Periodicity of bipartite walk on biregular graphs with conditional spectra

In this paper we study a class of discrete quantum walks, known as bipartite walks. These include the well-known Grover's walks. Any discrete quantum walk is given by the powers of a unitary matrix $U$ indexed by arcs or edges of the underlying graph. The walk is periodic if $U^k=I$ for some positive integer $k$. Kubota has given a characterization of periodicity of Grover's walk when the walk is defined on a regular bipartite graph with at most five eigenvalues. We extend Kubota's results--if a biregular graph $G$ has eigenvalues whose squares are algebraic integers with degree at most two, we characterize periodicity of the bipartite walk over $G$ in terms of its spectrum. We apply periodicity results of bipartite walks to get a characterization of periodicity of Grover's walk on regular graphs

Rational Cherednik algebras and Schubert cells

The representation theory of rational Cherednik algebras of type A at t = 0 gives rise, by considering supports, to a natural family of smooth Lagrangian subvarieties of the Calogero-Moser space. The goal of this article is to make precise the relationship between these Lagrangian families and Schubert cells in the adelic Grassmannian. In order to do this we show that the isomorphism, as constructed by Etingof and Ginzburg, from the spectrum of the centre of the rational Cherednik algebra to the Calogero-Moser space is compatible with the factorization property of both of these spaces. As a consequence, the space of homomorphisms between certain representations of the rational Cherednik algebra can be identified with functions on the intersection of some Schubert cells

Fusion Frame Homotopy and Tightening Fusion Frames by Gradient Descent

Finite frames, or spanning sets for finite-dimensional Hilbert spaces, are a ubiquitous tool in signal processing. There has been much recent work on understanding the global structure of collections of finite frames with prescribed properties, such as spaces of unit norm tight frames. We extend some of these results to the more general setting of fusion frames -- a fusion frame is a collection of subspaces of a finite-dimensional Hilbert space with the property that any vector can be recovered from its list of projections. The notion of tightness extends to fusion frames, and we consider the following basic question: is the collection of tight fusion frames with prescribed subspace dimensions path connected? We answer (a generalization of) this question in the affirmative, extending the analogous result for unit norm tight frames proved by Cahill, Mixon and Strawn. We also extend a result of Benedetto and Fickus, who defined a natural functional on the space of unit norm frames (the frame potential), showed that its global minimizers are tight, and showed that it has no spurious local minimizers, meaning that gradient descent can be used to construct unit-norm tight frames. We prove the analogous result for the fusion frame potential of Casazza and Fickus, implying that, when tight fusion frames exist for a given choice of dimensions, they can be constructed via gradient descent. Our proofs use techniques from symplectic geometry and Mumford's geometric invariant theory

Spherical Tropicalization

In this thesis, I extend tropicalization of subvarieties of algebraic tori over a trivially valued algebraically closed field to subvarieties of spherical homogeneous spaces. I show the existence of tropical compactifications in a general setting. Given a tropical compactification of a closed subvariety of a spherical homogeneous space, I show that the support of the colored fan of the ambient spherical variety agrees with the tropicalization of the closed subvariety. I provide examples of tropicalization of subvarieties of GL(n), SL(n), and PGL(n)

Quantization of nilpotent coadjoint orbits

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996.Includes bibliographical references (p. 59-60).by Diko Mihov.Ph.D

IST Austria Thesis

In the first part of the thesis we consider Hermitian random matrices. Firstly, we consider sample covariance matrices XX∗ with X having independent identically distributed (i.i.d.) centred entries. We prove a Central Limit Theorem for differences of linear statistics of XX∗ and its minor after removing the first column of X. Secondly, we consider Wigner-type matrices and prove that the eigenvalue statistics near cusp singularities of the limiting density of states are universal and that they form a Pearcey process. Since the limiting eigenvalue distribution admits only square root (edge) and cubic root (cusp) singularities, this concludes the third and last remaining case of the Wigner-Dyson-Mehta universality conjecture. The main technical ingredients are an optimal local law at the cusp, and the proof of the fast relaxation to equilibrium of the Dyson Brownian motion in the cusp regime. In the second part we consider non-Hermitian matrices X with centred i.i.d. entries. We normalise the entries of X to have variance N −1. It is well known that the empirical eigenvalue density converges to the uniform distribution on the unit disk (circular law). In the first project, we prove universality of the local eigenvalue statistics close to the edge of the spectrum. This is the non-Hermitian analogue of the TracyWidom universality at the Hermitian edge. Technically we analyse the evolution of the spectral distribution of X along the Ornstein-Uhlenbeck flow for very long time (up to t = +∞). In the second project, we consider linear statistics of eigenvalues for macroscopic test functions f in the Sobolev space H2+ϵ and prove their convergence to the projection of the Gaussian Free Field on the unit disk. We prove this result for non-Hermitian matrices with real or complex entries. The main technical ingredients are: (i) local law for products of two resolvents at different spectral parameters, (ii) analysis of correlated Dyson Brownian motions. In the third and final part we discuss the mathematically rigorous application of supersymmetric techniques (SUSY ) to give a lower tail estimate of the lowest singular value of X − z, with z ∈ C. More precisely, we use superbosonisation formula to give an integral representation of the resolvent of (X − z)(X − z)∗ which reduces to two and three contour integrals in the complex and real case, respectively. The rigorous analysis of these integrals is quite challenging since simple saddle point analysis cannot be applied (the main contribution comes from a non-trivial manifold). Our result improves classical smoothing inequalities in the regime |z| ≈ 1; this result is essential to prove edge universality for i.i.d. non-Hermitian matrices