Algebra and Number Theory

## Faculty members involved

Adam Clay (link). The study of group actions on partial ordered and ordered sets has experienced a revival in recent years due to new connections with topology and geometric group theory. As such, many modern problems of a topological flavour (problems related to braid groups, knot groups, and 3-manifold fundamental groups, for example) have arisen for which few classical theorems are applicable. My research therefore focuses on developing modern theorems and tools, specifically relating to orderable groups and their generalizations, for application to problems in low-dimensional topology.

Jaydeep Chipalkatti (link). The classical invariant theory of binary forms is a subject whose origins go back to the nineteenth century. It was then practised by several mathematicians in Europe; inter alia, Cayley, Salmon and Sylvester in England, Clebsch, Gordan and Hilbert in Germany and Hermite in France. Its central theme is to consider the action of the general linear group of 2 X 2 matrices on homogeneous forms of a fixed degree in two variables. This gives rise to covariants, i.e., polynomials which vary 'in step with' the group action.

These covariants show up prominently in many geometric questions involving the space of binary forms. For instance, a binary form has a double root exactly when its discriminant (which is a covariant) vanishes. My work involves several such geometric questions and the covariants implicated in them. In particular, I have worked on the Hermite invariant, as well as generalizations of the discriminant answering to more complicated distributions of roots.

Steve Kirkland (link) works in the areas of matrix theory and graph theory, with particular interest in the interplay between the two. His research in combinatorial matrix theory examines how key matrix properties, especially eigenvalue and eigenvector properties, are influenced by the zero-nonzero pattern of the entries of matrices in question. He also works in spectral graph theory, focusing on how spectral properties of various matrices associated with a graph (such as the Laplacian and signless Laplacian matrices) reflect structural properties of the graph such as connectivity and biparticity. He is also interested in the use of spectral graph theory in applied settings, including food webs, protein interaction networks, and quantum walks on graphs.

He also has an ongoing interest in entrywise nonnegative matrices and their applications, and has worked extensively on spectral properties of stochastic matrices. The latter are central to the analysis of discrete time, time homogeneous Markov chains on finite state spaces. His research in this direction includes results on stationary vectors and coefficients of ergodicity for stochastic matrices with specified directed graphs.

Thomas Kucera (link) works in mathematical logic (model theory) and applications of model theory to module theory and ring theory. His particular interest is in the structure of injective modules and related objects, such as pure-injectives, and their connections to parts of ring theory, especially non-commutative localization.

The key concept is that of a pure-injective module, which has natural, well-motivated definitions both as an object of interest in mathematical logic and as an object of interest in algebra. Injective modules over (even one-sided) noetherian rings are natural examples of a special kind of pure-injective module, called "totally transcendental". It is hoped that model-theoretic techniques for studying structural questions will lead to interesting and useful purely algebraic results.

Siddarth Sankaran (link) Fermat's last theorem says that there are no solutions to the equation $x^n + y^n = z^n$ with $x$, $y$, $n$, $z$ integers and $n>2$; its proof was completed by Andrew Wiles more than 350 years after the problem first appeared. The significance of the proof, however, went far beyond settling this long-standing conjecture: it illustrated a profound connection between the arithmetic of certain geometric ojects (elliptic curves) and objects from number theory (automorphic forms). Relations of this spirit are at the heart of many of the key questions and conjectures in modern number theory, including the Birch-Swinnerton-Dyer conjectures and their generalizations and the Langlands programme.

Shimura varieties are geometric objects that encode a wealth of number-theoretic information; they form a bridge between the arithmetic-geometric and automorphic worlds, and are therefore particularly useful in understanding the links beetween these two domains. Of partiular interest in the Department is a broad system of conjectures, at present still quite mysterious, relating arithmetic cycles on some Shimura varieties to the Fourier coefficients of automorphic forms.

Yang Zhang (link) interests are algebra, computer algebra and applications. Computer algebra, and symbolic computing in general, is a relatively recent area in computer science and mathematics, though its roots go back at least forty years. Whereas numerical computations are well-established to be efficient in many cases, computer algebra strives to address the different and arguably more difficult problem of providing exact solutions, or solutions parameterized by variables in the input. There has been an enormous amount of research on this over the past few decades, in computer science, mathematics, as well as in engineering and education. There have also been commercial and non-commercial software successes, the most ubiquitous being Maple and Mathematica.

## Current students and postdoctoral fellows

• Adriana-Stefania Ciupeanu (Ph.D.) - Adam Clay
• Serhii Dovhyi (M.Sc.) - A. Clay
• Daniel Johnson (M.Sc.) - Yang Zhang
• Yijian Liu (M.Sc.) - Yang Zhang
• Kenneth Onuma (Ph.D.) - Yang Zhang

## Representative recent publications

• S. Kirkland and M. Neumann, Group Inverses of M-matrices and their Applications, CRC Press, 2013.
• C. Godsil, S. Kirkland, S. Severini and J. Smith, Number-theoretic nature of communication in quantum spin systems, Physical Review Letters 109 (2012), 050502.
• C. Enns and T. G. Kucera. Purity and pure-injectivity for topological modules. Models, Logics and Higher Dimensional Categories: a Tribute to the Work of Mihály Makkai, pp55-77, 2011.
• S. Kirkland, Sign patterns of eigenmatrices of nonnegative matrices, Linear and Multilinear Algebra 59 (2011), 999-1018.
• J. Chipalkatti. On the ideals of general binary orbits. Journal of Algebra, vol. 324, 2010.
• J. Chipalkatti. The higher transvectants are redundant (with A. Abdesselam). Ann. Inst. Fourier, vol. 59, 2009.
• T. G. Kucera. The structure of indecomposable Ʃ-pure-injective modules (a survey of basic results and problems). Proceedings of the International Conference on Modules and Representation Theory, pp85--102, 2009.
• J. Chipalkatti. On Hermite's invariant for binary quintics, Journal of Algebra, vol. 317, 2007.
• J. Chipalkatti. Brill-Gordan loci, transvectants and an analogue of the Foulkes conjecture (with A. Abdesselam), vol. 208, 2007.
• T. G. Kucera. Explicit descriptions of the indecomposable injective modules over Jategaonkar's rings. Communications in Algebra, vol. 30, 2002.
• T. G. Kucera and Ph. Rothmaler. Pure-projective modules and positive constructibility, Journal of Symbolic Logic, vol. 65(1), 2000.

### News

MathCamp 2017 information is online.

Manitoba Workshop on Mathematical Imaging Science Friday, May 5, all day, Robert Schultz Lecture Theatre, details.

In an effort to help students, the Math department has put together the LevelUp program. See details here. Video explaining registration process is here.

### Events

Thursday, May 4th, 2017 at 15:30, 418 Machray Hall
Tommy Kucera
Fibonacci and The Liber Abaci
(Seminar series : Colloquium)

Friday, May 12th, 2017 at 15:30, 418 Machray Hall
John Dallon
Modeling Amoeboidal Cell Motion -- Force vs Speed
(Seminar series : Colloquium)

Friday, August 4th, 2017 at 15:30, 418 Machray Hall
Jose Aguayo
TBA
(Seminar series : Colloquium)