Kadanoff’s block idea pioneers the renormalization group (RG) theory and clarifies the scaling hypothesis in critical phenomena. Nevertheless, it has difficulty as a quantitatively reliable RG method due to uncontrolled approximations when formulated in the spin language. Reformulated in a modern tensor-network language, the block idea is equipped with a natural measure of RG errors. In 2D, the RG errors are typically smaller than 1% and decrease systematically when more coupling constants are retained in the RG map. The relative error of the estimated free energy of the 2D Ising model can easily go down to about 10-9 using a personal computer.
In 3D, due to the linear growth of entanglement entropy, the RG errors are too large for the block-tensor map to be reliable. For the 3D Ising model, the RG errors grow to more than 10% just after one RG step, and then keep growing to more than 30% near the critical fixed point. Even worse, the estimated scaling dimensions fail to converge with respect to the RG step. We propose an entanglement filtering (EF) scheme to cleanse the redundant entanglement. Enhanced by the proposed EF, the RG errors near the critical fixed point goes down to 6%; they decrease slowly to 2% when more couplings are retained. The estimated scaling dimensions become stable respect to the RG step. The relative errors of the first two relevant fields are 0.4% and 0.1% in the best case. The proposed RG is promising as a systematically-improvable real space RG method in 3D.
 
Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_physique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

The most universal kind of linear algebra is based not on abelian groups, but on homotopy-theoretic objects known as spectra. According to chromatic homotopy theory, one can systematically organize spectra into periodic families.  On the other hand, a natural source of spectra is provided by algebraic K-theory, a highly refined cohomological invariant of rings (or schemes, etc).  This leads to the subject of this course: the interaction of the chromatic theory with algebraic K-theory.  The story begins with classical theorems of Thomason, Mitchell, and Hesselholt-Madsen.  Bold generalizations of these theorems were conjectured by Rognes and Ausoni-Rognes, under the umbrella term of « redshift ».  Several of these conjectures are now theorems due to recent work of many people.  Remarkably, this work has applications to « pure » chromatic homotopy theory: Burklund-Hahn-Levy-Schlank used it to settle (in the negative) the « telescope conjecture », the last of Ravenel’s conjectures.
Lecture 1: Introduction to chromatic homotopy theory.Lecture 2: Descent and « soft redshift ».
 Lecture 3: « Hard redshift », a.k.a. the Lichtenbaum-Quillen property.
 Lecture 4: The telescope conjecture.

The most universal kind of linear algebra is based not on abelian groups, but on homotopy-theoretic objects known as spectra. According to chromatic homotopy theory, one can systematically organize spectra into periodic families.  On the other hand, a natural source of spectra is provided by algebraic K-theory, a highly refined cohomological invariant of rings (or schemes, etc).  This leads to the subject of this course: the interaction of the chromatic theory with algebraic K-theory.  The story begins with classical theorems of Thomason, Mitchell, and Hesselholt-Madsen.  Bold generalizations of these theorems were conjectured by Rognes and Ausoni-Rognes, under the umbrella term of « redshift ».  Several of these conjectures are now theorems due to recent work of many people.  Remarkably, this work has applications to « pure » chromatic homotopy theory: Burklund-Hahn-Levy-Schlank used it to settle (in the negative) the « telescope conjecture », the last of Ravenel’s conjectures.
Lecture 1: Introduction to chromatic homotopy theory.Lecture 2: Descent and « soft redshift ».
 Lecture 3: « Hard redshift », a.k.a. the Lichtenbaum-Quillen property.
 Lecture 4: The telescope conjecture.

