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Post-edited  Zeta functions and monodromy
Veys, Wim (Auteur de la Conférence) | CIRM (Editeur )

The $p$-adic Igusa zeta function, topological and motivic zeta function are (related) invariants of a polynomial $f$, reflecting the singularities of the hypersurface $f = 0$. The first one has a number theoretical flavor and is related to counting numbers of solutions of $f = 0$ over finite rings; the other two are more geometric in nature. The monodromy conjecture relates in a mysterious way these invariants to another singularity invariant of $f$, its local monodromy. We will discuss in this survey talk rationality issues for these zeta functions and the origins of the conjecture. The $p$-adic Igusa zeta function, topological and motivic zeta function are (related) invariants of a polynomial $f$, reflecting the singularities of the hypersurface $f = 0$. The first one has a number theoretical flavor and is related to counting numbers of solutions of $f = 0$ over finite rings; the other two are more geometric in nature. The monodromy conjecture relates in a mysterious way these invariants to another singularity invariant of ...

14D05 ; 11S80 ; 11S40 ; 14E18 ; 14J17

Post-edited  Wrapping in exact real arithmetic
Müller, Norbert (Auteur de la Conférence) | CIRM (Editeur )

A serious problem common to all interval algorithms is that they suffer from wrapping effects, i.e. unnecessary growth of approximations during a computation. This is essentially connected to functional dependencies inside vectors of data computed from the same inputs. Reducing these effects is an important issue in interval arithmetic, where the most successful approach uses Taylor models.
In TTE Taylor models have not been considered explicitly, as they use would not change the induced computability, already established using ordinary interval computations. However for the viewpoint of efficiency, they lead to significant improvements.
In the talk we report on recent improvements on the iRRAM software for exact real arithmetic (ERA) based on Taylor models. The techniques discussed should also easily be applicable to other software for exact real computations as long as they also are based on interval arithmetic.
As instructive examples we consider the one-dimensional logistic map and a few further discrete dynamical systems of higher dimensions
Joint work with Franz Brauße, Trier, and Margarita Korovina, Novosibirsk.
A serious problem common to all interval algorithms is that they suffer from wrapping effects, i.e. unnecessary growth of approximations during a computation. This is essentially connected to functional dependencies inside vectors of data computed from the same inputs. Reducing these effects is an important issue in interval arithmetic, where the most successful approach uses Taylor models.
In TTE Taylor models have not been considered ...

68Q25 ; 03D60 ; 65Y15

This talk sketches connections between Whitney problems and e.g. the problem of deciding whether a given rational function on $\mathbb{R}^n$ belongs to $C^m$.

26Bxx ; 46E10 ; 58A20 ; 14Qxx

Post-edited  Which geodesic flows are left-handed?
Dehornoy, Pierre (Auteur de la Conférence) | CIRM (Editeur )

Left-handed flows are 3-dimensional flows which have a particular topological property, namely that every pair of periodic orbits is negatively linked. This property (introduced by Ghys in 2007) implies the existence of as many Bikrhoff sections as possible, and therefore allows to reduce the flow to a suspension in many different ways. It then becomes natural to look for examples. A construction of Birkhoff (1917) suggests that geodesic flows are good candidates. In this conference we determine on which hyperbolic orbifolds is the geodesic flow left-handed: the answer is that yes if the surface is a sphere with three cone points, and no otherwise.
dynamical system - geodesic flow - knot - periodic orbit - global section - linking number - fibered knot
Left-handed flows are 3-dimensional flows which have a particular topological property, namely that every pair of periodic orbits is negatively linked. This property (introduced by Ghys in 2007) implies the existence of as many Bikrhoff sections as possible, and therefore allows to reduce the flow to a suspension in many different ways. It then becomes natural to look for examples. A construction of Birkhoff (1917) suggests that geodesic flows ...

37C27 ; 37C15 ; 37C10 ; 57M25

I will discuss work in progress aimed towards defining contact homology using "virtual" holomorphic curve counting techniques.

37J10 ; 53D35 ; 53D40 ; 53D42 ; 53D45 ; 57R17

Starting with a personal tribute to Victor Havin (1933-2015), I discuss a dozen achievements of Great Havin's Analysis Seminar, as well as some challenging still unsolved problems.
The Havin publications list is available in the PDF file at the bottom of the page.

30H10 ; 01A60 ; 01A70

Variational formulas for limit shapes of directed last-passage percolation models. Connections of minimizing cocycles of the variational formulas to geodesics, Busemann functions, and stationary percolation.

