We discuss several structural properties of functions belonging to a parabolic energy class, reminiscent of the elliptic De Giorgi class. In earlier works, sub-potential lower bounds, giving insight into the structural behavior of elements of these classes, were established for the linear case: Here, we extend these results to the nonlinear one. By showing that sub-potential lower bounds follow solely from the Harnack inequality, we show that positive solutions to Trudinger’s equation and elements of parabolic De Giorgi classes have a common lower bound. For both cases, we derive Liouville-type rigidity results in the parabolic setting.
Normal functions v provide a method for studying algebraic cycles Z(t) subset of X-t varying in a family of smooth projective varieties. Associated with v is an infinitesimal invariant 8v that reflects the first-order variation of pairs (X-t, Z(t)). Over the years, 8v has been widely used in the study of various geometric questions. We note that whereas v is a transcendental invariant, like periods of algebraic integrals, delta(v) has a natural filtration whose associated graded gives algebraic sections of coherent sheaves. In a number of interesting cases, these sections have had geometric interpretations. In this paper, we will discuss an identification between the singularities v and delta(v). The formal proof of this result will be given in a separate work.
We discuss several structural properties of functions belonging to a parabolic energy class, reminiscent of the elliptic De Giorgi class. In earlier works, sub-potential lower bounds, giving insight into the structural behavior of elements of these classes, were established for the linear case: Here, we extend these results to the nonlinear one. By showing that sub-potential lower bounds follow solely from the Harnack inequality, we show that positive solutions to Trudinger's equation and elements of parabolic De Giorgi classes have a common lower bound. For both cases, we derive Liouville-type rigidity results in the parabolic setting.
Normal functions v provide a method for studying algebraic cycles Z(t) subset of X-t varying in a family of smooth projective varieties. Associated with v is an infinitesimal invariant 8v that reflects the first-order variation of pairs (X-t, Z(t)). Over the years, 8v has been widely used in the study of various geometric questions. We note that whereas v is a transcendental invariant, like periods of algebraic integrals, delta(v) has a natural filtration whose associated graded gives algebraic sections of coherent sheaves. In a number of interesting cases, these sections have had geometric interpretations. In this paper, we will discuss an identification between the singularities v and delta(v). The formal proof of this result will be given in a separate work.
The large sieve inequality, although best possible in general, is subject to improvements for some basic special sequences. We exemplify how the ideas of Bombieri and Davenport can be exploited to this end.
We formulate a conjecture that is strictly equivalent to the conjunction of Schanuel's conjecture and the multiplicative Zilber-Pink conjecture.
Saddle solutions of the Allen-Cahn equation in R-2m are characterized by the property that they vanish precisely on the Simons cones, a family of classical minimal surfaces with one singularity at the origin. Their existence and uniqueness are known, by results of Cabre-Terra (2009-2012) and Dang (1992). Schatzman (1995) proved that the saddle solution is unstable for m = 1. Cabre-Terra (2009-2012) showed the instability form = 2, 3 and stability for m >= 7. This left open the case of m = 4, 5, 6, which is conjectured to be stable (even energy minimizing). Towards this conjecture, here we establish some pointwise estimates for the saddle solutions.
We develop the general theory of Airy sheaves of Laurent type, the local systems whose trace functions have a particular "Airy-Laurent" shape. The main goal is to provide tools for the later determination of their monodromy groups. See [Eur. J. Math. 10 (2024), article no. 65] for instances of such determinations.
Let L be a finite-dimensional semisimple Lie algebra of rank N over an algebraically closed field of characteristic 0. Associated with L is a family of polynomial folding maps F-n : A(N) -> A(N) for n >= 1 having the property that F(n )has topological degree n(N) and F-m o F-n = F(n )o F-m for all m; n >= 1: We derive formulas for the leading terms of the folding maps on A(2) associated with the Lie algebras A(2), B-2, and G(2), and we use these formulas to compute the affine automorphism group of each folding map.
When R is a rational (or even algebraic) function, it is well known how the height of R(alpha) behaves asymptotically as the height of alpha tends to infinity. Here we prove some analogous versions as the height of alpha tends instead to 0, also with explicit error terms. Our most general results, some with a trigonometrical flavour, are for roots of unity of large order. For example, the height of tan 2 pi/n tends to 2G/pi where G is the Catalan constant.
In this paper, we revisit some known results about stationary varifolds using simpler arguments. In particular, we obtain the height bound and the Lipschitz approximation along with its estimates, and as a consequence, the excess decay.
