A long-standing open conjecture of Branko Grünbaum from 1972 states that any simple arrangement of n pairwise intersecting pseudocircles in the plane can have at most \(2n-2\) digons. Agarwal et al. proved this conjecture for arrangements of pairwise intersecting pseudocircles in which there is a common point surrounded by all pseudocircles. Recently, Felsner, Roch and Scheucher showed that Grünbaum’s

For a finite abelian group G and a positive integer k, let \(\textsf{D}_k(G)\) denote the smallest integer \(\ell \) such that each sequence over G of length at least \(\ell \) has k disjoint nontrivial zero-sum subsequences. It is known that \(\mathsf D_k(G)=n_1+kn_2-1\) if \(G\cong C_{n_1}\oplus C_{n_2}\) is a rank 2 group, where \(1

We construct an explicit sequence V kn,an of crystalline representations of exceptional weights converging to a given irreducible two-dimensional semistable representation V k,ℒ of Gal ⁡ (ℚ ¯p/ℚp). The convergence takes place in the blow-up space of two-dimensional trianguline representations studied by Colmez and Chenevier. The process of blow-up is described in detail in the rigid-analytic setting

We develop new methods for counting integral orbits having bounded invariants that lie inside the cusps of fundamental domains for coregular representations. We illustrate these methods for a representation of cardinal interest in number theory, namely that of the split orthogonal group acting on the space of quadratic forms.

We provide an explicit and algorithmic version of a theorem of Momose classifying isogenies of prime degree of elliptic curves over number fields, which we implement in Sage and PARI/GP. Combining this algorithm with recent work of Box, Gajović and Goodman we obtain the first classifications of the possible prime degree isogenies of elliptic curves over cubic number fields, as well as for several quadratic

Let L be an affine Kac–Moody algebra, with central element c, and let λ ∈ ℂ. We study two-sided ideals in the central quotient Uλ(L) := U(L)∕(c − λ) of the universal enveloping algebra of L and prove: If λ≠0 then Uλ(L) is simple. The algebra U0(L) has just-infinite growth, in the sense that any proper quotient has polynomial growth. As an immediate corollary, we show that the annihilator of any nontrivial

We give a formula for the Chow rings of weighted blow-ups. Along the way, we also compute the Chow rings of weighted projective stack bundles, a formula for the Gysin homomorphism of a weighted blow-up, and a generalization of the splitting principle. In addition, in the Appendix we compute the Chern class of a weighted blow-up.

Suppose that k is a function of n and . We show that with probability \(1-O(1/n)\), a uniformly random \(k\times n\) Latin rectangle contains no proper Latin subsquare of order 4 or more, proving a conjecture of Divoux, Kelly, Kennedy and Sidhu. We also show that the expected number of subsquares of order 3 is bounded and find that the expected number of subsquares of order 2 is \(\left( {\begin{array}{c}k\\

The Liebeck–Nikolov–Shalev conjecture (Bull Lond Math Soc 44(3):469–472, 2012) asserts that, for any finite simple non-abelian group G and any set \(A\subseteq G\) with \(|A|\ge 2\), G is the product of at most \(N\frac{\log |G|}{\log |A|}\) conjugates of A, for some absolute constant N. For G of Lie type, we prove that for any \(\varepsilon >0\) there is some \(N_{\varepsilon }\) for which G is the

We obtain new results on the Turán number of any bounded degree uniform hypergraph obtained as the expansion of a hypergraph of bounded uniformity. These are asymptotically sharp over an essentially optimal regime for both the uniformity and the number of edges and solve a number of open problems in Extremal Combinatorics. Firstly, we give general conditions under which the crosscut parameter asymptotically

Motivated by the work of Lubotzky, we use Galois cohomology to study the difference between the number of generators and the minimal number of relations in a presentation of GS(k), the Galois group of the maximal extension of a global field k that is unramified outside a finite set S of places, as k varies among a certain family of extensions of a fixed global field Q. We define a group BS(k,A), for

This work concerns asymptotical stabilisation phenomena occurring in the moduli space of sections of certain algebraic families over a smooth projective curve, whenever the generic fibre of the family is a smooth projective Fano variety, or not far from being Fano. We describe the expected behaviour of the class, in a ring of motivic integration, of the moduli space of sections of given numerical class

A smooth cuboid can be identified with a 3×3 matrix of linear forms in three variables, with coefficients in a field K, whose determinant describes a smooth cubic in the projective plane. To each such matrix one can associate a group scheme over K. We produce isomorphism invariants of these groups in terms of their adjoint algebras, which also give information on the number of their maximal abelian

We show that the naive adaptation of Malle’s conjecture to fair counting functions is not true in general.

