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.

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.

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.

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 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.

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.

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 prove the correctness of the AKS algorithm [1] within the bounded arithmetic theory $T^{\text {count}}_2$ or, equivalently, the first-order consequences of the theory $\text {VTC}^0$ expanded by the smash function, which we denote by $\text {VTC}^0_2$. Our approach initially demonstrates the correctness within the theory $S^1_2 + \mathrm {iWPHP}$ augmented by two algebraic axioms and then shows that they are provable in $\text {VTC}^0_2$. The two axioms are: a generalized version of Fermat’s Little Theorem and an axiom adding a new function symbol which injectively maps roots of polynomials over a definable finite field to numbers bounded by the degree of the given polynomial. To obtain our main result, we also give new formalizations of parts of number theory and algebra:

• • In $\mathrm {PV}_1$: We formalize Legendre’s Formula on the prime factorization of $n!$, key properties of the Combinatorial Number System and the existence of cyclotomic polynomials over the finite fields $\mathbb {Z}/p$.

• • In $S^1_2$: We prove the inequality $\text {lcm}(1,\dots , 2n) \geq 2^n$.

• • In $\text {VTC}^0$: We verify the correctness of the Kung–Sieveking algorithm for polynomial division.

An orthomorphism of a finite group G is a bijection $\phi \colon G\to G$ such that $g\mapsto g^{-1}\phi (g)$ is also a bijection. In 1981, Friedlander, Gordon, and Tannenbaum conjectured that when G is abelian, for any $k\geq 2$ dividing $|G|-1$, there exists an orthomorphism of G fixing the identity and permuting the remaining elements as products of disjoint k-cycles as long as the Sylow $2$-subgroups of G are trivial or noncyclic. We prove this conjecture for all sufficiently large groups.

Let G be a connected reductive group over a non-archimedean local field. We say that an irreducible depth-zero (complex) G-representation is non-singular if its cuspidal support is non-singular. We establish a local Langlands correspondence for all such representations. We obtain it as a specialization from a categorical version: an equivalence between the category of finite-length non-singular depth-zero G-representations and the category of finite-length right modules of a direct sum of twisted affine Hecke algebras constructed from Langlands parameters. We also show that our LLC and our equivalence of categories have several nice properties, for example, compatibility with parabolic induction and with twists by depth-zero characters.

We prove that for polynomials $ f, g, h \in \mathbb {Z}[x] $ satisfying $ f = gh $ and $ f(0) \neq 0 $, the $\ell _2$-norm of the cofactor $ h $ is bounded by

$$ \begin{align*} \left\Vert {h} \right\Vert{}_2\leq \sqrt{\frac{\left\Vert {g} \right\Vert{}_0}{\deg g}}\cdot \left\Vert {f} \right\Vert{}_1\cdot \left(\widetilde{O}\left( \frac{\left\Vert {g} \right\Vert{}_0^{2.5} \cdot \deg^2{f}}{\sqrt{\deg{g}}} \right) \right)^{\left\Vert {g} \right\Vert{}_0-1} \ , \end{align*} $$

where $\left \Vert {g} \right \Vert {}_0$ is the number of nonzero coefficients of g (its sparsity). We also obtain similar results for polynomials over $\mathbb {C}$. These bounds are an improvement over the bounds presented in an earlier conference version of this paper [NS24].

This result significantly improves upon previously known exponential bounds (in $\deg {f}$) for general polynomials. It further implies that, under exact division, the polynomial division algorithm runs in quasi-linear time with respect to the input size and the number of terms in the quotient $ h $. This resolves a long-standing open problem concerning the exact divisibility of sparse polynomials.

In particular, our result demonstrates a quadratic separation between the runtime (and representation size) of exact and nonexact divisibility by sparse polynomials. Notably, prior to our work, it was not even known whether the representation size of the quotient polynomial could be bounded by a sub-quadratic function of its number of terms, or even by a subquadratic function of $\deg {f}$.

The aim of this paper is to extend the expanded degeneration construction of Li and Wu to obtain good degenerations of Hilbert schemes of points on semistable families of surfaces, as well as to discuss alternative stability conditions and parallels to the GIT construction of Gulbrandsen, Halle and Hulek and logarithmic Hilbert scheme constructions of Maulik and Ranganathan. We construct a good degeneration of Hilbert schemes of points as a proper Deligne-Mumford stack and show that it provides a geometrically meaningful example of a construction arising from the work of Maulik and Ranganathan.

We show that the $\infty $-category of normed algebras in genuine G-spectra, as introduced by Bachmann–Hoyois, is modeled by strictly commutative algebras in G-symmetric spectra for any finite group G. We moreover provide an analogous description of Schwede’s ultra-commutative global ring spectra in higher categorical terms.

Using these new descriptions, we exhibit the $\infty $-category of ultra-commutative global ring spectra as a partially lax limit of the $\infty $-categories of genuine G-spectra for varying G, in analogy with the nonmultiplicative comparison of Nardin, Pol, and the second author.

