Dimension Conjecture Research Articles

Winning of inhomogeneous bad for curves

AbstractWe prove the absolute winning property of weighted inhomogeneous badly approximable vectors on nondegenerate analytic curves. This answers a question by Beresnevich, Nesharim, and Yang. In particular, our result is an inhomogeneous version of the main result in (Duke Math J 171(14), 2022) by Beresnevich, Nesharim, and Yang. Also, the generality of the inhomogeneous part that we considered extends the previous result in An et al. (Adv Math 324:148–202, 2018). Moreover, our results even contribute to classical results, namely establishing the inhomogeneous Schmidt’s conjecture in arbitrary dimensions.

Generalised Lat-Igusa-Todorov Algebras and Morita Contexts

AbstractIn this paper we define (special) GLIT classes and (special) GLIT algebras. We prove that GLIT algebras, which generalise Lat-Igusa-Todorov algebras, satisfy the finitistic dimension conjecture and give several properties and examples. In addition we show that special GLIT algebras are exactly those that have finite finitistic dimension. Lastly we study Morita algebras arising from a Morita context and give conditions for them to be (special) GLIT in terms of the algebras and bimodules used in their definition. As a consequence we obtain simple conditions for a triangular matrix algebra to be (special) GLIT and also prove that the tensor product of a GLIT $$\mathbb {K}$$ K -algebra with a path algebra of a finite quiver without oriented cycles is GLIT.

Critical Lengths of Steklov Eigenvalues of Hypersurfaces of Revolution in Euclidean Space

We study the Steklov problem on hypersurfaces of revolution with two boundary components in Euclidean space and focus on the phenomenon of critical length, at which a Steklov eigenvalue is maximized. In this article, we conjecture that, in any dimension, there is a finite number of infinite critical length. To investigate this, we develop an algorithm to efficiently perform numerical experiments, providing support to our conjecture. Furthermore, we prove the conjecture in dimension n = 3 and n = 4.

Injective generation of the derived category and finitistic dimension conjecture

Injective generation of the derived category and finitistic dimension conjecture

Wilf inequality is preserved under gluing of semigroups

Wilf Conjecture on numerical semigroups is a question posed by Wilf in 1978 and is an inequality connecting the Frobenius number, embedding dimension and the genus of the semigroup. The conjecture is still open in general. We prove that this Wilf inequality is preserved under gluing of numerical semigroups. If the numerical semigroups minimally generated by [Formula: see text] and [Formula: see text] satisfy the Wilf inequality, then so does their gluing which is minimally generated by [Formula: see text]. We discuss the extended Wilf’s Conjecture in higher dimensions for certain affine semigroups and prove an analogous result.

Roudneff’s Conjecture in Dimension 4

J.-P. Roudneff conjectured in 1991 that every arrangement of n ≥ 2 d + 1 ≥ 5 pseudohyperplanes in the real projective space P d has at most ∑ i = 0 d − 2 ( n − 1 i ) complete cells (i.e., cells bounded by each hyperplane). The conjecture is true for d = 2, 3 and for arrangements arising from Lawrence oriented matroids. Our main contribution is to show the validity of Roudneff’s conjecture for d = 4. Moreover, based on computational data we show that for d ≤ 4 and n ≤ 2 d + 1 the maximum number of complete cells is only obtained (up to isomorphism) by cyclic arrangements.

On Fila-King conjecture in dimension four

We consider the following Cauchy problem for the four-dimensional energy critical heat equation{ut=Δu+u3 in R4×(0,∞),u(x,0)=u0(x) in R4. We construct a positive infinite time blow-up solution u(x,t) with the blow-up rate ‖u(⋅,t)‖L∞(R4)∼ln⁡t as t→∞ and establish the stability of the infinite time blow-up. This gives a rigorous proof of a conjecture by Fila and King [15, Conjecture 1.1].

Degenerating products of flag varieties and applications to the Breuil–Mézard conjecture

We consider closed subschemes in the affine grassmannian obtained by degenerating e-fold products of flag varieties, embedded via a tuple of dominant cocharacters. For G=GL2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G= {\ ext {GL}}_2$$\\end{document}, and cocharacters small relative to the characteristic, we relate the cycles of these degenerations to the representation theory of G. We then show that these degenerations smoothly model the geometry of (the special fibre of) low weight crystalline subspaces inside the Emerton–Gee stack classifying p-adic representations of the Galois group of a finite extension of Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}_p$$\\end{document}. As an application we prove new cases of the Breuil–Mézard conjecture in dimension two.

