Georgia Institute of Technology College of Sciences

Isospectral reductions is a network/graph reduction that preserves the
eigenvalues and the eigenvectors of the adjacency matrix. We analyze the
conditions under which the generalized eigenvectors would be preserved and
simplify the proof of the preservation of eigenvectors. Isospectral reductions
are associative and form a dynamical system on the set of all matrices/graphs.
We study the spectral equivalence relation defined by specific characteristics
of nodes under isospectral reductions and show some examples of the attractors.
Cooperation among antigens, cross-immunoreactivity (CR) has been observed in
various diseases. The complex viral population dynamics couldn't be explained
by traditional math models. A new math model was constructed recently with
promising numerical simulations. In particular, the numerical results recreated
local immunodeficiency (LI), the phenomenon where some viruses sacrifice
themselves while others are not attacked by the immune system. Here we analyze
small CR networks to find the minimal network with a stable LI. We also
demonstrate that you can build larger CR networks with stable LI using this
minimal network as a building block.

he KAM (Kolmogorov Arnold and Moser) theory studies
the persistence of quasi-periodic solutions under perturbations.
It started from a basic set of theorems and it has grown
into a systematic theory that settles many questions. 

The basic theorem is rather surprising since it involves delicate
regularity properties of the functions considered, rather
subtle number theoretic properties of the frequency as well
as geometric properties of the dynamical systems considered.

In these lectures, we plan to cover a complete proof of
a particularly representative theorem in KAM theory.

In the first lecture we will cover all the prerequisites
(analysis, number theory and geometry). In the second lecture
we will present a complete proof of Moser's twist map theorem
(indeed a generalization to more dimensions).

The proof also lends itself to very efficient numerical algorithms.
If there is interest and energy, we will devote a third lecture
to numerical implementations. 

The KAM (Kolmogorov Arnold and Moser) theory studies
the persistence of quasi-periodic solutions under perturbations.
It started from a basic set of theorems and it has grown
into a systematic theory that settles many questions. 

The basic theorem is rather surprising since it involves delicate
regularity properties of the functions considered, rather
subtle number theoretic properties of the frequency as well
as geometric properties of the dynamical systems considered.

In these lectures, we plan to cover a complete proof of
a particularly representative theorem in KAM theory.

In the first lecture we will cover all the prerequisites
(analysis, number theory and geometry). In the second lecture
we will present a complete proof of Moser's twist map theorem
(indeed a generalization to more dimensions).

The proof also lends itself to very efficient numerical algorithms.
If there is interest and energy, we will devote a third lecture
to numerical implementations. 

One of the classical problems in scissors congruence is
this: given two polytopes in $n$-dimensional Euclidean space, when is
it possible to decompose them into finitely many pieces which are
pairwise congruent via translations?  A complete set of invariants is
provided by the Hadwiger invariants, which measure "how much area is
pointing in each direction."  Proving that these give a complete set
of invariants is relatively straightforward, but determining the
relations between them is much more difficult.  This was done by
Dupont, in a 1982 paper. Unfortunately, this result is difficult to
describe and work with: it uses group homological techniques which
produce a highly opaque formula involving twisted coefficients and
relations in terms of uncountable sums.  In this talk we will discuss
a new perspective on Dupont's proof which, together with more
topological simplicial techniques, simplifies and clarifies the
classical results.  This talk is partially intended to be an
advertisement for simplicial techniques, and will be suitable for
graduate students and others unfamiliar with the approach.