About
This is an undergraduate student seminar on geometry and topology. Here is a course description for you to get
an idea of how it runs and particularly for those of you who wish to register for credit.
If you would like to propose a topic in geometry and topology for now or the future, write to Yifei Zhu.
Our theme for Fall 2024 is the geometry and physics of
Higgs bundles. References include Laura Schaposnik's lecture notes from a
summer school on quantum
field theory and manifold invariants (this volume also
contains other topics of interest) and her expository
article for the
Notices of the AMS. See here for
more motivations. Background materials can and should be
supplemented according to participant needs. If you're thinking about getting involved, let us know.
We meet on Thursdays 4:20–6:10 pm in Lecture Hall 3309. The first, organizational meeting is
tentatively scheduled on September 19. To facilitate
communication between participants, there is a QQ group: 811299006.
Talks
Sep 19, '24,
Yifei Zhu,
Organizational meeting / introduction: A partial, but motivated view of Higgs bundles

Why do we care about them?
 Vector bundles, parametrized objects, and moduli spaces: tangent bundle, line bundles
over $S^1$, moduli of elliptic curves

Motivation from physics (condensed matter and quantum materials): sensing and absorbing
devices (invisibility cloak,
antiobjects), optical
devices (holography,
laser), materials with exceptional properties not found in nature at the macroscopic
level realized by engineering and finetuning against
degeneracy at the microscopic/quantum level

Hamiltonian $\stackrel{\rm mathematically}{=}$ square matrix with prescribed symmetry
(Hermitian, reason), eigenvalue (energy) and eigenvector
(state), evolution of eigenstates (or rotation of eigenframe $\leadsto$ eigenbundle),
energy bands

Examples: Hermitian 2band system (orthogonal half Möbius bands, classified by
$\pi_1(S^1)\cong{\mathbb Z}$) and 3band system (classified by
$\pi_1\Big({\rm SO}(3)\big/\big({\rm O}(1)\times{\rm O}(1)\big)\Big)\cong Q=\{\pm1,\pm
i,\pm j,\pm k\}$, see
Wu et al.,
Science 2019), nonHermitian 2band system
(nonHermitian = Hermitian with respect to a nonEuclidean Riemannian metric form,
reason(s); stratified space,
kissing half
Möbius bands)

Higgs bundle $\approx$ family of matrices:
Peter Higgs, Nigel Hitchin, Carlos Simpson

Riemann–Hilbert correspondence (de Rham's theorem, the Poincaré lemma), Simpson
correspondence

Some specific goals (or rather questions to keep in mind):
 Background materials and a plan: course
description for MA423

Exercise: Compute the eigenbundle for the real matrices $$\begin{bmatrix} f_1 & f_2 \\ f_2 & f_1 \end{bmatrix}$$ [a solution] and find its relationship to the Hopf bundle $S^0\hookrightarrow S^1\to S^1$. What
about the complex and quaternionic (and octonionic) cases? Hint: See here.
Sep 26, '24,
Pengxu Zhang,
Overview of Higgs bundles through the lens of moduli spaces
Oct 10, '24,
TBD,
TBA
Previous Semesters
Fall 2023, Modular curves
Spring 2023, Knot theory and lowdimensional topology
Fall 2022, Modular forms and their applications to geometry and
topology
Spring 2022, Topological data analysis
Fall 2021, Differential forms in algebraic topology
Spring 2021, The wild world of 4manifolds
Even earlier, we worked on Manifolds, sheaves, and cohomology and more.