**
Andrew
Granville,
Université
de
Montréal**

"The Frobenius postage stamp problem and boundary turbulence"

The
Frobenius
postage
stamp
problem
asks
what
exact
postage
one
can
make
up
out
of
(arbitrarily
many)
stamps
of
a
given
finite
set
of
values;
but
how
many
stamps
are
really
needed?
A
knight
on
an
infinite
chess
board
can
(eventually)
get
to
any
point
on
the
board,
but
how
many
moves
are
really
needed?
These
are
examples
of
given
finite
sets *A* in **Z**^{d }(that
is,
the
stamp
values,
or
the
feasible
knight
moves)
for
which
we
are
asking
to
understand *NA* ={*a*_{1}+...+*a _{N}*:

*a*

_{1},...,

*a*∈

_{N }*A*}. It turns out that the set

*NA*is remarkably structured though boundary issues restrict what we can hope to prove. This is arguably surprising in light of Khovanskii's 1992 Theorem that there exists a polynomial

*f*(

*t*) such that |

*NA*|=

*f*(

*N*) for

*N*sufficiently large, but we will show that "sufficiently large" here can mean "not small".

This problem appears in different guises in different areas, for example in number theory, combinatorics, logic, algebraic geometry and the theory of dimensions of graded modules. This is joint work, via Zoom, with George Shakan and Aled Walker.

Zoom meeting: https://zoom.us/j/93240216981?pwd=aE0vbktRV1NvMTFzbFVaalVpb1pCdz09