Looking for an honours topic?

Deciding on a thesis topic and supervisor is an important task that you should spend some time on. Scout around, even if you have made up your mind that you wish to work with a particular professor on a particular area. It's a great excuse to meet more of the faculty and in the process, you may learn some interesting mathematics. There are many considerations, but perhaps the most important are that you pick a topic that is interesting and a supervisor who you can get along with and learn something from. Do have a chat with the Honours co-ordinator or myself for suggestions.

My research interests are in algebra and algebraic geometry, though I've had honours students write their theses in number theory, topology and geometry more generally. A good way to find out about the mathematics I'm interested in is to look at my YouTube channel. If anything there interests you, I can probably come up with a related honours project.

Below are some more suggestions for honours topics and possible source material.

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• Representation theory of finite dimensional algebras
• Algebraic Geometry
• Noncommutative Algebraic Geometry
• Noncommutative ring theory
• Homological Algebra and Category theory
• Commutative Algebra
• Number theory
• Geometry and topology

Representation theory of finite dimensional algebras

Algebra is ubiquitous in mathematics and science more generally, because it arises whenever you study a collection of linear operators or matrices. To pick one of many examples, Clifford algebras appear naturally in quantum physics where one needs to solve matrix equations. In this case, the equations define the Clifford algebra, whilst the choice of matrix solutions gives a representation of the algebra. Given the numerous applications of algebra, it is not surprising that the theory associated with them is surprisingly rich and varied. Often special classes of algebras are studied where interesting results can be obtained. One fairly large class is that of finite dimensional algebras. If you are interested in this area, you should also chat with Jie Du.
Diagrams of linear maps playlist.

• Auslander-Reiten theory: You learn in second year linear algebra, that every finite dimensional vector space is just a direct sum of copies of the field. In the honours "Modules and Representation theory course", you see that matrix rings and more generally, semisimple algebras have a similarly, elementary representation theory: all modules are direct sums of summands of the ring. But how do you study modules over more general rings? The answer lies in Auslander-Reiten Theory and is encoded in a mysterious graph called the Auslander-Reiten quiver. See the book Auslander, Reiten, Smalo. "Representation theory of Artin algebras" or Gabriel's "Auslander-Reiten sequences and representation-finite algebras".
• Functor categories: In mathematics, one often obtains greater understanding by enlarging the structure at hand. For example, the understanding of real solutions to polynomial equations is facilitated by enlarging the field of real numbers to the field of complex numbers. Auslander ingeniously applied this principle to representation theory, where he enlarged the category of modules to that of functor categories. This approach unlocks many of the mysteries in the representation theory of finite dimensional algebras. You may wish to check Auslander's "A functorial approach to representation theory" found in Lecture Notes in Mathematics 944 (1982).
• Hereditary algebras: Linear algebra studies linear maps answering questions such as: determine what any linear map T:V->W looks like up to change of basis. This naturally raises the question, what happens if you consider two linear maps from V to W, up to (a common) change of basis. More generally, we can consider several linear maps such as in the Four Subspace Problem: classify all quadruples of subspaces of a vector space W up to change of basis. It turns out the best way to approach these problems is to view these linear maps as defining a representation of a certain hereditary algebra arising from a directed graph or quiver. Their representation theory is beautifully connected to Lie theory and algebraic geometry. See Gabriel's lectures above or Crawley-Boevey's notes on quivers in http://www1.maths.leeds.ac.uk/~pmtwc/
• Canonical algebras:
• Tilting theory:
• Preprojective algebras

Algebraic geometry

Algebraic geometry is an old subject with deep connections with commutative algebra, number theory, complex analysis, representation theory, topology ... (you get the drift). Stated most simply, algebraic geometry is the study of solving polynomial equations. Viewing the solutions geometrically leads to an interesting interplay between algebraic, geometric, topological and complex analytic ideas. Typical questions in algebraic geometry include: Given the set X of solutions to a degree d polynomial in complex projective space, what is the topology of X? How many conics are tangential to 5 plane conics in general position? Perhaps one of the reasons why algebraic geometry is such an important discipline is that many geometric objects do arise as solutions of polynomial equations.

If you want to learn algebraic geometry seriously, you will have to learn about sheaves a la Serre's FAC paper (see Hartshorne's book below or Grothendieck's Elements de Geometrie Algebrique). This would fill up an honours thesis in itself but perhaps not one I would suggest. For your options I suggest one of three possibilities: i) learn about some interesting class of examples of algebraic varieties where you don't need to know about sheaves OR ii) obtain a working knowledge of sheaves (e.g. from Reid's Chapters on Surfaces below) and continue from there OR iii) work only in the affine case where things reduce to commutative algebra.

Check out my Youtube channel.
Algebraic geometry: First glimpses.
Projective geometry.
Affine algebraic geometry.

• Toric varieties: provide loads of elementary examples of algebraic varieties which can be studied via combinatorial data.
• Etale topology:
• Vector bundles
• Geometric invariant theory. Check out my Invariant theory playlist
• Moduli theory
• Singularity theory
• Theory of algebraic surfachttps://www.youtube.com/playlist?list=PLgAugiET8rrLD5kUh4-5STI9J0IlA-i9K"es:
• Luroths theorem
• Del Pezzo surfaces
• Intersection theory
• Derived categories of varieties
• Bott-Borel-Weil theorem

Noncommutative algebraic geometry

There is a fundamental duality between algebra and geometry which essentially tells you that every commutative algebra can be thought of as the ring of functions on some geometric space. This means to a large extent, one can convert questions about geometry, to ones about commutative algebra and vice versa. For example, in MATH1141, one already sees a strong relationship between the algebra of linear equations and linear subsets of affine space.

