I will give a definition of a certain category of « log quasicoherent » sheaves on a logarithmic variety which uses Falting’s « almost mathematics » and which has the property that log differential forms and log polyvector fields are the Hochshild homology (appropriately understood) and Hochschild cohomology, respectively, of this category. This implies a certain « noncommutative Hodge theory » associated to a log variety in mixed characteristic. I will also explain (if there is time left over) a relationship of the proof of the main results to mirror symmetry. 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. 

Séminaire Laurent Schwartz — EDP et applications 

Séminaire Laurent Schwartz — EDP et applications 

Séminaire Laurent Schwartz — EDP et applications 

Any finite-dimensional p-adic representation of the absolute Galois group of a p-adic local field with imperfect residue field is characterized by its arithmetic and geometric Sen operators defined by Sen and Brinon. We generalize their construction to the fundamental group of a p-adic affine variety with a semi-stable chart, and prove that the module of Sen operators is canonically defined, independently of the choice of the chart. Our construction relies on a descent theorem in the p-adic Simpson correspondence developed by Tsuji. When the representation comes from a Qp-representation of a p-adic analytic group quotient of the fundamental group, we describe its Lie algebra action in terms of the Sen operators, which is a generalization of a result of Sen and Ohkubo. These Sen operators can be extended continuously to certain infinite-dimensional representations. As an application, we prove that the geometric Sen operators annihilate locally analytic vectors, generalizing a result of Pan. 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. 

For $ell$-adic cohomology, the weight monodromy conjecture for complete intersections was proved by Scholze in his celebrated paper. Using his theory of perfectoid spaces, he reduced it to the equal characteristic case, which was already proved by Deligne. Considering that the equal characteristic case of the p-adic weight monodromy conjecture has been also formulated and proved (by Crew and Lazda–Pal), it is natural to try to reduce the p-adic weight monodromy conjecture to the equal characteristic case using Scholze’s technique. In this talk, I will discuss how to realize it (joint work with Federico Binda and Alberto Vezzani).  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. 

I will talk about a series of works with Artem Prikhodko where we develop a version of p-adic Hodge theory in the setting of Artin stacks. One of the main motivations for our project was a conjecture by Totaro: namely, based on his concrete computations, he suggested that the dimension of mod p de Rham cohomology of the classifying stack BG for G reductive might always be bounded from below by the dimension of the Fp-singular cohomology on the classifying space BG(C) of the Lie group G(C) of complex points of G.  For smooth and proper schemes such an inequality is a consequence of integral p-adic Hodge theory in the form proved by Bhatt-Morrow-Scholze; however, their results can not be applied here directly since BG is not proper.To prove Totaro’s conjecture, using the theory of prismatic cohomology, we develop integral p-adic Hodge theory in a more general setting of Hodge-proper stacks: these are stacks that only look proper from the point of view of its Hodge cohomology. However, one problem then still remains: namely, the étale comparison we get is with the étale cohomology of the Raynaud generic fiber, which a priori agrees with the algebraic generic fiber (and then complex points) only in the smooth proper setting. Nevertheless, we prove that the two étale cohomology theories still agree at least for quotient stacks [X/G] with X smooth and proper and G reductive. This then implies Totaro’s conjecture by plugging X=pt. In further work we also show that after inverting p the two étale cohomology agree for any Hodge-proper stack, which sets up rational p-adic Hodge theory (with the crystalline and de Rham comparisons, and Hodge-Tate decomposition) in this setting. If time permits, I will also tell about some explicit computations of cohomology of reductive groups in characteristic p that one can perform using the above comparison in the case of BG (this is a series of joint works with Federico Scavia and Anlong Chua). 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. 

