1. Lattice points in polytopes
A famous theorem of Pick states that if $P$ is a polygon in the plane

with integer vertices, with $I$ interior lattice points, $B$ boundary lattice

points,and area $A$, then $A=/frac 12(2I+B-2)$.How can this result

be extended to higher dimensions?

We will give a survey of this subject. Topics include Ehrhart polynomials of

integerpolytopes, reciprocity, magic squares, zonotopes, graphical degree

sequences,and Brion's theorem.

2. Alternating permutations
A permutation $a_1,a_2,/dots,a_n$ of $1,2,/dots,n$ is called /emph{alternating}

if $a_1>a_2 a_4

The number of alternating

permutations of $1,2,/dots,n$ is denoted $E_n$

and is called an /emph{Euler number}.

The most striking result aboutalternating permutations

is the generating function
$$ /sum_{n/geq 0}E_n/frac{x^n}{n!} = /sec x+/tan x, $$
found by D/'esir/'e Andr/'e in 1879.

We will discuss this result and how

it leadsto the subject of ``combinatorial trigonometry.''

We will then survey some further aspects of alternating

permutations,including some other objects that are counted by $E_n$,

a connection withthe$cd$-index of the symmetric group,

and the use of the representation theory of the symmetric

group tocount certainclasses of alternating permutations.