A brief overview of consonance by way of continued fractions and modular arithmetic.
This talk gives a survey on a series of work which I and co-authors have been doing for 10 years. I will start from the Feynman-Kac type formula for Dirichlet forms. Then a necessary and sufficient condition is given to characterize the killing transform of Markov processes. Lastly we shall discuss the regular subspaces of linear transform and answer some problems related to the Feynman-Kac formula
Counting paths, or walks, is an important ingredient in the classical representation theory of compact groups. Using Brownian paths gives a new flexible and intuitive approach, which allows to extend some of this theory to the non- cristallographic case. This is joint work with P. Biane and N. O'Connell
For the integers $a$ and $b$ let $P(a^b)$ be all partitions of the
set $N= {1,..., ab}$ into parts of size $a.$ Further, let
$\mathbb{C}P (a^b)$ be the corresponding permutation module for the
symmetric group acting on $N.$ A conjecture of Foulkes says
that $\mathbb{C}P (a^b)$ is isomorphic to a submodule of $\mathbb{C}P
(b^a)$ for all $a$ not larger than $b.$ The conjecture goes back to
the 1950's but has remained open. Nevertheless, for some values of
$b$ there has been progress. I will discuss some proofs and further
conjectures. There is a close correspondence between the
representations of the symmetric groups and those of the general
linear groups, via Schur-Weyl duality. Foulkes' conjecture therefore
has implications for $GL$-representations. There are interesting
connections to classical invariant theory which I hope to mention.
The Nielsen realisation problem asks when a collection of diffeomorphisms, which form a group up to isotopy, is isotopic to a collection of diffeomorphisms which form a group on the nose. For surfaces this problem is well-studied, I'll talk about this problem in the context of K3 surfaces.
In these (three) lectures, I will discuss the following topics:
1. The theorems of Ax on the elementary theory of finite and pseudo-finite fields, including decidability and quantifier-elimination, variants due to Kiefe, and connection to Diophantine problems.
2. The theorems on Chatzidakis-van den Dries-Macintyre on definable sets over finite and pseudo-finite fields, including their estimate for the number of points of definable set over a finite field which generalizes the Lang-Weil estimates for the case of a variety.
3. Motivic and p-adic aspects.
A Kuranishi space is a topological space equipped with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on many moduli spaces in differential geometry, and in particular, in moduli spaces of stable $J$-holomorphic curves in symplectic geometry.
Let $Y$ be an orbifold, and $R$ a commutative ring. We shall define four topological invariants of $Y$: two kinds of Kuranishi bordism ring $KB_*(Y;R)$, and two kinds of Kuranishi homology ring $KH_*(Y;R)$. Roughly speaking, they are spanned over $R$ by isomorphism classes $[X,f]$ with various choices of relations, where $X$ is a compact oriented Kuranishi space, which is without boundary for bordism and with boundary and corners for homology, and $f:X\rightarrow Y$ is a strong submersion. Our main result is that weak Kuranishi homology is isomorphic to the singular homology of $Y$.
These theories are powerful tools in symplectic geometry for several reasons. Firstly, using them eliminates the issues of virtual cycles and perturbation of moduli spaces, yielding technical simplifications. Secondly, as $KB_*,KH_*(Y;R)$ are very large, invariants defined in these groups contain more information than invariants in conventional homology. Thirdly, we can define Gromov-Witten type invariants in Kuranishi bordism or homology groups over $\mathbb Z$, not just $\mathbb Q$, so they can be used to study the integrality properties of Gromov-Witten invariants.
This is the first of two talks. Today we deal with motivation from symplectic geometry, and Kuranishi bordism. Next week's talk discusses Kuranishi homology.
Packings and coverings in graphs are related to two main problems of
graph theory, respectively error correcting codes and domination.
Given a set of words, an error correcting code is a subset such that
any two words in the subset are rather far apart, and can be
identified even if some errors occured during transmission. Error
correcting codes have been well studied already, and a famous example
of perfect error correcting codes are Hamming codes.
Domination is also a very old problem, initiated by some Chess problem
in the 1860's, yet Berge proposed the corresponding problem on graphs
only in the 1960's. In a graph, a subset of vertices dominates all the
graph if every vertex of the graph is neighbour of a vertex of the
subset. The domination number of a graph is the minimum number of
vertices in a dominating set. Many variants of domination have been
proposed since, leading to a very large literature.
During this talk, we will see how these two problems are related and
get into few results on these topics.