I will talk about a series of works with Artem Prikhodko where we develop a version of p-adic Hodge theory in the setting of Artin stacks. One of the main motivations for our project was a conjecture by Totaro: namely, based on his concrete computations, he suggested that the dimension of mod p de Rham cohomology of the classifying stack BG for G reductive might always be bounded from below by the dimension of the Fp-singular cohomology on the classifying space BG(C) of the Lie group G(C) of complex points of G.  For smooth and proper schemes such an inequality is a consequence of integral p-adic Hodge theory in the form proved by Bhatt-Morrow-Scholze; however, their results can not be applied here directly since BG is not proper.To prove Totaro’s conjecture, using the theory of prismatic cohomology, we develop integral p-adic Hodge theory in a more general setting of Hodge-proper stacks: these are stacks that only look proper from the point of view of its Hodge cohomology. However, one problem then still remains: namely, the étale comparison we get is with the étale cohomology of the Raynaud generic fiber, which a priori agrees with the algebraic generic fiber (and then complex points) only in the smooth proper setting. Nevertheless, we prove that the two étale cohomology theories still agree at least for quotient stacks [X/G] with X smooth and proper and G reductive. This then implies Totaro’s conjecture by plugging X=pt. In further work we also show that after inverting p the two étale cohomology agree for any Hodge-proper stack, which sets up rational p-adic Hodge theory (with the crystalline and de Rham comparisons, and Hodge-Tate decomposition) in this setting. If time permits, I will also tell about some explicit computations of cohomology of reductive groups in characteristic p that one can perform using the above comparison in the case of BG (this is a series of joint works with Federico Scavia and Anlong Chua). 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 Loewner energy for Jordan curves first arises from the large deviations of Schramm-Loewner evolution (SLE), a family of random fractal curves modeling interfaces in 2D statistical mechanics. In a certain way, this energy measures the roundness of a Jordan curve, and we show that it is finite if and only if the curve is a Weil-Petersson quasicircle and connect it to determinants of Laplacians. Furthermore, the Loewner energy is a Kahler potential on the Weil-Petersson Teichmueller space identified with the space of Weil-Petersson quasicircles. Intriguingly, this class of finite energy curves has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, spectral theory, and has been studied since the eighties with motivations from string theory. The myriad of perspectives on this class of curves is both luxurious and mysterious. I will overview the links between Loewner energy and SLE, Weil-Petersson quasicircles, and other branches of mathematics it touches on. I will highlight how ideas from random conformal geometry inspire new results on Weil-Petersson quasicircles and discuss further directions.​

The Loewner energy for Jordan curves first arises from the large deviations of Schramm-Loewner evolution (SLE), a family of random fractal curves modeling interfaces in 2D statistical mechanics. In a certain way, this energy measures the roundness of a Jordan curve, and we show that it is finite if and only if the curve is a Weil-Petersson quasicircle and connect it to determinants of Laplacians. Furthermore, the Loewner energy is a Kahler potential on the Weil-Petersson Teichmueller space identified with the space of Weil-Petersson quasicircles. Intriguingly, this class of finite energy curves has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, spectral theory, and has been studied since the eighties with motivations from string theory. The myriad of perspectives on this class of curves is both luxurious and mysterious. I will overview the links between Loewner energy and SLE, Weil-Petersson quasicircles, and other branches of mathematics it touches on. I will highlight how ideas from random conformal geometry inspire new results on Weil-Petersson quasicircles and discuss further directions.​

The Loewner energy for Jordan curves first arises from the large deviations of Schramm-Loewner evolution (SLE), a family of random fractal curves modeling interfaces in 2D statistical mechanics. In a certain way, this energy measures the roundness of a Jordan curve, and we show that it is finite if and only if the curve is a Weil-Petersson quasicircle and connect it to determinants of Laplacians. Furthermore, the Loewner energy is a Kahler potential on the Weil-Petersson Teichmueller space identified with the space of Weil-Petersson quasicircles. Intriguingly, this class of finite energy curves has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, spectral theory, and has been studied since the eighties with motivations from string theory. The myriad of perspectives on this class of curves is both luxurious and mysterious. I will overview the links between Loewner energy and SLE, Weil-Petersson quasicircles, and other branches of mathematics it touches on. I will highlight how ideas from random conformal geometry inspire new results on Weil-Petersson quasicircles and discuss further directions.​

The Loewner energy for Jordan curves first arises from the large deviations of Schramm-Loewner evolution (SLE), a family of random fractal curves modeling interfaces in 2D statistical mechanics. In a certain way, this energy measures the roundness of a Jordan curve, and we show that it is finite if and only if the curve is a Weil-Petersson quasicircle and connect it to determinants of Laplacians. Furthermore, the Loewner energy is a Kahler potential on the Weil-Petersson Teichmueller space identified with the space of Weil-Petersson quasicircles. Intriguingly, this class of finite energy curves has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, spectral theory, and has been studied since the eighties with motivations from string theory. The myriad of perspectives on this class of curves is both luxurious and mysterious. I will overview the links between Loewner energy and SLE, Weil-Petersson quasicircles, and other branches of mathematics it touches on. I will highlight how ideas from random conformal geometry inspire new results on Weil-Petersson quasicircles and discuss further directions.​