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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)