The slice decomposition is a bijective method for enumerating planar maps (graphs embedded in the sphere) with control over face degrees. In this paper, we extend the slice decomposition to the richer setting of hypermaps, naturally interpreted as properly face-bicolored maps, where the degrees of faces of each color can be controlled separately. This setting is closely related with the two-matrix model and the Ising model on random maps, which have been intensively studied in theoretical physics, leading to several enumerative formulas for hypermaps that were still awaiting bijective proofs.

Generally speaking, the slice decomposition consists in cutting along geodesics. A key feature of hypermaps is that the geodesics along which we cut are directed, following the canonical orientation of edges imposed by the coloring. This orientation requires us to introduce an adapted notion of slices, which admit a recursive decomposition that we describe.

Using these slices as fundamental building blocks, we obtain new bijective decompositions of several families of hypermaps: disks (pointed or not) with a monochromatic boundary, cylinders with monochromatic boundaries (starting with trumpets or cornets having one geodesic boundary), and disks with a “Dobrushin” boundary condition. In each case, the decomposition ultimately expresses these objects as sequences of slices whose increments correspond to downward skip-free (Łukasiewicz-type) walks subject to natural constraints.

Our approach yields bijective proofs of several explicit expressions for hypermap generating functions. In particular, we provide a combinatorial explanation of the algebraicity and of the existence of rational parametrizations for these generating functions when face degrees are bounded.

A set X is called Euclidean Ramsey if, for any k and sufficiently large n, every k-colouring of $\mathbb {R}^n$ contains a monochromatic congruent copy of X. This notion was introduced by Erdős, Graham, Montgomery, Rothschild, Spencer and Straus. They asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. It is not too difficult to show that if a set is not spherical then it is not Euclidean Ramsey either, but the converse is very much open despite extensive research over the years.

On the other hand, the block sets conjecture is a purely combinatorial, Hales-Jewett type of statement, concerning ‘blocks in large products’, introduced by Leader, Russell and Walters. If true, the block sets conjecture would imply that every transitive set (a set whose symmetry group acts transitively) is Euclidean Ramsey. As for the question above, the block sets conjecture remains very elusive, being known only in a few cases.

In this paper we show that the sizes of the blocks in the block sets conjecture cannot be bounded, even for templates over the alphabet of size 3. We also show that for the first nontrivial template, namely $123$, the blocks may be taken to be of size $2$ (for any number of colours). This is best possible; all previous bounds were ‘tower-type’ large.

We study extensions of the measure of maximal entropy to suitable compactifications of the parameter space and the moduli space of rational maps acting on the Riemann sphere. For parameter space, we consider a space which resolves the discontinuity of the iterate map. We show that the measure of maximal entropy extends continuously to this resolution space. For moduli space, we consider a space which resolves the discontinuity of the iterate map acting on its geometric invariant theory compactification. We show that the measure of maximal entropy, barycentered and modulo rotations, also extends continuously to this resolution, answering positively a question posed by DeMarco. A main ingredient is a description of limiting dynamics for some sequences.

We initiate a systematic study of nonnegative polynomials P such that $P^k$ is not a sum of squares for any odd $k\geq 1$, calling such P stubborn. We develop a new invariant of a real isolated zero of a nonnegative polynomial in the plane, that we call the SOS-invariant, and relate it to the well-known delta invariant of a plane curve singularity. Using the SOS-invariant we show that any polynomial that spans an extreme ray of the convex cone of nonnegative ternary forms of degree 6 is stubborn. We also show how to use the SOS-invariant to prove stubbornness of ternary forms in higher degree. Furthermore, we prove that in a given degree and number of variables, nonnegative polynomials that are not stubborn form a convex cone, whose interior consists of all strictly positive polynomials.

We derive exact formulas for the proportions of derangements and of derangements of p-power order in the affine classical groups $\operatorname {\mathrm {AU}}_m(q)$, $\operatorname {\mathrm {ASp}}_{2m}(q)$, $\operatorname {\mathrm {AO}}_{2m+1}(q)$ and $\operatorname {\mathrm {AO}}^{\pm }_{2m}(q)$, where p denotes the characteristic of the defining finite field.

In the unitary case, the proofs of the formulas rely on a result on partitions of independent interest: we obtain a generating function for integer partitions $\lambda =(\lambda _1, \dots , \lambda _m)$ into m parts, with $\lambda _1\ge \dots \ge \lambda _m$, such that either $\lambda _1=1$ or $\lambda _{k-1}>\lambda _k=k$ for some $k \in \{2, \dots ,m\}$.

In the symplectic and orthogonal cases, the proofs of the formulas reduce to verifying three q-polynomial identities conjectured by the author and later proved by Fulman and Stanton.

We compute the equivariant K-theoretic Donaldson–Thomas invariants of $[\mathbb {C}^2/\mu _r]\times \mathbb {C}$ using factorization and rigidity techniques. For this, we develop a generalization of Okounkov’s factorization technique that applies to Hilbert schemes of points on orbifolds. We show that the (twisted) virtual structure sheaves of Hilbert schemes of points on orbifolds satisfy the desired factorization property. We prove that the generating series of Euler characteristics of such factorizable systems are the plethystic exponential of a simpler generating series. For $[\mathbb {C}^2/\mu _r]\times \mathbb {C}$, the computation is then completed by a rigidity argument, involving an equivariant modification of Young’s combinatorial computation of the corresponding numerical Donaldson–Thomas invariants.

In this note we observe that the categorical structure of a flop occurs for some well-known noncommutative resolutions of a nodal curve. We describe the flop-flop spherical twists, and give a geometric interpretation in terms of Landau–Ginzburg models. The resolutions are all weakly crepant but not strongly crepant, and we formulate an intermediate condition that distinguishes the smaller ones.

We characterize sums of normal ultrafilters after the Magidor iteration of Prikry forcings over a discrete set of measurable cardinals. We apply this to show that the weak Ultrapower Axiom is not equivalent to the Ultrapower Axiom. We also construct a nonrigid ultrapower and two uniform ultrafilters on different cardinals that have the same ultrapower.

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.

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.

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 $2$.

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.

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.

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.

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.

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 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 $\deg {f}$.

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.

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.

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.