Based on the review paper in the area of stochastic network: Stochastic Processing Networks by Ruth J. Williams, I will recall some cornerstones in queueing networks. In addition, I will introduce some recent progress and open problems in this area if time allows.

We develop a simple semi-algebraic 2-level solver built
on traditional multigrid ideas. It is designed to be easily
incorporated into existing simulation software. It exhibits good
convergence for many classes of challenging problems including
discontinuous diffusion, convection-diffusion, and Helmholtz
equations. It has built-in structure that makes it simple to
generalize to a hierarchical basis multigrid solver.

Every area-minimizing hypercone having only an isolated singularity fits into a foliation by smooth, area-minimizing hypersurfaces asymptotic to the cone itself. In this talk I will present the following epsilon-regularity result: every minimal surfaces lying sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), is a perturbation of either the cone itself, or some leaf of its associated foliation. This result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation, and it also allows to study convergence to singular minimal hyper surfaces. This is a joint result with N. Edelen.

Let $\beta\equiv \beta^{(m)} = \{\beta_{i}\}_{i\in \mathbb{Z}_{+}^{n},
|i|\le m}$, $\beta_{0}>0$, denote a real $n$-dimensional multisequence of degree $m$.
The \textit{Truncated Moment Problem} for $\beta$ (TMP) concerns the existence
of a positive Borel measure $\mu$, supported in $\mathbb{R}^{n}$, such that
$
\beta_{i} = \int_{\mathbb{R}^{n}} x^{i}d\mu ~~~~~~~~( i\in \mathbb{Z}_{+}^{n},~~|i|\le m).
$
(Here, for $x\equiv (x_{1},\ldots,x_{n})\in \mathbb{R}^{n}$
and $i\equiv (i_{1},\ldots,i_{n})\in \mathbb{Z}_{+}^{n}$,
we set
$|i| = i_{1}+\cdots + i_{n}$ and
$x^{i} = x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}$.)
A measure $\mu$ as above is a {\it{representing measure}} for $\beta$.
We discuss three equivalent ``solutions" to TMP, based on: 1) flat extensions
of moment matrices, 2) positive extensions of Riesz functionals, and 3) the
\textit{core variety} of a multisequence. In work with G. Blekherman
[J. Operator Theory, to appear]
We proved that $\beta$ has a representing measure if and only if the core variety
is nonempty, in which case the core variety is the union of supports of all
finitely atomic representing measures. We discuss open questions concerning
difficulties in applying of any of the above solutions to TMP in special cases or in
numerical examples.

We consider general self-adjoint polynomials and rational expressions in several independent random matrices whose entries are centered and have constant variance. Under some numerically checkable conditions, we establish for these models the optimal local law, i.e., we show that the empirical spectral distribution on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We show that the above results can be applied to prove the optimal bulk local law for two concrete families of polynomials: general quadratic forms in Wigner matrices and symmetrized products of independent matrices with i.i.d. entries. Moreover, in the framework of the developed theory for rational expressions in random matrices, we study the density of transmission eigenvalues in the random matrix model for transport in quantum dots coupled to a chaotic environment.
This is a joint work with Laszlo Erd$\ddot{\text{o}}$s and Torben Kr$\ddot{\text{u}}$ger.

Faltings's isogeny theorem states that two abelian varieties
over a number field are isogenous precisely when the characteristic
polynomials associated to the reductions of the abelian varieties at all
prime ideals are equal. This implies that two abelian varieties defined
over the rational numbers with the same L-function are necessarily
isogenous, but this is false over a general number field.

In order to still use the L-function to determine the underlying field,
we extract more information from the L-function by "twisting": a twist
of an L-function is the L-function of the tensor of the underlying
representation with a character. We discuss a theorem stating that
abelian varieties over a general number field are characterized by their
L-functions twisted by Dirichlet characters of the underlying number field.

Calculus in normed vector spaces is the basis for several areas of
mathematics and physics, but it is not a topic that is often covered with
very much care or detail. This talk's focus is the theory of
differentiation in normed vector spaces, more specifically the Gateaux and
Fr\'{e}chet derivatives. Towards the end, we shall cover an
infinite-dimensional Taylor's Theorem, and we shall likely get to discuss
some applications. Plus, there will be plenty of examples throughout! This
talk is Part I of a (likely) three- or four-part series, with future
topics including integration and complex analysis.