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One of the most fruitful points of contact between mathematics and theoretical computer science is the theory of expander graphs, which are sparse yet well-connected graphs. Over the past decade, this interaction has deepened via the theories of high...
Expanding graphs are ubiquitous throughout mathematics and computer science and have cemented themselves a fundamental objects of interest. This minicourse joint with Sidhanth Mohanty will give a powerful framework from group theory (Kazhdan's property (T)) and operator algebras to construct explicit expanders in a wide variety of settings.
This talk will present Ozawa's remarkable sum of squares certificate for property (T), which subsumes Żuk's criterion presented previously. We then prove Ozawa's theorem showing that property (T) is in fact equivalent to possessing a sum of squares certificate. This gives a semidefinite programming approach towards establishing property (T), yielding computer based proofs of expansion. Time permitting, we will discuss (T) for automophism groups of free groups.
We will explore the pseudorandomness toolkit: the basic objects and techniques used to construct pseudorandom generators and prove various other derandomization results. Topics will include bounded independence, small-bias spaces, randomness extractors, expander graphs, and random restrictions. We will present some of the key constructions and their applications, and see how these ingredients fit together in the main challenges in pseudorandomness. Time permitting, we will also give some ideas being more advanced results, and discuss some open problems.
The first instance of the strong convergence phenomenon was established by Haagerup and Thorbjornsen in the context of operator algebras more than two decades ago. Since then, the strong convergence phenomenon has emerged naturally in connection with a variety of topics across mathematics, including expander graphs, spectral gaps for manifolds, and constrained minimal surfaces.
This mini-course will have the following objectives:
i) Introduce and motivate the notion of strong convergence.
ii) Discuss recent proof techniques.
iii) Showcase concrete applications.
We will explore the pseudorandomness toolkit: the basic objects and techniques used to construct pseudorandom generators and prove various other derandomization results. Topics will include bounded independence, small-bias spaces, randomness extractors, expander graphs, and random restrictions. We will present some of the key constructions and their applications, and see how these ingredients fit together in the main challenges in pseudorandomness. Time permitting, we will also give some ideas being more advanced results, and discuss some open problems.