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SUMMARY:Journée de l'équipe Math-Physique
DTSTART;VALUE=DATE-TIME:20210624T074500Z
DTEND;VALUE=DATE-TIME:20210624T143000Z
DTSTAMP;VALUE=DATE-TIME:20210922T090505Z
UID:indico-event-6765@indico.math.cnrs.fr
DESCRIPTION:La journée d'équipe se tiendra en hybride: en salle rené Ba
ire et sur TEAMS.\n\nProgramme:\n\n\n \n \n 9h45-10h10\n exposé de N
icolas Babinet\n \n \n 10h15-10h40\n exposé de Eddy Brandon De Leó
n\n \n \n 10h45-11h\n Pause (café)\n \n \n 11h-11h25\n expos
é de Sion ChanLang\n \n \n 11h30-11h55\n exposé de Edgar Gasperin-
Garcia\n \n \n 12h-13h30\n pic-nic (pelouses devant le bâtiment mir
ande)\n \n \n 13h30-14h\n présentation des membres de l'équipe\n
\n \n 14h-14h25\n exposé de Francisco Hernandez-Iglesias\n \n \n
14h30-14h55\n exposé de Ajinkya Kulkarni\n \n \n 15h-15h15\n Pau
se (café)\n \n \n 15h15-15h40\n exposé de Oscar Meneses Rojas\n \
n \n 15h45-16h10\n exposé de Nikola Stoilov\n \n \n\n\nTitres et R
ésumés:\n\n\n Nicolas Babinet: Supersymmetric matrix models\, ABJM theor
y and quantum curves\n Matrix models have a long and fruitful story both i
n physics and pure mathematics. In physics Wigner has introduced random ma
trix formalism to describe energy levels of heavy nuclei for instance\, wh
ile in mathematics there are strong evidences of a deep connection between
eigenvalues of random matrices and distribution of prime numbers\, thus l
eading to some conjectures about Riemann’s zeta function.\n Many propert
ies have been proven for the associated partition function and the corresp
onding free energy\, even if some are still conjectures. One which still l
eads to rich developments is the so-called quantum curve. In this framewor
k the partition function behaves as a wave function and the quantum curve
acts on it as an operator\, i.e. promoting classical coordinates to operat
ors.\n\n Starting from this framework and after presenting the basic rando
m matrix model\, I want first to focus on the supersymmetric case that I
’m interested in. I will present some properties that we have found and
structures that still need to be elucidated. A very similar model that I w
ill then present is the ABJM theory\, first introduced in string theory\,
whose partition function can be studied with matrix model. An interesting
method was developed to explore this theory yet some problems are still un
solved. I will finally review conjectures that I’m interesting in\, more
specifically those in relation with quantum curves.\n \n Eddy Brandon De
León: Computational approach to the Schottky problem\n \n The characteriz
ation of jacobians of algebraic curves among all principally polarized Abe
lian varieties (PPAV) is known as the Schottky problem. A PPAV can be cons
idered as the quotient ℂᵍ/λ\, where λ=ℤᵍ+Ωᵍ and Ω is a symme
tric complex matrix with a positive-definite imaginary part (Riemann matri
x). We aim to develop a computational tool such that given a PPAV A we can
tell for a given precision whether it is the jacobian of an algebraic cur
ve or not. For this purpose\, we make use of the Welter’s conjecture\, w
hich states that A=Jac(X)\, for some algebraic curve X\, if and only if th
ere exists a trisecant of its kummer variety\, which is the image of the e
mbedding Kum : A/{±1}↪ℙ²ˆᵍ⁻¹. This becomes an optimization pro
blem\, since we consider an auxiliary non-negative function f : A×A×A→
ℝ whose global minimum is zero if and only if A=Jac(X).\n Sion ChanLang:
To Be Announced\n Edgar Gasperin-Garcia: The Role of the Energy Scalar Pr
oduct in the QNM Spectral Instability Problem\n \n The impact of the scala
r product in the Quasinormal mode (QNM) spectral instability problem is di
scussed. It is illustrated in a simple example how the use of different no
rms can lead to different looking pseudospectra for the same operator. Wit
h this motivation\, we will study QNMs of a scalar field on a spherically
symmetric spacetime background. The relation between the physical energy a
nd the effective energy (used in a recent result of Jaramillo\, Panosso-Ma
cedo and Al Sheikh) is obtained and the boundary terms are identified. The
energy inner product is exploited to obtain a weak formulation of the ass
ociated eigenvalue problem which opens the possibility of using finite ele
ments methods to numerically explore the spectrum. Other applications such
