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Documents : Multi angle  Conférences Vidéo | enregistrements trouvés : 200

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Multi angle  Prime numbers with preassigned digits
Swaenepoel, Cathy (Auteur de la Conférence) | CIRM (Editeur )

Bourgain (2015) estimated the number of prime numbers with a proportion $c$ > 0 of preassigned digits in base 2 ($c$ is an absolute constant not specified). We present a generalization of this result in any base $g$ ≥ 2 and we provide explicit admissible values for the proportion $c$ depending on $g$. Our proof, which adapts, develops and refines Bourgain’s strategy, is based on the circle method and combines techniques from harmonic analysis together with results on zeros of Dirichlet $L$-functions, notably a very sharp zero-free region due to Iwaniec.
Bourgain (2015) estimated the number of prime numbers with a proportion $c$ > 0 of preassigned digits in base 2 ($c$ is an absolute constant not specified). We present a generalization of this result in any base $g$ ≥ 2 and we provide explicit admissible values for the proportion $c$ depending on $g$. Our proof, which adapts, develops and refines Bourgain’s strategy, is based on the circle method and combines techniques from harmonic analysis ...

11N05 ; 11A41 ; 11A63

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A number field is monogenic if its ring of integers is generated by a single element. It is conjectured that for any degree d > 2, the proportion of degree d number fields which are monogenic is 0. There are local obstructions that force this proportion to be < 100%, but beyond this very little is known. I’ll discuss work with Alpoge and Bhargava showing that a positive proportion of cubic fields (d = 3) have no local obstructions and yet are still not monogenic. This uses new results on ranks of Selmer groups of elliptic curves in twist families.
A number field is monogenic if its ring of integers is generated by a single element. It is conjectured that for any degree d > 2, the proportion of degree d number fields which are monogenic is 0. There are local obstructions that force this proportion to be < 100%, but beyond this very little is known. I’ll discuss work with Alpoge and Bhargava showing that a positive proportion of cubic fields (d = 3) have no local obstructions and yet are ...

11R16 ; 11G05

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In this talk I will discuss questions concerning the asymptotic behavior of the Epstein zeta function $E_{n}\left ( L, s \right )$ in the limit of large dimension $n$. In particular I will be interested in the behavior of $E_{n}\left ( L, s \right )$ for a random lattice $L$ of large dimension $n$ and $s$ a complex number in the critical strip. Along the way we will encounter certain random functions that are closely related to $E_{n}\left ( L, s \right )$ and interesting in their own right.
In this talk I will discuss questions concerning the asymptotic behavior of the Epstein zeta function $E_{n}\left ( L, s \right )$ in the limit of large dimension $n$. In particular I will be interested in the behavior of $E_{n}\left ( L, s \right )$ for a random lattice $L$ of large dimension $n$ and $s$ a complex number in the critical strip. Along the way we will encounter certain random functions that are closely related to $E_{n}\left ( L, ...

11E45 ; 11M41 ; 11P21 ; 60G55

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The geometric P=W conjecture is a conjectural description of the asymptotic behavior of a celebrated correspondence in non-abelian Hodge theory. In particular, it is expected that the dual boundary complex of the compactification of character varieties is a sphere. In a joint work with Enrica Mazzon and Matthew Stevenson, we manage to compute the first non-trivial examples of dual complexes in the compact case. This requires to develop a new theory of essential skeletons over a trivially-valued field. As a byproduct, inspired by these constructions, we show that certain character varieties appear in degenerations of compact hyper-Kähler manifolds. In this talk we will explain how these new non-archimedean techniques can shed new light into classical algebraic geometry problems.
The geometric P=W conjecture is a conjectural description of the asymptotic behavior of a celebrated correspondence in non-abelian Hodge theory. In particular, it is expected that the dual boundary complex of the compactification of character varieties is a sphere. In a joint work with Enrica Mazzon and Matthew Stevenson, we manage to compute the first non-trivial examples of dual complexes in the compact case. This requires to develop a new ...

14G22

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We study locally the representation varieties of fundamental groups of smooth complex algebraic varieties. These are schemes whose complex points parametrize such representations into linear algebraic groups. At a given representation, the structure of the formal local ring to the representation variety tells about the obstructions to deform formally this representation, which is ultimately related to topological obstructions to the possible fundamental groups of complex algebraic varieties. This was first described by Goldman and Millson in the case of compact Kähler manifold, using formal deformation theory and differential graded Lie algebras. We review this using methods of Hodge theory and of derived deformation theory and we are able to describe locally the representation variety for non-compact smooth varieties and representations underlying a variation of Hodge structure.
We study locally the representation varieties of fundamental groups of smooth complex algebraic varieties. These are schemes whose complex points parametrize such representations into linear algebraic groups. At a given representation, the structure of the formal local ring to the representation variety tells about the obstructions to deform formally this representation, which is ultimately related to topological obstructions to the possible ...