Along the way, we establish various new results in parametrized higher algebra, which we hope to be of independent interest.

In this paper, we develop canonical bundle formulas for fibrations of relative dimension one in characteristic $p>0$. For such a fibration from a log pair $f\colon (X, \Delta ) \to S$, if f is separable, we can obtain a formula similar to the one due to Witaszek [36]; if f is inseparable, we treat the case when S is of maximal Albanese dimension. As an application, we prove that for a klt pair $(X,\Delta )$ with $-(K_X+\Delta )$ nef, if the Albanese morphism $a_X\colon X \to A$ is of relative dimension one, then X is a fiber space over A.

If A and B are subsets of an abelian group, their sumset is $A+B:=\{a+b:a\in A, b\in B\}$. We study sumsets in discrete abelian groups, where at least one summand has positive upper Banach density.

Jin proved in [27] that if A and B are sets of integers having positive upper Banach density, then $A+B$ is piecewise syndetic. Bergelson, Furstenberg, and Weiss [4] improved the conclusion to “$A+B$ is piecewise Bohr.” In [2] this was shown to be qualitatively optimal, in the sense that if $C\subseteq \mathbb Z$ is piecewise Bohr, then there are $A, B\subseteq \mathbb Z$ having positive upper Banach density such that $A+B\subseteq C$.

We improve these results by establishing a strong correspondence between sumsets in discrete abelian groups, level sets of convolutions in compact abelian groups, and sumsets in compact abelian groups. Our proofs avoid measure preserving dynamics and nonstandard analysis, and our results apply to discrete abelian groups of any cardinality.

This paper employs the Riemann-Hilbert problem and nonlinear steepest descent method of Deift-Zhou to provide a comprehensive analysis of the asymptotic behavior of the genus two Korteweg-de Vries soliton gases. It is demonstrated that the genus two soliton gas is related to the two-phase Riemann-Theta function as $x \to +\infty $, and approaches zero as $x \to -\infty $. Additionally, the long-time asymptotic behavior of this genus two soliton gas can be categorized into five distinct regions in the x-t plane, which from left to right are quiescent region, modulated one-phase wave, unmodulated one-phase wave, modulated two-phase wave, and unmodulated two-phase wave. Moreover, an innovative method is introduced to solve the model problem associated with the high-genus Riemann surface, leading to the determination of the leading terms, which is also related to the multiphase Riemann-Theta function. A general discussion on the case of arbitrary genus N soliton gas is also presented.

A monoid presentation is called special if the right-hand side of each defining relation is equal to 1. We prove results which relate the two-sided homological finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid enjoys the homological finiteness property bi- if its group of units is of type . We also obtain results which relate the Hochschild cohomological dimension of the monoid to the cohomological dimension of its group of units. In particular we show that the Hochschild cohomological dimension of the monoid is bounded above by the maximum of $2$ and the cohomological dimension of its group of units. We apply these results to prove a Lyndon’s Identity type theorem for the two-sided homology of one-relator monoids of the form $\langle A \mid r=1 \rangle $. In particular, we show that all such monoids are of type bi-. Moreover, we show that if r is not a proper power then the one-relator monoid has Hochschild cohomological dimension at most $2$, while if r is a proper power then it has infinite Hochschild cohomological dimension. For any nonspecial one-relator monoid M with defining relation $u=v$ we show that if there is no nonempty word w such that $u,v \in A^*w \cap w A^*$ then M is of type bi-$\mathrm {FP}_\infty $ and has Hochschild cohomological dimension at most $2$.

Under suitable conditions on a family of logarithmic curves, we endow the tropicalization of the family with an affine structure in a neighborhood of the sections in such a way that the tropical $\psi $ classes from [6] arise as tropicalizations of algebraic $\psi $ classes.

Let A be a unital $\mathrm {C}^*$-algebra containing a closed two-sided ideal J and an operator system X. We enlarge X to an operator system $\mathcal{S} (X,J)$ in ${\mathbb {M}}_2(A)$, and show that in order for $\mathcal{S} (X,J)$ to be hyperrigid, each $*$-representation of $\mathrm {C}^*(X)$ annihilating $\mathrm {C}^*(X)\cap J$ must admit a unique contractive completely positive extension from X to the larger $\mathrm {C}^*$-algebra $\mathrm {C}^*(X)+J$. We leverage this implicit additional rigidity constraint to construct counterexamples to Arveson’s hyperrigidity conjecture. A key condition in our construction is the mutual orthogonality of the atomic projection of $\mathrm {C}^*(X)$ and the support projection of J, which we interpret as a new obstruction to the conjecture. Specializing to the case where J is the ideal of compact operators on a Hilbert space, we recover as a by-product of our general construction the recent counterexample of Bilich and Dor-On. On the other hand, we find that such a pathology cannot be implemented using our construction when A admits only finite-dimensional irreducible $*$-representations, thereby illustrating that the obstruction only manifests itself in noncommutative settings.