Bender, Coley, Robbins, and Rumsey posed the problem of counting the number of subspaces which have a given profile with respect to a linear endomorphism defined on a finite vector space. We settle this problem in full generality by giving an explicit counting formula involving symmetric functions. This formula can be expressed compactly in terms of a Hall scalar product involving dual q-Whittaker functions and another symmetric function that is determined by conjugacy class invariants of the endomorphism. As corollaries, we obtain new combinatorial interpretations for the coefficients in the q-Whittaker expansions of several symmetric functions. These include the power sum, complete homogeneous, products of modified Hall–Littlewood functions, and certain products of q-Whittaker functions. These results are used to derive a formula for the number of anti-invariant subspaces (as defined by Barría and Halmos) with respect to an arbitrary operator. We also give an application to an open problem in Krylov subspace theory.

We investigate trace and observability inequalities for Laplace eigenfunctions on the d-dimensional torus $\mathbb {T}^d$, with respect to arbitrary Borel measures $\mu $. Specifically, we characterize the measures $\mu $ for which the inequalities

$$ \begin{align*} \int |u|^2 \mathop{}\!\mathrm{d}\mu \lesssim \int |u|^2 \mathop{}\!\mathrm{d} x \quad \text{(trace)}, \qquad \int |u|^2 \mathop{}\!\mathrm{d}\mu \gtrsim \int |u|^2 \mathop{}\!\mathrm{d} x \quad \text{(observability)} \end{align*} $$

hold uniformly for all eigenfunctions u of the Laplacian. Sufficient conditions are derived based on the integrability and regularity of $\mu $, while necessary conditions are formulated in terms of the dimension of the support of the measure. These results generalize classical theorems of Zygmund and Bourgain–Rudnick to higher dimensions. Applications include results in the spirit of Cantor–Lebesgue theorems, constraints on quantum limits, and control theory for the Schrödinger equation. Our approach combines several tools: the cluster structure of lattice points on spheres; decoupling estimates; and the construction of eigenfunctions exhibiting strong concentration or vanishing behavior, tailored respectively to the trace and observability inequalities.

We describe a family of hyperplane arrangements depending on a positive integer parameter r, which we refer to as the r-braid arrangements, and which can be viewed as a generalization of the classical braid arrangement. The wonderful compactification of the braid arrangement (with respect to its minimal building set) is well-known to yield the moduli space $\overline {\mathcal {M}}_{0,n}$, and, in this work, we generalize this result, constructing a moduli space $\overline {\mathcal {M}}^r_{n}$ of certain genus-zero curves with an order-r automorphism that we identify with the corresponding wonderful compactification of the r-braid arrangement. The resulting space is a variant of the previously studied moduli space $\overline {\mathcal {L}}^r_n$ of [CDH+22], related via a change of weights on the markings.

In this paper, we consider two variations on Mann’s $\infty $-categorical definition of abstract six-functor formalisms. We consider Nagata six-functor formalisms, that have the additional requirement of having Grothendieck and Wirthmüller contexts. We also consider local six-functor formalisms, which in addition to this, take values in presentable stable $\infty $-categories, and have recollements. Using Nagata’s compactification theorem, we show that Nagata six-functor formalisms on varieties can be given by just specifying adjoint triples for open immersions and for proper morphisms, satisfying certain compatibilities. The existence of recollements is (almost) equivalent to a hypersheaf condition for a Grothendieck topology on the category of “varieties and spans consisting of an open immersion and a proper map”. Using this characterization, we show that the category of local six-functor formalisms embeds faithfully into the category of lax symmetric monoidal functors from the category of smooth and complete varieties to the category of presentable stable $\infty $-categories and adjoint triples. We characterize which lax symmetric monoidal functors on complete varieties, taking values in the category of presentable stable $\infty $-categories and adjoint triples, extend to local six-functor formalisms.

We study the locus of smooth hypersurfaces inside the Hilbert scheme of a smooth projective complex variety. In the spirit of scanning, we construct a map to a continuous section space of a projective bundle, and show that it induces an isomorphism in integral homology in a range of degrees growing with the ampleness of the hypersurfaces. When the ambient variety is a curve, this recovers a result of McDuff about configuration spaces. We compute the rational cohomology of the section space and exhibit a homological stability phenomenon for hypersurfaces with first Chern class going to infinity. For simply connected varieties, the rational cohomology is shown to agree with the stable cohomology of a moduli space of hypersurfaces, with a peculiar tangential structure, as studied by Galatius and Randal-Williams.

