Seminar of Differential Equations and Numerical Analysis 2023-2024
Thursday 25/4/2024, 12:05-12:55
Room: E.324
Speaker:
Elias Kiritsis (UoC)
Title: An introduction to AdS/CFT
Abstract: I will present an informal introduction to the AdS/CFT correspondence presenting its
most important aspects
as well as the problems where mathematicians can importantly contribute.
Tuesday 16/4/2024, 12:05-12:55
Room: E.324
Speaker:
Arthur Touati (IHES)
Title: High-frequency limits in general relativity
Abstract: In this talk I will give an overview of the construction of high-frequency solutions
of the Einstein vacuum equations, starting with the pioneering work of Choquet-Bruhat. I will
make the connection with the Burnett conjecture in general relativity and present the three main
settings where it has been addressed, ie. the elliptic gauge in U(1) symmetry, the double null gauge
and the generalized wave gauge. If time permits, I will sketch the proof of the reverse Burnett
conjecture for null dusts obtained recently and present some perspectives.
Tuesday 2/4/2024, 12:05-12:55
Room: E.324
Speaker:
Spyridon Filippas (University of Helsinki)
Title: On unique continuation for Schrödinger operators
Abstract: We are interested in the following question: a solution of the linear time dependent
Schrödinger equation vanishing in a small open set during a small time does it vanishes everywhere?
In the case where the operator includes a potential the answer to this question depends on its regularity.
We will present a result under the assumption that the potential has a Gevrey 2 regularity with respect
to time. This relaxes the analyticity assumption known previously. This is a joint work with Camille
Laurent and Matthieu Léautaud.
Tuesday 5/3/2024, 12:05-12:55
Room: E.324
Speaker:
Spyros Sotiriadis (University of Crete)
Title: Expansion of a one-dimensional interacting Bose gas (part II)
Abstract: The analytical study of quantum many-body dynamics is a challenging mathematical problem.
Considering a process in which a far from equilibrium initial state is let to evolve unitarily under
an interacting Hamiltonian, the physically relevant quantities are the asymptotics of observables in
the thermodynamic limit and at large times. However, even when the expansion of the initial state in
the energy eigenstate basis is known, the evaluation of sums over all energy eigenstates and the
derivation of the asymptotics is a formidable task. I will present a strategy to tackle this problem
in a case study, the expansion of a one-dimensional Bose gas in the limit of point-like interactions,
the Lieb-Liniger model, which offers the advantage of integrability.
Tuesday 13/2/2024, 12:05-12:55
Room: E.324
Speaker:
Spyros Sotiriadis (University of Crete)
Title: Expansion of a one-dimensional interacting Bose gas (part I)
Abstract: The analytical study of quantum many-body dynamics is a challenging mathematical problem.
Considering a process in which a far from equilibrium initial state is let to evolve unitarily under
an interacting Hamiltonian, the physically relevant quantities are the asymptotics of observables in
the thermodynamic limit and at large times. However, even when the expansion of the initial state in
the energy eigenstate basis is known, the evaluation of sums over all energy eigenstates and the
derivation of the asymptotics is a formidable task. I will present a strategy to tackle this problem
in a case study, the expansion of a one-dimensional Bose gas in the limit of point-like interactions,
the Lieb-Liniger model, which offers the advantage of integrability.
Teusday 19/12/2023, 13:05-13:55
Room: E.324
Speaker: Georgios Athanasopoulos (University of Warwick)
Title: Ising model and Kac-Ward method
Abstract: Onsager proposed a closed-form expression for the free energy of the Ising model in 1944.
In 1952, Kac and Ward introduced an alternative elegant method of combinatorial nature which became
rigorous by Kager, Lis and Meester in 2013 followed by Aizenman and Warzel in 2018. We extend the
result of isotropic ferromagnetic case to the anisotropic ferromagnetic and antiferromagnetic case.
Furthermore, we show that the above holds true in a more general setting.
This is joint work with Daniel Ueltschi.
Tuesday 28/11/2023, 13:05-13:55
Room: E.324
Speaker:
Bruno Barton-Singer (Heraklion)
Title: Elliptical Instability of Magnetic Skyrmions
Abstract: Magnetic Skyrmions are robust emergent particle-like textures in chiral magnets,
which we model as topological solitons in a continuum field theory of the magnetisation.
