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Research talks

The productivity of the $\kappa $-chain condition, where $\kappa $ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970’s, consistent examples of $kappa-cc$ posets whose squares are not $\kappa-cc$ were constructed by Laver, Galvin, Roitman and Fleissner. Later, ZFC examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which $\kappa = \aleph{_2}$, was resolved by Shelah in 1997.
In the first part of this talk, we shall present analogous results regarding the infinite productivity of chain conditions stronger than $\kappa-cc$. In particular, for any successor cardinal $\kappa$, we produce a ZFC example of a poset with precaliber $\kappa$ whose $\omega ^{th}$ power is not $\kappa-cc$. To do so, we introduce and study the principle $U(\kappa , \mu , \theta , \chi )$ asserting the existence of a coloring $c:\left [ \kappa \right ]^{2}\rightarrow \theta $ satisfying a strong unboundedness condition.
In the second part of this talk, we shall introduce and study a new cardinal invariant $\chi \left ( \kappa \right )$ for a regular uncountable cardinal $\kappa$ . For inaccessible $\kappa$, $\chi \left ( \kappa \right )$ may be seen as a measure of how far away $\kappa$ is from being weakly compact. We shall prove that if $\chi \left ( \kappa \right )> 1$, then $\chi \left ( \kappa \right )=max(Cspec(\kappa ))$, where:
(1) Cspec$(\kappa)$ := {$\chi (\vec{C})\mid \vec{C}$ is a sequence over $\kappa$} $\setminus \omega$, and
(2) $\chi \left ( \vec{C} \right )$ is the least cardinal $\chi \leq \kappa $ such that there exist $\Delta\in\left [ \kappa \right ]^{\kappa }$ and
b : $\kappa \rightarrow \left [ \kappa \right ]^{\chi }$ with $\Delta \cap \alpha \subseteq \cup _{\beta \in b(\alpha )}C_{\beta }$ for every $\alpha < \kappa$.
We shall also prove that if $\chi (\kappa )=1$, then $\kappa$ is greatly Mahlo, prove the consistency (modulo the existence of a supercompact) of $\chi (\aleph_{\omega +1})=\aleph_{0}$, and carry a systematic study of the effect of square principles on the $C$-sequence spectrum.
In the last part of this talk, we shall unveil an unexpected connection between the two principles discussed in the previous parts, proving that, for infinite regular cardinals $\theta< \kappa ,\theta \in Cspec(\kappa )$ if there is a closed witness to $U_{(\kappa ,\kappa ,\theta ,\theta )}$.
This is joint work with Chris Lambie-Hanson.
The productivity of the $\kappa $-chain condition, where $\kappa $ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970’s, consistent examples of $kappa-cc$ posets whose squares are not $\kappa-cc$ were constructed by Laver, Galvin, Roitman and Fleissner. Later, ZFC examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which $\kappa = \aleph{_2}$, ...

03E35 ; 03E05 ; 03E75 ; 06E10

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Research talks

This talk focuses on challenges that we address when designing linear solvers that aim at achieving scalability on large scale computers, while also preserving numerical robustness. We will consider preconditioned Krylov subspace solvers. Getting scalability relies on reducing global synchronizations between processors, while also increasing the arithmetic intensity on one processor. Achieving robustness relies on ensuring that the condition number of the preconditioned matrix is bounded. We will discuss two different approaches for this. The first approach relies on enlarged Krylov subspace methods that aim at computing an enlarged subspace and obtain a faster convergence of the iterative method. The second approach relies on a multilevel Schwarz preconditioner, a multilevel extension of the GenEO preconditioner, that is basedon constructing robustly a hierarchy of coarse spaces. Numerical results on large scale computers, in particular for linear systems arising from solving linear elasticity problems, will discuss the efficiency of the proposed methods. This talk focuses on challenges that we address when designing linear solvers that aim at achieving scalability on large scale computers, while also preserving numerical robustness. We will consider preconditioned Krylov subspace solvers. Getting scalability relies on reducing global synchronizations between processors, while also increasing the arithmetic intensity on one processor. Achieving robustness relies on ensuring that the condition ...

