Faculty of Mathematics, University of Waterloo

WebNotice Postings: 29-Jun-2015 through 05-Jul-2015

Postings for all Departments

Monday, 29 June 2015, 1:30PM -- MC 5479
Graph Algebras Seminar -- Pure Mathematics
Speaker: Ehsaan Hossain, Department of Pure Mathematics University of Waterloo
Title: "“ELI5: Leavitt Path Algebras”"
Abstract: If E is a directed graph, you can construct a ring L(E) generated by the vertices and edges of E subject to some relations. This is called a Leavitt path algebra. These rings were introduced around 2004, generalizing an earlier construction of Leavitt from the 80’s. L(E) has a very rich algebraic structure, being a locally unital, Z-graded, semiprimitive ring. Changing the graph E results in changing the algebraic structure of L(E), and as of today this is a predictable process in many cases. For example, there are necessary-and-sufficient geometric conditions on E so that L(E) is finite-dimensional, simple, prime, and so on. There are also some “moves” you can perform on E which result in Morita equivalent Leavitt path algebras. But the converse is currently unknown — if two Leavitt path algebras are Morita equivalent, how can you relate their underlying graphs? In this talk I hope to give an overview of the theory of Leavitt path algebras, including known results and some open questions. As the title suggests, you should bring your five year-old.

Monday, 29 June 2015, 4:15PM -- MC 6486
Seminar -- Combinatorics and Optimization
Speaker: Cameron Marcott, University of Waterloo
Title: "Combinatorial geometries, convex polyhedra, and Schubert cells"
Abstract: This is a reminder for our next meeting that will take place on Monday (June 29th) in MC6486.

Door will open at 4:15pm. As usual we will have 15 mins for socializing and for eating pizza before the talk starts (at 4:30pm sharp).

This paper illustrates the connections between matroid theory, algebraic geometry, and the theory of convex polytopes. A realization of a matroid of rank k on n elements is a point in the Grassmannian of k dimensional planes in n dimensional space. We obtain a stratification of the Grassmannian by looking at the sets of points corresponding to different realizable matroids. This stratification agrees with two other stratifications of the Grassmannian: one coming from a torus action and the other coming from permuted Schubert cells. To each cell, we may associate certain convex polytopes coming from the torus action and from the matroid defining the cell.

The presentation will be based on the following manuscript: http://www.math.ias.edu/~goresky/pdf/combinatorial.jour.pdf

For more information about our reading group, please visit our webpage http://www.math.uwaterloo.ca/~k2georgi/reading.htm

Tuesday, 30 June 2015, 3:30PM *** CANCELLED ***
Test Events Master's Essay Presentation -- Computer Science
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Friday, 3 July 2015, 3:30PM -- MC 5501
Tutte Colloquium Seminar -- Combinatorics and Optimization
Speaker: Gary Au, Milwaukee School of Engineering
Title: "Generalized de Bruijn words for primitive words and powers"
Abstract: In this talk, we will discuss some problems in combinatorics on words that are related to de Bruijn words. We show that for every $n \geq 1$ and over any finite alphabet, there is a word whose circular subwords of length $n$ have a one-to-one correspondence with the set of primitive (aperiodic) words of length $n$, and provide several ways to generate such a word. We also look into connections between de Bruijn graphs of primitive words and Lyndon graphs.

We also show that the shortest word that contains every $p$-power of length $pn$ over a $k$-letter alphabet has length between $pk^n$ and roughly $(p+ \frac{1}{k}) k^n$, for all integers $p \geq 1$, and sketch an algorithm that generates a word which achieves the upper bound.

This talk does not assume any prior knowledge in combinatorics on words, and should be accessible to grad students in all areas of C&O.

(Joint work with Jeffrey Shallit.)

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