In this talk, I will survey some recent progress towards understanding the slopes of modular forms, with or without level structures. This has direct application to the conjecture of Breuil-Buzzard-Emerton on the slopes of Kisin’s crystabelline deformation spaces. In particular, we obtain certain refined version of the spectral halo conjecture, where we may identify explicitly the slopes at the boundary when given a reducible non-split generic residual local Galois representation. This is a joint work in progress with Ruochuan Liu, Nha Truong, and Bin Zhao.

Let W be a Coxeter group. We provide a precise description of the conjugacy classes in W, yielding an analogue of Matsumoto’s theorem for the conjugacy problem in arbitrary Coxeter groups. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. In particular, we describe the cyclically reduced elements of W, thereby proving a conjecture of A. Cohen from 1994.

Along with a strict determinism of early embryogenesis in most living organisms, the variability of cell fates and developmental pathways exhibits in some of them. We propose that both determinism and variability are established through cell interactions, with the important role played by cell potency, cell sensitivity and cell signaling. The sensitivity of embryonic stem cells is considered to be strong, while the sensitivity of fully differentiated cells is small. This means that high potency correlates with high sensitivity, and vice versa. Experimental data were obtained and analyzed for the early developmental stages of plant species with regular and  irregular types of embryogenesis, which explicit correspondingly determinism or variability of developmental pathways. For the irregular type the species with underdeveloped embryos in mature seeds were examined. As the result, we propose three conjectures for explanation the phenomenon of variability, leading to the invariant final embryo shape, and support each of  these cases with the example(s) of actual developmental pathways.

For an automorphism of an algebraic variety, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, related to the notion of entropy. This is a report on joint work with Hélène Esnault, and some subsequent work of K. Shuddhodan.

In this talk we will discuss the Besson-Courtois-Gallot (BCG) theorem in the context of convex projective geometry. The BCG theorem is a rigidity statement relating the volume and entropy of a negatively curved Riemannian manifold, and has many applications including Mostow rigidity. In the world of convex real projective structures, the natural Hilbert geometry on these objects is only Finsler and the geometry is generally not even $C^2$. We discuss our analogous BCG theorem and some applications in the case where the manifold is closed. We will include some ongoing work to extend the result to finite volume. This is based on joint work with Ilesanmi Adeboye and David Constantine.

Consider a reductive group G over a p-adic field F. The local Langlands conjecture relates the irreducible smooth representations of G(F) with the set of (local) L-parameters, which are maps from the Weil group of F to the L-group of G; refinements of the conjecture relate the fibres of this map with the automorphism group of the L-parameter. Based on ideas from V. Lafforgue's work in the global function field case, I outlined a strategy for attaching (semisimple) L-parameters to irreducible smooth representations of G(F) in my 2014 Berkeley course. At the same time and place, L. Fargues formulated a conjecture relating the local Langlands conjecture with a geometric Langlands conjecture on the Fargues-Fontaine curve. The goal of this course will be to discuss some of the developments since then. On the foundational side, this concerns basics on the etale cohomology of diamonds including smooth and proper base change and Poincare duality, leading up to a good notion of "constructible" sheaves on the stack of G-bundles on the Fargues-Fontaine curve. On the applied side, this concerns the construction of (semisimple) L-parameters, the conjecture of Harris (as modified by Viehmann) on the cohomology of non-basic Rapoport-Zink spaces, and the conjecture of Kottwitz on the cohomology of basic Rapoport-Zink spaces.

 

 

Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :

https://www.fondation-hadamard.fr/fr/evenements/lecons-hadamard

Consider a reductive group G over a p-adic field F. The local Langlands conjecture relates the irreducible smooth representations of G(F) with the set of (local) L-parameters, which are maps from the Weil group of F to the L-group of G; refinements of the conjecture relate the fibres of this map with the automorphism group of the L-parameter. Based on ideas from V. Lafforgue's work in the global function field case, I outlined a strategy for attaching (semisimple) L-parameters to irreducible smooth representations of G(F) in my 2014 Berkeley course. At the same time and place, L. Fargues formulated a conjecture relating the local Langlands conjecture with a geometric Langlands conjecture on the Fargues-Fontaine curve. The goal of this course will be to discuss some of the developments since then. On the foundational side, this concerns basics on the etale cohomology of diamonds including smooth and proper base change and Poincare duality, leading up to a good notion of "constructible" sheaves on the stack of G-bundles on the Fargues-Fontaine curve. On the applied side, this concerns the construction of (semisimple) L-parameters, the conjecture of Harris (as modified by Viehmann) on the cohomology of non-basic Rapoport-Zink spaces, and the conjecture of Kottwitz on the cohomology of basic Rapoport-Zink spaces.