Their research is motivated by practical uses, such as mobile data storage, and yet the commonly
considered model is very amenable to analytical investigation, displaying an interesting interplay
of gauge theory, symmetry and topology. The phase diagram of the chiral magnet is rich with different
ground states, and there are also critical couplings where we can find many explicit solutions.
In this talk I will give a general introduction to the area, before focusing on the question of
predicting the region of the phase diagram where skyrmions and other related topological solitons
exhibit an 'elliptical instability', stretching out indefinitely into a domain-wall like solution.
Based on work with Bernd Schroers at Heriot-Watt University and Nikolai Kiselev and Vlad Kuchkin at
Forschungszentrum Jülich.
Tuesday 14/11/2023, 13:05-13:55
Room: E.324
Speaker: Angeliki Menegaki
(Imperial College London)
Title: L2-stability for the 4-waves kinetic equation around the
Rayleigh-Jeans equilibrium
Abstract: 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 solutions.
In particular, I will present an L2 stability of mild solutions on the whole frequency space
for initial data close to Rayleigh-Jeans when assuming radial solutions for the equation,
as well as a stability of the same kind without the radial solution-assumption but after
introducing a cut-off on the frequencies.
Parts of this talk are joint works with Miguel Escobedo (UPV/EHU).
Thursday 9/11/2023, 13:05-13:55
Room: E.324
Speaker: Filippos Oikonomidis
(University of Athens)
Title: Black hole uniqueness
Abstract: After reviewing uniqueness theorems for black holes in general relativity,
I will present a proof by Bunting and Masood-ul-Alam of the black hole uniqueness theorem
for static, vacuum, asymptotically Euclidean spacetimes. I will explain the overall method and
focus on certain details of the proof.
Tuesday 31/10/2023, 13:05-13:55
Room: E.324
Speaker: Stathis Filippas
(University of Crete)
Title: Body motion in a fluid (part II)
Absract: I will discuss various questions and results related to the motion of a body inside
an incompressible fluid. In the end I will present some estimates for the velocity of the body,
as it approaches the boundary of the fluid container. The talk is based on joint work with
Alkis Tersenov.
Tuesday 24/10/2023, 11:05-11:55
Room: E.324
Speaker: Stathis Filippas
(University of Crete)
Title: Body motion in a fluid (part I)
Absract: I will discuss various questions and results related to the motion of a body inside
an incompressible fluid. In the end I will present some estimates for the velocity of the body,
as it approaches the boundary of the fluid container. The talk is based on joint work with
Alkis Tersenov.
Seminar of Differential Equations and Numerical Analysis 2022-2023
Tuesday 13/6/2023, 13:05-13:55
Room: E.324
Speaker:
Maxime Van de Moortel (Rutgers University)
Title: Impulsive gravitational waves
Abstract: In a 1972 influential paper, Penrose introduced the concept of an impulsive gravitational wave,
defined as a localized and singular solution of the Einstein equations.
This is a model for the spacetime distortions created by a strongly gravitating source such as, say,
a black hole merger.
An abundant literature, both in Physics and Mathematics, has since built up on Penrose’s work,
with a particular interest in the interaction of several impulsive gravitational waves.
The understanding was, until recently, however, limited to binary interactions.
I will review these works and finish with a new theory (obtained with J. Luk)
that also allows for ternary interactions of such impulsive gravitational waves
under translation symmetry.
Tuesday 9/5/2023, 13:05-13:55
Room: E.324
Speaker:
Warren Li (Princeton University)
Title: The structure of singularities for spherically symmetric solutions of
Einstein's equations with scalar matter
Abstract: By the Hawking-Penrose singularity theorems,
there is a large class of 'singular' (i.e. incomplete) spacetimes solving the
Einstein field equations. However, these theorems do not provide a mechanism of how
such 'singularities' are formed, or any description of the structure of spacetime near singularity.
In this talk, upon reducing to the spherically symmetric case and adding a scalar field,
we provide a study of the types of singularity that arise for general initial data,
including naked singularities, spacelike singularities and (weak) null singularities.
Tuesday 25/4/2023, 13:05-13:55
Room: E.324
Speaker:
Olena Gomonay (Johanes Gutenberg University)
Title: Elastic manipulation of antiferromagnetic domain structure
Abstract: Since the recent prediction [1] and discovery [2] of the Neel
spin-orbit torques, antiferromagnets became active elements of spintronic
devices in which antiferromagnetic states are manipulated electrically.
