We show that, for a constant-degree algebraic curve γ in R, every set of n points on γ spans at least Ω(n) distinct distances, unless γ is an algebraic helix (see Definition 1.1). This improves the earlier bound Ω(n) of Charalambides [2]. We also show that, for every set P of n points that lie on a d-dimensional constantdegree algebraic variety V in R, there exists a subset S ⊂ P of size at lea...

By Northcott’s Theorem there are only finitely many algebraic points in affine n-space of fixed degree e over a given number field and of height at most X. Finding the asymptotics for these cardinalities as X becomes large is a long standing problem which is solved only for e = 1 by Schanuel, for n = 1 by Masser and Vaaler, and for n “large enough” by Schmidt, Gao, and the author. In this paper...

We prove that the function field of an algebraic variety of dimension ≥ 2 over an algebraically closed field is completely determined by its first and second Milnor K-groups.

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston’s Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincaré Conjecture. The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture,...

where the implied constant is absolute. This is proved in [1, §13, 14] (see also [5, Th. 11.42] for an outline of the proof). In this note, we sketch a different proof, based on the general philosophy of reduction to one-variable sums and on the use of the powerful form of Deligne’s proof of the Riemann Hypothesis over finite fields involving sums of trace functions of general sheaves [3] (see ...

Let G be an adjoint simple algebraic group over an algebraically closed field of characteristic p; let be the root system of G, and take t ∈ N. Lawther has proven that the dimension of the set G[t] = {g ∈G : g = 1} depends only on and t . In particular the value is independent of the characteristic p; this was observed for t small and prime by Liebeck. Since G[t] is clearly a disjoint union of ...

In this paper we give an application of equivariant moduli spaces to the study of smooth group actions on certain 4-manifolds. A rich source of examples for such actions is the collection of algebraic surfaces (compact and nonsingular) together with their groups of algebraic automorphisms. From this collection, further examples of smooth but generally nonalgebraic actions can be constructed by ...

We compare the domain of the assembly map in algebraic K – theory with respect to the family of finite subgroups with the domain of the assembly map with respect to the family of virtually cyclic subgroups and prove that the former is a direct summand of the later. AMS Classification 19D50; 19A31, 19B28

We produce two separate algebraic descriptions of the isomorphism classes of the solvable subgroups of the group PLo(I) of piecewise-linear orientation-preserving homeomorphisms of the unit interval under the operation of composition, and also of the generalized R. Thompson groups Fn. The first description is as a set of isomorphism classes of groups which is closed under three algebraic operat...

In this note we give a different and direct short proof to a previous result of Nastasescu and Torrecillas in [NT] stating that if the rational part of any right C∗ module M is a direct sumand in M then C must be finite dimensional (the splitting problem for coalgebras).