Volume 2023, Issue 804

We give an alternative proof of the Michael–Simon–Sobolev inequality using techniques from optimal transport. The inequality is sharp for submanifolds of codimension 2.

This paper provides the first variational proof of the existence of periodic nonlocal-CMC surfaces. These are nonlocal analogues of the classical Delaunay cylinders. More precisely, we show the existence of a set in R n \mathbb{R}^{n} which is periodic in one direction, has a prescribed (but arbitrary) volume within a slab orthogonal to that direction, has constant nonlocal mean curvature, and minimizes an appropriate periodic version of the fractional perimeter functional under the volume constraint. We show, in addition, that the set is cylindrically symmetric and, more significantly, that it is even as well as nonincreasing on half its period. This monotonicity property solves an open problem and an obstruction which arose in an earlier attempt, by other authors, to show the existence of minimizers.

For a constructible étale sheaf on a smooth variety of positive characteristic ramified along an effective divisor, the largest slope in Abbes and Saito’s ramification theory of the sheaf gives a divisor with rational coefficients called the conductor divisor . In this article, we prove decreasing properties of the conductor divisor after pull-backs. The main ingredient behind is the construction of étale sheaves with pure ramifications. As applications, we first prove a lower semi-continuity property for conductors of étale sheaves on relative curves in the equal characteristic case, which supplement Deligne and Laumon’s lower semi-continuity property of Swan conductors ([33]) and is also an ℓ {\ell} -adic analogue of André’s semi-continuity result of Poincaré–Katz ranks for meromorphic connections on complex relative curves. ([6]). Secondly, we give a ramification bound for the nearby cycle complex of an étale sheaf ramified along the special fiber of a regular scheme semi-stable over an equal characteristic henselian trait, which extends a main result in a joint work with Teyssier ([20]) and answers a conjecture of Leal ([35]) in a geometric situation.

We study the Borel–Moore homology of stacks of representations of preprojective algebras Π Q \Pi_{Q} , via the study of the DT theory of the undeformed 3-Calabi–Yau completion Π Q ⁢ [ x ] \Pi_{Q}[x] . Via a result on the supports of the BPS sheaves for Π Q ⁢ [ x ] ⁢ -mod \Pi_{Q}[x]\textup{-mod} , we prove purity of the BPS cohomology for the stack of Π Q ⁢ [ x ] \Pi_{Q}[x] -modules and define BPS sheaves for stacks of Π Q \Pi_{Q} -modules. These are mixed Hodge modules on the coarse moduli space of Π Q \Pi_{Q} -modules that control the Borel–Moore homology and geometric representation theory associated to these stacks. We show that the hypercohomology of these objects is pure and thus that the Borel–Moore homology of stacks of Π Q \Pi_{Q} -modules is also pure. We transport the cohomological wall-crossing and integrality theorems from DT theory to the category of Π Q \Pi_{Q} -modules. We use our results to prove positivity of a number of “restricted” Kac polynomials, determine the critical cohomology of Hilb n ⁡ ( A 3 ) \operatorname{Hilb}_{n}(\mathbb{A}^{3}) , and the Borel–Moore homology of genus one character stacks, as well as providing various applications to the cohomological Hall algebras associated to Borel–Moore homology of stacks of modules over preprojective algebras, including the PBW theorem, and torsion-freeness.

We study viscosity solutions to the classical one-phase problem and its thin counterpart. In low dimensions, we show that when the free boundary is the graph of a continuous function, the solution is the half-plane solution. This answers, in the salient dimensions, a one-phase free boundary analogue of Bernstein’s problem for minimal surfaces. As an application, we also classify monotone solutions of semilinear equations with a bump-type nonlinearity.

We prove a mixed version of a conjecture of Griffiths: that the closure of the image of any admissible mixed period map is quasi-projective, with a natural ample bundle. Specifically, we consider the map from the image of the mixed period map to the image of the period map of the associated graded. On the one hand, we show in a precise manner that the parts of this map parametrizing extension data of non-adjacent-weight pure Hodge structures are quasi-affine. On the other hand, extensions of adjacent-weight pure polarized Hodge structures are parametrized by a compact complex torus (the intermediate Jacobian) equipped with a natural theta bundle which is ample in Griffiths transverse directions. Our proof makes heavy use of o-minimality, and recent work with B. Klingler associating an ℝ an , exp {\mathbb{R}_{\mathrm{an},\exp}} -definable structure to mixed period domains and admissible mixed period maps.

We study the family of irreducible modules for quantum affine 𝔰 ⁢ 𝔩 n + 1 {\mathfrak{sl}_{n+1}} whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to A m {A_{m}} with m ≤ n {m\leq n} . These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category 𝒞 - {\mathscr{C}^{-}} . This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov–Reshetikhin module with its dual always contains an imaginary module in its Jordan–Hölder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc, A cluster algebra approach to q -characters of Kirillov–Reshetikhin modules, J. Eur. Math. Soc. (JEMS) 18 2016, 5, 1113–1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. Mínguez, Geometric conditions for □ \square -irreducibility of certain representations of the general linear group over a non-archimedean local field, Adv. Math. 339 2018, 113–190]. Finally, we use our methods to give a family of imaginary modules in type D 4 {D_{4}} which do not arise from an embedding of A r {A_{r}} with r ≤ 3 {r\leq 3} in D 4 {D_{4}} .

In this paper, we give an algorithm to determine all local 𝐴-packets containing a given irreducible representation of a split special odd orthogonal group or a symplectic group over a 𝑝-adic field. Especially, we can determine whether a given irreducible representation is of Arthur type or not.