14D07 ; 14C30 ; 14D15 ; 18D50

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Multi angle  Rational curves on K3 surfaces
Gounelas, Frank (Auteur de la Conférence) | CIRM (Editeur )

Bogomolov and Mumford proved that every complex projective K3 surface contains a rational curve. Since then, a lot of progress has been made by Bogomolov, Chen, Hassett, Li, Liedtke, Tschinkel and others, towards the stronger statement that any such surface in fact contains infinitely many rational curves. In this talk I will present joint work with Xi Chen and Christian Liedtke completing the remaining cases of this conjecture, reproving some of the main previously known cases more conceptually and extending the result to arbitrary genus in a suitable sense.
Bogomolov and Mumford proved that every complex projective K3 surface contains a rational curve. Since then, a lot of progress has been made by Bogomolov, Chen, Hassett, Li, Liedtke, Tschinkel and others, towards the stronger statement that any such surface in fact contains infinitely many rational curves. In this talk I will present joint work with Xi Chen and Christian Liedtke completing the remaining cases of this conjecture, reproving some ...

14J28

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I will present some results about the momentum polytopes of the multiplicity-free Hamiltonian compact manifolds acted on by a compact group which are Kählerizable. I shall give a characterization of these polytopes, explain how much they determine these manifolds and sketch some applications of this characterization - most of these results have been obtained jointly with G. Pezzini and B. Van Steirteghem.

14M27 ; 53D20 ; 32Q15

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Let $X$ be a Banach space of holomorphic functions on the unit disk. A linear polynomial approximation scheme for $X$ is a sequence of bounded linear operators $T_{n} :X\rightarrow X$ with the property that, for each $f\in X$, the functions $T_{n}\left ( f \right )$ are polynomials converging to $f$ in the norm of the space. We completely characterize those spaces $X$ that admit a linear polynomial approximation scheme. In particular, we show that it is not sufficient merely that polynomials be dense in $X$. (Joint work with Javad Mashreghi).
Let $X$ be a Banach space of holomorphic functions on the unit disk. A linear polynomial approximation scheme for $X$ is a sequence of bounded linear operators $T_{n} :X\rightarrow X$ with the property that, for each $f\in X$, the functions $T_{n}\left ( f \right )$ are polynomials converging to $f$ in the norm of the space. We completely characterize those spaces $X$ that admit a linear polynomial approximation scheme. In particular, we show ...

41A10 ; 46B15 ; 46B28

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Let $V$ be an analytic subvariety of a domain $\Omega$ in $\mathbb{C}^{n}$. When does $V$ have the property that every bounded holomorphic function $f$ on $V$ has an extension to a bounded holomorphic function on $\Omega$ with the same norm?
An obvious sufficient condition is if $V$ is a holomorphic retract of $\Omega$. We shall discuss for what domains $\Omega$ this is also necessary.
This is joint work with Łukasz Kosiński.

47-XX ; 46-XX ; 32-XX

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Multi angle  The Szegö minimum problem
Borichev, Alexander (Auteur de la Conférence) | CIRM (Editeur )

Given a finite positive measure $\mu$ on the unit circle, we consider the distance $e_{n}\left ( \mu \right )$ from $z^{n}$ to the analytic polynomials of degree less than $n$ in $L^{2}\left ( \mu \right )$. We study the asymptotic behavior of $e_{n}\left ( \mu \right )$ for $n\rightarrow \infty$ when the logarithmic integral of the density of $\mu$ diverges for different classes of measures $\mu$.

42C05

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This talk will discuss how to study singular rational inner functions (RIFs) using their zero set behaviors. In the two-variable setting, zero sets can be used to define a quantity called contact order, which helps quantify derivative integrability and non-tangential regularity. In the three-variable and higher setting, the RIF singular sets (and corresponding zero sets) can be much more complicated. We will discuss what holds in general, what holds for simple three-variable RIFs, and some examples illustrating why some of the nice two-variable behavior is lost in higher dimensions. This is joint work with James Pascoe and Alan Sola.
This talk will discuss how to study singular rational inner functions (RIFs) using their zero set behaviors. In the two-variable setting, zero sets can be used to define a quantity called contact order, which helps quantify derivative integrability and non-tangential regularity. In the three-variable and higher setting, the RIF singular sets (and corresponding zero sets) can be much more complicated. We will discuss what holds in general, ...

32A20 ; 14C17 ; 14H20 ; 32A35 ; 32A40

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Multi angle  Closed $G_{2}$-structures
Fino, Anna (Auteur de la Conférence) | CIRM (Editeur )

I will review known examples of compact 7-manifolds admitting a closed $G_{2}$-structure. Moreover, I will discuss some results on the behaviour of the Laplacian $G_{2}$-flow starting from a closed $G_{2}$-structure whose induced metric satisfies suitable extra condition.