65F08 ; 65F10 ; 65N55

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Research talks

Twistor Theory was proposed in the late 1960s by Roger Penrose as a potential geometric unification of general relativity and quantum mechanics. During the past 50 years, there have been many mathematical advances and achievements in twistor theory. In physics, however, there are aspirations yet to be realised. Twistor Theory and Loop Quantum Gravity (LQG) share a common background. Their aims are very much related. Is there more to it? This talk will sketch the geometry and symmetry behind twistor theory with the hope that links with LQG can be usefully strengthened. We believe there is something significant going on here: what could it be? Twistor Theory was proposed in the late 1960s by Roger Penrose as a potential geometric unification of general relativity and quantum mechanics. During the past 50 years, there have been many mathematical advances and achievements in twistor theory. In physics, however, there are aspirations yet to be realised. Twistor Theory and Loop Quantum Gravity (LQG) share a common background. Their aims are very much related. Is there more to it? This ...

32L25 ; 53A30 ; 53C28

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Research schools

Bruhat-Tits theory applies to a semisimple group G, defined over an henselian discretly valued field K, such that G admits a Borel K-subgroup after an extension of K. The construction of the theory goes then by a deep Galois descent argument for the building and also for the parahoric group scheme. In the case of unramified extension, that descent has been achieved by Bruhat-Tits at the end of [BT2]. The tamely ramified case is due to G. Rousseau [R]. Recently, G. Prasad found a new way to investigate the descent part of the theory. This is available in the preprints [Pr1, Pr2] dealing respectively with the unramified case and the tamely ramified case. It is much shorter and the method is based more on fine geometry of the building (e.g. galleries) than algebraic groups techniques. Bruhat-Tits theory applies to a semisimple group G, defined over an henselian discretly valued field K, such that G admits a Borel K-subgroup after an extension of K. The construction of the theory goes then by a deep Galois descent argument for the building and also for the parahoric group scheme. In the case of unramified extension, that descent has been achieved by Bruhat-Tits at the end of [BT2]. The tamely ramified case is due to G. ...

20G15 ; 20E42 ; 51E24

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Research schools

We give some results about tree-indexed random walks aka branching random walks. In particular, we investigate the growth of the maximum of such a walk.
Based on joint work with Piotr Dyszewski and Thomas Hofelsauer.

60G50 ; 60J10 ; 60J80

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Research talks

Markov chain Monte Carlo methods have become ubiquitous across science and engineering to model dynamics and explore large combinatorial sets. Over the last 20 years there have been tremendous advances in the design and analysis of efficient sampling algorithms for this purpose. One of the striking discoveries has been the realization that many natural Markov chains undergo phase transitions, whereby they abruptly change from being efficient to inefficient as some parameter of the system is modified. Generating functions can offer an alternative approach to sampling and they play a role in showing when certain Markov chains are efficient or not. We will explore the interplay between Markov chains, generating functions, and phase transitions for a variety of combinatorial problems, including graded posets, Boltzmann sampling, and 3-colorings on $Z^{2}$. Markov chain Monte Carlo methods have become ubiquitous across science and engineering to model dynamics and explore large combinatorial sets. Over the last 20 years there have been tremendous advances in the design and analysis of efficient sampling algorithms for this purpose. One of the striking discoveries has been the realization that many natural Markov chains undergo phase transitions, whereby they abruptly change from being efficient to ...

60C05 ; 68R05 ; 60J20

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Research talks

(joint work with Michael Handel) $Out(F_{n}) := Aut(F_{n})/Inn(F_{n})$ denotes the outer automorphism group of the rank n free group $F_{n}$. An element $f$ of $Out(F_{n})$ is polynomially growing if the word lengths of conjugacy classes in Fn grow at most polynomially under iteration by $f$. The existence in $Out(F_{n}), n > 2$, of elements with non-linear polynomial growth is a feature of $Out(F_{n})$ not shared by mapping class groups of surfaces.
To avoid some finite order behavior, we restrict attention to the subset $UPG(F_{n})$ of $Out(F_{n})$ consisting of polynomially growing elements whose action on $H_{1}(F_{n}, Z)$ is unipotent. In particular, if $f$ is polynomially growing and acts trivially on $H_{1}(F_{n}, Z_{3})$ then $f $ is in $UPG(F_{n})$ and further every polynomially growing element of $Out(F_{n})$ has a power that is in $UPG(F_{n})$. The goal of the talk is to describe an algorithm to decide given $f,g$ in $UPG(F_{n})$ whether or not there is h in $Out(F_{n})$ such that $hf h^{-1} = g$.
The conjugacy problem for linearly growing elements of $UPG(F_{n})$ was solved by Cohen-Lustig. Krstic-Lustig-Vogtmann solved the case of linearly growing elements of $Out(F_{n})$.
A key technique is our use of train track representatives for elements of $Out(F_{n})$, a method pioneered by Bestvina-Handel in the early 1990s that has since been ubiquitous in the study of $Out(F_{n})$.
(joint work with Michael Handel) $Out(F_{n}) := Aut(F_{n})/Inn(F_{n})$ denotes the outer automorphism group of the rank n free group $F_{n}$. An element $f$ of $Out(F_{n})$ is polynomially growing if the word lengths of conjugacy classes in Fn grow at most polynomially under iteration by $f$. The existence in $Out(F_{n}), n > 2$, of elements with non-linear polynomial growth is a feature of $Out(F_{n})$ not shared by mapping class groups of ...

