# Seminars at SMU Delhi

## Forthcoming Seminar(s)

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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