53C30 ; 53C10 ; 22E25

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We show that the spectrum of fundamental particles of matter and their symmetries can be encoded in a finite quantum geometry equipped with a supplementary structure connected with the quark-lepton symmetry. The occurrence of the exceptional quantum geometry for the description of the standard model with 3 generations is suggested. We discuss the field theoretical aspect of this approach taking into account the theory of connections on the corresponding Jordan modules.
We show that the spectrum of fundamental particles of matter and their symmetries can be encoded in a finite quantum geometry equipped with a supplementary structure connected with the quark-lepton symmetry. The occurrence of the exceptional quantum geometry for the description of the standard model with 3 generations is suggested. We discuss the field theoretical aspect of this approach taking into account the theory of connections on the ...

81R10 ; 17C90 ; 20G41 ; 81Q35 ; 17C40

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Multi angle  Christian Mauduit in ergodic theory
Weiss, Benjamin (Auteur de la Conférence) | CIRM (Editeur )

While most of Christian’s work was in number theory he made important contributions to several aspects of ergodic theory throughout his career. I will discuss some of these and their impact on later developments.

28D05 ; 37A45 ; 37B99

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Multi angle  Non-normal sets
Queffélec, Martine (Auteur de la Conférence) | CIRM (Editeur )

The talk will have two main parts:
At the beginning of his mathematical career, Christian inherits from his advisor, Gérard Rauzy, some appetite for distributions of arithmetical sequences, with a special interest for those obtained by simple algorithmic constructions. I will discuss his result on “normal sets associated with substitutions" [?] and continue with recent developments and open questions on sets of non-normal numbers, in a more general setting.
We have written 48 joint papers with Christian. As a tribute to his memory I will present a short survey of our most important papers and recall some of the memorable moments of our cooperation.
The talk will have two main parts:
At the beginning of his mathematical career, Christian inherits from his advisor, Gérard Rauzy, some appetite for distributions of arithmetical sequences, with a special interest for those obtained by simple algorithmic constructions. I will discuss his result on “normal sets associated with substitutions" [?] and continue with recent developments and open questions on sets of non-normal numbers, in a more ...

11Kxx ; 37EXX ; 11Jxx

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Multi angle  Symbolic bounded remainder sets
Berthé, Valérie (Auteur de la Conférence) | CIRM (Editeur )

Discrepancy is a measure of equidistribution for sequences of points. We consider here discrepancy in the setting of symbolic dynamics and we discuss the existence of bounded remainder sets for some families of zero entropy subshifts, from a topological dynamics viewpoint. A bounded remainder set is a set which yields bounded discrepancy, that is, the number of times it is visited differs by the expected time only by a constant. Bounded discrepancy provides particularly strong convergence properties of ergodic sums. It is also closely related to the notions of balance in word combinatorics.
Discrepancy is a measure of equidistribution for sequences of points. We consider here discrepancy in the setting of symbolic dynamics and we discuss the existence of bounded remainder sets for some families of zero entropy subshifts, from a topological dynamics viewpoint. A bounded remainder set is a set which yields bounded discrepancy, that is, the number of times it is visited differs by the expected time only by a constant. Bounded ...

37B10 ; 11K50 ; 37A30 ; 28A80 ; 11J70 ; 11K38

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Multi angle  $L^p$-theory for Schrödinger systems
Rhandi, Abdelaziz (Auteur de la Conférence) | CIRM (Editeur )

In this talk we study for $p\in \left ( 1,\infty \right )$ the $L^{p}$-realization of the vector-valued Schrödinger operator $\mathcal{L}u:= div\left ( Q\triangledown u \right )+Vu$. Using a noncommutative version of the Dore-Venni theorem due to Monniaux and Prüss, and a perturbation theorem by Okazawa, we prove that $L^{p}$, the $L^{p}$-realization of $\mathcal{L}$, defined on the intersection of the natural domains of the differential and multiplication operators which form $\mathcal{L}$, generates a strongly continuous contraction semigroup on $L^{p}\left ( \mathbb{R}^{d} ;\mathbb{C}^{m}\right )$. We also study additional properties of the semigroup such as positivity, ultracontractivity, Gaussian estimates and compactness of the resolvent. We end the talk by giving some generalizations obtained recently and several examples.
In this talk we study for $p\in \left ( 1,\infty \right )$ the $L^{p}$-realization of the vector-valued Schrödinger operator $\mathcal{L}u:= div\left ( Q\triangledown u \right )+Vu$. Using a noncommutative version of the Dore-Venni theorem due to Monniaux and Prüss, and a perturbation theorem by Okazawa, we prove that $L^{p}$, the $L^{p}$-realization of $\mathcal{L}$, defined on the intersection of the natural domains of the differential and ...

47D06 ; 35J15 ; 47D08

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