20F65 ; 57M07

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Research talks

In this talk I will describe a systematic investigation into congruences between the mod $p$ torsion modules of elliptic curves defined over $\mathbb{Q}$. For each such curve $E$ and prime $p$ the $p$-torsion $E[p]$ of $E$, is a 2-dimensional vector space over $\mathbb{F}_{p}$ which carries a Galois action of the absolute Galois group $G_{\mathbb{Q}}$. The structure of this $G_{\mathbb{Q}}$-module is very well understood, thanks to the work of J.-P. Serre and others. When we say the two curves $E$ and $E'$ are ”congruent” we mean that $E[p]$ and $E'[p]$ are isomorphic as $G_{\mathbb{Q}}$-modules. While such congruences are known to exist for all primes up to 17, the Frey-Mazur conjecture states that p is bounded: more precisely, that there exists $B$ > 0 such that if $p > B$ and $E[p]$ and $E'[p]$ are isomorphic then $E$ and $E'$ are isogenous. We report on work toward establishing such a bound for the elliptic curves in the LMFDB database. Secondly, we describe methods for determining whether or not a given isomorphism between $E[p]$ and $E'[p]$ is symplectic (preserves the Weil pairing) or antisymplectic, and report on the results of applying these methods to the curves in the database.
This is joint work with Nuno Freitas (Warwick).
In this talk I will describe a systematic investigation into congruences between the mod $p$ torsion modules of elliptic curves defined over $\mathbb{Q}$. For each such curve $E$ and prime $p$ the $p$-torsion $E[p]$ of $E$, is a 2-dimensional vector space over $\mathbb{F}_{p}$ which carries a Galois action of the absolute Galois group $G_{\mathbb{Q}}$. The structure of this $G_{\mathbb{Q}}$-module is very well understood, thanks to the work of ...

11G05 ; 14H52 ; 11A07

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Research talks

Very large networks linking dynamical agents are now ubiquitous and there is significant interest in their analysis, design and control. The emergence of the graphon theory of large networks and their infinite limits has recently enabled the formulation of a theory of the centralized control of dynamical systems distributed on asymptotically infinite networks [Gao and Caines, IEEE CDC 2017, 2018]. Furthermore, the study of the decentralized control of such systems has been initiated in [Caines and Huang, IEEE CDC 2018] where Graphon Mean Field Games (GMFG) and the GMFG equations are formulated for the analysis of non-cooperative dynamical games on unbounded networks. In this talk the GMFG framework will be first be presented followed by the basic existence and uniqueness results for the GMFG equations, together with an epsilon-Nash theorem relating the infinite population equilibria on infinite networks to that of finite population equilibria on finite networks. Very large networks linking dynamical agents are now ubiquitous and there is significant interest in their analysis, design and control. The emergence of the graphon theory of large networks and their infinite limits has recently enabled the formulation of a theory of the centralized control of dynamical systems distributed on asymptotically infinite networks [Gao and Caines, IEEE CDC 2017, 2018]. Furthermore, the study of the decentralized ...

91A13 ; 49N70 ; 93E20 ; 93E35

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Research talks

For surfaces immersed into a compact Riemannian manifold, we consider the curvature functional given by the $L^{2}$ integral of the second fundamental form. We discuss an area bound in terms of the energy, with application to the existence of minimizers. This is joint work with V. Bangert.

53C44 ; 53C45

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Research talks

Some convergence properties for the approximation of second order elliptic problems with a variety of boundary conditions (homogeneous Dirichlet, homogeneous or non-homogeneous Neumann or Fourier boundary conditions), using a given discretisation method, can be obtained when this method is plugged into the Gradient Discretisation Method (GDM) framework.
Instead of defining one GDM framework for each of these boundary conditions, we show that these properties can be stated using the same abstract tools for all the above boundary conditions. Then these tools enable the application of the GDM to a larger class of elliptic problems.
Some convergence properties for the approximation of second order elliptic problems with a variety of boundary conditions (homogeneous Dirichlet, homogeneous or non-homogeneous Neumann or Fourier boundary conditions), using a given discretisation method, can be obtained when this method is plugged into the Gradient Discretisation Method (GDM) framework.
Instead of defining one GDM framework for each of these boundary conditions, we show that ...