 

 

Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :

https://www.fondation-hadamard.fr/fr/evenements/lecons-hadamard

Consider a reductive group G over a p-adic field F. The local Langlands conjecture relates the irreducible smooth representations of G(F) with the set of (local) L-parameters, which are maps from the Weil group of F to the L-group of G; refinements of the conjecture relate the fibres of this map with the automorphism group of the L-parameter. Based on ideas from V. Lafforgue's work in the global function field case, I outlined a strategy for attaching (semisimple) L-parameters to irreducible smooth representations of G(F) in my 2014 Berkeley course. At the same time and place, L. Fargues formulated a conjecture relating the local Langlands conjecture with a geometric Langlands conjecture on the Fargues-Fontaine curve. The goal of this course will be to discuss some of the developments since then. On the foundational side, this concerns basics on the etale cohomology of diamonds including smooth and proper base change and Poincare duality, leading up to a good notion of "constructible" sheaves on the stack of G-bundles on the Fargues-Fontaine curve. On the applied side, this concerns the construction of (semisimple) L-parameters, the conjecture of Harris (as modified by Viehmann) on the cohomology of non-basic Rapoport-Zink spaces, and the conjecture of Kottwitz on the cohomology of basic Rapoport-Zink spaces.

 

 

Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :

https://www.fondation-hadamard.fr/fr/evenements/lecons-hadamard

Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In these pedagogical lectures, I will explore causality constraints on conformal field theory. First, I will show how causality is encoded in crossing symmetry and reflection positivity of Euclidean correlators, and derive constraints on the interactions of low-lying operators directly from the conformal bootstrap. Then, I will explain the connection between these causality constraints and the averaged null energy condition. Finally, I will use causality to show that the averaged null energy is positive in interacting quantum field theory in flat spacetime. Based on arXiv:1509.00014, arXiv:1601.07904, arXiv:1610.05308.

Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In these pedagogical lectures, I will explore causality constraints on conformal field theory. First, I will show how causality is encoded in crossing symmetry and reflection positivity of Euclidean correlators, and derive constraints on the interactions of low-lying operators directly from the conformal bootstrap. Then, I will explain the connection between these causality constraints and the averaged null energy condition. Finally, I will use causality to show that the averaged null energy is positive in interacting quantum field theory in flat spacetime. Based on arXiv:1509.00014, arXiv:1601.07904, arXiv:1610.05308.

Hydrogen bonds stabilize the crystal structures of most proteins. In the aqueous environment, they lie at the edge of stability and hence are easily formed and broken, especially those with high free energy. Using a Boltzmann-like formalism, empirical sampling of their geometry leads to free energy estimates, thus allowing a priori predictions of protein regions primed for structural rearrangement. One among diverse applications involves viral glycoproteins and capsids, namely those proteins employed by viruses to fuse with or penetrate the host cell.  Here the methods are used to identify sites of interest for vaccine/drug/testing development, in particular for the SARS CoV-2 virus that causes COVID-19.

Please click the link below to register to the webinar:

https://us02web.zoom.us/webinar/register/WN_j0sc3dGdSqW-7NcSffuwIg

By quantum matrix algebras I mean these related to braidings (solutions to Quantum Yang-Baxter Equation) and in a sense similar to the classical matrix algebras. In first turn, I am interested in the so-called Reflection Equation algebra. By using it, me (in collaboration with P.Saponov) have introduced the notion of partial derivatives on the enveloping algebra U(gl(m)). This leads to a new type of Noncommutative Geometry (we call it Quantum Geometry), which is deformation of the classical one. In my talk I plan to consider a way of defining some dynamical models on U(u(2)) background.