Research in the Mathematical Sciences最新文献

Dimension interpolation is a novel programme of research which attempts to unify the study of fractal dimension by considering various spectra which live in between well-studied notions of dimension such as Hausdorff, box, Assouad and Fourier dimension. These spectra often reveal novel features not witnessed by the individual notions and this information has applications in many directions. In this survey article, we discuss dimension interpolation broadly and then focus on applications to the dimension theory of orthogonal projections. We focus on three distinct applications coming from three different dimension spectra, namely, the Fourier spectrum, the intermediate dimensions, and the Assouad spectrum. The celebrated Marstrand-Mattila projection theorem gives the Hausdorff dimension of the orthogonal projection of a Borel set in Euclidean space for almost all orthogonal projections. This result has inspired much further research on the dimension theory of projections including the consideration of dimensions other than the Hausdorff dimension, and the study of the exceptional set in the Marstrand-Mattila theorem.

A "tropical ideal" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal ideal to a tropical ideal. We call this the Macaulay tropical ideal. It has a universal property: any other extension of the given principal ideal to a tropical ideal with the expected Hilbert function is a weak image of the Macaulay tropical ideal. For each $n\ge 2$ and $d\ge 1$ , our construction yields a non-realizable degree d hypersurface scheme in $P^n$ . Maclagan-Rincón produced a non-realizable line in $P^n$ for each n, and for $(d,n)=(1,2)$ the two constructions agree. An appendix by Mundinger compares the Macaulay construction with another method for canonically extending ideals to tropical ideals.

These are lecture notes for a mini-course given in Banff in June 2024. They discuss the problem of bounding the number of $\delta$ -incidences $I_{\delta }(P,L):=\{(p,\ell )\in P\times L:p\in {\left[\ell \right]}_{\delta }\}$ under various hypotheses on $P\subset R^2$ and $L\subset A(2,1)$ . The main focus will be on hypotheses relevant for the Furstenberg set problem.

We prove specific biases in the number of occurrences of parts belonging to two different residue classes a and b, modulo a fixed nonnegative integer m, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size n that belong to these sets of partitions and have a symmetric residue class bias (i.e., for $1\le a<m/2$ and $b=m-a$ ), as n tends to infinity.

We examine hypergeometric functions in the finite field, p-adic and classical settings. In each setting, we prove a formula which splits the hypergeometric function into a sum of lower order functions whose arguments differ by roots of unity. We provide multiple applications of these results, including new reduction and summation formulas for finite field hypergeometric functions, along with classical analogues; evaluations of special values of these functions which apply in both the finite field and p-adic settings; and new relations to Fourier coefficients of modular forms.

Several methods are used to evaluate finite trigonometric sums. In most cases, either the sum had not previously been evaluated, or it had been evaluated, but only by analytic means, e.g., by complex analysis or modular transformation formulas. We establish both reciprocity and three sum relations for trigonometric sums. Motivated by certain sums that we have evaluated, we add coprime conditions to the summands and thereby define analogues of Ramanujan sums, which we in turn evaluate. One of these analogues leads to a criterion for the Riemann Hypothesis, analogous to the Franel–Landau criterion.

Toward building tropical analogues of adic spaces, we study certain spaces of prime congruences as a topological semiring replacement for the space of continuous valuations on a topological ring. This requires building the theory of topological idempotent semirings, and we consider semirings of convergent power series as a primary example. We consider the semiring of convergent power series as a topological space by defining a metric on it. We check that, in tropical toric cases, the proposed objects carry meaningful geometric information. In particular, we show that the dimension behaves as expected. We give an explicit characterization of the points in terms of classical polyhedral geometry in a follow-up paper.

Matrix-valued holomorphic quantum modular forms are intricate objects associated to 3-manifolds (in particular to knot complements) that arise in successive refinements of the volume conjecture of knots and involve three holomorphic, asymptotic and arithmetic realizations. It is expected that the algebraic properties of these objects can be deduced from the algebraic properties of descendant state integrals, and we illustrate this for the case of the ((-2,3,7))-pretzel knot.

A promising approach to investigating high-dimensional problems is to identify their intrinsically low-dimensional features, which can be achieved through recently developed techniques for effective low-dimensional representation of functions such as machine learning. Based on available finite-dimensional approximate solution manifolds, this paper proposes a novel model reduction framework for kinetic equations. The method employs projections onto tangent bundles of approximate manifolds, naturally resulting in first-order hyperbolic systems. Under certain conditions on the approximate manifolds, the reduced models preserve several crucial properties, including hyperbolicity, conservation laws, entropy dissipation, finite propagation speed, and linear stability. For the first time, this paper rigorously discusses the relation between the H-theorem of kinetic equations and the linear stability conditions of reduced systems, determining the choice of Riemannian metrics involved in the model reduction. The framework is widely applicable for the model reduction of many models in kinetic theory.