Yu (Leon) Liu
yuleonliu*at*math.harvard.edu
Hi! I am a first-year Harvard math gradudate student. Before grad school, I did my undergraduate at UT Austin.
Besides
math,
I love playing sports (basketball, ping pong, and tennis), dancing (social and ballroom), and guitar (mostly
fingerpicking). Feel free to reach out!
Research Interests
My primary research interests are mathematical physics, representation theories, and homotopy theory. Lately I
am interested in topological abelian (electromagnetic)
duality (see below). Besides math, I am also interested in physics (QFT) and CS (type theory and programming
language).
Paper and Preprints
- Abelian Duality in Topological Field Theories:
my senior thesis
following Freed-Teleman. Any comment is appreciated! A
preprint, which reinterprets the results here in a more conceptual way,
is in preparation.
- Abstract: We investigate abelian duality in topological field theories. The theories are finite
homotopy TFTs, generalizations of finite gauge theories to $\pi$-finite spaces and spectra. In $d$
dimension,
We proof that the theories associated to $K(A,n)$ and $K(\hat{A}, d-1-n)$ are equivalent up to an
invertible field theory, where $A$ is a finite abelian group and $\hat{A}$ its Pontryagin dual
group.
This is a version of abelian duality for $p$-form gauge theories, where the gauge groups are finite
abelian.
In low dimensions, the duality recovers discrete Fourier transform and character theory for finite
abelian groups. In addition, using Brown-Comenetz duality, we extend our results to $\pi$-finite
spectra.
Quick links
Seminars organized
Other projects
- Arend documentation of HoTT. This is an ongoing project
to code/verify all the theorems in Homotopy Type Theory: an Univalent Foundation
in Arend, which a formal verification system that uses homotopy type theory (it assumes the univalence
axiom).
-
Aplite. A purely functional statically typed programming language with emphasis on composibility and
run-time
guarantees by type-checking.