Recent and Upcoming Talks
Dec 28, '23 (Colloquium, Thursday, 4:30–5:30 pm, M1001),
Neil Strickland (University of Sheffield),
Recent progress in chromatic homotopy theory
Chromatic homotopy theory is a branch of algebraic topology which has seen an explosion of new activity in recent years.
In particular, in the summer of 2023 Burklund, Hahn, Levy and Schlank announced the negative resolution of Ravenel's Telescope Conjecture, which had been open since 1984.
That conjecture said that two categories are the same; now we know that they are different, and we expect to find a rich and intricate structure in the gap between them.
In this talk I will attempt to survey, for a general pure mathematical audience, the main ideas of chromatic homotopy theory and some highlights of recent progress.
Slides available at the Speaker's website.
Dec 27, '23 (Wednesday, 4:20–5:20 pm, Taizhou Hall 240A),
Shicheng Wang (Peking University),
Some connections between topology and number theory
We will discuss some connections between topology and number theory inspired by the studies of mapping degrees and achirality of manifolds.
Dec 20, '23 (Wednesday, 4:20–5:10 pm, M1001),
Zhipeng Duan (Nanjing Normal University),
Moduli spaces of trees and complexes of not 2-connected graphs
In this talk we will discuss two kinds of symmetric spaces. The first one is the moduli space of trees which were introduced by Whitehouse for studying $\Gamma$-homology and $\mathbb{E}_{\infty}$-obstruction theory. The second one is the complex of not $2$-connected graphs introduced by Vassiliev on studying knot invariants. Interestingly, their homology as $\Sigma_n$-modules are isomorphic up to a sign representation of the symmetric group $\Sigma_n$ and a degree shift.
During this talk, I will show that this isomorphism can be lifted into topological level. Specifically, I will show that these two spaces are in fact $\Sigma_n$-equivariant homotopy equivalent to each other. This is joint work with Greg Arone and Guchuan Li.
Nov 9, '23 (Colloquium, Thursday, 4:30–5:30 pm, M1001),
Zhouli Xu (University of California San Diego),
The Adams differentials on the classes $h_j^3$
In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams computed the differentials on the classes $h_j$,
resolving the Hopf invariant one problem. In Adams filtration 2, using equivariant and chromatic homotopy theory, Hill–Hopkins–Ravenel
proved that the classes $h_j^2$ support non-trivial differentials for $j \geq 7$, resolving the celebrated Kervaire invariant one problem.
I will talk about joint work with Robert Burklund: In Adams filtration 3, we prove an infinite family of non-trivial $d_4$-differentials on the
classes $h_j^3$ for $j \geq 6$, confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory –
$\mathbb C$-motivic stable homotopy theory and $\mathbb F_2$-synthetic homotopy theory – both in an essential way.
Nov 2, '23 (Colloquium, Thursday, 4:30–5:30 pm, M1001),
Ruochuan Liu (Peking University),
Congruences between modular forms and geometry of the eigencurve
The study of congruences between modular forms is a classic and vital subject in number theory.
Hida was the first to establish the theory of $p$-adic modular forms with slope zero; Hida theory
is a core tool in Wiles' proof of the Iwasawa main conjecture, and also one of the main tools in
proving Fermat's Last Theorem. Coleman extended Hida's theory to $p$-adic modular forms of
general finite slopes and, together with Mazur, constructed the eigencurve. The geometry of the
eigencurve is essential for understanding the congruence properties of modular forms, yet our
knowledge about geometry of the eigencurve is very limited. In a recent joint work with Nha Xuan
Truong, Xiao Liang and Zhao Bin, we prove the finiteness of irreducible components of the
eigencurve under a generic condition.
Oct 12, '23 (Colloquium, Thursday, 4:30–5:30 pm, M1001),
Don Zagier (Max Planck Institute for Mathematics),
Modular forms and quantum modular forms
Modular forms are one of the most beautiful and fundamental notions in modern number theory and
play a role in almost every domain of pure mathematics and physics. The famous specialist
Martin Eichler once said, "There are five fundamental operations of mathematics:
addition, subtraction, multiplication, division, and modular forms."
In the first talk I will explain this notion and present some highlights of the theory and its applications.
The last part, about the relatively new notion of "quantum modular forms" and its variants, is based on
joint work with Stavros Garoufalidis (SUSTech) during the last several years in which
many beautiful interconnections between number theory and 3-dimensional topology, and in particular the
so-called quantum invariants of 3-dimensional topology, were found.
Sep 27, '23 (Wednesday, 4:20–5:10 pm, M5024),
Guozhen Wang (Fudan University),
Chromatic families in the EHP sequence, II
We will try to explain the patterns of chromatic families in the EHP sequence and the applications in unstable homotopy theory. This is joint work with Y. Zhu.
Sep 25, '23 (Monday, 4:20–5:10 pm, M5024),
Guozhen Wang (Fudan University),
Chromatic families in the EHP sequence
We will try to explain the patterns of chromatic families in the EHP sequence and the applications in unstable homotopy theory. This is joint work with Y. Zhu.
Sep 21, '23 (Colloquium, Thursday, 4:30–5:30 pm, M1001),
Guozhen Wang (Fudan University),
Structures and computations in the motivic stable homotopy categories
Motivic homotopy theory is an application of abstract notions of homotopy in the world of algebraic varieties. It turns out that motivic homotopy theory gives us powerful tools in understanding classical homotopy theory. In this talk, we will show how structures in the motivic stable homotopy categories can be used to compute both classical and motivic stable homotopy groups.
Aug 9, '23 (Wednesday, 5–6 pm, M5024),
Jiaqi Fu (Paul Sabatier University),
A derived bridge between Lie algebroids and foliations
Lie algebroid and algebraic foliation are two natural algebraic analogues of foliation in differential geometry, which are nevertheless not equivalent without smooth condition. In this project, we disregard this disharmony by considering their derived analogues, and use a refined Koszul duality to establish an equivalence (of sub-$\infty$-categories) between partition Lie algebroids and infinitesimal derived foliations.