# Thesis defense: Two-parameter non-commutative Gaussian processes

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Natasha Blitvic

## Event Location:

Allen Room, 36-462

## Event Date/Time:

Monday, July 2, 2012 - 10:00am

## Research Area:

The continued explosion of experimental data leads to the seeming conundrum of seeking to derive meaningful features from prohibitively large structures. Yet modern mathematical theories are providing analysis tools specifically adapted to such large-dimensional settings. "Non-commutative probability", which broadly refers to frameworks in which probabilistic intuition is ascribed to operators on Hilbert spaces, is one such area of interdisciplinary interest. In particular, it describes in a tractable and intuitive manner the behavior of large random objects, such as certain classes of large random matrices or random graphs. This thesis develops a new non-commutative probabilistic framework that is both a natural generalization of several existing frameworks (viz. free probability, q-deformed probability) and a richer setting in which to describe scaling limits of large random objects. Specifically, we introduce (1) a generalized Non-commutative Central Limit Theorem giving rise to a two-parameter deformation of the classical Gaussian statistics and (2) a two-parameter continuum of non-commutative probability theories in which to realize these statistics. The probabilistic framework that emerges serves as a setting in which to realize a broader class of random matrix limits (explicit constructions provided), such as a correlated version" of Wigner's semicircle law. Thesis Supervisor: Todd Kemp