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Math 5010 -- (Topics in Analysis) Analysis on graphs with applications -- Spring 2013

Prerequisites: MATH 5110 Intro to Modern Analysis and MATH 5111 Real Analysis (the latter can be taken concurrently); the usual undergraduate background in linear algebra, differential equations, probability, applied math, PDE, physics may be helpful but is not required.

The course is going to have four components (everything will be flexible and dependent on student's background and interests):

1. there will be a review of analysis on combinatorial graphs: relation between geometry and eigenvalues and eigenvectors of the graph Laplacian with emphasis on simple computational techniques and symmetries; we also will discuss such topics as relation between probability theory and theory of electrician networks, and the notion of energy that connect all of the above;

2. after that we will discuss analysis on metric graphs, which are graphs with edges of different lengths and masses. In this case simple differential equations along edges together with matching conditions at vertices allow to develop an elementary but fruitful theory (recently called "quantum graphs" although one does not have to know quantum physics to understand everything);

3. we will discuss in elementary terms how differential equations may lead to notions of differential geometry on graphs (for instance, directional derivatives are well defined);

4. if we have time, we will try to use this background to understand some simple quantum geometries, such as in this paper

There will be projects offered of various difficulty (from elementary to wide open question), some theoretical and some involving computations. Similar courses were offered before at our department, and resulted in several published papers, such as
M. Begue, D. J. Kelleher, A. Nelson, H. Panzo, R. Pellico and A. Teplyaev, Random walks on barycentric subdivisions and Strichartz hexacarpet, arXiv:1106.5567 Experimental Mathematics, 21(4):402417, 2012

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