For all integers \(n \ge k > d \ge 1\), let \(m_{d}(k,n)\) be the minimum integer \(D \ge 0\) such that every k-uniform n-vertex hypergraph \({\mathcal {H}}\) with minimum d-degree \(\delta _{d}({\mathcal {H}})\) at least D has an optimal matching. For every fixed integer \(k \ge 3\), we show that for \(n \in k \mathbb {N}\) and \(p = \Omega (n^{-k+1} \log n)\), if \({\mathcal {H}}\) is an n-vertex

A storage code on a graph G is a set of assignments of symbols to the vertices such that every vertex can recover its value by looking at its neighbors. We consider the question of constructing large-size storage codes on triangle-free graphs constructed as coset graphs of binary linear codes. Previously it was shown that there are infinite families of binary storage codes on coset graphs with rate

In the first part of this paper we study the coaction dual to the action of the cosmic Galois group on the motivic lift of the sunrise Feynman integral with generic masses and momenta, and we express its conjugates in terms of motivic lifts of Feynman integrals associated to related Feynman graphs. Only one of the conjugates of the motivic lift of the sunrise, other than itself, can be expressed in

We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated using tools from homological algebra and the theory of $D$-modules. We present an overview and uncover new relations between these approaches. We also provide new

In this paper, we prove a conjecture by Gukov–Pei–Putrov–Vafa for a wide class of plumbed $3$-manifolds. Their conjecture states that Witten–Reshetikhin–Turaev (WRT) invariants are radial limits of homological blocks, which are $q$-series introduced by them for plumbed $3$-manifolds with negative definite linking matrices. The most difficult point in our proof is to prove the vanishing of weighted

Dubrovin $\href{https://doi.org/10.1007/s00023-015-0449-2}{[10]}$ has shown that the spectrum of the quantization (with respect to the first Poisson structure) of the dispersionless Korteweg–de Vries (KdV) hierarchy is given by shifted symmetric functions; the latter are related by the Bloch–Okounkov Theorem $\href{https://doi.org/10.1007/JHEP07(2014)141}{[1]}$ to quasimodular forms on the full modular

We develop the theory of colored bosonic models (initiated by Borodin and Wheeler). We will show how a family of such models can be used to represent the values of Iwahori vectors in the “spherical model” of representations of $\mathrm{GL}_r (F)$, where $F$ is a nonarchimedean local field. Among our results are a monochrome factorization, which is the realization of the Boltzmann weights by fusion

We show that, for any prime p and integer \(k \ge 2\), a simple \({{\,\textrm{GF}\,}}(p)\)-representable matroid with sufficiently high rank has a rank-k flat which is either independent in M, or is a projective or affine geometry. As a corollary we obtain a Ramsey-type theorem for \({{\,\textrm{GF}\,}}(p)\)-representable matroids. For any prime p and integer \(k\ge 2\), if we 2-colour the elements

We formulate and prove the weight part of Serre’s conjecture for three-dimensional mod p Galois representations under a genericity condition when the field is unramified at p. This removes the assumption made previously that the representation be tamely ramified at p. We also prove a version of Breuil’s lattice conjecture and a mod p multiplicity one result for the cohomology of U(3)-arithmetic manifolds

We combine two of Igusa’s conjectures with recent semicontinuity results by Mustaţă and Popa to form a new, natural conjecture about bounds for exponential sums. These bounds have a deceivingly simple and general formulation in terms of degrees and dimensions only. We provide evidence consisting partly of adaptations of already known results about Igusa’s conjecture on exponential sums, but also some

We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less arithmetic information is required to prove nontrivial lower bounds. As an application of our method, assuming the existence of infinitely many exceptional characters

A flag positroid of ranks r := (r1 < ⋯ < rk) on [n] is a flag matroid that can be realized by a real rk × n matrix A such that the ri × ri minors of A involving rows 1,2,… ⁡,ri are nonnegative for all 1 ≤ i ≤ k. In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when r := (a,a + 1,… ⁡,b) is a sequence of consecutive numbers. In this case we show that the

Given a finite group G and a prime p, let 𝒜p(G) be the poset of nontrivial elementary abelian p-subgroups of G. The group G satisfies the Quillen dimension property at p if 𝒜p(G) has nonzero homology in the maximal possible degree, which is the p-rank of G minus 1. For example, D. Quillen showed that solvable groups with trivial p-core satisfy this property, and later, M. Aschbacher and S. D. Smith

