## Current students and postdoctoral fellows

- Zehui Chen (Undergraduate Research Project)
- Eduardo Mendonça (Mitacs Globalink)
- Samuel Nyobe Likeng (Ph.D.)
- Mariam Shlepchyan (NSERC USRA)
- Yananan Wang (Co-op Work-Study)

## Former students and postdoctoral fellows

### Undergraduate research students

- Luke Volk (Work-Study 2016)
- Project: Crystals from 5-vertex ice models (with Javier Lorca Espiro), submitted for publication.

- Ian Dewan (NSERC USRA 2016)
- Project: Graph homology and cohomology

- Edward Poon (Undergraduate Research Project 2016)
- Project: Skew group rings

- Dene Lepine (Co-op Work-Study 2016)
- Jonah Robotham (Work-Study 2016)
- Project: Highest weight categories

- Marley Liu (High school enrichment 2016)
- Project: Proof and logical argument in abstract mathematics

- Edward Poon (NSERC USRA 2016)
- Paper: Nested Frobenius extensions of graded superrings (with A. Savage), Involve, to appear.

- Chad Couture (Work-Study 2015)
- Paper: Skew-zigzag algebras, SIGMA (Symmetry, Integrability and Geometry: Methods and Applications)
**12**(2016), 062, 19 pages.

- Paper: Skew-zigzag algebras, SIGMA (Symmetry, Integrability and Geometry: Methods and Applications)
- Diego Berdeja Suarez (Mitacs Globalink 2015)
- Paper: Integral presentations of quantum Heisenberg algebras, Contemporary Mathematics
**683**(2017) pp. 247–260.

- Paper: Integral presentations of quantum Heisenberg algebras, Contemporary Mathematics
- Rabib Islam (Work-Study 2015)
- Project: Morita equivalence

- Edward Poon (Co-op Work-Study 2014)
- Paper: Category theoretic interpretation of rings, American Journal of Undergraduate Research
**12**(2015), no. 4, pp. 5–29.

- Paper: Category theoretic interpretation of rings, American Journal of Undergraduate Research
- Matteo Copelli (Undergraduate Research Project 2014)
- Paper: Categorification of the nonnegative rational numbers, Rose-Hulman Undergraduate Mathematics Journal
**16**(2015), no. 2, pp. 19–37.

- Paper: Categorification of the nonnegative rational numbers, Rose-Hulman Undergraduate Mathematics Journal
- Wei Lu (Work-Study 2013)
- Aaron McBride (Work-Study 2013)
- Pablo Gonzalez Pagotto (Mitacs Globalink 2013)
- Jeffrey Pike (Summer Project 2012)
- Stéphane Guérin (NSERC USRA 2009)
- Andrew Sirjoosingh (NSERC USRA 2008)
- Project: An elementary proof of Gabriel's theorem

### Graduate students

- Gang Wu (M.Sc. 2016)
- Lucas Calixto (Ph.D. 2015, UNICAMP, FAPESP BEFE Award)
- Thesis: Representations of map superalgebras
- Paper: Equivariant map queer Lie superalgebras (with A. Moura and A. Savage),
*Canadian Journal of Mathematics***68**(2016), pp. 258–279. - Paper: Weyl modules for Lie superalgebras (with J. Lemay and A. Savage), Proceedings of the American Mathematical Society, to appear.

- Aaron McBride (M.Sc. 2015)
- Joel Lemay (Ph.D. 2015, NSERC Postgraduate Scholarship, Ontario Graduate Scholarship)
- Thesis: Geometric realizations of the basic representation of the affine general linear Lie algebra
- Paper: Geometric realizations of the basic representation of affine
*gl*,_{r}*Selecta Mathematica***22**(2016), no. 1, pp. 341–387. Winner of the Department of Mathematics & Statistics Best Student Paper Prize 2014–15. - Paper: Weyl modules for Lie superalgebras (with L. Calixto and A. Savage), Proceedings of the American Mathematical Society, to appear.

- Jeffrey Pike (M.Sc. 2015)
- Thesis: Quivers and three-dimensional Lie algebras
- Paper: Quivers and three-dimensional solvable Lie algebras, Journal of Lie Theory
**27**(2017), no. 3, pp. 707–726. - Paper: Twisted Frobenius extensions of graded superrings (with A. Savage), Algebras and Representation Theory
**19**(2016), no. 1, pp. 113–133.

- Daniel Cicala (M.Sc. 2014)
- Joel Lemay (M.Sc. 2011)
- Thesis: Valued graphs and the representation theory of Lie algebras
- Paper: Valued graphs and the representation theory of Lie algebras,
*Axioms***2012**, 1(2), pp. 111–148.

- Caroline El-Chaâr (M.Sc. 2010)
- Thesis: The Onsager algebra

### Visiting students

- Javier Lorca Espiro (Universidade de São Paulo 2016)
- Project: Crystals from 5-vertex ice models (with Luke Volk), submitted for publication.

- Tantely Rakotoarisoa (AIMS HeadStart 2014)
- Bea Schumann (University of Cologne 2012)
- Project: Representations of quivers

### Postdoctoral fellows

- Tiago Macedo (CNPq Grant), 2015–2016
- Daniele Rosso, 2013–2014
- Nathan Manning, 2012–2014
- Alex Hoffnung, 2010–2012
- Prasad Senesi, 2008–2009

## Prospective students and postdoctoral fellows

### Undergraduate students

There are many opportunities for undergraduate students to gain experience in mathematical research. Applicants wishing to work with me should have a background in linear algebra (MAT 1341, 2141) and group theory (MAT 2143). Some examples of potential projects can be found here. Projects can be completed in the context of URSA awards, the Co-op Program, the Work-Study Program, UROP, or Undergraduate Research Projects (see below for details). Although most projects are completed in the summer, it is also possible to do them during the fall or winter terms. Please contact me if you are interested in learning more about these possibilities.

