In the so called light bulb process of Rao, Rao and Zhang (2007), on days r =
1, . . . , n, out of n light bulbs, all initially off, exactly r bulbs, selected uniformly and
independent of the past, have their status changed from off to on or vice versa. With
X the number of bulbs on at the terminal time n, an even integer and = n/2, σ2 =
varX, we have
sup
∈R 􏰐
􏰐
P ( X −
σ ≤ z ) − P (Z ≤ z )
􏰐􏰐 ≤
n
2σ2 ∆0 + 1.64
n
σ3 +
2
σ
where Z is a
N (0, 1) random variable and
∆0
≤
1
2√n +
1
2n + e−
n/2
, for n
≥ 4,
yielding a bound of order O(n−1/2 ) as n
→ ∞.
The results are shown using a version of Steins method for bounded, monotone
size bias couplings. The argument for even n depends on the construction of a variable
X s on the same space as X which has the X size bias distribution, that is, which
satisfies
E[X g(X )] = E[g(X s )], for all bounded continuous g
and for which there exists a B
≥ 0, in this case, B = 2, such that X ≤ X
s
≤ X + B
almost surely. The argument for odd n is similar to that for n even, but one first
couples X closely to V , a symmetrized version of X, for which a size bias coupling of
V to V s can proceed as in the even case.

I will talk about the rigidity for a local holomorphic isometric embedding
from ${\BB}^n$ into ${\BB}^{N_1} \times\cdots \times{\BB}^{N_m}$ with
respect to the normalized Bergman metrics. Each component of the map is a
multi-valued holomorphic map between complex Euclidean spaces by Mok's
algebraic extension theorem. By using the method of the holomorphic
continuation and analyzing real analytic subvarieties carefully, we show
that a component is either a constant map or a proper holomorphic map
between balls. Hence the total geodesy of non-constant components follows
from a linearity criterion of Huang. In fact, the rigidity is derived in a
more general setting for a local holomorphic conformal embedding. This is
a joint work with Y. Zhang.

Fourier coefficients of automorphic forms are the building blocks for automorphic L-functions. While these coefficients are often quite mysterious, there is one family of automorphic forms whose Fourier coefficients do have an explicit and rather uniform description -- Eisenstein series. In fact, Langlands' initial study of Eisenstein series' coefficients in the 1960's led him to make conjectures about equalities of L-functions which inform much of modern number theory. I'll discuss two new explicit descriptions for Fourier coefficients of Eisenstein series which hold in great generality and hint at undiscovered connections among automorphic forms, representation theory, and physics. One description makes use of Kashiwara crystal graphs and the other uses the 6-vertex model in statistical mechanics. Both objects possess beautiful combinatorial structure that deserves to be more widely known, though we do not assume familiarity with either and all concepts mentioned above will be defined over the course of the talk.

We examine methods to produce triples of integers which are the sides of a right triangle (i.e., (3,4,5), (5,12,13), or (8,15,17)). Multiplication of complex numbers will make an appearance, the first example of the interplay between algebra and geometry. We will then learn about elliptic curves, which provide a similar, but much more intricate, synthesis of algebra and geometry. Elliptic curves are a current area of intense research in mathematics and computer science, playing a central role in modern cryptology and in the recent proof of Fermat's Last Theorem.