We aim at extending the Fourier transform on the Heisenberg group ${ℍ}^d$, to tempered distributions. Our motivation is to provide the reader with a hands-on approach that allows for further investigating Fourier analysis and PDEs on ${ℍ}^d$.

As in the Euclidean setting, the strategy is to show that the Fourier transform is an isomorphism on the Schwartz space, then to define the extension by duality. To achieve it, the Fourier transform of an integrable function is viewed as a uniformly continuous mapping on the set ${\widetilde{ℍ}}^d={\mathbb{N}}^d\times {\mathbb{N}}^d\times ℝ\setminus \left\{0\right\}$, that may be completed to a larger set ${\widetilde{ℍ}}^d$ for some suitable distance. This viewpoint provides a user friendly description of the range of the Schwartz space on ${ℍ}^d$ by the Fourier transform, which makes the extension to the whole set of tempered distributions straightforward.

To highlight the strength of our approach, we give examples of computations of Fourier transforms of tempered distributions that do not correspond to integrable or square integrable functions. The most striking one is a formula for the Fourier transform of functions on ${ℍ}^d$ that are independent of the vertical variable, an open question, to the best of our knowledge.

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We investigate the long-time behavior of solutions to the isothermal Euler, Korteweg or quantum Navier–Stokes equations, as well as generalizations of these equations where the convex pressure law is asymptotically linear near vacuum. By writing the system with a suitable time-dependent scaling we prove that the densities of global solutions display universal dispersion rate and asymptotic profile. This result applies to weak solutions defined in an appropriate way. In the exactly isothermal case, we establish the compactness of bounded sets of such weak solutions, by introducing modified entropies adapted to the new unknown functions.

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We prove that geodesic rays in the Uniform Infinite Planar Triangulation (UIPT) coalesce in a strong sense using the skeleton decomposition of random triangulations discovered by Krikun. This implies the existence of a unique horofunction measuring distances from infinity in the UIPT. We then use this horofunction to define the skeleton “seen from infinity” of the UIPT and relate it to a simple Galton–Watson tree conditioned to survive, giving a new and particularly simple construction of the UIPT. Scaling limits of perimeters and volumes of horohulls within this new decomposition are also derived, as well as a new proof of the two point function formula for random triangulations in the scaling limit due to Ambjørn and Watabiki.

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Bromberg and Ulcigrai constructed piecewise smooth functions on the circle such that the set of $\alpha$ for which the sum ${\sum }_{k=0}^{n-1}f(x+k\alpha mod1)$ satisfies a temporal distributional limit theorem along the orbit of a.e. $x$ has Hausdorff dimension one. We show that the Lebesgue measure of this set is equal to zero.

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We study random typical minimal factorizations of the $n$-cycle into transpositions, which are factorizations of $(1,...,n)$ as a product of $n-1$ transpositions. By viewing transpositions as chords of the unit disk and by reading them one after the other, one obtains a sequence of increasing laminations of the unit disk (i.e. compact subsets of the unit disk made of non-intersecting chords).

When an order of $\sqrt{n}$ consecutive transpositions have been read, we establish, roughly speaking, that a phase transition occurs and that the associated laminations converge to a new one-parameter family of random laminations, constructed from excursions of specific Lévy processes.

Our main tools involve coding random minimal factorizations by conditioned two-type Bienaymé–Galton–Watson trees. We establish in particular limit theorems for two-type BGW trees conditioned on having given numbers of vertices of both types, and with an offspring distribution depending on the conditioning size. We believe that this could be of independent interest.

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In the moduli space of polarized varieties $(X,L)$ the same unpolarized variety $X$ can occur more than once. However, for K3 surfaces, compact hyperkähler manifolds, and abelian varieties the ‘orbit’ of $X$, i.e. the subset $\{(X_i,L_i)\mid X_i\cong X\}$, is known to be finite, which may be viewed as a consequence of the Kawamata–Morrison cone conjecture. In this note we provide a proof of this finiteness not relying on the cone conjecture and, in fact, not even on the global Torelli theorem. Instead, it uses the geometry of the moduli space of polarized varieties to conclude the finiteness by means of Baily–Borel type arguments. We also address related questions concerning finiteness in twistor families associated with polarized K3 surfaces of CM type.

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Let ${\left\{U_t^N\right\}}_{t\ge 0}$ be a standard Brownian motion on $𝕌\left(N\right)$. For fixed $N\in \mathbb{N}$ and $t>0$, we give explicit almost-sure bounds on the $L_1$-Wasserstein distance between the empirical spectral measure of $U_t^N$ and the large-$N$ limiting measure. The bounds obtained are tight enough that we are able to use them to study the evolution of the eigenvalue process in time, bounding the rate of convergence of paths of the measures on compact time intervals. The proofs use tools developed by the first author to obtain rates of convergence of the empirical spectral measures in classical random matrix ensembles, as well as recent estimates for the rates of convergence of moments of the ensemble-averaged spectral distribution.

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We study the simplicial ${\ell }^{q,p}$ cohomology of Carnot groups $G$. We show vanishing and non-vanishing results depending of the range of the $(p,q)$ gap with respect to the weight gaps in the Lie algebra cohomology of $G$.

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We consider the free Klein–Gordon equation with periodic damping. We show on this simple model that if the usual geometric condition holds then the decay of the energy is uniform with respect to the oscillations of the damping, and in particular the sizes of the derivatives do not play any role. We also show that without geometric condition the polynomial decay of the energy is even slightly better for a highly oscillating damping. To prove these estimates we provide a parameter dependent version of well known results of semigroup theory.

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We study the (covering) gonality of abelian varieties and their orbits of zero-cycles for rational equivalence. We show that any orbit for rational equivalence of zero-cycles of degree $k$ has dimension at most $k-1$. Building on the work of Pirola, we show that very general abelian varieties of dimension $g$ have (covering) gonality at least $f\left(g\right)$, where $f\left(g\right)$ grows like $\log g$. This answers a question asked by Bastianelli, De Poi, Ein, Lazarsfeld and B. Ullery. We also obtain results on the Chow ring of very general abelian varieties $A$ of dimension $g$, e.g., if $g\ge 2k-1$, the set of divisors $D\in {\mathrm{Pic}}^0\left(A\right)$ such that $D^k=0$ in ${\mathrm{CH}}^k\left(A\right)$ is at most countable.

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