Theoretical Statistics and Mathematics Unit, ISI Delhi

January 17, 2018,
3:30 PM at Seminar Room 2

Speaker:
Anish Mallick,
ICTS-TIFR, Bangalore

Title:
Multiplicity of spectrum for certain class of random operators

Abstract of Talk

Random operators are an important field of study because of their role in
the theory of disordered media. One of the early models that used
randomness is the Anderson tight binding model, which was developed to
study spin wave diffusion in doped semiconductors. To study the random
operator is same as understanding the spectrum of the operator, and part of
the spectral theorem deals with multiplicity of the operator. In case of
Anderson type operator there are many results identifying pure point
spectrum and in some cases singular continuous and absolutely continuous
spectrum, but except for Anderson tight binding model multiplicity of
spectrum is unknown. Here we focus on the multiplicity problem for Anderson
type random operators and provide bound on multiplicity of singular
spectrum using the Green's function associated with each of the
perturbation (disorder is viewed as series of perturbation). In general
these type of result are false for fixed operator and these analysis works
because of disorder. Using the conclusions obtained, simplicity and bound
on multiplicity is also obtained for certain family of random operators.

January 24, 2018,
3:30 PM at Seminar Room 2

Speaker:
Fouzul Atik,
ISI Delhi

Title:
On distance and Laplacian matrices of trees with matrix weights

Abstract of Talk

The \emph{distance matrix} of a simple connected graph $G$ is
$D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the vertices
$i$ and $j$ in $G$. We consider a weighted tree $T$ on $n$ vertices
with edge weights are square matrix of same size. The distance
$d_{ij}$ between the vertices $i$ and $j$ is the sum of the weight
matrices of the edges in the unique path from $i$ to $j$. We
establish a characterization for the trees in terms of rank of
(matrix) weighted Laplacian matrix associated with it. We present a
necessary and sufficient condition for the distance matrix $D$, with
matrix weights, to be invertible and the formula for the inverse of
$D$, if it exists. Then we study some of the properties of the
distance matrices of matrix weighted trees in connection with the
Laplacian matrices, incidence matrices and g-inverses. Finally, we
derive an interlacing inequality of the eigenvalues of distance and
Laplacian matrices for the case of positive definite matrix weights.

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