Reconstruction and Higher Dimensional Geometry

In this paper, we give a new proof on a Theorem of Tutte which says that the determinants of the adjacency matrices of two hypomorphic graphs are the same. We also study the lowest eigenvectors.Comment: 9 pages, to appear in Journal of Combinatorial Theory Series

A Deep Network with Visual Text Composition Behavior

While natural languages are compositional, how state-of-the-art neural models achieve compositionality is still unclear. We propose a deep network, which not only achieves competitive accuracy for text classification, but also exhibits compositional behavior. That is, while creating hierarchical representations of a piece of text, such as a sentence, the lower layers of the network distribute their layer-specific attention weights to individual words. In contrast, the higher layers compose meaningful phrases and clauses, whose lengths increase as the networks get deeper until fully composing the sentence.Comment: accepted to ACL201

On Matrix-Valued Square Integrable Positive Definite Functions

In this paper, we study matrix valued positive definite functions on a unimodular group. We generalize two important results of Godement on square integrable positive definite functions to matrix valued square integrable positive definite functions. We show that a matrix-valued continuous $L^2$ positive definite function can always be written as a convolution of a $L^2$ positive definite function with itself. We also prove that, given two $L^2$ matrix valued positive definite functions $\Phi$ and $\Psi$, $\int_G Trace(\Phi(g) \bar{\Psi(g)}^t) d g \geq 0$. In addition this integral equals zero if and only if $\Phi * \Psi=0$. Our proofs are operator-theoretic and independent of the group.Comment: 11 page

Gan-Gross-Prasad Conjecture for U(p,q)

In this paper, we give a proof of the Gan-Gross-Prasad conjecture for the discrete series of U(p,q). Given a discrete series representation $D(\lambda)$ in terms of the Harish-Chandra parameter, the restriction of $D(\lambda)$ to U(p-1,q) contains $D(\mu)$ as a subrepresentation if and only if $\lambda$ and $\mu$ interlaces in a very special way.Comment: 34 pages, to appear in Inventiones Mathematica

Functions on Symmetric Spaces and Oscillator Representations

We study square integrable functions on the metaplectic group and functions on the space of unitary symmetric matrices. We relate them using the oscillator representations.Comment: 25 pages, to appear in Journal of Functional Analysi