For a smooth affine algebraic group G over an algebraically closed field, we consider several two-variables generalizations of the affine Grassmannian , given by quotients of the double loop group $G(\!(x)\!)(\!(y)\!)$. We prove that they are representable by ind-schemes if G is solvable. Given a smooth surface X and a flag of subschemes of X, we provide a geometric interpretation of the two-variables Grassmannians, in terms of bundles and trivialisation data defined on appropriate loci in X, which depend on the flag.

Building on a recent construction of Plebanek and Salguero-Alarcón, which solved the Complemented Subspace Problem for $C(K)$-spaces, and the subsequent work of De Hevia, Martínez-Cervantes, Salguero-Alarcón, and Tradacete solving the Complemented Subspace Problem for Banach lattices, we show that the class of Banach lattices is not primary. Specifically, we exhibit a compact Hausdorff space L such that $C(L) \simeq X \oplus \tilde {X}$ and neither X nor $\tilde {X}$ is isomorphic to a Banach lattice. In particular, it also follows that the class of $C(K)$-spaces is not primary.

In this paper, we give a complete list of strongly tempered anomaly-free hyperspherical Hamiltonian spaces-those that are dual to symplectic vector spaces under the relative Langlands duality. We show that the period integrals attached to the list contain many previously studied Rankin-Selberg integrals and period integrals, thus giving a new conceptual understanding of these integrals. The list also proposes many new interesting period integrals to study.

For Lebesgue generic $({x}_1,x_2)\in \mathbb {R}^2$, we investigate the distribution of small values of products $q\cdot \|qx_1\| \cdot \|qx_2\|$ with $q\in \mathbb {N}$, where $\|\cdot \|$ denotes the distance to the closest integer. The main result gives an asymptotic formula for the number of $1\le q< T$ such that

$$ \begin{align*}a <q\cdot \|qx_1\| \cdot \|qx_2\|\le b\quad \text{and} \quad \|qx_1\|, \|qx_2\|\le c \end{align*} $$

for given parameters $a,b, c$ satisfying certain growth conditions.

We consider a toy model of rate-independent droplet motion on a surface with contact angle hysteresis based on the one-phase Bernoulli free boundary problem. We introduce a notion of solutions based on an obstacle problem. These solutions jump “as late and as little as possible”, a physically natural property that energy solutions do not satisfy. When the initial data is star-shaped, we show that obstacle solutions are uniquely characterized by satisfying the local stability and dynamic slope conditions. This is proved via a novel comparison principle, which is one of the main new technical results of the paper. In this setting we can also show the (almost) optimal $C^{1,1/2-}$-spatial regularity of the contact line. This regularity result explains the asymptotic profile of the contact line as it de-pins via tangential motion similar to de-lamination. Finally we apply our comparison principle to show the convergence of minimizing movements schemes to the same obstacle solution, again in the star-shaped setting.

Let G be a simple algebraic group over an algebraically closed field $\Bbbk $ of positive characteristic. We consider the questions of when the tensor product of two simple G-modules is multiplicity free or completely reducible. We develop tools for answering these questions in general, and we use them to provide complete answers for the groups $G = \mathrm {SL}_3(\Bbbk )$ and $G = \mathrm {Sp}_4(\Bbbk )$.

We prove a version of Sylvester’s law of inertia for the Reflection Equation Algebra (=REA). We will only be concerned with the REA constructed from the R-matrix associated to the standard q-deformation of $GL(N,\mathbb {C})$. For q positive, this particular REA comes equipped with a natural $*$-structure, by which it can be viewed as a q-deformation of the $*$-algebra of polynomial functions on the space of self-adjoint N-by-N-matrices. We will show that this REA satisfies a type I-condition, so that its irreducible representations can in principle be classified. Moreover, we will show that, up to the adjoint action of quantum $GL(N,\mathbb {C})$, any irreducible representation of the REA is determined by its extended signature, which is a classical signature vector extended by a parameter in $\mathbb {R}/\mathbb {Z}$. It is this latter result that we see as a quantized version of Sylvester’s law of inertia.

We show that the moduli spaces of Scholze’s p-adic shtukas with framing satisfy a p-adic rigid analytic version of Borel’s extension theorem. In particular, this holds for local Shimura varieties, for all local Shimura data $(G, [b],\{\mu \})$, even for exceptional groups G, and extends work of Oswal-Shankar-Zhu-Patel who proved a p-adic Borel extension property for Rapoport-Zink spaces. As a corollary, we deduce that all these spaces satisfy a p-adic rigid analytic version of Brody hyperbolicity.