The most universal kind of linear algebra is based not on abelian groups, but on homotopy-theoretic objects known as spectra. According to chromatic homotopy theory, one can systematically organize spectra into periodic families.  On the other hand, a natural source of spectra is provided by algebraic K-theory, a highly refined cohomological invariant of rings (or schemes, etc).  This leads to the subject of this course: the interaction of the chromatic theory with algebraic K-theory.  The story begins with classical theorems of Thomason, Mitchell, and Hesselholt-Madsen.  Bold generalizations of these theorems were conjectured by Rognes and Ausoni-Rognes, under the umbrella term of « redshift ».  Several of these conjectures are now theorems due to recent work of many people.  Remarkably, this work has applications to « pure » chromatic homotopy theory: Burklund-Hahn-Levy-Schlank used it to settle (in the negative) the « telescope conjecture », the last of Ravenel’s conjectures.
Lecture 1: Introduction to chromatic homotopy theory.Lecture 2: Descent and « soft redshift ».
 Lecture 3: « Hard redshift », a.k.a. the Lichtenbaum-Quillen property.
 Lecture 4: The telescope conjecture.

The most universal kind of linear algebra is based not on abelian groups, but on homotopy-theoretic objects known as spectra. According to chromatic homotopy theory, one can systematically organize spectra into periodic families.  On the other hand, a natural source of spectra is provided by algebraic K-theory, a highly refined cohomological invariant of rings (or schemes, etc).  This leads to the subject of this course: the interaction of the chromatic theory with algebraic K-theory.  The story begins with classical theorems of Thomason, Mitchell, and Hesselholt-Madsen.  Bold generalizations of these theorems were conjectured by Rognes and Ausoni-Rognes, under the umbrella term of « redshift ».  Several of these conjectures are now theorems due to recent work of many people.  Remarkably, this work has applications to « pure » chromatic homotopy theory: Burklund-Hahn-Levy-Schlank used it to settle (in the negative) the « telescope conjecture », the last of Ravenel’s conjectures.
Lecture 1: Introduction to chromatic homotopy theory.Lecture 2: Descent and « soft redshift ».
 Lecture 3: « Hard redshift », a.k.a. the Lichtenbaum-Quillen property.
 Lecture 4: The telescope conjecture.

I will explain how convex projective geometry over non-Archimedean ordered fields may be used to study large scale properties of individual real Hilbert geometries and degenerations of convex projective actions, using a projective geometry version of ultralimits. Non-Archimedean convex subsets have a naturally associated quotient Hilbert metric space. In the case of ultralimits, we show that it is the ultralimit of the real Hilbert metric spaces under a natural non-degeneracy condition. I will present some examples and give a full description of the Hilbert metric space for non-Archimedean polytopes defined over R, which correspond to the asymptotic cones of a fixed real polytope. This is joint work with Xenia Flamm.
 

Khintchine’s theorem is a key result in Diophantine approximation. Given a positive non-increasing function f defined over the integers, it states that the set of real numbers that are f-approximable has zero or full Lebesgue measure depending on whether the series of terms (f(n))n converges or diverges. I will present a recent work in collaboration with Weikun He and Han Zhang in which we extend Khintchine’s theorem to any self-similar probability measure on the real line. The argument involves the quantitative equidistribution of upper triangular random walks on SL(2,R)/SL(2,Z). 

Séminaire de géométrie arithmétique
From the 1980s to the 1990s, Jean-Marc Fontaine introduced the theory of (phi, Gamma)-modules to study p-adic Galois representations. They are simpler than p-adic Galois representations, but he showed an equivalence between them. Among p-adic Galois representations, some classes are particularly important in number theory. Main examples are crystalline representations, semi-stable representations and de Rham representations. In this talk, I will explain how we can determine the (phi, Gamma)-modules corresponding to these representations. These results can be seen, in a sense, as generalizations of Wach modules.
========
Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

