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.

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 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.

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 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 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.

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.

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.

Given a polynomial $\sum _\nu a_\nu X^\nu $ of degree $<d$, bounded by one on the unit disk, how large can $ \left \lvert {a_0+a_1+\ldots +a_n} \right \rvert $ ($n<d$) get? This question dates back at least to the 1952 thesis work of H. S. Shapiro. In 1978, D. J. Newman gave an exact answer for $d=2(n+1)$, but there does not seem to have been further progress on the question since. We study variations on exact answers for some related coefficient sums, and answer the original question in an asymptotic sense, provided that n is ‘not too large’ in terms of d. The latter is achieved via a ‘quantitative’ Eneström–Kakeya theorem, while the former is based on certain identities for carefully selected Lagrange interpolators. From the interpolation approach we also obtain a general inequality for coefficient sums $ \left \lvert { t_0 a_0 + \ldots + t_{d-1} a_{d-1} } \right \rvert $ for arbitrary complex numbers $t_0,\ldots ,t_{d-1}$. This inequality fails to be sharp in general, yet it is in some cases and also yields non-trivial bounds for Shapiro’s problem for some choices of n and d.

Saturated fusion systems are categories modeling properties of conjugacy of p-elements in finite groups. It was shown by Chermak that there are group-like structures called regular localities associated to saturated fusion systems. Both the theory of fusion systems and the theory of regular localities are developed in analogy to the theory of finite groups. In this paper we focus on a classical theorem of Wielandt, which states that any two subnormal subgroups of a finite group G generate a subnormal subgroup of G. We prove versions of this theorem for regular localities and for fusion systems. Along the way we prove also a purely group-theoretical result which may be of independent interest.

Liouville field theory has long been a cornerstone of two-dimensional quantum field theory and quantum gravity, which has attracted much recent attention in the mathematics literature. Timelike Liouville field theory is a version of Liouville field theory where the kinetic term in the action appears with a negative sign, which makes it closer to a theory of quantum gravity than ordinary (spacelike) Liouville field theory. Making sense of this “wrong sign” requires a theory of Gaussian random variables with negative variance. Such a theory is developed in this paper, and is used to prove the timelike DOZZ formula for the $3$-point correlation function when the parameters satisfy the so-called “charge neutrality condition.” Expressions are derived also for the k-point correlation functions for all $k\ge 3$, and it is shown that these functions approach the correct semiclassical limits as the coupling constant is sent to zero.

With a mere usage of well-established properties of paradifferential operators, the conjugacy equations in several model KAM problems are converted to parahomological equations solvable by standard fixed point argument. Such discovery greatly simplifies KAM proofs, renders the traditional KAM iteration steps unnecessary, and may suggest a systematic scheme of finding quasi-periodic solutions of realistic magnitude.

We provide a natural definition of an elliptic arrangement, extending the classical framework to an elliptic curve $\mathcal {E}$ with complex multiplication. We analyze the intersections of elements of the arrangement and their connected components as $\operatorname {End}(\mathcal {E})$-modules. Furthermore, we prove that the combinatorial data of elliptic arrangements define both an arithmetic matroid and a matroid over the ring $\operatorname {End}(\mathcal {E})$. In this way, we obtain a class of arithmetic matroids that is different from the class of arithmetic matroids realizable via toric arrangements. Finally, we show that the Euler characteristic of the complement is an evaluation of the arithmetic Tutte polynomial.

We consider a generalized model of random walk in dynamical random environment, and we show that the multiplicative-noise stochastic heat equation (SHE) describes the fluctuations of the quenched density at a certain precise spatial location in the tail called the critical scale. The distribution of transition kernels is fixed rather than changing under the diffusive rescaling of space-time, that is, there is no tuning of the model parameters needed to observe the stochastic PDE limit. The proof is done by pushing the methods developed in [DDP24a, DDP24b] to their maximum, substantially weakening the assumptions and obtaining fairly sharp conditions under which one expects to see the SHE arise in a wide variety of random walk models in random media. In particular we are able to get rid of conditions such as nearest-neighbor interaction as well as spatial independence of quenched transition kernels. Moreover, we observe an entire hierarchy of moderate deviation exponents at which the SHE can be found, confirming a physics prediction of [Has25] and mirroring a result from [HQ18] in the context of this model.