We show sharp well-posedness with analytic data-to-solution mapping in the semilinear regime for dispersion-generalized KP-I equations on $\mathbb {R}^2$ and $\mathbb {R} \times \mathbb {T}$. On $\mathbb {R}^2$ we cover the full subcritical range, whereas on $\mathbb {R} \times \mathbb {T}$ the sharp well-posedness is strictly subcritical. We rely on linear and bilinear Strichartz estimates which are proved using decoupling techniques and square function estimates. Nonlinear Loomis-Whitney inequalities are a further ingredient. These are presently proved for Borel measures with growth condition reflecting the different geometries of the plane $\mathbb {R}^2$, the cylinder $\mathbb {R} \times \mathbb {T}$, and the torus $\mathbb {T}^2$. Finally, we point out that on tori $\mathbb {T}^2_\gamma $, KP-I equations are never semilinear.

The moduli space of bundle stable pairs $\overline {M}_C(2,\Lambda )$ on a smooth projective curve C, introduced by Thaddeus, is a smooth Fano variety of Picard rank two. Focusing on the genus two case, we show that its K-moduli space is isomorphic to a GIT moduli of lines in quartic del Pezzo threefolds. Additionally, we construct a natural forgetful morphism from the K-moduli of $\overline {M}_C(2,\Lambda )$ to that of the moduli spaces of stable vector bundles $\overline {N}_C(2,\Lambda )$. In particular, Thaddeus’ moduli spaces for genus two curves are all K-stable.

In this work, we develop a unified framework for establishing sharp threshold results for various Ramsey properties. To achieve this, we view such properties as noncolourability of auxiliary hypergraphs. Our main technical result gives sufficient conditions on a sequence of such hypergraphs that guarantee that this noncolourability property has a sharp threshold in subhypergraphs induced by random subsets of the vertices.

Furthermore, we verify these conditions in several cases of interest. In the classical setting of Ramsey theory for graphs, we show that the property of being Ramsey for a graph H in r colours has a sharp threshold in $G_{n,p}$, for all $r \geqslant 2$ and all H in a class of graphs that includes all cliques and cycles. In the arithmetic setting, we establish sharpness of thresholds for the properties corresponding to van der Waerden’s theorem and Schur’s theorem, also in any number of colours.

We show that the set of locally finite Borel graphs with finite Borel asymptotic dimension is $\boldsymbol {\Sigma }^1_2$-complete. The result is based on a combinatorial characterization of finite Borel asymptotic dimension for graphs generated by a single Borel function. As an application of this characterization, we classify the complexities of digraph homomorphism problems for this class of graphs.

A hyperbinary partition of the nonnegative integer n is a partition where every part is a power of $2$ and every power of $2$ appears at most twice. We give three applications of the length generating function for such partitions, denoted by $h_q(n)$. Morier-Genoud and Ovsienko defined the q-analogue of a rational number $[r/s]_q$ in various ways, most of which depend directly or indirectly on the continued fraction expansion of $r/s$. As our first application we show that $[r/s]_q=q\,h_q(n-1)/h_q(n)$ where $r/s$ occurs as the nth entry in the Calkin-Wilf enumeration of the non-negative rationals. Next we consider fence posets which are those which can be obtained from a sequence of chains by alternately pasting together maxima and minima. For every n we show there is a fence poset ${\cal F}(n)$ whose lattice of order ideals is isomorphic to the poset of hyperbinary partitions of n ordered by refinement. For our last application, Morier-Genoud and Ovsienko also showed that $[r/s]_q$ can be computed by taking products of certain matrices which are q-analogues of the standard generators for the special linear group $\operatorname {\mathrm {SL}}(2,{\mathbb Z})$. We express the entries of these products in terms of the polynomials $h_q(n)$.

We introduce the notions of u-amenability and hyper-u-amenability for countable Borel equivalence relations and we show that treeable, hyper-u-amenable countable Borel equivalence relations are hyperfinite. As corollaries of this result, we obtain that if a countable Borel equivalence relation is either:

1. 1. measure-hyperfinite and equal to the orbit equivalence relation of a free continuous action of a virtually free group on a $\sigma $-compact Polish space,

2. 2. treeable and equal to the orbit equivalence relation of a Borel action of an amenable group on a standard Borel space,

3. 3. treeable, amenable and Borel bounded,

then it is hyperfinite.