as using Keldysh theorem to obtain asymptotic resonant expansions exploit
ing the scalar product are also discussed.\n Francisco Hernandez-Iglesias:
Dubrovin equation and canonical coordinates for the 2D-Toda Frobenius man
ifold.\n \n Integrable systems of N evolutionary PDEs that admit a bi-Hami
ltonian formulation\, i.e.\, that are endowed with a pair of compatible Po
isson brackets\, have an underlying geometric structure known as a Frobeni
us manifold. This N-dimensional Frobenius manifold has two compatible flat
metrics on its tangent space\, which define the Poisson brackets of the s
ystem. Extending this theory to the 2D-Toda hierarchy\, a 2+1 integrable s
ystem of bi-Hamiltonian type\, we define an infinite-dimensional Frobenius
manifold and study its main elements: the product and the metric on the t
angent spaces\, the unit and Euler vector fields. Then we introduce the de
formed flat connection and derive from this definition the Dubrovin equa
tion. The main difference with respect to the finite-dimensional case is t
hat the cotangent spaces are no longer isomorphic to the tangent spaces\,
meaning the Dubrovin equation will take a weak form. Finally\, we will ana
lyze the (weak) solutions of the Dubrovin equation around infinity and lin
k them to the canonical coordinates.\n Ajinkya Kulkarni:Algebraic and topo
logical invariants of fusion categories\n \n I will talk about invariants
of fusion categories coming from finite groups. A topological invariant ca
lled the B-tensor (along with the T-matrix) is shown to suffice to disting
uish twisted Drinfeld doubles of square-free groups of odd order. We also
define a character theory for objects in group-theoretical fusion categori
es (GTCs) and use it to compute fusion rings of GTCs of dimensions less th
an 21. We find that there are 34 non-pointed fusion rings coming from GTCs
of dimension less than 21\, of which 26 are singly generated and only 3 a
re non-hyperrings. In the process\, we obtain weak lower bounds for non-eq
uivalent categorifications of each of these non-pointed fusion rings. Furt
her\, we find several examples of Morita equivalent GTCs which have the sa
me fusion ring but distinct Frobenius-Schur indicators (including one case
where both the GTCs are pointed)\, thereby refuting a conjecture of H. Tu
cker.\n Oscar Meneses Rojas: Caustics and fronts in General Relativity\n \
n The concept of caustic naturally arises when systems of light rays are s
tudied. They are defined as the envelope of a family of light rays. In gra
vitational lensing for example\, when light rays issuing from a surface te
nds to re converge after passing close to a strong gravitational field\, l
ight rays will start to intersect each other and the surfaces equidistants
to the initial surface generically will have singularities. The propagati
on of a front can be described by the study of a single hypersurface in th
e space time\, which generically has singularities\, is a null hypersurfac
e and is called Big Front. In a general way caustics and fronts belong to
the symplectic and contact worlds respectively and there is a caustic asso
ciated to the propagation of the front.\n I will introduce what a front is
and its generic singularities to finally conclude what is the caustic ass
ociated to the propagating front. This constitutes a preliminary step in d
eveloping a setting for understanding the stability properties and pattern
s of generic caustics in the birth of event horizons in general relativity
.\n Nikola Stoilov: Numerical studies of the Zakharov-Kuznetsov equations\
n \n In this work we look at the behaviour of the Zakharov Kuznetsov (ZK)
equations\, using ad- vanced numerical tools. As a nonlinear dispersive PD
E\, initially used to model magnetized plasma\, ZK has solutions that deve
lop a singularity in finite time from smooth initial data. We demonstrate
its behaviour and will look at phenomena including blow-up\, soliton resol
ution and soliton interaction and discuss how the non-integrability transp
ires in these cases. We propose several conjectures for the long term beha
viour.\n Based on joint works with Christian Klein and Svetlana Roudenko.\
n\n\nhttps://indico.math.cnrs.fr/event/6765/
LOCATION: Salle René Baire
URL:https://indico.math.cnrs.fr/event/6765/
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