The study of of algebraic structures, or sets with a finite number of operations that satisfy a number of fixed axioms.
Algebra is the study of algebraic structures, that is, sets with a finite number of operations that satisfy a number of fixed axioms. Historically, algebra developed from the study of solutions of equations defined by polynomials. Algebraic Geometry refers to the area of mathematics devoted to the powerful correspondence between algebraic and geometric problems. Number Theory is the branch of mathematics that studies the algebraic structure of the integers.
At McMaster the research in these areas focus mainly on problems in combinatorial commutative algebra, equivariant algebraic and symplectic/Poisson geometry, and number theory, including (but not limited to) topics such as: Hessenberg varieties, L-functions, minimal free resolutions and syzygies, modular forms, Newton-Okounkov bodies, quiver loci, Schubert calculus, Schubert varieties, toric degenerations, toric varieties, and vertex operator algebras.
For more information about doing research in Number Theory, see the webpage of C. Franc. The research of M. Harada, J. Rajchgot, and A. Van Tuyl is broadly grouped under the umbrella of Combinatorial Algebraic Geometry; for more information on this research group, including information for potential students, visit the CAG website.
This research area runs the Algebra and Algebraic Geometry seminar at McMaster.
Associate Professor
Professor
On Leave
Research Area: geometry-topology
Research Profile: Geometry and Topology More specifically, I compute topological invariants, such as equivariant cohomology theories, of spaces with such structure. Symplectic geometry is the mathematical framework of classical physics; hyperkahler manifolds are symplectic manifolds wiht extra structure, are of particular recent interest due to their connections to theoretical physics. I am mainly concerned with the theory of symmetries of manifolds with these structures, as encoded by a Hamiltonian Lie group action, i.e. there exists a moment map on M encoding the action by Hamiltonian flows. Such group actions on symplectic and hyperkahler manifolds arise naturally in the context of physics, representation theory, and algebraic geometry. To a Hamiltonian space, one associates a symplectic (hyperkahler) quotient, which inherits a symplectic (hyperkahler) structure from the original manifold. The main theme of my recent research is the study of the topology and equivariant topology of these quotients, in particular the computation of their cohomology and complex K-theory rings.
On leave until January 2027
Professor and Associate Chair (Undergraduate)
Research Area: Algebra
Research Interests: My research area is commutative algebra and its connection to other areas like combinatorics and algebraic geometry. I am interested in homological invariants, edge ideals, simplicial complexes, sets of points in (multi)-projective spaces, toric ideals, symbolic powers of ideals, and combinatorial matrix theory