Discrete Groups, Expanding Graphs and Invariant Measures


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Most of the theory we need we will develop in class. An extreme example is a "Ramanujan graph". Foundational and inspirational articles include:. Attendance: Our class is small and someone's absence can greatly impact the rest of the class.

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Therefore you are expected to attend class or to let me know in advance when you are unable to attend. I will suggest problems over the course of the semester, and you can decide which you want to do. You are encouraged to share your journal with me during the semester so that I can give you feedback.


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You must turn in an electronic copy to me by Dec 8. Shorter lecture due to Grad Student Seminar on constructing Almost Invariant sequences in amenable topological groups. Rate of escape of random walks. Kazhdan's property T.

Isomorphic Graphs - Example 1 (Graph Theory)

Harmonic Dirichlet functions. Free and Wired Uniform Spanning Forests.


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Percolation on infinite and finite groups. Random walks on percolation clusters. Bootstrap percolation. Local approximation of infinite groups.

Discrete Groups, Expanding Graphs and Invariant Measures - torhlefdeotusa.ml

Quasi-isometric rigidity. Here is the LaTeX template for the lecture write-ups , and here is the compiled pdf. Revised on Oct 23! Looking at these two files will reveal a few technical details, like how to write conditional probabilities, or how to refer to lemmas and equations. Please note that the two of you are equally responsible for the entire file you submit to me, so you both should agree with everything, and help each other with both math and LaTeX.

Summary : Probability is one of the fastest developing areas of mathematics today, finding new connections to other branches constantly. One example is the rich interplay between large-scale geometric properties of a space and the behaviour of stochastic processes like random walks and percolation on the space. The obvious best source of discrete metric spaces are the Cayley graphs of finitely generated groups, especially that their large-scale geometric and hence, probabilistic properties reflect the algebraic properties.

A famous example is the construction of expander graphs using group representations, another one is Gromov's theorem on the equivalence between a group being almost nilpotent and the polynomial volume growth of its Cayley graphs. The course will contain a large variety of interrelated topics in this area, with an emphasis on open problems.

A tentative non-exhaustive list of topics : Recurrence, transience, spectral radius of random walks. Free groups, presentations, nilpotent and solvable groups.

Volume growth versus isoperimetric inequalities. Here are further useful links and documents. Expander graphs and their applications by S.

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Hoory, N. Linial and A.


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This is the best current reference on expanders in general, with emphasis in particular on their applications in theoretical computer science. Graph partitioning and expanders by L.

Discrete Groups, Expanding Graphs and Invariant Measures Discrete Groups, Expanding Graphs and Invariant Measures
Discrete Groups, Expanding Graphs and Invariant Measures Discrete Groups, Expanding Graphs and Invariant Measures
Discrete Groups, Expanding Graphs and Invariant Measures Discrete Groups, Expanding Graphs and Invariant Measures
Discrete Groups, Expanding Graphs and Invariant Measures Discrete Groups, Expanding Graphs and Invariant Measures
Discrete Groups, Expanding Graphs and Invariant Measures Discrete Groups, Expanding Graphs and Invariant Measures
Discrete Groups, Expanding Graphs and Invariant Measures Discrete Groups, Expanding Graphs and Invariant Measures
Discrete Groups, Expanding Graphs and Invariant Measures Discrete Groups, Expanding Graphs and Invariant Measures

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