60K35 ; 60K37 ; 82C22 ; 82C43 ; 82D60

Post-edited  Unramified graph covers of finite degree
Li, Winnie (Auteur de la Conférence) | CIRM (Editeur )

Given a finite connected undirected graph $X$, its fundamental group plays the role of the absolute Galois group of $X$. The familiar Galois theory holds in this setting. In this talk we shall discuss graph theoretical counter parts of several important theorems for number fields. Topics include
(a) Determination, up to equivalence, of unramified normal covers of $X$ of given degree,
(b) Criteria for Sunada equivalence,
(c) Chebotarev density theorem.
This is a joint work with Hau-Wen Huang.
Given a finite connected undirected graph $X$, its fundamental group plays the role of the absolute Galois group of $X$. The familiar Galois theory holds in this setting. In this talk we shall discuss graph theoretical counter parts of several important theorems for number fields. Topics include
(a) Determination, up to equivalence, of unramified normal covers of $X$ of given degree,
(b) Criteria for Sunada equivalence,
(c) Chebotarev density ...

05C25 ; 05C50 ; 11R32 ; 11R44 ; 11R45

Post-edited  Unirational varieties - Part 1
Mella, Massimiliano (Auteur de la Conférence) | CIRM (Editeur )

The aim of these talks is to give an overview to unirationality problems. I will discuss the behaviour of unirationality in families and its relation with rational connectedness. Then I will concentrate on hypersurfaces and conic bundles. These special classes of varieties are a good place where to test different techniques and try to approach the unirationality problem via rational connectedness.

14M20 ; 14G05 ; 14E05

Post-edited  Une deuxième révolution galiléenne ?
Dowek, Gilles (Auteur de la Conférence) | CIRM (Editeur )

L'introduction d'un nouveau concept scientifique permet souvent de donner de nouvelles réponses à des questions anciennes qui n'avaient jusqu'alors reçu que des réponses imparfaites. Cet exposé présente quelques questions qui ont trouvé de nouvelles réponses depuis que nous comprenons mieux la notion d'algorithme : qu'est-ce qu'un aéroport ?, qu'est-ce qu'une cellule, qu'est-ce qu'une loi physique ?, ... La prise de conscience du caractère algorithmique de ces objets scientifiques nous amène à considérer de nouveaux langages pour les décrire. Cette révolution, dans le langage dans lequel la science s'écrit, peut-être comparée à la révolution qui s'est produite, au début du XVIIe siècle, quand le langage mathématique a commencé à être utilisé pour décrire des phénomènes physiques. L'introduction d'un nouveau concept scientifique permet souvent de donner de nouvelles réponses à des questions anciennes qui n'avaient jusqu'alors reçu que des réponses imparfaites. Cet exposé présente quelques questions qui ont trouvé de nouvelles réponses depuis que nous comprenons mieux la notion d'algorithme : qu'est-ce qu'un aéroport ?, qu'est-ce qu'une cellule, qu'est-ce qu'une loi physique ?, ... La prise de conscience du caractère ...

00A30 ; 03B35 ; 68T15

We show that two-player stochastic games with perfect-information and shift-invariant submixing payoff functions are half-positional, i.e. in these games the maximizer has a positional optimal strategy. This extension of our previous result for one-player games relies on an interesting existence result about the existence of epsilon-subgame-perfect strategies.

68Q60 ; 91AXX

We consider "higher dimensional Teichmüller discs", by which we mean complex submanifolds of Teichmüller space that contain the Teichmüller disc joining any two of its points. We prove results in the higher dimensional setting that are opposite to the one dimensional behavior: every "higher dimensional Teichmüller disc" covers a "higher dimensional Teichmüller curve" and there are only finitely many "higher dimensional Teichmüller curves" in each moduli space. The proofs use recent results in Teichmüller dynamics, especially joint work with Eskin and Filip on the Kontsevich-Zorich cocycle. Joint work with McMullen and Mukamel as well as Eskin, McMullen and Mukamel shows that exotic examples of "higher dimensional Teichmüller discs" do exist. We consider "higher dimensional Teichmüller discs", by which we mean complex submanifolds of Teichmüller space that contain the Teichmüller disc joining any two of its points. We prove results in the higher dimensional setting that are opposite to the one dimensional behavior: every "higher dimensional Teichmüller disc" covers a "higher dimensional Teichmüller curve" and there are only finitely many "higher dimensional Teichmüller curves" in ...

30F60 ; 32G15

La théorie des valeurs extrêmes décrit le comportement du maximum d'une suite de variables aléatoires i.i.d. à valeurs réelles. L'une des distributions limites possibles, la loi de Gumbel, apparaît également dans l'asymptotique en bruit faible du temps de transition réactive pour des équations différentielles stochastiques métastables. Nous décrivons des résultats récents en dimension 1 et leur interprétation, et donnons un résultat en dimension 2, motivé par le phénomène de synchronisation d'oscillateurs couplés. La théorie des valeurs extrêmes décrit le comportement du maximum d'une suite de variables aléatoires i.i.d. à valeurs réelles. L'une des distributions limites possibles, la loi de Gumbel, apparaît également dans l'asymptotique en bruit faible du temps de transition réactive pour des équations différentielles stochastiques métastables. Nous décrivons des résultats récents en dimension 1 et leur interprétation, et donnons un résultat en dimension ...

