This thesis consists of two parts. In the first part of the thesis, we prove an index theorem for Dirac operators of conic singularities with codimension 2. One immediate corollary is the generalized Rohklin congruence ...

In the first part we study Double Affine Hecke algebra of type An-1 which is important tool in the theory of orthogonal polynomials. We prove that the spherical subalgebra eH(t, 1)e of the Double Affine Hecke algebra H(t, ...

We examine the dynamics of the mitogen-activated protein kinase (MAPK) multi-step enzymatic switching cascade, a highly conserved architecture utilised in cellular signal transduction. In treating the equations of motion, ...

It is well known that if a finite graded lattice of rank n is supersolvable, then it has an EL-labelling where the labels along any maximal chain form a permutation of [1, 2,..., n]. We call such a labelling an Sn EL-labelling ...

Random matrix theory is a maturing discipline with decades of research in multiple fields now beginning to converge. Experience has shown that many exact formulas are available for certain matrices with real, complex, or ...

This thesis consists of three projects on the enumerative and algebraic properties of matroids and hyperplane arrangements. In particular, a central object of study is the Tutte polynomial, which stores much of the enumerative ...