Action potential solitons and waves in axons

We show that the action potential signals generated inside axons propagate as reaction-diffusion solitons or as reaction-diffusion waves, refuting the Hodgkin and Huxley (HH) hypothesis that action potentials propagate along axons with an elastic wave mechanism. Action potential signals are solitary propagating spikes along the axon, occurring in a type I intermittency regime of the HH model. Reaction-diffusion action potential wave fronts annihilate at collision and at the boundaries of axons with zero flux, in contrast with elastic waves, where amplitudes add up and reflect at boundaries. We calculate numerically the values of the speed of the action potential spikes, as well as the dispersion relations.
These findings suggest several experiments as validating and falsifying tests for the HH model.

Super McShane Identity

The McShane identity for the once-punctured
super torus is derived following Bowditch's proof in the
bosonic case using techniques in super Teichmueller
theory developed by the two latter-named authors.

An electrophysiology model for cells and tissues

We introduce a kinetic model to study the dynamics of ions in aggregates of cells and tissues. Different types of communication channels between adjacent cells and between cells and intracellular space are considered (ion channels, pumps and gap junctions). We shows that stable transmembrane ionic Nernst potentials are due to the coexistence of both specialised ion pumps and channels. Ion pumps or channels alone do not contribute to an equilibrium transmembrane potential drop. The kinetic parameters of the model straightforwardly calibrate with the Nernst potentials and ion concentrations. The model is based on the ATPase enzymatic mechanism for the ions $\hbox{Na}^+$, $\hbox{K}^+$, and it can be generalised for other ion pumps.
We extend the model to account for electrochemical effects, where transmembrane gating mechanism are introduced.
In this framework, axons can be seen as the evolutionary result of the aggregation of cells through gap junctions, which can be identified as the Ranvier nodes. In this kinetic framework, the injection of current in an axon induces the modification of the potassium equilibrium potential along the axon.

McShane identities for Higher Teichmuller theory and the Goncharov-Shen potential

In [GS15], Goncharov and Shen introduce a family of mapping class group invariant regular functions on their A-moduli space to explicitly formulate a particular homological mirror symmetry conjecture. Using these regular functions, we obtain McShane identities for general rank positive surface group representations with loxodromic boundary monodromy and (non-strict) McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple ratios. Moreover, we obtain McShane identities for finite-area cusped convex real projective surfaces by generalizing the Birman--Series geodesic scarcity theorem. We apply our identities to derive the simple spectral discreteness of unipotent bordered positive representations, collar lemmas, and generalizations of the Thurston metric.

Crystal Volumes and Monopole Dynamics

Third Kind Elliptic Integrals and 1-Motives

In [5] we have showed that the Generalized Grothendieck's Conjecture of Periods applied to 1-motives, whose underlying semi-abelian variety is a product of elliptic curves and of tori, is equivalent to a transcendental conjecture involving elliptic integrals of the first and second kind, and logarithms of complex numbers.
In this paper we investigate the Generalized Grothendieck's Conjecture of Periods in the case of 1-motives whose underlying semi-abelian variety is a non trivial extension of a product of elliptic curves by a torus. This will imply the introduction of elliptic integrals of the third kind for the computation of the period matrix of M and therefore the Generalized Grothendieck's Conjecture of Periods applied to M will be equivalent to a transcendental conjecture involving elliptic integrals of the first, second and third kind.

Large genus behavior of topological recursion

We show that for a rather generic set of regular spectral curves, the {\it Topological--Recursion} invariants $F_g$ grow at most like $O((\beta g)! r^{-g}) $ with some $r>0$ and $\beta\leq 5$

Standard conjectures in model theory, and categoricity of comparison isomorphisms. A model theory perspective.

On Phases Of Melonic Quantum Mechanics

We explore in detail the properties of two melonic quantum mechanical theories which can be formulated either as fermionic matrix quantum mechanics in the new large D limit, or as disordered models. Both models have a mass parameter m and the transition from the perturbative large m region to the strongly coupled "black-hole" small m region is associated with several interesting phenomena. One model, with U(n)^2 symmetry and equivalent to complex SYK, has a line of first-order phase transitions terminating, for a strictly positive temperature, at a critical point having non-trivial, non-mean-field critical exponents for standard thermodynamical quantities. Quasi-normal frequencies, as well as Lyapunov exponents associated with out-of-time-ordered four-point functions, are also singular at the critical point, leading to interesting new critical exponents. The other model, with reduced U(n) symmetry, has a quantum critical point at strictly zero temperature and positive critical mass m∗. For 0

The lure of conformal symmetry

The Clifford algebra ${\rm Cl} (4,1) \simeq {\mathbb C} [4]$, generated by the real (Majorana) $\gamma$-matrices and by a hermitian $\gamma_5$, gives room to the reductive Lie algebra $u(2,2)$ of the conformal group extended by the $u(1)$ helicity operator. Its unitary positive energy ladder representations, constructed by Gerhard Mack and the author 50 years ago, opened the way to a better understanding of zero-mass particles and fields and their relation to the space of bound states of the hydrogen atom. They became a prototypical example of a minimal representation of a non-compact reductive group introduced during the subsequent decade by Joseph.