However, many of antiferromagnets that are interesting for applications show
strong magnetoelastic coupling which cannot be ignored in description and
interpretation of switching processes. On the other hand, this coupling can
be used for effective and energy consuming manipulation of antiferromagnetic
states via external stresses and strains.
In this presentation we discuss magnetoelastic origin of the domain structure in a
tetragonal antiferromagnet and formulate general criteria which define formation of
the particular domain shapes depending on the sample geometry and the external stress.
By comparing domain structures in presence of the electrical and stress-induced torques
we derive equivalence between both stimuli and demonstrate possibility of strain-induced
switching. We illustrate our predictions with recent experimental observations which
underline importance and efficiency of strain-induced switching in antiferromagnetic-based films.
References
[1] J. Železný, et al, Phys. Rev. Lett. 113, 157201 (2014).
[2] P. Wadley, et al, Science 351 (2016).
Teusday 21/3/2023, 13:05-13:55
Room: E.324
Speaker:
Nikolaos Athanasiou (University of Crete)
Title: Formation of trapped surfaces in general relativity
Abstract: Trapped surfaces play an important role in general relativity as they typically signal
the formation of singularities which are enclosed inside black holes. I will discuss their basic properties
and give a historical overview of the trapped surface formation problem. Time-permitting, I will
present recent progress concerning the Einstein-Yang-Mills system.
Tuesday 7/3/2023, 13:05-13:55
Room: E.324
Speaker:
Spyridon Filippas (Université Paris-Saclay)
Title: Οn unique continuation for waves in singular media
Abstract: The question of unique continuation consists in asking whether a partial observation
of a wave on a small set is sufficient to determine the whole wave. After presenting some fundamental results of the theory we will explain how one
can obtain quantitative uniqueness results in singular media. The key ingredient will be a Carleman estimate, which will be combined with the recent techniques
of Laurent-Létautaud who obtained similar results in a smooth context.
Tuesday 28/2/2023, 13:05-13:55
Room: E.324
Speaker:
Alkis Tersenov (University of Crete)
Title: Καινούργιες a priori εκτιμήσεις στη μελέτη των μη γραμμικών εξισώσεων παραβολικού τύπου
Thursday 15/12/2022, 12:05-12:55
Room: E.324
Speaker:
Jonatan Lenells (KTH Royal Institute of Technology)
Title: Boundary value problems for the Ernst equation
Abstract: For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a
single nonlinear PDE in two dimensions called the (elliptic) Ernst equation. By solving this equation
with Dirichlet boundary conditions imposed along a disk, Neugebauer and Meinel in the 1990s derived an
explicit expression for the spacetime metric corresponding to the Bardeen-Wagoner uniformly rotating disk
of dust. I will discuss some other boundary value problems for the Ernst equation that can also be solved
exactly. The exact solutions are obtained by using the integrable structure of the Ernst equation.
I will also discuss a boundary value problem for the hyperbolic version of the Ernst equation that is
relevant for the collision of two plane gravitational waves.
Thursday 8/12/2022, 12:05-12:55
Room: E.324
Speaker:
Stavros Komineas (University of Crete)
Title: Topological skyrmions and dynamics
Abstract: The Dzyaloshinskii-Moriya (DM) interaction breaks chiral symmetry in magnets and it is instrumental
for the stabilisation of a periodic solution (termed the spiral) in one space dimension, and of topological
solitons (termed skyrmions) in two space dimensions. In addition, the chiral DM interaction has a profound
effect on the magnetization field dynamics. A skyrmion breathing mode (i.e., oscillations of the skyrmion
radius) is due to the breaking of the conservation of the total magnetization by the DM interaction. We
describe nonlinear breathing oscillations in an antiferromagnet (AFM). We show that an expansion of the
skyrmion can lead to its collapsing into a singularity. The process can be efficient even when the skyrmion
is only mildly excited.
Thursday 1/12/2022, 12:05-12:55
Room: E.324
Speaker:
Rembert Duine (Eindhoven University of Technology)
Title: Antimagnons, the bosonic Klein paradox, and magnonic black holes
Abstract: The collective excitations of the magnetic order in the ground state of ferromagnets are called
spin waves, or, quantum-mechanically, magnons. In this talk I will discuss how spin currents may be used
to dynamically stabilize magnetic configurations at an energy maximum. The collective excitations of such
configurations are negative energy excitations that are most conveniently described in terms of antimagnons.