Irreducible modules of modular Lie superalgebras and super version of the first Kac–Weisfeiler conjecture

Abstract Suppose $\mathfrak {g}=\mathfrak {g}_{\bar 0}+\mathfrak {g}_{\bar 1}$ is a finite-dimensional restricted Lie superalgebra over an algebraically closed field $\mathbf {k}$ of characteristic $p>2$ . In this article, we propose a conjecture for maximal dimensions of irreducible modules over the universal enveloping algebra $U(\mathfrak {g})$ of $\mathfrak {g}$ , as a super generalization of the celebrated first Kac–Weisfeiler conjecture. It is demonstrated that the conjecture holds for all basic classical Lie superalgebras and all completely solvable restricted Lie superalgebras. In this process, we investigate irreducible representations of solvable Lie superalgebras.

Left–right symmetry of finite finitistic dimension

AbstractWe show that the finitistic dimension conjecture for finite‐dimensional algebras is equivalent to the left–right symmetry of finite finitistic dimension for finite‐dimensional algebras. We also prove the equivalent statement for injective generation.

A short proof of a strong form of the three dimensional Gaussian product inequality

We prove a strong form of the Gaussian product conjecture in dimension three. Our purely analytical proof simplifies previously known proofs based on combinatorial methods or computer-assisted methods, and allows us to solve the case of any triple of even positive integers which remained open so far.

Geometry of static perfect fluid space-time

In this article, we investigate the geometry of static perfect fluid space-time (SPFST) on compact manifolds with boundary. We use the generalized Reilly’s formula to establish a geometric inequality for a SPFST involving the area of the boundary and its volume. Moreover, we obtain new boundary estimates for SPFST. One of the boundary estimates is obtained in terms of the Brown–York mass and another one related to the first eigenvalue of the Jacobi operator. In addition, we provide a new (simply connected) counterexample to the Cosmic no-hair conjecture for arbitrary dimension

Localization for random quasi-one-dimensional models

In this Review Article, we review the results of Anderson localization for different random families of operators that enter the framework of random quasi-one-dimensional models. We first recall what is Anderson localization from both physical and mathematical points of view. From the Anderson–Bernoulli conjecture in dimension 2, we justify the introduction of quasi-one-dimensional models. Then, we present different types of these models: the Schrödinger type in the discrete and continuous cases, the unitary type, the Dirac type, and the point interaction type. We present tools coming from the study of dynamical systems in dimension one: the transfer matrix formalism, the Lyapunov exponents, and the Furstenberg group. We then prove a criterion of localization for quasi-one-dimensional models of Schrödinger type involving only geometric and algebraic properties of the Furstenberg group. Then, we review results of localization, first for Schrödinger-type models and then for unitary type models. Each time, we reduce the question of localization to the study of the Furstenberg group and show how to use more and more refined algebraic criteria to prove the needed properties of this group. All the presented results for quasi-one-dimensional models of Schrödinger type include the case of Bernoulli randomness.

Deformations of corank 1 frontals

We develop a Thom–Mather theory of frontals analogous to Ishikawa's theory of deformations of Legendrian singularities but at the frontal level, avoiding the use of the contact setting. In particular, we define concepts like frontal stability, versality of frontal unfoldings or frontal codimension. We prove several characterizations of stability, including a frontal Mather–Gaffney criterion, and of versality. We then define the method of reduction with which we show how to construct frontal versal unfoldings of plane curves and show how to construct stable unfoldings of corank 1 frontals with isolated instability which are not necessarily versal. We prove a frontal version of Mond's conjecture in dimension 1. Finally, we classify stable frontal multigerms and give a complete classification of corank 1 stable frontals from $\mathbb {C}^3$ to $\mathbb {C}^4$ .