60G70 ; 37H10

We give a summary of a joint work with Giovanni Landi (Trieste University) on a non commutative generalization of Henri Cartan's theory of operations, algebraic connections and Weil algebra.

81R10 ; 81R60 ; 16T05

We present a novel methodology for causal inference based on an invariance principle. It exploits the advantage of heterogeneity in larger datasets, arising from different experimental conditions (i.e. an aspect of "Big Data"). Despite fundamental identifiability issues, the method comes with statistical confidence statements leading to more reliable results than alternative procedures based on graphical modeling. We also discuss applications in biology, in particular for large-scale gene knock-down experiments in yeast where computational and statistical methods have an interesting potential for prediction and prioritization of new experimental interventions. We present a novel methodology for causal inference based on an invariance principle. It exploits the advantage of heterogeneity in larger datasets, arising from different experimental conditions (i.e. an aspect of "Big Data"). Despite fundamental identifiability issues, the method comes with statistical confidence statements leading to more reliable results than alternative procedures based on graphical modeling. We also discuss applications in ...

62H12 ; 62Fxx ; 62Pxx

The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint work with Mircea Mustata and Chenyang Xu. The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint ...

14B05 ; 14D06 ; 14E30 ; 14E18 ; 14G10 ; 14G22

We review basic properties of the moment-LP and moment-SOS hierarchies for polynomial optimization and compare them. We also illustrate how to use such a methodology in two applications outside optimization. Namely :
- for approximating (as claosely as desired in a strong sens) set defined with quantifiers of the form
$R_1 =\{ x\in B : f(x,y)\leq 0 $ for all $y$ such that $(x,y) \in K \}$.
$D_1 =\{ x\in B : f(x,y)\leq 0 $ for some $y$ such that $(x,y) \in K \}$.
by a hierarchy of inner sublevel set approximations
$\Theta_k = \left \{ x\in B : J_k(x)\leq 0 \right \}\subset R_f$.
or outer sublevel set approximations
$\Theta_k = \left \{ x\in B : J_k(x)\leq 0 \right \}\supset D_f$.
for some polynomiales $(J_k)$ of increasing degree :
- for computing convex polynomial underestimators of a given polynomial $f$ on a box $B \subset R^n$.
We review basic properties of the moment-LP and moment-SOS hierarchies for polynomial optimization and compare them. We also illustrate how to use such a methodology in two applications outside optimization. Namely :
- for approximating (as claosely as desired in a strong sens) set defined with quantifiers of the form
$R_1 =\{ x\in B : f(x,y)\leq 0 $ for all $y$ such that $(x,y) \in K \}$.
$D_1 =\{ x\in B : f(x,y)\leq 0 $ for ...

44A60 ; 90C22

Spherical Hecke algebra, Satake transform, and an introduction to local Langlands correspondence.

20C08 ; 22E50 ; 11S37

At the end of the 70', Littlejohn [1, 2, 3] shed new light on what is called the Gyro-Kinetic Approximation. His approach incorporated high-level mathematical concepts from Hamiltonian Mechanics, Differential Geometry and Symplectic Geometry into a physical affordable theory in order to clarify what has been done for years in the domain. This theory has been being widely used to deduce the numerical methods for Tokamak and Stellarator simulation. Yet, it was formal from the mathematical point of view and not directly accessible for mathematicians.
This talk will present a mathematically rigorous version of the theory. The way to set out this Gyro-Kinetic Approximation consists of the building of a change of coordinates that decouples the Hamiltonian dynamical system satisfied by the characteristics of charged particles submitted to a strong magnetic field into a part that concerns the fast oscillation induced by the magnetic field and a other part that describes a slower dynamics.
This building is made of two steps. The goal of the first one, so-called "Darboux Algorithm", is to give to the Poisson Matrix (associated to the Hamiltonian system) a form that would achieve the goal of decoupling if the Hamiltonian function does not depend on one given variable. Then the second change of variables (which is in fact a succession of several ones), so-called "Lie Algorithm", is to remove the given variable from the Hamiltonian function without changing the form of the Poisson Matrix.
(Notice that, beside this Geometrical Gyro-Kinetic Approximation Theory, an alternative approach, based on Asymptotic Analysis and Homogenization Methods was developed in Frenod and Sonnendrücker [5, 6, 7], Frenod, Raviart and Sonnendrücker [4], Golse and Saint-Raymond [9] and Ghendrih, Hauray and Nouri [8].)
At the end of the 70', Littlejohn [1, 2, 3] shed new light on what is called the Gyro-Kinetic Approximation. His approach incorporated high-level mathematical concepts from Hamiltonian Mechanics, Differential Geometry and Symplectic Geometry into a physical affordable theory in order to clarify what has been done for years in the domain. This theory has been being widely used to deduce the numerical methods for Tokamak and Stellarator s...

70H05 ; 82D10 ; 58Z05 ; 58J37 ; 58J45 ; 58D10

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