# Alexandru Nica

I am a Professor in the Pure Mathematics Department of the University of Waterloo. I did my undergraduate studies in the Mathematics Faculty of the University of Bucharest, Romania, and the PhD Degree in the Mathematics Department of the University of California at Berkeley, under the supervision of Dan-Virgil Voiculescu. My research interests are in noncommutative probability and operator algebras, particularly their connections to combinatorics and classical probability. My current line of research is in an area called ``free probability.''

 Here is a picture of me (while teaching a course of introduction to free probability, in Winter Term 2011).

Here is my list of Publications. My recent papers are posted on the arXiv database.

Note: the Pure Math deadline for applications is January 7, 2020. For details of the application process, check this Pure Math Undergraduate link.

During a USRA Term under my supervision, you are expected to read and make presentations of expository and/or research papers on a given topic, and to write an essay demonstrating the insights which you gained while studying that topic.
Some examples of such essays, written by USRA students in preceding years:
Fan Huang, Spring Term 2014 (supervised jointly with Bruce Richmond).
Alex Gatea and Simon Huang, Spring Term 2017 (supervised jointly with Ian Goulden).

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Brief descriptions for a couple of possible USRA topics in Spring Term 2020.

1. Algebras defined by diagrammatic structures.

For this topic, the main object of study are the Temperley-Lieb algebras. A good way of getting introduced to them is via a "learning seminar" based on expository papers about these algebras which were previously written by undergraduate students, e.g. [1] or [2] below. Further reading could include Chapter 2 of the monograph [3], which would take you towards inclusions of finite dimensional multi-matrix algebras (or of finite dimensional C*-algebras, if you should be familiar with that notion).

[1] Anne Moore. Representations of the Temperley-Lieb algebra, Honors Paper at Macalester College, 2008.

[2] Jim de Groot. An introduction to the representation theory of Temperley-Lieb algebras, Bachelor Thesis at the University of Amsterdam, 2015.

[3] F.M. Goodman, P. de la Harpe and V.F.R. Jones. Coxeter graphs and towers of algebras, Springer Verlag, 1989.

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2. Random permutations and random Young diagrams.

The main reading for this topic will be Chapter 1 of the monograph [4] below. Very beautiful stuff -- but a lot to learn!
The limiting shape of a random Young diagram, in various regimes, is a very interesting topic of research, which is considered in particular in the research literature on "free probability". (But what papers might be appropriate for you to read would be determined during the USRA term.)

[4] Dan Romik. The surprising mathematics of longest increasing subsequences, Cambridge University Press, 2015.