In this survey, we outline the role of G-functions in arithmetic geometry, notably their link with Picard-Fuchs differential equations and periods. We explain how polynomial relations between special values of G-functions arising from a pencil of algebraic varieties may occur at a parameter where the fiber has more "motivic" symmetries; and how Bombieri's principle of global relations can be used to control the height of such parameters (which was also one of the origins of the Andr & eacute;-Oort conjecture). At the end, we sketch the recent revival of the G-function method in the context of unlikely intersections and the Zilber-Pink conjecture.
We introduce a notion of quasiconvexity for continuous functions f defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold (M, g) and R-m, naturally generalizing the classical Euclidean definition. We prove that this condition characterizes the sequential lower semicontinuity of the associated integral functional F(u,Omega) = integral(Omega )f(du)d mu with respect to the weak & lowast; topology of W-1,W- infinity(Omega, R-m), for every bounded open subset Omega subset of M.
We prove that the log canonical threshold of the base ideal of a complete linear system on a complex abelian variety is >= 1, and that equality holds if and only if the base locus has divisorial components. Consequently, the same assertions hold for the ideal of the intersection of translates of theta divisors by the points of a finite subgroup.
We study the optimal control of an infinite-dimensional stochastic system governed by an SDE in a separable Hilbert space driven by cylindrical stable noise. We establish the existence and uniqueness of a mild solution to the associated HJB equation. This result forms the basis for the proof of the Verification Theorem, which is the subject of ongoing research and will provide a sufficient condition for optimality.
In this paper, we investigate how weakening the classical hydrostatic balance hypothesis impacts the well-posedness of the stochastic LU primitive equations. The models we consider are intermediate between the incompressible 3D LU Navier-Stokes equations and the LU primitive equations with standard hydrostatic balance. As such, they are expected to be numerically tractable, while accounting well for phenomena within the grey zone between hydrostatic balance and non-hydrostatic processes. Our main result is the well-posedness of a low-pass filtering-based stochastic interpretation of the LU primitive equations, with rigid-lid type boundary conditions, in the limit of “quasi-barotropic” flow. This assumption is linked to the structure assumption proposed in the work of Agresti et al., which can be related to the dynamical regime where the primitive equations remain valid. Furthermore, we present and study two eddy-(hyper)viscosity-based models.
We prove a homogenization result in terms of two-scale Young measures for non-local integral functionals. The result is obtained by means of a characterization of two-scale Young measures.
The Fourier coefficients of a Maass form phi for SL(n, Z) are complex numbers A(phi)(M), where M = (m(1), m(2), ..., m(n-1)) and m(1), m(2), ..., m(n-1) are non-zero integers. It is well known that coefficients of the form A(phi)(m(1), 1, ..., 1) are eigenvalues of the Hecke algebra and are multiplicative. We prove that the more general Fourier coefficients A(phi)(m(1), ..., m(n-1)) are also eigenvalues of the Hecke algebra and satisfy the multiplicativity relations A(phi)(m(1)m(1)', m(2)m(2)', ..., m(n-1)m(n-1)') = A(phi)(m(1), m(2), ..., m(n-1)).A(phi)(m(1)', m(2)', ..., m(n-1)') provided the products & prod;(n-1)(i=1) m(i) and & prod;(n-1)(i=1) m(i)' are relatively prime to each other.
In this survey, we outline the role of G-functions in arithmetic geometry, notably their link with Picard-Fuchs differential equations and periods. We explain how polynomial relations between special values of G-functions arising from a pencil of algebraic varieties may occur at a parameter where the fiber has more “motivic" symmetries; and how Bombieri's principle of global relations can be used to control the height of such parameters (which was also one of the origins of the André-Oort conjecture). At the end, we sketch the recent revival of the G-function method in the context of unlikely intersections and the Zilber-Pink conjecture.
We derive the expressions of the local Maxwellians that solve the Boltzmann equation in the interior of a regular open domain, without assuming the boundedness of the domain. We investigate separately, on the one hand, the case of the bounce-back boundary condition in any dimension, and, on the other hand, the case of the specular reflection boundary condition, in dimension d=2 and d=3. In the case of the bounce-back boundary condition, we prove that the only local Maxwellians solving the Boltzmann equation with boundary condition are the global Maxwellians. In the case of the specular reflection, we provide a complete classification of the domains for which only the global Maxwellians solve the Boltzmann equation with boundary condition, and we describe all the local Maxwellians that solve the equation for the domains presenting symmetries.