We give simple geometric proofs of Aprodu, Farkas, Papadima, Raicu and Weyman’s theorem on syzygies of tangent-developable surfaces of rational normal curves and Raicu and Sam’s result on syzygies of K3 carpets. As a consequence, we obtain a quick proof of Green’s conjecture for general curves of genus g over an algebraically closed field k ⁡ with char ⁡ (k ⁡ ) = 0 or char ⁡ (k ⁡ ) ≥⌊(g − 1)∕2⌋. Our

Given an n-element set \(C\subseteq \mathbb {R}^d\) and a (sufficiently generic) k-element multiset \(V\subseteq \mathbb {R}^d\), we can order the points in C by ranking each point \(c\in C\) according to the sum of the distances from c to the points of V. Let \(\Psi _k(C)\) denote the set of orderings of C that can be obtained in this manner as V varies, and let \(\psi ^{\textrm{max}}_{d,k}(n)\) be

A subset S of real numbers is called bi-Sidon if it is a Sidon set with respect to both addition and multiplication, i.e., if all pairwise sums and all pairwise products of elements of S are distinct. Imre Ruzsa asked the following question: What is the maximum number f(N) such that every set S of N real numbers contains a bi-Sidon subset of size at least f(N)? He proved that \(f(N)\geqslant cN^{\frac{1}{3}}\)

Jordán and Tanigawa recently introduced the d-dimensional algebraic connectivity \(a_d(G)\) of a graph G. This is a quantitative measure of the d-dimensional rigidity of G which generalizes the well-studied notion of spectral expansion of graphs. We present a new lower bound for \(a_d(G)\) defined in terms of the spectral expansion of certain subgraphs of G associated with a partition of its vertices

In this paper, we investigate the hypergraph Turán number \(\textrm{ex}(n,K^{(r)}_{s,t})\). Here, \(K^{(r)}_{s,t}\) denotes the r-uniform hypergraph with vertex set \(\left( \cup _{i\in [t]}X_i\right) \cup Y\) and edge set \(\{X_i\cup \{y\}: i\in [t], y\in Y\}\), where \(X_1,X_2,\cdots ,X_t\) are t pairwise disjoint sets of size \(r-1\) and Y is a set of size s disjoint from each \(X_i\). This study

In the max–min allocation problem a set P of players are to be allocated disjoint subsets of a set R of indivisible resources, such that the minimum utility among all players is maximized. We study the restricted variant, also known as the Santa Claus problem, where each resource has an intrinsic positive value, and each player covets a subset of the resources. Bezáková and Dani (SIGecom Exch 5(3):11–18

For two graphs F, H and a positive integer n, the function \(f_{F,H}(n)\) denotes the largest m such that every H-free graph on n vertices contains an F-free induced subgraph on m vertices. This function has been extensively studied in the last 60 years when F and H are cliques and became known as the Erdős–Rogers function. Recently, Balogh, Chen and Luo, and Mubayi and Verstraëte initiated the systematic

Montgomery and Soundararajan showed that the distribution of ψ(x + H) − ψ(x), for 0 ≤ x ≤ N, is approximately normal with mean ∼ H and variance ∼ Hlog ⁡ (N∕H), when Nδ ≤ H ≤ N1−δ . Their work depends on showing that sums Rk(h) of k-term singular series are μk(−hlog ⁡ h + Ah)k∕2 + Ok(hk∕2−1∕(7k)+𝜀), where A is a constant and μk are the Gaussian moment constants. We study lower-order terms in the size

A Thue–Mahler equation is a Diophantine equation of the form F(X,Y ) = a ⋅ p1z1 ⋯pvzv ,gcd ⁡ (X,Y ) = 1 where F is an irreducible binary form of degree at least 3 with integer coefficients, a is a nonzero integer and p1,… ⁡,pv are rational primes. Existing algorithms for resolving such equations require computations in the field L = ℚ(𝜃,𝜃′,𝜃′′), where 𝜃, 𝜃′, 𝜃′′ are distinct roots of F(X,1) =

We classify the automorphism groups of del Pezzo surfaces of degrees 1 and 2 over an algebraically closed field of characteristic 2. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics.