It is thus tempting to view noncommutative algebras as rings of functions on some putative "noncommutative" space and to try to understand noncommutative algebra geometrically. This tantalising proposal gives rise to the subject of noncommutative algebraic geometry.

• Grothendieck categories
• Twisted coordinate rings
• Weighted projective spaces
• Orders on curves and surfaces
• Zhang twists
• Noncommutative curves
• Geometry of quantum planes
• Noncommutative projective line bundles

Noncommutative ring theory

• Artin-Procesi theorem and PI-algebras. PI-algebras are close to commutative and so one expects that there should be a decent noncommutative algebraic geometry associated to them.
• Maximal orders. The study of these is sometimes described as noncommutative arithmetic.
• Dimension theory for noncommutative rings
• The Nullstellensatz is one of the pillars of (commutative) algebraic geometry. There are various noncommutative Nullstellensatze which suggest a type of noncommutative algebraic geometry. They have applications to the study of Lie algebras.
• Koszul algebras

Homological algebra and Category theory

Category theory is a higher order abstraction, where the mathematical objects of study are things like the collection of all groups, or the collection of all topological spaces. The language of category theory was developed in part to allow mathematicians to apply one area of mathematics to study another, the original example being the study of topology through algebra via the fundamental group. Those interested in this subject may also wish to chat with Pinhas Grossman and Mircea Voineagu. You can check out my Category theory playlist on Youtube

Homological algebra has its roots in the homology theory of algebraic topology. To illustrate, consider a compact connected surface (such as a sphere or torus). We can triangulate it i.e. express it in a nice way as homeomorphic to a union of triangles (e.g. a sphere is homeomorphic to a tetrahedron). The Euler characteristic of the surface is
Euler = no. of triangular faces - no. edges + no. vertices
The surprise is that this number is independent of the choice of triangulation. This topological invariant can be obtained from more subtle invariants called homology groups. The algebra arising from the study of these homology groups is known as homological algebra. It has since found applications to numerous parts of mathematics from geometry (sheaf, de Rham cohomology) to number theory (Galois cohomology) to (non)commutative algebra (Ext and Tor).

• Adjoint functor theorems and representability theorems: see Maclanes "Categories for the working mathematician". See also Krause's "Derived categories, resolutions and Brown representability" on https://arxiv.org/abs/math/0511047.
• Categorification: Some categories can be studied by looking at the set of isomorphism classes of objects studying algebraic structures that arise from some functorial constructions. The simplest example is the category of finite dimensional vector spaces, whose set of isomorphism classes is determined by its dimension, and the direct sum induces addition on the set of natural numbers. In categorification, we reverse this procedure and try to study algebraic structures by lifting them to some categorical setting. See the article by Baez and Dolan arXiv:9802029.
• Triangulated categories: Students meet the fundamental notions of kernel and image in first year. These concepts appear not only in linear algebra, but across a wide spectrum of algebra, most generally in any abelian category. Unfortunately, there are many categories where we would like to use this notion, but which are not abelian. (The derived category which arises in algebra and geometry, and captures homological information is one such example.) Is there a good substitute? Often the notion of a triangulated category is what's required. Quillen's notion of an exact category is yet another possiblity.
• Monoidal categories: Tannakian duality
• Hilbert syzygies theorem
• Auslander-Reiten triangles

Commutative algebra

Commutative algebras arise naturally in mathematics in two ways: firstly as functions on some geometric object and secondly, as systems of numbers, or more precisely, rings of integers. Consequently, the study of commutative algebra is intimately related to both algebraic geometry and to number theory. Moreover, commutative algebra serves as a conduit to pass number theoretic ideas (such as integral closure) into algebraic geometry and also geometric ideas (such as ramification) into number theory. To get an idea of some of the basics of commutative algebra, check out the lecture notes on my webpage.

• Polynomial invariants of finite groups. See my Invariant Theory playlist.
• Auslander-Buchsbaum-Serre theorem. This gives a homological characterisation of smoothness. It is a fantastic example of how fairly abstract algebraic machinery can be used to prove geometric results.
• Grobner Bases. These allow you to compute in commutative algebra. The theory is essentially algorithmic.
• Quillen-Suslin theorem.
• Homological algebra and algebraic geometry. Homological algebra can be used to answer questions such as when certain varieties are determinantal. See Buchsbaum D., Eisenbud, D., Algebra Structures for Finite Free Resolutions and Some Structure Theorems for Ideals of Codimension 3, American Journal of Mathematics, vol. 99 (1977) p.447-85.
• Shephard-Todd theorem: When is the invariant ring of a the polymonial ring itself a polynomial ring? The answer is given by the Shephard-Todd theorem as those from reflection groups.

Number theory

My main interest in number theory is via the Brauer group where arithmetic invariants can be constructed using noncommutative algebra.
Playlist: Brauer groups and number theory
Playlist: Diophandtine equations and p-adic numbers

• Merkurjev-Suslin theorem
• Class field theory

Geometry and Topology

I'm not a topologist but if you insist, I think K-theory and the theory of characteristic classes are extremely important parts of topology which are well worth learning. You should also talk to Mircea Voineagu.