an important role. The talk is based on joint work with Lyndsay Kerr and Wilson Lamb.
Wednesday 8 May 2024
4:00 PM Jan Lang (Ohio SU)
Different degrees of non-compactness of optimal Sobolev embeddings
In this talk we will look at some examples of non-compact "optimal" Sobolev embeddings and try to study their behaviour (using different tools like entropy numbers, s-numbers and concept of strict singularity).
Wednesday 15 May 2024
2:30 PM Ulisse Stefanelli (Vienna)
A free-boundary problem related to accretive growth
Growth is a fundamental process in biological systems, as well as in various technological contexts, including epitaxial deposition and additive manufacturing. The interaction between growth and mechanics in deformable bodies gives rise to a wealth of very challenging mathematical questions. I will provide a brief overview of the fundamental concepts of morphoelasticity, namely, the theory of elastic deformations in growing bodies. In contrast to the classical case, the reference state of a growing body evolves over time, also in response to external stimuli and stress. In some situations, this calls for free-boundary formulations, for the actual shape of the undeformed body is also unknown. I plan to discuss the case of surface accretion, which poses specific challenges. The focus will be the development of a variational framework where the existence of three-dimensional morphoelastic evolution can be proved, both in the infinitesimal and in the finite deformation case. This is work in collaboration with Andrea Chiesa (University of Vienna), Elisa Davoli (TU Vienna), Katerina Nik (TU Delft), and Giuseppe Tomassetti (Roma 3).