Let Z be the germ of a complex hypersurface isolated singularity of equation f, with Z at least of dimension 2. We consider the family of analytic D-modules generated by the powers of 1∕f and describe it in terms of the pole order filtration on the de Rham cohomology of the complement of {f = 0} in the neighbourhood of the singularity.

We investigate a generalization of the classical notion of a Schur functor associated to a ribbon diagram. These functors are defined with respect to an arbitrary algebra, and in the case that the underlying algebra is the symmetric/exterior algebra, we recover the classical definition of Schur/Weyl functors, respectively. In general, we construct a family of 3-term complexes categorifying the classical

In 2002, D. Fon-Der-Flaass constructed a prolific family of strongly regular graphs. In this paper, we prove that for infinitely many natural numbers n and a positive constant c, this family contains at least \(n^{c\cdot n^{2/3}}\) strongly regular n-vertex graphs X with the same parameters, which satisfy the following condition: an isomorphism between X and any other graph can be verified by the 4-dimensional

We generalize the (signed) Varchenko matrix of a hyperplane arrangement to complexes of oriented matroids and show that its determinant has a nice factorization. This extends previous results on hyperplane arrangements and oriented matroids.

We prove that in every metric space where no line contains all the points, there are at least \(\Omega (n^{2/3})\) lines. This improves the previous \(\Omega (\sqrt{n})\) lower bound on the number of lines in general metric space, and also improves the previous \(\Omega (n^{4/7})\) lower bound on the number of lines in metric spaces generated by connected graphs.

Nešetřil and Ossona de Mendez recently proposed a new definition of graph convergence called structural convergence. The structural convergence framework is based on the probability of satisfaction of logical formulas from a fixed fragment of first-order formulas. The flexibility of choosing the fragment allows to unify the classical notions of convergence for sparse and dense graphs. Since the field

An r-regular graph is an r-graph, if every odd set of vertices is connected to its complement by at least r edges. Let G and H be r-graphs. An H-coloring of G is a mapping \(f:E(G) \rightarrow E(H)\) such that each r adjacent edges of G are mapped to r adjacent edges of H. For every \(r\ge 3\), let \(\mathcal H_r\) be an inclusion-wise minimal set of connected r-graphs, such that for every connected

As a first step towards a conjecture of Kahle and Newman, we prove that if \(T_n\) is a random 2-dimensional determinantal hypertree on n vertices, then $$\begin{aligned} \frac{\dim H_1(T_n,\mathbb {F}_2)}{n^2} \end{aligned}$$ converges to zero in probability. Confirming a conjecture of Linial and Peled, we also prove the analogous statement for the 1-out 2-complex. Our proof relies on the large deviation

The supersaturation problem for a given graph F asks for the minimum number \(h_F(n,q)\) of copies of F in an n-vertex graph with \(\textrm{ex}(n,F)+q\) edges. Subsequent works by Rademacher, Erdős, and Lovász and Simonovits determine the optimal range of q (which is linear in n) for cliques F such that \(h_F(n,q)\) equals the minimum number \(t_F(n,q)\) of copies of F obtained from a maximum F-free

Given integers \(n> k > 0\), and a set of integers \(L \subset [0, k-1]\), an L-system is a family of sets \(\mathcal {F}\subset \left( {\begin{array}{c}[n]\\ k\end{array}}\right) \) such that \(|F \cap F'| \in L\) for distinct \(F, F'\in \mathcal {F}\). L-systems correspond to independent sets in a certain generalized Johnson graph G(n, k, L), so that the maximum size of an L-system is equivalent

We prove that, for any \(B \subset {\mathbb {R}}\), the Cartesian product set \(B \times B\) determines \(\Omega (|B|^{2+c})\) distinct angles.

