The Heriot-Watt Analysis Seminar takes place on Wednesdays during term time. Our venue for Semester 2 is in Appleton Tower second floor, but the room changes, so see below. The building is located next to 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.
Wednesday 1 October 2025: room TBC
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 TBC
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).
