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The operation of sub-cellular processes in living organisms often require the transfer of biopolymers across impermeable lipid membranes. The emergence of new experimental techniques for manipulation of single molecules at nanometer scales have made possible in vitro experiments that can directly probe such translocation processes in cells as well as in synthetic systems. Some of these ideas have spawned novel bio-technologies with many more likely to emerge in the near future. In this talk I would review some of these experiments and attempt to provide a quantitative understanding of the data in terms of physical laws, primarily mechanics and electrostatics.

I will present an efficient algorithm for computing certain special values of Rankin triple product $p$-adic L-functions and give an application of this to the explicit construction of rational points on elliptic curves.

The $p$-adic Gross-Zagier formula for diagonal cycles and the $p$-adic Beilinson formulae described in the lectures of Rotger and Bertolini respectively suggest a connection between certain  {\em $p$-adic iterated integrals} attached to modular forms and rational points on elliptic curves. I will describe an ongoing project (in collaboration with Alan Lauder and Victor Rotger) whose goal is to explore these relationships numerically, with the goal of better understanding the notion of {\em Stark-Heegner points}. It is hoped that these experiments might suggest new perspectives on Stark-Heegner points based on suitable {\em $p$-adic deformations} of the global objects--diagonal cycles, Beilinson-Kato and Beilinson-Flach elements-- described in the lectures of Rotger, Bertolini, Dasgupta, and Loeffler, following  the influential approach to $p$-adic $L$-functions pioneered by Coates-Wiles, Kato, and Perrin-Riou.

We define an integral version of Sczech cocycle on ${\rm GL}_n(\mathbf{Z})$ by raising the level at a prime $\ell$.As a result, we obtain a new construction of the $p$-adic L-functions of Barsky/Cassou-Nogu\`es/Deligne-Ribet. This cohomological construction further allows for a study of the leading term of these L-functions at $s=0$:\\1) we obtain a new proof that the order of vanishing is at least the oneconjectured by Gross. This was already known as result of Wiles.\\2) we deduce an analog of the modular symbol algorithm for ${\rm GL}_n$ from the cocyclerelation and LLL. It enables for the efficient computation of the special values of these $p$-adic L-functions.\\When combined  with a refinement of the Gross-Stark conjecture, we obtain some examples of numerical construction of $\mathfrak p$-units in class fields of totally real (cubic) fields.This is joint work with Samit Dasgupta.

I will describe a construction of special cohomology classes over the cyclotomic tower for the product of the Galois representations attached to two modular forms, which $p$-adically interpolate the "Beilinson--Flach elements" of Bertolini, Darmon and Rotger. I will also describe some applications to the Iwasawa theory of modular forms over imaginary quadratic fields.

We show that the $p$-adic L-function associated to the tensor square of a $p$-ordinary eigenform factors as the product of the symmetric square $p$-adic L-function of the form with a Kubota-Leopoldt $p$-adic L-function.  Our method of proof follows that of Gross, who proved a factorization for Katz's $ p$-adic L-function for a character arising as the restriction of a Dirichlet character.  We prove certain special value formulae for classical and $p$-adicRankin L-series at non-critical points.  The formula of Bertolini, Darmon, and Rotger in the $p$-adic setting is a key element of our proof.  As demonstrated by Citro, we obtain as a corollary of our main result a proof of the exceptional zero conjecture of Greenberg for the symmetric square.

I will report on $p$-adic Beilinson's formulas, relating the values of certain Rankin $p$-adic L-functions outside their range of classical interpolation, to $p$-adic syntomic regulators of Beilinson-Kato and Beilinson-Flach elements. Applications to the theory of Euler systems and to the Birch and Swinnerton-Dyer conjecture will also be discussed. This is joint work with Henri Darmon and Victor Rotger.

In this lecture I shall introduce certain generalised Gross-Kudla-Schoen diagonal cycles in the product of three Kuga-Sato varieties and a $p$-adic formula of Gross-Zagier type which relates the images of these diagonal cycles under the $p$-adic Abel-Jacobi map to special values of the $p$-adic L-function attached to the Garrett triple convolution of three  Hida families of  modular forms. This formula has applications to the Birch--Swinnerton-Dyer conjecture and the theory of Stark-Heegner points. (Joint work with Henri Darmon.)

The main result of the talk is that two curves over a finite field are isomorphic, up to automorphisms of the ground field, if and only if there is an isomorphism of groups of Dirichlet characters such that the corresponding L-series are all equal. This can be shown by combining Uchida's proof of the anabelian theorem for global function fields with methods from (noncommutative) dynamical systems. I will also discuss how to turn this theorem into a theoretical algorithm that, given a listing of L-functions, determines an equation for the corresponding curve(s).

I will present an overview of a series of joint works with Akio Tamagawa about l-adic representations of etale fundamental group of curves (to simplify, over finitely generated fields of characteristic 0).
More precisely, when the generic representation is GLP (geometrically Lie perfect) i.e. the Lie algebra of the geometric etale fundamental group is perfect, we show that the associated local $\ell$-adic Galois representations satisfies a strong uniform open image theorem (ouside a `small' exceptional locus). Representations on l-adic cohohomology provide an important example of GLP representations. In that case, one can even provethat the exceptional loci that appear in the statement of our stronguniform open image theorem are independent of $\ell$, which was predicted by motivic conjectures.
Without the GLP assumption, we prove that the  associated local l-adic Galois representations still satisfy remarkable rigidity properties: the codimension of the image at the special fibre in the image at the generic fibre is at most 2 (outside a 'small' exceptional locus) and its Lie algebra is controlled by the first terms of the derived series of the Lie algebra of the image at the generic fibre.
I will state the results precisely, mention a few applications/open questions and draw a general picture of the proof in the GLP case (which,in particular, intertwins via the formalism of Galois categories, arithmetico-geometric properties of curves and $\ell$-adic geometry). If time allows, I will also give a few hints about the $\ell$-independency of the exceptional loci or the non GLP case.