We develop obstruction theory for lifting characteristic-p local Galois representations valued in reductive groups of type Bl, Cl, Dl or G2. An application of the Emerton–Gee stack then reduces the existence of crystalline lifts to a purely combinatorial problem when p is not too small. As a toy example, we show for all local fields K∕ℚp, with p > 3, all representations ρ¯ : GK → G2(𝔽¯p) admit a crystalline

In this paper, we begin the study of the Fermat equation xn + yn = zn over real biquadratic fields. In particular, we prove that there are no nontrivial solutions to the Fermat equation over ℚ(2,3) for n ≥ 4.

We find lower bounds on higher moments of the error term in the Chebotarev density theorem. Inspired by the work of Bellaïche, we consider general class functions and prove bounds which depend on norms associated to these functions. Our bounds also involve the ramification and Galois theoretical information of the underlying extension L∕K. Under a natural condition on class functions (which appeared

Over a finite field 𝔽q, abelian varieties with commutative endomorphism rings can be described by using modules over orders in étale algebras. By exploiting this connection, we produce four theorems regarding groups of rational points and self-duality, along with explicit examples. First, when End ⁡ (A) is locally Gorenstein, we show that the group structure of A(𝔽q) is determined by End ⁡ (A). In

We study instances of Beilinson–Tate conjectures for automorphic representations of PGSp ⁡ 6 whose spin L-function has a pole at s = 1. We construct algebraic cycles of codimension 3 in the Siegel–Shimura variety of dimension 6 and we relate its regulator to the residue at s = 1 of the L-function of certain cuspidal forms of PGSp ⁡ 6. Using the exceptional theta correspondence between the split group

Let G be a bridgeless cubic graph. In 2023, the three authors solved a conjecture (also known as the \(S_4\)-Conjecture) made by Mazzuoccolo in 2013: there exist two perfect matchings of G such that the complement of their union is a bipartite subgraph of G. They actually show that given any \(1^+\)-factor F (a spanning subgraph of G such that its vertices have degree at least 1) and an arbitrary edge

We formulate an analogue of the Breuil–Mézard conjecture for the group of units of a central division algebra over a p-adic local field, and we prove that it follows from the conjecture for GL ⁡ n. To do so we construct a transfer of inertial types and Serre weights between the maximal compact subgroups of these two groups, in terms of Deligne–Lusztig theory, and we prove its compatibility with mod

We prove the Pappas–Rapoport conjecture on the existence of canonical integral models of Shimura varieties with parahoric level structure in the case where the Shimura variety is defined by a torus. As an important ingredient, we show, using the Bhatt–Scholze theory of prismatic F-crystals, that there is a fully faithful functor from 𝒢-valued crystalline representations of Gal ⁡ (K¯∕K) to 𝒢-shtukas

We establish a geometrization of the Breuil–Mézard conjecture for potentially Barsotti–Tate representations, as well as of the weight part of Serre’s conjecture, for moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field. These results are first proved for the stacks of our earlier papers, and then transferred to the stacks of Emerton and Gee by

An arc space of an affine cone over a projective toric variety is known to be nonreduced in general. It was demonstrated recently that the reduced scheme structure of arc spaces is very meaningful from algebro-geometric, representation-theoretic and combinatorial points of view. In this paper we develop a general machinery for the description of the reduced arc spaces of affine cones over toric varieties

Let k be a positive integer. We show that, as n goes to infinity, almost every entry of the character table of Sn is divisible by k. This proves a conjecture of Miller.

We study the index of log Calabi–Yau pairs (X,B) of coregularity 0. We show that 2λ(KX + B) ∼ 0, where λ is the Weil index of (X,B). This is in contrast to the case of klt Calabi–Yau varieties, where the index can grow doubly exponentially with the dimension. Our sharp bound on the index extends to the context of generalized log Calabi–Yau pairs, semi-log canonical pairs, and isolated log canonical

Given a residually connected incidence geometry \(\Gamma \) that satisfies two conditions, denoted \((B_1)\) and \((B_2)\), we construct a new geometry \(H(\Gamma )\) with properties similar to those of \(\Gamma \). This new geometry \(H(\Gamma )\) is inspired by a construction of Lefèvre-Percsy, Percsy and Leemans (Bull Belg Math Soc Simon Stevin 7(4):583–610, 2000). We show how \(H(\Gamma )\) relates

