Márton Hablicsek

On the Number of Rich Lines in High Dimensional Real Vector Spaces

In this short note we use the Polynomial Ham Sandwich Theorem to strengthen a recent result of Dvir and Gopi about the number of rich lines in high dimensional Euclidean spaces. Our result shows that if there are sufficiently many rich lines incident to a set of points then a large fraction of them must be contained in a hyperplane.

Kakeya sets over non-Archimedean local rings

In a recent paper of Ellenberg, Oberlin, and Tao, the authors asked whether there are Besicovitch phenomena in F_q[[t]]^n. In this paper, we answer their question in the affirmative by explicitly constructing a Kakeya set in F_q[[t]]^n of measure 0. Furthermore, we prove that any Kakeya set in F_q[[t]]^2 or Z_p^2 is of Minkowski dimension 2.

Explicit computations of Hida families via overconvergent modular symbols

In Pollack and Stevens (Ann Sci Éc Norm Supér 44(1):1–42, 2011), efficient algorithms are given to compute with overconvergent modular symbols. These algorithms then allow for the fast computation of p-adic L-functions and have further been applied to compute rational points on elliptic curves (e.g. Darmon and Pollack in Israel J Math 153:319–354, 2006, Trifkovic in Duke Math J 135(3):415–453, 2006). In this paper, we generalize these algorithms to the case of families of overconvergent modular symbols. As a consequence, we can compute p-adic families of Hecke-eigenvalues, two-variable p-adic L-functions, L-invariants, as well as the shape and structure of ordinary Hida–Hecke algebras.

POWER MAP PERMUTATIONS AND SYMMETRIC DIFFERENCES IN FINITE GROUPS

Let G be a finite group. For all a ∈ ℤ, such that (a, |G|) = 1, the function ρa: G → G sending g to ga defines a permutation of the elements of G. Motivated by a recent generalization of Zolotarevs proof of classic quadratic reciprocity, due to Duke and Hopkins, we study the signature of the permutation ρa. By introducing the group of conjugacy equivariant maps and the symmetric difference method on groups, we exhibit an integer dG such that

Algebraic 3D graphic statics: Reciprocal constructions

{\rm sgn}(\rho_a) = \left(\frac{d_G}{a}\right)

Algebraic geometry over the residue field of the infinite place

for all G in a large class of groups, containing all finite nilpotent and odd order groups.

AN INCIDENCE CONJECTURE OF BOURGAIN OVER FIELDS OF POSITIVE CHARACTERISTIC

Abstract The recently developed 3D graphic statics (3DGS) lacks a rigorous mathematical definition relating the geometrical and topological properties of the reciprocal polyhedral diagrams as well as a precise method for the geometric construction of these diagrams. This paper provides a fundamental algebraic formulation for 3DGS by developing equilibrium equations around the edges of the primal diagram and satisfying the equations by the closeness of the polygons constructed by the edges of the corresponding faces in the dual/reciprocal diagram. The research provides multiple numerical methods for solving the equilibrium equations and explains the advantage of using each technique. The approach of this paper can be used for compression-and-tension combined form-finding and analysis as it allows constructing both the form and force diagrams based on the interpretation of the input diagram. Besides, the paper expands on the geometric/static degrees of (in)determinacies of the diagrams using the algebraic formulation and shows how these properties can be used for the constrained manipulation of the polyhedrons in an interactive environment without breaking the reciprocity between the two.

Formality of derived intersections and the orbifold HKR isomorphism

ABSTRACT Nikolai Durov introduced the theory of generalized rings and schemes to study Arakelov geometry in an alternative algebraic framework, and introduced the residue field at the infinite place, 𝔽∞. We show an elementary algebraic approach to modules and algebras over this object, define prime congruences, show that the polynomial ring of n variables is of Krull dimension n, and derive a prime decomposition theorem for these primes.