We consider the Brown measure of $a+\mathfrak {c}$, where a lies in a commutative tracial von Neumann algebra $\mathcal {B}$ and $\mathfrak {c}$ is a $\mathcal {B}$-valued circular element. Under certain regularity conditions on a and the covariance of $\mathfrak {c}$ this Brown measure has a density with respect to the Lebesgue measure on the complex plane which is real analytic apart from jump discontinuities at the boundary of its support. With the exception of finitely many singularities this one-dimensional spectral edge is real analytic. We provide a full description of all possible edge singularities as well as all points in the interior, where the density vanishes. The edge singularities are classified in terms of their local edge shape while internal zeros of the density are classified in terms of the shape of the density locally around these points. We also show that each of these countably infinitely many singularity types occurs for an appropriate choice of a when $\mathfrak {c}$ is a standard circular element. The Brown measure of $a+\mathfrak {c}$ arises as the empirical spectral distribution of a diagonally deformed non-Hermitian random matrix with independent entries when its dimension tends to infinity.

Pólya trees are unlabeled rooted trees on n vertices. This paper gives a new way to generate Pólya trees, that conjecturally is very efficient. This allows comparing typical unlabeled and labeled tree statistics and comparing asymptotic theorems with “reality.”

Our method is an application of the Burnside process, alternating two steps: from a labeled rooted tree, produce a uniform permutation fixing it; and from a permutation, produce a uniform labeled rooted tree fixed by it. This last step is linked to a product formula, refining Cayley’s, for the number of rooted labeled trees preserved by a given permutation.

We introduce and study different compactifications of the moduli space of n distinct weighted labeled points in a flag of affine spaces. We construct these spaces via the weighted and generalized Fulton-MacPherson compactifications of Routis and Kim-Sato. For certain weights, our compactifications are toric and isomorphic to the polypermutohedral and polystellahedral varieties, which arise in the work of Crowley-Huh-Larson-Simpson-Wang and Eur-Larson on polymatroids, a generalization of matroids. Moreover, we show that these toric compactifications have a fibration structure, with fibers isomorphic to the Losev-Manin space, and are related to each other via a geometric quotient.

Let $R \to S$ be a cyclically pure map of Noetherian $\mathbb {Q}$-algebras. In this paper, we show that if S has Du Bois singularities, then R has Du Bois singularities. Our result is new even when $R \to S$ is faithfully flat. Our proof also yields interesting results in prime characteristic and in mixed characteristic. As a consequence, we show that if $R \to S$ is a cyclically pure map of rings essentially of finite type over the complex numbers $\mathbb {C}$, S has log canonical type singularities, and $K_R$ is Cartier, then R has log canonical singularities. Along the way, we prove a version of the key injectivity theorem of Kovács and Schwede for Noetherian schemes of equal characteristic zero that have isolated non-Du Bois points. Throughout the paper, we use the characterization of the complex $\underline {\Omega }^0_X$ and of Du Bois singularities in terms of sheafification with respect to Grothendieck topologies.

We describe the K-moduli spaces of weighted hypersurfaces of degree $2(n+3)$ in $\mathbb {P}(1,2,n+2,n+3)$. We show that the K-polystable limits of these weighted hypersurfaces are also weighted hypersurfaces of the same degree in the same weighted projective space. This is achieved by an explicit study of the wall crossing for K-moduli spaces $M_w$ of certain log Fano pairs with coefficient w whose double cover gives the weighted hypersurface. Moreover, we show that the wall crossing of $M_w$ coincides with variation of GIT except at the last K-moduli wall which gives a divisorial contraction. Our K-moduli spaces provide new birational models for some natural loci in the moduli space of marked hyperelliptic curves.

Let $H_g$ denote the 4-dimensional handlebody of genus g and $U_g$ its boundary. We show that for all $g \ge 0$ the map from $B\!\operatorname {\mathrm {Homeo}}(H_g)$ to $B\!\operatorname {\mathrm {Homeo}}(U_g)$ induced by restriction to the boundary admits a section.