This work is devoted to the study of uncertainty principles for finite combinations of Hermite functions. We establish some spectral inequalities for control subsets that are thick with respect to some unbounded densities growing almost linearly at infinity, and provide quantitative estimates, with respect to the energy level of the Hermite functions seen as eigenfunctions of the harmonic oscillator, for the constants appearing in these spectral estimates. These spectral inequalities allow us to derive the null-controllability in any positive time for evolution equations enjoying specific regularizing effects. More precisely, for a given index $\frac {1}{2} \leq \mu <1$, we deduce sufficient geometric conditions on control subsets to ensure the null-controllability of evolution equations enjoying regularizing effects in the symmetric Gelfand–Shilov space $S^{\mu }_{\mu }(\mathbb {R}^{n})$. These results apply in particular to derive the null-controllability in any positive time for evolution equations associated to certain classes of hypoelliptic non-self-adjoint quadratic operators, or to fractional harmonic oscillators.

We define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the one defined by [1, 18, 32, 33, 34] using Nakajima quiver varieties.

We determine reductions of $2$-dimensional, irreducible, semistable, and non-crystalline representations of $\mathrm {Gal}\left (\overline {\mathbb {Q}}_p/\mathbb {Q}_p\right )$ with Hodge–Tate weights $0 < k-1$ and with $\mathcal L$-invariant whose p-adic norm is sufficiently large, depending on k. Our main result provides the first systematic examples of the reductions for$k \geq p$.

Let p be a rational prime. Let F be a totally real number field such that F is unramified over p and the residue degree of any prime ideal of F dividing p is $\leq 2$. In this paper, we show that the eigenvariety for $\mathrm {Res}_{F/\mathbb {Q}}(\mathit {GL}_{2})$, constructed by Andreatta, Iovita, and Pilloni, is proper at integral weights for $p\geq 3$. We also prove a weaker result for $p=2$.

Given a quiver with potential $(Q,W)$, Kontsevich–Soibelman constructed a cohomological Hall algebra (CoHA) on the critical cohomology of the stack of representations of $(Q,W)$. Special cases of this construction are related to work of Nakajima, Varagnolo, Schiffmann–Vasserot, Maulik–Okounkov, Yang–Zhao, etc. about geometric constructions of Yangians and their representations; indeed, given a quiver Q, there exists an associated pair $(\widetilde{Q}, \widetilde{W})$ whose CoHA is conjecturally the positive half of the Maulik–Okounkov Yangian $Y_{\text {MO}}(\mathfrak {g}_{Q})$.

For a quiver with potential $(Q,W)$, we follow a suggestion of Kontsevich–Soibelman and study a categorification of the above algebra constructed using categories of singularities. Its Grothendieck group is a K-theoretic Hall algebra (KHA) for quivers with potential. We construct representations using framed quivers, and we prove a wall-crossing theorem for KHAs. We expect the KHA for $(\widetilde{Q}, \widetilde{W})$ to recover the positive part of quantum affine algebra $U_{q}(\widehat {\mathfrak {g}_{Q}})$ defined by Okounkov–Smirnov.

This article deals with kinetic Fokker–Planck equations with essentially bounded coefficients. A weak Harnack inequality for nonnegative super-solutions is derived by considering their log-transform and adapting an argument due to S. N. Kružkov (1963). Such a result rests on a new weak Poincaré inequality sharing similarities with the one introduced by W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.

We calculate the extension groups between simple modules of pro-p-Iwahori Hecke algebras.

In this paper, we prove some convexity results associated to orbit projection of noncompact real reductive Lie groups.

Let p be a prime number. In the present paper, we prove that the moduli of hyperbolic curves of genus $0$ over an algebraic closure of the field of p-adic numbers may be completely determined by their tempered fundamental groups.

Using degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree d and dimension r on a general curve of genus g satisfying prescribed incidence conditions at n points. We determine these numbers completely for linear series of arbitrary dimension when d is sufficiently large, and for all d when either $r=1$ or $n=r+2$. Our formulas generalise and give new proofs of recent results of Tevelev and of Cela, Pandharipande and Schmitt.

Let G be a complex semisimple Lie group and H a complex closed connected subgroup. Let and be their Lie algebras. We prove that the regular representation of G in $L^2(G/H)$ is tempered if and only if the orthogonal of in contains regular elements by showing simultaneously the equivalence to other striking conditions, such as has a solvable limit algebra.

Let $\Gamma $ be a finite set, and $X\ni x$ a fixed kawamata log terminal germ. For any lc germ $(X\ni x,B:=\sum _{i} b_iB_i)$, such that $b_i\in \Gamma $, Nakamura’s conjecture, which is equivalent to the ascending chain condition conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor E over $X\ni x$, such that $a(E,X,B)=\mathrm {mld}(X\ni x,B)$, and $a(E,X,0)$ is bounded from above. We extend Nakamura’s conjecture to the setting that $X\ni x$ is not necessarily fixed and $\Gamma $ satisfies the descending chain condition, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of $a(E,X,0)$ for any such E.

Let M be an irreducible $3$-manifold M with empty or toroidal boundary which has at least one hyperbolic piece in its geometric decomposition, and let A be a finite abelian group. Generalizing work of Sun [20] and of Friedl–Herrmann [7], we prove that there exists a finite cover $M' \to M$ so that A is a direct factor in $H_1(M',{\mathbb Z})$.

We show that the theory of Galois actions of a torsion Abelian group A is companionable if and only if, for each prime p, the p-primary part of A is either finite or it coincides with the Prüfer p-group. We also provide a model-theoretic description of the model companions we obtain.

In this paper, we formulate and present ample evidence towards the conjecture that the partition function (i.e. the exponential of the generating series of intersection numbers with monomials in psi classes) of the Pixton class on the moduli space of stable curves is the topological tau function of the noncommutative Korteweg-de Vries hierarchy, which we introduced in a previous work. The specialisation of this conjecture to the top degree part of Pixton’s class states that the partition function of the double ramification cycle is the tau function of the dispersionless limit of this hierarchy. In fact, we prove that this conjecture follows from the double ramification/Dubrovin–Zhang equivalence conjecture. We also provide several independent computational checks in support of it.