Group and Lie algebra filtrations and homotopy groups of spheres

AbstractWe establish a bridge between homotopy groups of spheres and commutator calculus in groups, and solve in this manner the “dimension problem” by providing a converse to Sjogren's theorem: every abelian group of bounded exponent can be embedded in the dimension quotient of a group. This is proven by embedding for arbitrary the torsion of the homotopy group into a dimension quotient, via a result of Wu. In particular, this invalidates some long‐standing results in the literature, as for every prime , there is some ‐torsion in by a result of Serre. We explain in this manner Rips's famous counterexample to the dimension conjecture in terms of the homotopy group . We finally obtain analogous results in the context of Lie rings: for every prime there exists a Lie ring with ‐torsion in some dimension quotient.

A Zariski dense exceptional set in Manin’s Conjecture: dimension 2

Recently, Lehmann, Sengupta, and Tanimoto proposed a conjectural construction of the exceptional set in Manin’s Conjecture, which we call the geometric exceptional set. We construct a del Pezzo surface of degree 1 whose geometric exceptional set is Zariski dense. In particular, this provides the first counterexample to the original version of Manin’s Conjecture in dimension 2 in characteristic 0. Assuming the finiteness of Tate-Shafarevich groups of elliptic curves over $${\mathbb Q}$$ with j-invariant 0, we show that there are infinitely many such counterexamples.

Topological Bounds on Hyperkähler Manifolds

We conjecture that certain curvature invariants of compact hyperkähler manifolds are positive/negative. We prove the conjecture in complex dimension four, give an “experimental proof” in higher dimensions, and verify it for all known hyperkähler manifolds up to dimension eight. As an application, we show that our conjecture leads to a bound on the second Betti number in all dimensions.

$\mathcal{I}^{\prime}$-curvatures in higher dimensions and the Hirachi conjecture

We construct higher-dimensional analogues of the $\mathcal{I}^{\prime}$-curvature of Case and Gover in all CR dimensions $n \geq 2$. Our $\mathcal{I}^{\prime}$-curvatures all transform by a first-order linear differential operator under a change of contact form and their total integrals are independent of the choice of pseudo-Einstein contact form on a closed CR manifold. We exhibit examples where these total integrals depend on the choice of general contact form, and thereby produce counterexamples to the Hirachi conjecture in all CR dimensions $n \geq 2$.

On Thakur’s basis conjecture for multiple zeta values in positive characteristic

Abstract In this paper, we study multiple zeta values (abbreviated as MZV’s) over function fields in positive characteristic. Our main result is to prove Thakur’s basis conjecture, which plays the analogue of Hoffman’s basis conjecture for real MZV’s. As a consequence, we derive Todd’s dimension conjecture, which is the analogue of Zagier’s dimension conjecture for classical real MZV’s.

Sharpening the Distance Conjecture in diverse dimensions

The Distance Conjecture holds that any infinite-distance limit in the scalar field moduli space of a consistent theory of quantum gravity must be accompanied by a tower of light particles whose masses scale exponentially with proper field distance ‖ϕ‖ as m ~ exp(−λ‖ϕ‖), where λ is order-one in Planck units. While the evidence for this conjecture is formidable, there is at present no consensus on which values of λ are allowed. In this paper, we propose a sharp lower bound for the lightest tower in a given infinite-distance limit in d dimensions: λ ≥ 1/sqrt{d-2} . In support of this proposal, we show that (1) it is exactly preserved under dimensional reduction, (2) it is saturated in many examples of string/M-theory compactifications, including maximal supergravity in d = 4 – 10 dimensions, and (3) it is saturated in many examples of minimal supergravity in d = 4 – 10 dimensions, assuming appropriate versions of the Weak Gravity Conjecture. We argue that towers with λ < 1/sqrt{d-2} discussed previously in the literature are always accompanied by even lighter towers with λ ≥ 1/sqrt{d-2} , thereby satisfying our proposed bound. We discuss connections with and implications for the Emergent String Conjecture, the Scalar Weak Gravity Conjecture, the Repulsive Force Conjecture, large-field inflation, and scalar field potentials in quantum gravity. In particular, we argue that if our proposed bound applies beyond massless moduli spaces to scalar fields with potentials, then accelerated cosmological expansion cannot occur in asymptotic regimes of scalar field space in quantum gravity.