As an international student from New Zealand, the graduate school application process was very confusing! I've compiled a short document here which contains some advice based on my experiences. Hopefully you find it useful! I recommend looking for advice from other sources as well, as my experience is now 7 years out of date. The aforementioned advice together with many of my raw application materials can be found in this zip file.
MATH 283 A (Heegaard Floer homology) - Topics in Topology, Fall 2020, taught by Ciprian Manolescu. An introduction to Heegaard Floer homology. The first half of the course was dedicated to defining Heegaard Floer homology in dimensions 3 and 4. The next quarter of the course emphasised the combinatoral nature of Heegaard Floer homology as we worked through applications. Some notable results in chronological order are: the proof that "hat" Heegaard Floer homology (with mod 2 coefficients) has a combinatorial algorithm, that Heegaard Floer homology detects the Thurston norm, and that the integral homology cobordism group has a Z^infty summand. In the final quarter we learned about knot Floer homology, proved that it was also combinatorial, and studied applications to the knot concordance group.
MATH 283 A (4-manifold topology) - Topics in topology, Spring 2020, taught by Ciprian Manolescu. An introduction to the topology of 4-manifolds. The course focused on four main topics. First we studied the classification of simply connected topological 4-manifolds based on Freedman and Donaldson's work. Next we studied visual representations of smooth 3 and 4-manifolds, focusing primarily on Kirby diagrams. The next topic was the main focus of the class: Seiberg-Witten gauge theory. We used it to prove the existence of exotic R⁴s, Donaldson's diagonalisability theorem, and the Thom and Milnor conjectures. Finally we developed Khovanov homology, and in particular Rasmussen's s-invariant, to give combinatorial proofs of some results originally obtained via gauge theory.
MATH 282 B - Homotopy theory, Winter 2020, taught by Chris Ohrt. A general introduction to homotopy theory. Main topics homotopy groups of spheres, fibre sequences, cohomology and obstruction theory, and spectral sequences. The last topic (model categories) was cut short due to the COVID 19 outbreak.
MATH 257 A - Symplectic geometry, Fall 2019, taught by Umut Varolgunes. A general introduction to symplectic geometry. Main topics include the interplay between complex and symplectic geometry, Lagrangian submanifolds, Hamiltonian dynamics, and local and global invariants.
Slice Bennequin inequality. The Bennequin inequality relates the classical "geometric" knot invariants of transverse and Legendrian knots (in contact 3-manifolds) to "topology" (the maximum Euler characteristic of a Seifert surface of the knots). A remarkable fact is that the inequality remains true when we move from Seifert surfaces to slice surfaces. These notes contain the historic contact-geometric background, describe how transverse links relate to braids, and provides a proof of the slice Bennequin inequality which is stated in terms of braids (using purely combinatorial methods - Khovanov homology). A large chunk of the notes is dedicated to defining and developing Khovanov homology and Rasmussen's s-invariant.
Algebraic topology. Exploration of several classical topics from algebraic topology, assuming a knowledge of the first two chapters of Hatcher. We start with fibrations and fibre bundles, and introduce classifying spaces to try to understand bundles over a space. Next we introduce characteristic classes, which are usually used to classify vector bundles in particular. Another classification of vector bundles uses K-theory, which we study in depth with an emphasis on Bott periodicity and its applications. We also find that K-theory defines an extraordinary cohomology theory. Another important example of an extraordinary cohomology theory is cobordism theory, which is the final standalone topic. In the last two chapters we introduce concepts and tools that help us better understand K-theory and cobordism theory (or more generally any extraordinary cohomology theory), namely equivariant cohomology and the Atiyah-Hirzebruch spectral sequence. These notes were compiled during Summer 2020 with guidance from Ciprian Manolescu.
Knot theory. Introductory knot theory notes, largely following Lickorish. Knot polynomials are given the most emphasis, but many other invariants are described. Some solutions to exercises are also given here. These notes were compiled during a reading course with Ciprian Manolescu in Spring 2020.
Homology 3-spheres. Introduction to the topology of homology 3-spheres, largely following Saveliev. The reader is expected to be familiar with some knot theory - any low dimensional topology concepts that are not explained in these notes should be explained in the knot theory notes above. The main topics of these notes are the Rokhlin invariant, the triangulation conjecture, and the Casson invariant. These notes were compiled during a reading course with Ciprian Manolescu in Spring 2020.
Morse theory. Introductory Morse theory notes, largely following Audin and Damian. The focus is on developing Morse homology and exploring some applications (such as the Morse inequalities). Some solutions to exercises are also given here. At the end of these notes we give a proof outline of the h-cobordism theorem (and prove the generalised Poincaré conjecture) following Milnor's lecture notes. Finally we explore the status of the generalised Poincaré conjecture and h-cobordism theorem (for each dimension) in several categories of manifolds.
Hilbert's Nullstellensatz. This document contains results culminating in a proof of Hilbert's Nullstellensatz. The notes are mostly self-contained, relying only on basic algebra (integral domains, prime ideals, modules etc). However, some knowledge of localisations is assumed when developing preliminary dimension theory results.
A non-visual proof that higher homotopy groups are abelian. This document is a short self-contained proof that higher homotopy groups are abelian using only algebra. By using the Eckmann-Hilton argument, we avoid having to construct any homotopies.
The uncertainty principal. This document is a short self-contained proof of the Heisenberg uncertainty principal (in a general mathematical setting). It will be easier to follow given some familiarity with Fourier transforms.