### NSERC Undergraduate Student Research Award (USRA)

Each summer, I often have an opening for an undergraduate student as part of the NSERC USRA program. Students interested in a project to take place in the summer should contact me in December or early January. The application deadline is typically February 1 each year. See here for internal deadlines and information.

### Work-Study Program

Students who meet the requirements may apply for the Work-Study Program and then apply for positions through the Work-Study Navigator. Postings for summer positions are usually made in early spring. A work-study position can also qualify as a work term under the university's Co-op Program.

### Undergraduate Research Opportunity Program (UROP)

Selected students receive a one-time $1,000 award and devote, during one academic year, at least 50 hours to a research project conducted by the faculty sponsor.

### Undergraduate Research Project

Qualified undergraduate students can work on a one-semester project under the supervision of one or more faculty members in the department. At the end of the semester, the student writes a report and/or gives a seminar talk on some aspect of the research. This counts as a course (MAT4900) and is worth 3 credits.

### Graduate students

I am currently accepting graduate students. There are many intriguing problems in geometric, combinatorial, and categorical representation theory accessible at the master's and Ph.D. level (see below for potential projects). Students interested in working in this area should have completed courses in abstract algebra (uOttawa courses MAT3141 and MAT3143 or their equivalents). Descriptions of these uOttawa courses can be found here.

Canadian students (citizens and permanent residents) with an admission average of at least 8.0/10 are eligible for an Admission Scholarship. Excellent foreign students (those with an admission average of at least 9.0/10) are eligible for an International Scholarship. In addition, francophone students can apply for a differential tuition fee exemption that covers the difference between international and domestic student tuition. Further information can be found on the following websites:

- Admission Scholarships
- International Scholarships (M.Sc.)
- International Scholarships (Ph.D.)
- Differential tuition fee exemption

In addition to those mentioned above, there are many other internal and external scholarships available to graduate students. For a comprehensive list, please visit the following website:

### Visiting student researchers

I welcome applications from undergraduate or graduate students at other institutions who would like to spend some time working under my supervision at the University of Ottawa. Such students must typically have their own funding (e.g. from their home institution) and would have the status of visiting student researcher while here at the University of Ottawa. More information on this program can be found here. For graduate students, there is also the possibility of such visits leading to co-supervision of the graduate degree. Interested students should contact me by email.

### Postdoctoral fellows

I often have an opening for a postdoctoral fellow. Applicants should be recent Ph.D.s (or those expecting to receive their Ph.D. before the start of the appointment) working in a field closely related to my research areas. Those interested should submit an application to the department as well as contact me personally by email.

## Research areas

Below are some examples of areas in which students under my supervision can work (as well as areas connected to the ones listed below). If you are interested in learning more about possible projects, please send me an email.

### Algebra

Abstract algebra is a very broad area of mathematics that includes linear algebra, group theory, ring theory, the theory of modules, representation theory, and much more. Most projects completed under my supervision (including those in the areas listed below) involve some field of abstract algebra. They often also involve other topics, such as geometry, topology, combinatorics, or category theory.

### Lie theory

Lie groups and Lie algebras are indispensable tools in modern mathematics and mathematical physics. Lie groups are mathematical objects that have both geometric and algebraic structure. In particular, they are simultaneously differentiable manifolds and groups. They are a mathematically precise way of studying the continuous symmetry of mathematical and physical structures. For example, in quantum physics, particles correspond to representations of Lie groups. There are many potential projects in Lie theory for undergraduate and graduate students in both classical Lie theory as well as connections between Lie theory and other subjects (see below).

### Categorification

Category theory is a mathematical formalism that allows one to organize and study in an abstract way structure that is common to many different subjects. For instance, one can study properties of maps between objects and prove results that apply to maps between sets or groups. The advantage is that one does not need to reprove the results in each setting (in this case, for sets and for groups) because one has proven them in the more general setting of categories.

Recently, there has been considerable interest in using categories in a different way. This involves a process called categorification. Seeing as this field is relatively young, there are many good projects at the undergraduate and graduate level. These projects can involve connections to geometry, algebra, topology, and knot theory.

### Geometric representation theory

In general terms, geometric representation theory involves reformulating often classical results in algebra in new geometric terms. For example, one realizes a representation of a group or Lie algebra (this is a vector space on which a group or Lie algebra acts – think of a group of rotations of a plane or 3-dimensional space) as the homology of some topological space. Such realizations can yield new insight because they allow one to use geometric tools in the study of representation theory as well as representation theoretic tools in the study of geometry and topology.

### Quivers

A quiver is a directed graph consisting of vertices and arrows, and so the study of quivers is naturally related to graph theory. However, there are many surprising connections to other areas of mathematics, including geometry (e.g. the McKay correspondence) and Lie theory. The starting point of the connection between quivers and Lie theory is Gabriel's Theorem, which states a beautiful relationship between representations of quivers and root systems of Lie algebras. This connection has flourished into an extensive theory including the subjects of Hall algebras and quiver varieties. Projects in this area can involve many different fields including graph theory, Lie theory, representation theory, and algebraic geometry.

## Jobs in mathematics

Are you interested in mathematics and wondering what type of career opportunities there are for mathematics majors and those with advanced degrees (M.Sc. and Ph.D.) in mathematics? Take a look at the following websites to learn about the many opportunities available to mathematicians.

- WeUseMath.org
- Math at Work, Canadian Mathematical Society
- Career Information, American Mathematical Society
- Mathematics Careers, Mathematical Association of America