For an odd integer \(n = 2d-1\), let \({\mathcal {B}}_d\) be the subgraph of the hypercube \(Q_n\) induced by the two largest layers. In this paper, we describe the typical structure of proper q-colorings of \(V({\mathcal {B}}_d)\) and give asymptotics on the number of such colorings when q is an even number. The proofs use various tools including information theory (entropy), Sapozhenko’s graph container

A spherical L-code, where \(L \subseteq [-1,\infty )\), consists of unit vectors in \(\mathbb {R}^d\) whose pairwise inner products are contained in L. Determining the maximum cardinality \(N_L(d)\) of an L-code in \(\mathbb {R}^d\) is a fundamental question in discrete geometry and has been extensively investigated for various choices of L. Our understanding in high dimensions is generally quite poor

A permutation \(\pi \in \mathbb {S}_n\) is k-balanced if every permutation of order k occurs in \(\pi \) equally often, through order-isomorphism. In this paper, we explicitly construct k-balanced permutations for \(k \le 3\), and every n that satisfies the necessary divisibility conditions. In contrast, we prove that for \(k \ge 4\), no such permutations exist. In fact, we show that in the case \(k

Abstract polytopes are combinatorial objects that generalise geometric objects such as convex polytopes, maps on surfaces and tilings of the space. Chiral polytopes are those abstract polytopes that admit full combinatorial rotational symmetry but do not admit reflections. In this paper we build chiral polytopes whose facets (maximal faces) are isomorphic to a prescribed regular cubic tessellation

We determine the excluded minors characterising the class of countable graphs that embed into some compact surface.

In this paper, we prove that the Fourier entropy of an n-dimensional boolean function f can be upper-bounded by \(O(I(f)+ \sum \limits _{k\in [n]}I_k(f)\log \frac{1}{I_k(f)})\), where I(f) is its total influence and \(I_k(f)\) is the influence of the k-th coordinate. There is no strict quantitative relationship between our bound with the known bounds for the Fourier-Min-Entropy-Influence conjecture

A subset \(\mathcal {C}\subseteq \{0,1,2\}^n\) is said to be a trifferent code (of block length n) if for every three distinct codewords \(x,y, z \in \mathcal {C}\), there is a coordinate \(i\in \{1,2,\ldots ,n\}\) where they all differ, that is, \(\{x(i),y(i),z(i)\}\) is same as \(\{0,1,2\}\). Let T(n) denote the size of the largest trifferent code of block length n. Understanding the asymptotic behavior

A majority edge-coloring of a graph without pendant edges is a coloring of its edges such that, for every vertex v and every color \(\alpha \), there are at most as many edges incident to v colored with \(\alpha \) as with all other colors. We extend some known results for finite graphs to infinite graphs, also in the list setting. In particular, we prove that every infinite graph without pendant edges

In this paper we give anticoncentration bounds for sums of independent random vectors in finite-dimensional vector spaces. In particular, we asymptotically establish a conjecture of Leader and Radcliffe (SIAM J Discrete Math 7:90–101, 1994) and a question of Jones (SIAM J Appl Math 34:1–6, 1978). The highlight of this work is an application of the strong perfect graph theorem by Chudnovsky et al. (Ann

We introduce a modification of the linear sieve whose weights satisfy strong factorization properties, and consequently equidistribute primes up to size x in arithmetic progressions to moduli up to x10∕17. This surpasses the level of distribution x4∕7 with the linear sieve weights from well-known work of Bombieri, Friedlander, and Iwaniec, and which was recently extended to x7∕12 by Maynard. As an

Let A be an abelian variety over a number field F, and suppose that ℤ[ζn] embeds in End ⁡ F¯A, for some root of unity ζn of order n = 3m. Assuming that the Galois action on the finite group A[1 − ζn] is sufficiently reducible, we bound the average rank of the Mordell–Weil groups Ad(F), as Ad varies through the family of μ2n-twists of A. Combining this result with the recently proved uniform Mordell–Lang

Let X be a K3 surface over a number field. We prove that X has infinitely many specializations where its Picard rank jumps, hence extending our previous work with Shankar, Shankar and Tang to the case where X has bad reduction. We prove a similar result for generically ordinary nonisotrivial families of K3 surfaces over curves over 𝔽¯p which extends previous work of Maulik, Shankar and Tang. As a