65J05 ; 65Nxx ; 47A58

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Research school

The 1983 discovery of the fractional quantum Hall effect marks a milestone in condensed matter physics: systems of “ordinary particles at ordinary energies” displayed highly exotic effects, most notably fractional quantum numbers. It was later recognized that this was due to emergent quasi-particles carrying a fraction of the charge of an electron. It was also conjectured that these quasi-particles had fractional statistics, i.e. a behavior interpolating between that of bosons and fermions, the only two types of fundamental particles.
These lectures will be an introduction to the basic physics of the fractional quantum Hall effect, with an emphasis on the challenges to rigorous many-body quantum mechanics emerging thereof. Some progress has been made on some of these, but lots remains to be done, and open problems will be mentioned.

After the lectures a few references regarding the spectrum of the magnetic Schrödinger operator were suggested to me.
See the bibiography below.

Thanks to Alix Deleporte, Frédéric Faure, Stéphane Nonnenmacher and others for discussions relative to the magnetic Weyl law.
The 1983 discovery of the fractional quantum Hall effect marks a milestone in condensed matter physics: systems of “ordinary particles at ordinary energies” displayed highly exotic effects, most notably fractional quantum numbers. It was later recognized that this was due to emergent quasi-particles carrying a fraction of the charge of an electron. It was also conjectured that these quasi-particles had fractional statistics, i.e. a behavior ...

81Sxx ; 81V70

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Research talks

In this talk we will discuss a new geodesic beam approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of $L^{2}$ mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Using the description of concentration, we obtain quantitative improvements on the known bounds in a wide variety of settings. In this talk we will discuss a new geodesic beam approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of $L^{2}$ mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along ...

35P20 ; 58J50 ; 53C22 ; 53C40 ; 53C21

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- iii; 174 p.
ISBN 978-1-57146-384-5

Localisation : Colloque 1er étage (CAMB)

conjecture de Kannan-Lovasz-Simonovits # versalité dans la symétrie du miroir # variété de codimension 1 # varifold

68Q87 ; 68W20 ; 53D37 ; 49Q20

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- 259 p.
ISBN 978-83-86806-44-7

Banach center publications , 0118

Localisation : Périodique 1er étage

théorie des nombres # théorie des groupes # Jerzy Kaczorowski # fonction Z(t) de Hardy # convolution de Rankin-Selberg # séquence de Stern # fonction zêta de Riemann # fonction zêta de Lerch # groupe de Ttae-Shafarevich

11-06 ; 20-06 ; 44-06 ; 00B25

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- viii; 490 p.
ISBN 978-1-108-48580-7

Mathematical sciences research institute publications , 0070

Localisation : Colloque 1er étage (BANF)

théorie des jeux # analyse combinatoire # jeux de stratégie # dissuasion # échecs

91-06 ; 91A46 ; 91-03 ; 01A60 ; 00B25

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- xvi; 929 p.

Les Houches

Localisation : Colloque 1er étage (HOUC)

relativité générale # topologie # rayonnement gravitationnel # théorie des groupes

83-XX ; 22-XX

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- 192 p.
ISBN 978-93-82006-17-6

Localisation : Colloque 1er étage (GORA)

fonctions spéciales # théorie de Lie # fonction de Bessel # fraction continue # polynôme de Jacobi # fonction hypergéométrique # algèbre de Racah-Wigner # transformation intégrale

33-XX ; 33Cxx ; 33E20 ; 33D60 ; 33D15 ; 35R11 ; 33C60 ; 33C45 ; 44-XX

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- vii; 322 p.
ISBN 978-81-7764-169-1

Localisation : Colloque 1er étage (JODH)

fonctions spéciales # série hypergéométrique # fonction de Bessel # fraction continue

33-XX ; 33Cxx ; 33D15 ; 33C05 ; 11A55

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- ii; 105 p.

Localisation : Colloque 1er étage (JODH)

fonctions spéciales # intégrale de Kratzel # calcul fractionnel d'une fonction I # série hypergéométrique thêta # équation de Fokker-Planck # fonction hypergéométrique # expansion asymptotique # fonction W de Lambert

33E20 ; 33D60 ; 33D15 ; 35R11 ; 33C60 ; 33B20

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