Institut de recherche mathématique avancée

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La fonction de Möbius est une fonction multiplicative sur les entiers qui apparaît en théorie des nombres comme les coefficients de l'inverse de la fonction Zeta de Riemann. On peut en fait définir la fonction de Möbius sur n'importe quel poset (ensemble partiellement ordonné). Elle peut s'interpréter comme la caractéristique d'Euler réduite d'un espace topologique naturellement associé au poset, le complexe d'ordre. D'un point de vue plus combinatoire, dans certains cas, cette fonction de Möbius peut également compter des choses intéressantes, notamment dans des configurations d'hyperplans.

In this talk we will discuss the problem of profinite rigidity of finitely generated groups. We will examine connections between algebraic fibring, twisted Alexander invariants, and profinite rigidity. If time permits we will mention an application to Kähler groups. Based on joint work with Dawid Kielak.

The symmetric group admits two natural covering groups: the braid group and the twin group. These are obtained, respectively, by removing the involution and cubic relations in the Coxeter presentation of the symmetric group. There is a natural Burau representation for each group, which for the braid group is a q-analog of the permutation representation of the symmetric group and for the twin group is a related orthogonal representation generated by complex reflections. New instances of Schur-Weyl duality are found by examining the diagonal action of these groups on tensor powers of the Burau representation. The centralizer algebra of the action of each group is described diagramatically in terms of partial permutation and partial Brauer algebras. As a result, we obtain many representations of the braid and twin groups that can be combinatorially constructed. This is joint work with Stephen Doty.

The total number of nodal domains of a Laplace-Beltrami eigenfunction on a closed manifold is bounded from above by an appropriate power of the corresponding eigenvalue. This is a consequence of Courant's nodal domain theorem combined with Weyl's law. In general, bounds of this type do not exist for linear combinations of eigenfunctions. We will show how, by coarsely counting nodal domains, i.e. by ignoring small oscillations, we may obtain a similar upper bound for linear combinations as well. The proof uses the theory of persistence modules and barcodes combined with multiscale polynomial approximation of functions in Sobolev spaces. Using the same method, we may study coarse topology of a zero set of a function, as well as coarse topology of the set of common zeros of a number of different functions. This allows us to prove a coarse version of Bézout's theorem for linear combination of Laplace-Beltrami eigenfunctions. The talk is based on a joint work with L. Buhovsky, J. Payette, I. Polterovich, L. Polterovich and E. Shelukhin.