I will describe how the coupling of ordinary magnons with these antimagnons paves the way for schemes for
magnon amplification via the bosonic Klein paradox, and for black-hole-horizon analogues for magnons.
Tuesday 15/11/2022, 14:05-14:55
Room: E.324
Speaker: Panos Karagiorgos (University of Crete)
Title: Ημικλασική Ανάλυση της Εξίσωσης Schrödinger Φασικού Χώρου
Abstract: Παρουσιάζουμε δύο μεθόδους κατασκευής ασυμπτωτικών λύσεων του ημικλασικού προβλήματος Cauchy
για την εξίσωση Schrödinger φασικού χώρου, για αρχικά δεδομένα Lagrange φασικού χώρου και μικρά χρονικά
διαστήματα. Η πρώτη μέθοδος πρόκειται για κατασκευή ασυμπτωτικής λύσης μέσω της κατασκευής ημικλασικού
διαδότη, στη βάση της ανισοτροπικής ημικλασικής χρονικής εξέλιξης κυματοδεσμών Gauss. Η δεύτερη μέθοδος
πρόκειται για κατασκευή ασυμπτωτικής λύσης μέσω της Μιγαδικής Θεωρίας WKB, στη βάση κανονικού συστήματος και
ροής χαρακτηριστικών στον διπλό φασικό χώρο.
Πρόκειται για καινοτόμα προσπάθεια βαθέματος στην κατανόηση της αναπαράστασης φασικού χώρου της ημικλασικής
χρονικής εξέλιξης μικροσκοπικών φυσικών συστημάτων, στην κατανόηση εξισώσεων χρονικής εξέλιξης επαγόμενων από
ψευδοδιαφορικούς τελεστές φασικού χώρου ή τελεστές Weyl φασικού χώρου. Η θεωρία που αναπτύσσεται μπορεί να
πλαισιώσει την ανάπτυξη μεθόδων αριθμητικής επίλυσης σειράς προβλημάτων, σε αναπαράσταση φασικού χώρου, στην
Ατομική Φυσική, στην Κβαντική Οπτική και στη Θεωρητική Χημεία, τα οποία είναι πλέον πειραματικά διερευνήσιμα.
Thursday 3/11/2022, 12:05-12:55
Room: E.324
Speaker: Ioannis Athanasopoulos (University of Crete)
Title: The Stefan problem
Abstract: The Stefan problem is one of the simplest free boundary problems of parabolic type.
It is at the center of interest for more than 100 years. Although it is consider to be simple,
its resolution is still far away even in its simplest form. We will focus our attention on
the local regularity of the solution, as well as that of the free boundary, mainly in recent results.
Thursday 27/10/2022, 12:00-13:00
Room: E.324
Speaker:
Nikolaos Athanasiou (University of Crete)
Title: Results on the formation of singularities for the 1-dimensional Relativistic Euler equations
Abstract: An archetypal phenomenon in the study of hyperbolic systems of conservation laws is the development
of singularities (in particular shocks) in finite time, no matter how smooth or small the initial data are.
A series of works by Lax, John et al confirmed that for some important systems, when the initial data set
is a smooth small perturbation of a constant state, singularity formation in finite time is equivalent to
the existence of compression in the initial data (this being appropriately defined in terms of spatial
gradients of the Riemann invariants). Our talk will address the question of whether this dichotomy persists
for large data problems, possibly containing a far-field vacuum, at least for the system of the Relativistic
Euler equations in (1+1) dimensions. I shall discuss results on both the isentropic and the non-isentropic
cases.
The talk will be based on the following joint works with Shengguo Zhu (Shanghai Jiao-Tong) and
Tianrui Bayles-Rea (Oxford):
arXiv:1903.03355,
arXiv:2106.07467.
Thursday 13/10/2022, 11:00-12:00
Room: E.324
Speaker: Grigorios Fournodavlos (University of Crete)
Title: Mathematical problems of General Relativity
Abstract: We will go over some classical topics in general relativity and discuss problems that are of interest to mathematicians. This is intended to be an introductory talk and anyone is welcome to attend.