Given a graph G with a set F(v) of forbidden values at each \(v \in V(G)\), an F-avoiding orientation of G is an orientation in which \(\deg ^+(v) \not \in F(v)\) for each vertex v. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if \(|F(v)| < \frac{1}{2} \deg (v)\) for each \(v \in V(G)\), then G has an F-avoiding orientation, and they showed that this statement is true when \(\frac{1}{2}\)

$\def\M{\mathscr{M}}\def\Rscr{\mathscr{R}}\def\Rsf{\mathsf{R}}\def\Tsf{\mathsf{T}}\def\tildeM{\widetilde{\M}}$For any positive integer $n$, we introduce a modular vector field $\Rsf$ on a moduli space $\Tsf$ of enhanced Calabi–Yau $n$-folds arising from the Dwork family. By Calabi–Yau quasi-modular forms associated to $\Rsf$ we mean the elements of the graded $\mathbb{C}$-algebra $\tildeM$ generated

related to Gopakumar-Vafa (GV) invariants, and rank 0 Donaldson-Thomas (DT) invariants countingD4-D2-D0 BPS bound states, we rigorously compute the first few terms in the generating series of Abelian D4-D2-D0 indices for compact one-parameter Calabi-Yau threefolds of hypergeometric type. In all cases where GV invariants can be computed to sufficiently high genus, we find striking confirmation that

In recent years, a version of enumerative geometry over arbitrary fields has been developed and studied by Kass-Wickelgren, Levine, and others, in which the counts obtained are not integers but quadratic forms. Aiming to understand the relation to other “refined invariants”, and especially their possible interpretation in quantum theory, we explain how to obtain a quadratic version of Donaldson-Thomas

We introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obtained. In particular, the scalar curvature density and the total scalar curvature

One of the first key examples of a quantum modular form, which unifies the Witten-Reshetikhin-Turaev (WRT) invariants of the Poincaré homology sphere, appears in work of Lawrence and Zagier. We show that the series they construct is one instance in an infinite family of quantum modular invariants of negative definite plumbed 3‑manifolds whose radial limits toward roots of unity may be thought of as

Products of shifted characteristic polynomials, and ratios of such products, averaged over the classical compact groups are of great interest to number theorists as they model similar averages of L-functions in families with the same symmetry type as the compact group. We use Toeplitz and Toeplitz plus Hankel operators and the identities of Borodin–Okounkov–Case–Geronimo, and Basor–Ehrhardt to prove

We provide a dynamical study of a model of multiplicative perturbation of a unitary matrix introduced by Fyodorov. In particular, we identify a flow of deterministic domains that bound the spectrum with high probability, separating the outlier from the typical eigenvalues at all sub-critical timescales. These results are obtained under generic assumptions on U that hold for a variety of unitary random

It is well known that the semi-circle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko–Pastur theorems. In this work, we shed the light on an intriguing phenomenon suggesting that this functional is monotonic along the mean empirical spectral distribution in

In this paper, we study high-dimensional behavior of empirical spectral distributions {LN(t),t∈[0,T]} for a class of N×N symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic differential equation driven by fractional Brownian motion with Hurst parameter H∈(1/2,1). For Wigner-type matrices, we obtain almost sure relative compactness of {LN(t),t∈[0,T]}N∈ℕ in

A family \(\mathcal {F} \subset \mathcal {P}(n)\) is r-wise k-intersecting if \(|A_1 \cap \dots \cap A_r| \ge k\) for any \(A_1, \dots , A_r \in \mathcal {F}\). It is easily seen that if \(\mathcal {F}\) is r-wise k-intersecting for \(r \ge 2\), \(k \ge 1\) then \(|\mathcal {F}| \le 2^{n-1}\). The problem of determining the maximum size of a family \(\mathcal {F}\) that is both \(r_1\)-wise \(k_1\)-intersecting

The problem of Zarankiewicz asks for the maximum number of edges in a bipartite graph on n vertices which does not contain the complete bipartite graph \(K_{k,k}\) as a subgraph. A classical theorem due to Kővári, Sós, and Turán says that this number of edges is \(O\left( n^{2 - 1/k}\right) \). An important variant of this problem is the analogous question in bipartite graphs with VC-dimension at most

