Camille Jordan Institute

We consider Gelfand pairs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper G comma upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(G,K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a compact Lie group and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a subgroup of fixed points of an involutive automorphism. We study the regularity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bi-invariant matrix coefficients of unitary representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The results rely on the analysis of the spherical functions of the Gelfand pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper G comma upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(G,K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . When the symmetric space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash upper K"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">G/K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or isomorphic to a Lie group, we find the optimal regularity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bi-invariant matrix coefficients of unitary representations. Furthermore, in rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we also find the optimal regularity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bi-invariant Herz-Schur multipliers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript p Baseline left-parenthesis upper L squared left-parenthesis upper G right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">S_p(L^2(G))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also give a lower bound for the optimal regularity in some families of higher rank symmetric spaces. From these results, we make a conjecture in the general case involving the root system of the symmetric space. Finally, we prove that if all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bi-invariant matrix coefficients of unitary representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Hölder continuous for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" al

Abstract The rheology of spheroids has been studied intensively recently and it was shown that the presence of the Extra-Cellular Matrix (ECM) can have significant effects on the overall behaviour of these biological systems. Collagen I can indeed be a proxy between cells and bring new intriguing effects, as its content increases. To investigate these effects further, a two-phase emulsion model is proposed including interactions between cells and the ECM. Starting with the single cell and collagen individual viscoelastic properties, the model can be tested against previously obtained data for spheroids. The model has interesting features and capabilities for it covers a variety of behaviours and uses fitting parameters such as collagen and cell concentration, as well as adhesion energy. It is shown that the final intercellular collagen content can be large as compared to the initial one, and that this increase in collagen content induces a larger packing of cells, together with a larger adhesion energy.

This article presents some results obtained recently in the history of mathematics using the tools of textometry.The results pertain to two case-studies, devoted to Charles Hermite's style and to the theory of algebraic surfaces at the end of the 19th century, respectively.

It is well-known that any Lennard–Jones type potential energy must have a periodic ground state given by a triangular lattice in dimension 2. In this paper, we describe a computer-assisted method that rigorously shows such global minimality result among 2-dimensional lattices once the exponents of the potential have been fixed. The method is applied to the widely used classical (12,6) Lennard–Jones potential, which is the main result of this work. Furthermore, a new bound on the inverse density (i.e. the co-volume) for which the triangular lattice is minimal is derived, improving those found in (Bétermin and Zhang 2015 Commun. Contemp. Math. 17 1450049) and (Bétermin 2016 SIAM J. Math. Anal. 48 3236–3269). The same results are also shown to hold for other exponents as additional examples and a new conjecture implying the global optimality of a triangular lattice for any parameters is stated.

We prove that Calogero–Moser cells coincide with Kazhdan–Lusztig cells for dihedral groups in the equal parameter case.

The emergence of new variants of concern (VOCs) of the SARS-CoV-2 infection is one of the main factors of epidemic progression. Their development can be characterized by three critical stages: virus mutation leading to the appearance of new viable variants; the competition of different variants leading to the production of a sufficiently large number of copies; and infection transmission between individuals and its spreading in the population. The first two stages take place at the individual level (infected individual), while the third one takes place at the population level with possible competition between different variants. This work is devoted to the mathematical modeling of the first two stages of this process: the emergence of new variants and their progression in the epithelial tissue with a possible competition between them. The emergence of new virus variants is modeled with non-local reaction–diffusion equations describing virus evolution and immune escape in the space of genotypes. The conditions of the emergence of new virus variants are determined by the mutation rate, the cross-reactivity of the immune response, and the rates of virus replication and death. Once different variants emerge, they spread in the infected tissue with a certain speed and viral load that can be determined through the parameters of the model. The competition of different variants for uninfected cells leads to the emergence of a single dominant variant and the elimination of the others due to competitive exclusion. The dominant variant is the one with the maximal individual spreading speed. Thus, the emergence of new variants at the individual level is determined by the immune escape and by the virus spreading speed in the infected tissue.

Abstract A reduction formula for the branching coefficients of the restrictions of representations of a semisimple group to a semisimple subgroup is proved in [9, 17, 33]. This formula holds when the highest weights of the representations belong to a codimension $1$ face of the Horn cone, which by [32] corresponds to some Schubert coefficient equal to $1$. We prove a similar reduction formula when this Schubert coefficient is equal to $2$ and show some properties of the class of the branch divisor corresponding to a generically finite morphism naturally defined in this context.

Abstract According to Waddington’s epigenetic landscape concept, the differentiation process can be illustrated by a cell akin to a ball rolling down from the top of a hill (proliferation state) and crossing furrows before stopping in basins or “attractor states” to reach its stable differentiated state. However, it is now clear that some committed cells can retain a certain degree of plasticity and reacquire phenotypical characteristics of a more pluripotent cell state. In line with this dynamic model, we have previously shown that differentiating cells (chicken erythrocytic progenitors (T2EC)) retain for 24 hours the ability to self-renew when transferred back in self-renewal conditions. Despite those intriguing and promising results, the underlying molecular state of those “reverting” cells remains unexplored. The aim of the present study was therefore to molecularly characterize the T2EC reversion process by combining advanced statistical tools to make the most of single cell transcriptomic data. For this purpose, T2EC, initially maintained in a self-renewal medium (0H), were induced to differentiate for 24h (24H differentiating cells); then a part of these cells was transferred back to the self-renewal medium (48H reverting cells) and the other part was maintained in the differentiation medium for another 24h (48H differentiating cells). For each time point, cell transcriptomes were generated using scRT-qPCR and scRNAseq. Our results showed a strong overlap between 0H and 48H reverting cells when applying dimensional reduction. Moreover, the statistical comparison of cell distributions and differential expression analysis indicated no significant differences between these two cell groups. Interestingly, gene pattern distributions highlighted that, while 48H reverting cells have gene expression pattern more similar to 0H cells, they retained traces of their engagement in the differentiation process. Finally, Sparse PLS analysis showed that only the expression of 3 genes discriminates 48H reverting and 0H cells. Altogether, we show that reverting cells return to an earlier molecular state almost identical to undifferentiated cells and demonstrate a previously undocumented physiological and molecular plasticity during the differentiation process, which most likely results from the dynamic behavior of the underlying molecular network.