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

Let K be a number field, and S a finite set of nonarchimedean places of K, and write 𝒪× for the group of S-units of K. A famous theorem of Siegel asserts that the S-unit equation 𝜀 + δ = 1, with 𝜀, δ ∈𝒪×, has only finitely many solutions. A famous theorem of Shafarevich asserts that there are only finitely many isomorphism classes of elliptic curves over K with good reduction outside S. Now instead

We prove vanishing results for the coherent cohomology of the good reduction modulo p of the Siegel modular variety with coefficients in some automorphic bundles. We show that for an automorphic bundle with highest weight λ near the walls of the antidominant Weyl chamber, there is an integer e ≥ 0 such that the cohomology is concentrated in degrees [0,e]. The accessible weights with our method are

Let E be a field of characteristic p. In a previous paper of ours, we defined and studied super-Hölder vectors in certain E-linear representations of ℤp. In the present paper, we define and study super-Hölder vectors in certain E-linear representations of a general p-adic Lie group. We then consider certain p-adic Lie extensions K∞∕K of a p-adic field K, and compute the super-Hölder vectors in the

We study the variation of linear sections of hypersurfaces in ℙn. We completely classify all plane curves, necessarily singular, whose line sections do not vary maximally in moduli. In higher dimensions, we prove that the family of hyperplane sections of any smooth degree d hypersurface in ℙn varies maximally for d ≥ n + 3. In the process, we generalize the classical Grauert–Mülich theorem about lines

We describe separating G2-invariants of several copies of the algebra of octonions over an algebraically closed field of characteristic two. We also obtain a minimal separating and a minimal generating set for G2-invariants of several copies of the algebra of octonions in case of a field of odd characteristic.

We construct scattering diagrams for Chekhov–Shapiro generalized cluster algebras where exchange polynomials are factorized into binomials, generalizing the cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. They turn out to be natural objects arising in Fock and Goncharov’s cluster duality. Analogous features and structures (such as positivity and the cluster complex structure) in

We study a family of exponential sums that arises in the study of expanding horospheres on GL ⁡ n. We prove an explicit version of general purity and find optimal bounds for these sums.

We prove that the Galois equidistribution of torsion points of the algebraic torus 𝔾md extends to the singular test functions of the form log ⁡ |P|, where P is a Laurent polynomial having algebraic coefficients that vanishes on the unit real d-torus in a set whose Zariski closure in 𝔾md has codimension at least 2. Our result includes a power-saving quantitative estimate of the decay rate of the equidistribution

We show the image of rooted tree maps forms a subspace of the kernel of the evaluation map of multiple L-values. To prove this, we define the diamond product as a modified harmonic product and describe its properties. We also show that τ-conjugate rooted tree maps are their antipodes.

The local structure of terminal Brauer classes on arithmetic surfaces was classified (2021), generalising the classification on geometric surfaces (2005). Part of the interest in these classifications is that it enables the minimal model program to be applied to the noncommutative setting of orders on surfaces. We give étale local structure theorems for terminal orders on arithmetic surfaces, at least

Fix a finite field K of order q and a word w in a free group F on r generators. A w-random element in GL ⁡ N(K) is obtained by sampling r independent uniformly random elements g1,… ⁡,gr ∈ GL ⁡ N(K) and evaluating w(g1,… ⁡,gr). Consider 𝔼w[fix ⁡ ], the average number of vectors in KN fixed by a w-random element. We show that 𝔼w[fix ⁡ ] is a rational function in qN. If w = ud with u a nonpower, then

We investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants |d|≤ x. In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime

We explain how logarithmic structures select principal components in an intersection of schemes. These manifest in Chow homology and can be understood using strict transforms under logarithmic blowups. Our motivation comes from Gromov–Witten theory. The toric contact cycles in the moduli space of curves parameterize curves that admit a map to a fixed toric variety with prescribed contact orders. We

Spectral moment formulae of various shapes have proven very successful in studying the statistics of central L-values. We establish, in a completely explicit fashion, such formulae for the family of GL ⁡ (3) × GL ⁡ (2) Rankin–Selberg L-functions using the period integral method. Our argument does not rely on either the Kuznetsov or Voronoi formulae. We also prove the essential analytic properties and

We prove that the double (or canonical unramified) wavefront set of an irreducible depth-0 supercuspidal representation of a reductive p-adic group is a singleton provided p > 3(h − 1), where h is the Coxeter number. We deduce that the geometric wavefront set is also a singleton in this case, proving a conjecture of Mœglin and Waldspurger. When the group is inner to split and the representation belongs

We associate with a generalised deep hole of the Leech lattice vertex operator algebra a generalised hole diagram. We show that this Dynkin diagram determines the generalised deep hole up to conjugacy and that there are exactly 70 such diagrams. In an earlier work we proved a bijection between the generalised deep holes and the strongly rational, holomorphic vertex operator algebras of central charge

Given a smooth projective toric variety X of Picard rank 2, we resolve the diagonal sheaf on X × X by a linear complex of length dim ⁡ X consisting of finite direct sums of line bundles. As applications, we prove a new case of a conjecture of Berkesch, Ermana and Smith that predicts a version of Hilbert’s syzygy theorem for virtual resolutions, and we obtain a Horrocks-type splitting criterion for