The theory of De Giorgi (1958) and Nash (1959) solved Hilbert’s 19th problem and was a major contribution to 20th century PDE analysis. It is concerned with the Hölder regularity of solutions to elliptic and parabolic PDEs with rough (merely measurable) coefficients; it was developed by Moser (1960-1964) to include the Harnack inequality. These lectures  are an introduction to a recent active research area (Pascucci-Polidoro, Wang-Zhang, Golse-Imbert-M-Vasseur, Imbert-Silvestre, Imbert-Guerand, Guerand-M, Anceschi-Rebucci, Loher, Niebel-Zacher…): the extension of this theory to the hypoelliptic PDEs, local and nonlocal, that appear naturally in kinetic theory. The simpler prototypical case is the Kolmogorov equation (aka kinetic Fokker-Planck equation) with a rough matrix of coefficients in the kinetic diffusion. The course will in particular emphasize the recent quantitative robust methods based on the construction of trajectories and their connexions to control theory and hypocoercivity (works with Dieter, Hérau, Hutridurga, Niebel, Zacher).

The theory of De Giorgi (1958) and Nash (1959) solved Hilbert’s 19th problem and was a major contribution to 20th century PDE analysis. It is concerned with the Hölder regularity of solutions to elliptic and parabolic PDEs with rough (merely measurable) coefficients; it was developed by Moser (1960-1964) to include the Harnack inequality. These lectures  are an introduction to a recent active research area (Pascucci-Polidoro, Wang-Zhang, Golse-Imbert-M-Vasseur, Imbert-Silvestre, Imbert-Guerand, Guerand-M, Anceschi-Rebucci, Loher, Niebel-Zacher…): the extension of this theory to the hypoelliptic PDEs, local and nonlocal, that appear naturally in kinetic theory. The simpler prototypical case is the Kolmogorov equation (aka kinetic Fokker-Planck equation) with a rough matrix of coefficients in the kinetic diffusion. The course will in particular emphasize the recent quantitative robust methods based on the construction of trajectories and their connexions to control theory and hypocoercivity (works with Dieter, Hérau, Hutridurga, Niebel, Zacher).

The theory of De Giorgi (1958) and Nash (1959) solved Hilbert’s 19th problem and was a major contribution to 20th century PDE analysis. It is concerned with the Hölder regularity of solutions to elliptic and parabolic PDEs with rough (merely measurable) coefficients; it was developed by Moser (1960-1964) to include the Harnack inequality. These lectures  are an introduction to a recent active research area (Pascucci-Polidoro, Wang-Zhang, Golse-Imbert-M-Vasseur, Imbert-Silvestre, Imbert-Guerand, Guerand-M, Anceschi-Rebucci, Loher, Niebel-Zacher…): the extension of this theory to the hypoelliptic PDEs, local and nonlocal, that appear naturally in kinetic theory. The simpler prototypical case is the Kolmogorov equation (aka kinetic Fokker-Planck equation) with a rough matrix of coefficients in the kinetic diffusion. The course will in particular emphasize the recent quantitative robust methods based on the construction of trajectories and their connexions to control theory and hypocoercivity (works with Dieter, Hérau, Hutridurga, Niebel, Zacher).

The theory of De Giorgi (1958) and Nash (1959) solved Hilbert’s 19th problem and was a major contribution to 20th century PDE analysis. It is concerned with the Hölder regularity of solutions to elliptic and parabolic PDEs with rough (merely measurable) coefficients; it was developed by Moser (1960-1964) to include the Harnack inequality. These lectures  are an introduction to a recent active research area (Pascucci-Polidoro, Wang-Zhang, Golse-Imbert-M-Vasseur, Imbert-Silvestre, Imbert-Guerand, Guerand-M, Anceschi-Rebucci, Loher, Niebel-Zacher…): the extension of this theory to the hypoelliptic PDEs, local and nonlocal, that appear naturally in kinetic theory. The simpler prototypical case is the Kolmogorov equation (aka kinetic Fokker-Planck equation) with a rough matrix of coefficients in the kinetic diffusion. The course will in particular emphasize the recent quantitative robust methods based on the construction of trajectories and their connexions to control theory and hypocoercivity (works with Dieter, Hérau, Hutridurga, Niebel, Zacher).