Reflection groups occur all over representation theory and geometry. We want to begin with a quick survey of finite reflection groups, talk a little about classifying them and their connections to some other areas of mathematics. Then we want to focus on two examples of hyperbolic reflection groups; one real and one complex. Both examples involve the Leech lattice; the lattice that produces the best packing of spheres in 24 dimensional Euclidean space. Both examples are (probably) related to the largest sporadic finite simple group known as the monster. The connection in the complex case is still a conjecture. We will not assume any previous familiarity with hyperbolic reflection groups or the Leech lattice or the Monster.

Partial differential equations (PDE) and their boundary value problems (BVP) arise naturally in a number of applications. Typically the system of interest is modelled by a PDE/BVP. In this talk, I will present two unexpected applications of BVPs where the original system does not immediately indicate their importance. The first application comes from the study of a particle moving in a fluid whose motion is modelled by a finite dimensional system. The analysis will imply a natural interpretation to the half derivative in terms of boundary-value problems. The second application comes from the classical study of dispersive shock waves (DSWs). DSWs are specific solutions to nonlinear dispersive equations. However, I will present a BVP for a linear equation which reproduces a number of DSW features. This raises an important question on how to match experimental DSWs with particular nonlinear models: qualitative comparisons do not suffice.

In 1971, Graham and Pollak showed that if $D_T$ is the distance matrix of a tree $T$ on $n$ nodes, then $\det(D_T)$ depends only on $n$, not $T$. This independence from the tree structure has been verified for many different variants of weighted bi-directed trees. In my talk: 1. I will present a general setting which strictly subsumes every known variant, and where we show that $\det(D_T)$ - as well as another graph invariant, the cofactor-sum - depends only on the edge-data, not the tree-structure. 2. More generally - even in the original unweighted setting - we strengthen the state-of-the-art, by computing the minors of $D_T$ where one removes rows and columns indexed by equal-sized sets of pendant nodes. (In fact, we go beyond pendant nodes.) 3. We explain why our result is the "most general possible", in that allowing greater freedom in the parameters leads to dependence on the tree-structure. 4. Our results hold over an arbitrary unital commutative ring. This uses Zariski density, which seems to be new in the field, yet is richly rewarding. We then discuss related results for arbitrary strongly connected graphs, including a third, novel invariant. If time permits, a formula for $D_T^{-1}$ will be presented for trees $T$, whose special case answers an open problem of Bapat-Lal-Pati (Linear Alg. Appl. 2006), and which extends to our general setting a result of Graham-Lovasz (Advances in Math. 1978). (Joint with Apoorva Khare)

We define and give elementary properties of triangulated categories. We also give an application of triangulated categories to linkage theory in commutative algebra.

In this talk we will discuss about the group structure on orbit spaces of unimodular rows over smooth real affine algebras. With a few definition and some results to start, we will prove a structure theorem of elementary orbit spaces of unimodular rows over aforementioned ring with the help of similar kind results on Euler class group. As a consequences, we will prove that : Let $X=Spec(R)$ be a smooth real affine variety of even dimension $d > 1$, whose real points $X(R)$ constitute an orientable manifold. Then the set of isomorphism classes of (oriented) stably free $R$ of rank $d > 1$ is a free abelian group of rank equal to the number of compact connected components of $X(R)$. In contrast, if $d > 2$ is odd, then the set of isomorphism classes of stably free $R$-modules of rank $d$ is a $Z/2Z$-vector space (possibly trivial). We will end this talk by giving a structure theorem of Mennicke symbols.

For a finite field F, a Kakeya set is a subset of F^n that contains a line in every direction. We will discuss a result of Dvir showing a lower bound of C_n*q^n on the size of any Kakeya set over F^n, where C_n only depends on n and F is a finite field of size q.

Vector bundles in geometry generalize projective modules in algebra. They are the simplest sort of coherent sheaves. We will introduce the all important notion of stability of vector bundles. The stable bundles have nice properties, and luckily, most bundles are stable. But even those bundles that are not stable can be analysed in terms of stable bundles. This is done by the notion of the Harder-Narasimhan filtration of a bundle. We will give a sketch of the theory and illustrate it with examples.

Fo a two-colouring of the vertex set of a simple graph G = (V,E), consider the following colour-change rule: a red vertex is converted to blue if it is the only red neighbour of some blue vertex. A vertex set S ⊆ V is called zero-forcing if, starting with the vertices in S blue and the vertices in the complement V \ S red, all the vertices can be converted to blue by repeatedly applying the colour-change rule. The minimum cardinality of a zero-forcing set for the graph G is called the zero-forcing number of G, denoted by Z(G). This concept was introduced by the AIM Minimum Rank –Special Graphs Work Group in [1] as a tool to bound the minimum rank of matrices associated with the graph G. In this talk, I shall give an overview of the zero forcing problem along with some of the results that we have obtained during my Ph.D candidature. To conclude, I shall state few open problems that I intend to tackle along with my mentors. References [1] AIM Minimum Rank –Special Graphs Work Group. Zero forcing sets and the minimum rank of graphs. Linear Algebra and its Applications, 428(7):16281648, 2008.