The Math Union at the University of Toronto is pleased to invite undergraduate students to participate in the mini directed reading program. You will be put in a group of 3-5 with an upper year student who will guide you to read some math literature of their choice. If you are interested, please use this form to sign up and rank your top 3 projects that you’d like to participate in. The program will start mid October and last until the end of this semester. There will be a final presentation that you are encouraged to take part and share what you learned with your peers.
Expected Commitment: You are expected to meet your mentor either in person or online for one hour a week for about 6 weeks, and read the literature assigned. The final presentation is optional and the exact date will be announced closer to the end of the semester.
The forms closes at Oct 5, Sunday, 23:59.
We will cover a selection of topics from Adams Knot Book, starting from the basic definitions of knots including Reidemeister moves, some invariants, and complements. An ambitious goal would be to talk about the Poincaré Conjecture.
Some familiarity with topology may be useful, although we will develop all the topology we need.
Any students interested in knot theory or topology.
What is a topology? Open/closed sets, basis for a topology, continuous functions, product topology, metric spaces… other topics time permitting
Mathematical maturity.
First or second year math spec.
A few notions from topology: open and closed sets, compactness, connectedness, continuity, metric spaces.
Topology of \(\mathbb{R}^n\): Heine-Borel, completeness, Bolzano-Weinerstrass, topological formulations of IVT and EVT.
High school calculus, familiarity with quantifiers, ideally: the first month of MAT137 or MAT157.
First year students, enrolled in either MAT137 or MAT157, who are planning on taking higher math courses.
Based mainly on A. Mishchenko’s book ‘Vector Bundles and Threir Applications’ and J. Milnor and J. Stasheff’s book ‘Characteristic Classes’.
Participants of this group will become acquainted with vector bundles with structures and principal bundles and their basic topological invariants, characteristic classes. Both the topological ‘obstruction’ viewpoint and the smooth ‘connection’ viewpoint on characteristic classes will be discussed, in proportion depending on the participants’ preference. To the necessary extent, singular and de Rham cohomology theories will be discussed as well, but those are not strictly required as prerequisites.
Good understanding of linear algebra (real and complex vector spaces, matrices, eigenvalues, trace, characteristic polynomial; inner products, groups GL(n), O(n), U(n)); basic group theory (groups, subgroups, cosets, group actions); basic point-set topology (familiarity with topological spaces and continuous maps); basic smooth manifold theory (smooth manifolds, smooth functions, vector fields, differential forms and the exterior differential).
Second- (and higher-) year specialists, interested upper-year majors.
Linear representation of finite groups, focused on character theory. Mostly following Serre’s Linear Representation of Finite Groups.
Familiar with group theory. Taking 301/347 concurrently is fine.
Second (and higher) year math spec.
Classical Set Theory by Derek Goldrei chapters 1-6.
Interest in math, willingness to sit and think.
First and second year math spec.
DOLGACHEV, Classical Algebraic Geometry: a modern view or A Royal Road to Algebraic Geometry by Holme or others.
high school algebra, abstract algebra (rings and modules) preferred. Basic geometry.
People with basic knowledge of algebra and geometry.
An introduction to \(p\)-adic numbers, group theory, and ring theory. From here, either I will propose participants classify quadratic forms over \(\mathbb{Q}_p\) (including \(p=2\)), or anisotropic tori for \(SL(2)\) or \(SO(4)/SO(5)\), all the tori coming from quadratic forms. Time permitting, we will use these tori to construct a bulk of the irreducible representations of these groups, under mild restrictions on the prime \(p\). This is connected to a recent problem of describing representations through a representation theoretic version of the local character expansion.
Mat224/247, MAT137/157, (Preferably enrolled in: Mat301/347).
Second year math spec and up or third year math major and up.
Geodesics, Isometries of the plane, Poincare disc model, triangles.
Know what \(\mathbb{C}\) is. Integration on manifolds useful but not at all required.
Anyone. There will be some repetition if you’ve already taken complex analysis.