# Math Colloquium

Speaker: Maxim Burke

(University of Prince Edward Island)

Title: Approximation by entire functions in the construction of

order-isomorphisms and large cross-sections

Abstract: A theorem of Hoischen states that given ε > 0 and a closed discrete set T ⊆ R

t

,

any C

n

function g : R

t → R can be approximated by an entire function f so that for

k = 1, . . . , n, for all x ∈ R

t and for each multi-index α such that |α| ≤ n,

(a) |(Dα

f)(x) − (Dα

g)(x)| < ε(x);

(b) (Dα

f)(x) = (Dα

g)(x) if x ∈ T.

This theorem has been useful in helping to analyze the existence of entire orderisomorphisms

of everywhere non-meager subsets of R, analogous to the Barth-Schneider

theorem, which gives entire order-isomorphisms of countable dense sets, and the existence

for a given everywhere non-meager subsets A of R

t+1 ∼= R

t × R, of entire

functions f so that {x ∈ R

t

: (x, f(x)) ∈ A} is everywhere non-meager. Conversely,

the insights gained from this work have also led to variations on the Hoischen theorem

that incorporate the ability to choose the approximating function so that the graphs

of its derivatives cut a small section through a given null set or a given meager set.

#### Category

Lectures, Seminars

#### Time

Starts:

Ends:

#### Location

Room 319, Chase Building

#### Cost

Free

#### Contact

Jenny Edison (jbenv@dal.ca)