Matthias Aschenbrenner

3-Manifold Groups

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

Ideal membership in polynomial rings over the integers

We will reproduce a proof, using Hermanns classical method, in Section 3 below. Note that the computable character of this bound reduces the question of whether fo G (fi,..., fn) for given fj G F[X] to solving an (enormous) system of linear equations over F. Hence, in this way one obtains a (naive) algorithm for solving the ideal membership problem for F[X] (provided F is given in some explicitly computable manner). Later, Buchberger in his Ph.D. thesis (1965) introduced the important concept of a Gr?bner basis and gave an algorithm for deciding ideal membership for F[X] which is widely used today (see, e.g., [6]). The doubly exponential nature of ? above is essentially unavoidable, as a family of examples due to Mayr and Meyer [27] shows. In fact, they prove that ideal membership for Q[X] is exponential-space hard: the amount of space needed by any algorithm to decide ideal membership for Q[X] (or Z[X]) grows exponentially in the size of the input. If we restrict to/o,...,/n ofa special form, often dramatic

Finite generation of symmetric ideals

Let be a commutative Noetherian ring, and let be the polynomial ring in an infinite collection of indeterminates over . Let be the group of permutations of . The group acts on in a natural way, and this in turn gives the structure of a left module over the group ring . We prove that all ideals of invariant under the action of are finitely generated as -modules. The proof involves introducing a certain well-quasi-ordering on monomials and developing a theory of Grobner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finite-dimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it.

DEFINABLE VERSIONS OF THEOREMS BY KIRSZBRAUN AND HELLY

Kirszbrauns Theorem states that every Lipschitz map S ! R n , where S R m , has an extension to a Lipschitz map R m ! R n with the same Lipschitz constant. Its proof relies on Hellys Theorem: every family of compact subsets of R n , having the property that each of its subfamilies consisting of at most n + 1 sets share a common point, has a non-empty intersection. We prove versions of these theorems valid for denable maps and sets in arbitrary denably complete expansions of ordered elds.

Toward a Model Theory for Transseries

The differential field of transseries extends the field of real Laurent series, and occurs in various context: asymptotic expansions, analytic vector fields, o-minimal structures, to name a few. We give an overview of the algebraic and model-theoretic aspects of this differential field, and report on our efforts to understand its first-order theory.

Vapnik–Chervonenkis Density in Some Theories without the Independence Property, II

We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and

An algorithm for finding symmetric Grobner bases in infinite dimensional rings

P

Erratum for “Finite generation of symmetric ideals”

-minimal theories.

Logarithms of iteration matrices, and proof of a conjecture by Shadrin and Zvonkine

A symmetric ideal I ⊂ R = K[x1,x2,...] is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Grobner bases for symmetric ideals in the infinite dimensional polynomial ring R. This allows for symbolic computation in a new class of rings. In particular, we solve the ideal membership problem for symmetric ideals of R.

Maximal immediate extensions of valued differential fields

Multiplication is given by fσ · gτ = f(σg)(στ) for f, g ∈ R, σ, τ ∈ SX , and extended by linearity. The natural multiplication in R[SX ] does not make R into an R[SX ]-module as claimed in [1], which is why one must use R ∗SX instead. This change affects none of the results in the paper since the multiplicative structure of R[SX ] was never used except to simplify the statement of our main result [1, Theorem 1.1]. The proper statement is as follows.