Starting from Montgomery’s conjecture, there has been a substantial interest on the connections of random matrix theory and the theory of L-functions. In particular, moments of characteristic polynomials of random matrices have been considered in various works to estimate the asymptotics of moments of L-function families. In this paper, we first consider joint moments of the characteristic polynomial

We prove that every properly edge-colored n-vertex graph with average degree at least \(32(\log 5n)^2\) contains a rainbow cycle, improving upon the \((\log n)^{2+o(1)}\) bound due to Tomon. We also prove that every properly edge-colored n-vertex graph with at least \(10^5 k^3 n^{1+1/k}\) edges contains a rainbow 2k-cycle, which improves the previous bound \(2^{ck^2}n^{1+1/k}\) obtained by Janzer.

For a d-dimensional regular proper variety X over the function field of a smooth variety B over a field k and for i ≥ 0, we define a subgroup CH ⁡ i(X)(0) of CH ⁡ i(X) and construct a “refined height pairing” CH ⁡ i(X)(0) × CH ⁡ d+1−i(X)(0) → CH ⁡ 1(B) in the category of abelian groups up to isogeny. For i = 1,d, CH ⁡ i(X)(0) is the group of cycles numerically equivalent to 0. This pairing relates

The Balmer spectrum of a monoidal triangulated category is an important geometric construction which is closely related to the problem of classifying thick tensor ideals. We prove that the forgetful functor from the Drinfeld center of a finite tensor category C to C extends to a monoidal triangulated functor between their corresponding stable categories, and induces a continuous map between their Balmer

We establish functoriality of higher Coleman theory for certain unitary Shimura varieties and use this to construct a p-adic analytic function interpolating unitary Friedberg–Jacquet periods.

We study the conjugacy classes of the classical affine groups. We derive generating functions for the number of classes analogous to formulas of Wall and the authors for the classical groups. We use these to get good upper bounds for the number of classes. These naturally come up as difficult cases in the study of the noncoprime k(GV ) problem of Brauer.

We prove that every additive set A with energy \(E(A)\ge |A|^3/K\) has a subset \(A'\subseteq A\) of size \(|A'|\ge (1-\varepsilon )K^{-1/2}|A|\) such that \(|A'-A'|\le O_\varepsilon (K^{4}|A'|)\). This is, essentially, the largest structured set one can get in the Balog–Szemerédi–Gowers theorem.

The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc uv in a tournament T is the set of vertices that form a directed triangle with arc uv. We show that if the neighborhood of

We prove the ordinary Hecke orbit conjecture for Shimura varieties of Hodge type at primes of good reduction. We make use of the global Serre–Tate coordinates of Chai as well as recent results of D’Addezio about the monodromy groups of isocrystals. The new ingredients in this paper are a general monodromy theorem for Hecke-stable subvarieties for Shimura varieties of Hodge type, and a rigidity result

We develop a version of Sen theory for equivariant vector bundles on the Fargues–Fontaine curve. We show that every equivariant vector bundle canonically descends to a locally analytic vector bundle. A comparison with the theory of (φ,Γ)-modules in the cyclotomic case then recovers the Cherbonnier–Colmez decompletion theorem. Next, we focus on the subcategory of de Rham locally analytic vector bundles

The equation xm = 0 defines a fat point on a line. The algebra of regular functions on the arc space of this scheme is the quotient of k[x,x′,x(2),… ⁡] by all differential consequences of xm = 0. This infinite-dimensional algebra admits a natural filtration by finite-dimensional algebras corresponding to the truncations of arcs. We show that the generating series for their dimensions equals m∕(1 −

By using results on poles of L-functions and theta correspondence, we give a bound on b for (χ,b)-factors of the global Arthur parameter of a cuspidal automorphic representation π of a classical group or a metaplectic group where χ is a conjugate self-dual automorphic character and b is an integer which is the dimension of an irreducible representation of SL ⁡ 2(ℂ). We derive a more precise relation

This paper is an extension of Kim et al. (2020a), and we prove equidistribution theorems for families of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur’s invariant trace formula in terms of Shintani zeta functions in a uniform way. Several applications, including the

When does a graph admit a tree-decomposition in which every bag has small diameter? For finite graphs, this is a property of interest in algorithmic graph theory, where it is called having bounded “tree-length”. We will show that this is equivalent to being “boundedly quasi-isometric to a tree”, which for infinite graphs is a much-studied property from metric geometry. One object of this paper is to