\indent Since 2005 a class of algebras, the Leavitt path algebras $L_K(E)$ (for $K$ any field and $E$ any directed graph), has been a focus of investigation by both algebraists and $C*-$analysts. In this talk I'll define these algebras, and give some insight regarding the ideas which prompted the initial description of these structures.

\indent I'll briefly describe some results of the expected form, namely, results of the form: $E$ has property $P$ if and only if $L_K(E)$ has property $P'$.

\indent However, the main goal of the talk will be to show how Leavitt path algebras have been used to answer various questions outside the subject per se. For example, results about von Neumann regular rings; about prime or primitive algebras; about $C*-$algebras; and about Lie algebras have been gleaned from these structures.

\indent A discretization scheme for space-time domains and two second order solvers for convection diffusion reaction partial differential equations.

\indent Categorical quantum mechanics seeks to distill quantum mechanics to minimal assumptions, based on categories with tensor products. We address the question of how to usefully represent observables in this setting. Orthonormal bases in the category of finite-dimensional Hilbert spaces turn out to correspond to Frobenius algebras. We show that for arbitrary dimensions one needs H*-algebras instead, which can be defined in any monoidal category. Finally we compare the notion of H*-algebra to that of nonunital Frobenius algebra in various categories

Non-uniform circuit lower bounds are among the strongest
impossibility results attainable in complexity theory, but they are
also among the most difficult to prove. The circuit class ACC consists
of circuit families with constant depth over unbounded fan-in AND, OR,
NOT, and MODm gates, where $m > 1$ is an arbitrary constant. Despite the
apparent simplicity of such circuits, the power of MODm has been very
hard to reason about. For instance, it was not known whether a
complexity class as large as $EXP^{NP}$ (the class of languages
recognized in $2^{O(n^k)}$ time with an NP oracle) could be simulated
with depth-$3$ polynomial size circuits made out of only MOD6 gates.

We prove:
- There are functions computable in Nondeterministic Exponential Time
that cannot be simulated with non-uniform ACC circuits of polynomial
size. The size lower bound can be slightly strengthened to
quasi-polynomials and other less natural functions.
- There are functions in $EXP^{NP}$ that cannot be simulated with
non-uniform ACC circuits of $2^{n^{o(1)}}$ size. The lower bound gives
an exponential size-depth tradeoff: for every $d$ and $m$ there is a $b > 0$
such that the relevant function doesn't have depth-$d$ ACC circuits of
size $2^{n^b}$ with MODm gates.

The proofs are more interesting than the results. The high-level
strategy is to design faster algorithms for the circuit satisfiability
problem over ACC circuits, then show how such algorithms can be
applied to yield lower bound proofs against ACC circuits, via a more
general "algorithm-lower bound" connection. This connection provides a
new direction for further progress in circuit complexity.

Given any complex power series in n-variables one can always
ask what is the largest negative power which is integrable. This
number is the log canonical threshold (its reciprocal is called the
Arnold multiplicity) of the underlying hypersurface. It is a measure
of the complexity of the singularity at the origin, which carries more
information than the multiplicity.

I describe some recent work with Hacon and Xu, where we prove some
conjectures of Kollár and Shokurov, which state that the set of log
canonical thresholds satisfies the ascending chain condition and which
identifies the accumulation points.

Given a curve (Riemann surface), one can construct an abelian variety: its Jacobian. Abelian varieties are quotients of vector spaces by lattices. The classical Torelli theorem states that the Jacobian determines the curve. We discuss some generalizations of this and their history.

\indent Given a fixed graph $G$ on $n$ vertices, we can create a random coloring of $G$ in the following way: randomly pick an edge, then randomly pick a color, and then color both endpoints of that edge that color. We can continue this process on a graph that is already colored by simply overwriting any vertices that have already been assigned a color. This gives rise to a random walk on the $2^n$ colorings of $G$, and it is this random walk that we will investigate. The eigenvalues of the transition matrix are known and have a simple form. We discuss these and other quantities as well as several related problems.

\indent We give a survey of what is known about how many symmetries an algebraic variety can possess. We start with some classical results, including those of Hurwitz, Noether and Riemann, to do with the automorphism group of the plane and the automorphism group of curves (or equivalently Riemann surfaces), and we end with some more recent results to do with the automorphism group of threefolds of low degree and varieties with finite automorphism group.

Perhaps the most studied varieties in algebraic geometry are
the moduli spaces of smooth and stable curves. A stable curve is a
nodal curve whose canonical divisor is ample. Adding stable curves
gives a geometrically meaning compactification to the space of smooth
curves.

A motivating problem in higher dimensional geometry is to construct
the moduli space of varieties of general type, in any dimension. Just
as with curves, we need to consider non-normal varieties, possibly
with more than one component. Unlike the case of curves, if we fix
the degree, it is not at all clear how to bound the number of
components.