Heriot-Watt Analysis Seminar
The Heriot-Watt Analysis Seminar takes place on Wednesdays 4:30-5:30 PM (unless specified otherwise) during term time in 40 George Square, Lecture Theatre A. The latter 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 organisers: Lyonell Boulton and Matteo Capoferri.
Programme
Winter/Spring 2024
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 about1:45 PM Gabriel Schmid (Genova)
The Quantization of Maxwell Theory in the Cauchy Radiation Gauge: Hodge Decomposition and Hadamard States2:20 PM Claudia Garetto (Queen Mary)
Higher order hyperbolic equations with multiplicities3:10 PM Coffee Break
3:40 PM Claudio Dappiaggi (Pavia)
Interplay between Stochastic Partial Differential Equations and microlocal analysis4:30 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
2:00 PM Ulisse Stefanelli (Vienna)