Talks in 2023
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.
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.
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 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.
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.
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.
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.
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.
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 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.
Talks in 2022
Sep 16, '22 (Friday, 16:30–17:30, Tencent Meeting 865-807-509),
Jian Liu (Hebei Normal University),
On the Cayley-persistence algebra
In this talk, we introduce a persistent (co)homology theory for Cayley digraph grading. We give the algebraic
structures of Cayleypersistence object. Specifically, we consider the module structure of persistent (co)homology
and show the decomposition of a finitely generated Cayleypersistence module. Moreover, we introduce the persistence-cup
product on the Cayley-persistence module and study the twisted structure with respect to the persistence-cup product.
As an application on manifolds, we show that the persistent (co)homology is closely related to the persistent map of
fundamental classes.
Slides
Sep 19, '22 (Monday, 16:30–17:30, Tencent Meeting 865-807-509),
Xiang Liu (Nankai University),
Persistent function based machine learning for drug design
Artificial intelligence (AI) based drug design has demonstrated great potential to fundamentally change the pharmaceutical industries.
However, a key issue in all AI-based drug design models is efficient molecular representation and featurization.
Recently,topological data analysis (TDA) has been used for molecular representations and its combination with machine learning models
have achieved great success in drug design.
In this talk, we will introduce our recently proposed persistent models for molecular representation and featurization.
In our persistent models, molecular interactions and structures are characterized by various topological objects,
including hypergraph, Dowker complex, Neighborhood complex, Hom-complex.
Then mathematical invariants can be calculated to give quantitative featurization of the molecules.
By considering a filtration process of the representations, various persistent functions can be constructed from the mathematical
invariants of the representations through the filtration process, like the persistent homology and persistent spectral.
These persistent functions are used as molecular descriptors for the machine learning models.
The state-of-art results can be obtained by these persistent function based machine learning models.
Slides
Sep 26, '22 (Monday, 16:20–17:20, Tencent Meeting 402-5413-1062),
Xing Gu (Westlake University),
The Ordinary and Motivic Cohomology of BPGLn(ℂ)
For an algebraic group G over the field ℂ of complex numbers, we have the classifying space BG in the sense of Totaro and Voevodsky,
which is an object in the unstable motivic homotopy category that plays a similar role in algebraic geometry as
the classifying space of a Lie group in topology. The motivic cohomology (in particular, the Chow ring) of BG is
closely related, via the cycle map, to the singular cohomology of the topological realization of BG, which is the
classifying space of G(ℂ), the underlying Lie group of the complex algebraic group G. In this talk we present
a work which exploits the above connection between topological and motivic theory and yields new results on both the
ordinary and the motivic cohomology of the complex projective linear group.
Oct 10, '22 (Monday, 16:20–17:20, Tencent Meeting 402-5413-1062),
Su Yang (Chinese Academy of Sciences),
The mapping class group of projective-plane-like manifolds
A closed simply-connected smooth manifold is projective plane like if its homology groups are the same as the homology groups of
a projective plane over the complex numbers, the quaternions or the octonions. A smooth classification of projective-plane-like
manifolds in dimension>4 was obtained by Kramer and Stolz in 2007. In a joint work with WANG Wei we computed the mapping class
group of these manifolds. In this talk I will give a brief report of this computation.
Oct 24, '22 (Monday, 16:20–17:20, Tencent Meeting 402-5413-1062),
Zhongjian Zhu (Wenzhou University),
Homotopy decomposition of An2-complexes
An (n-1)-connected finite CW-complex of dimension ≤n+k is called an Ank-complex. The homotopy types of indecomposable (up to wedge product)
An2-complexes, classified by Prof. Zhang Sucheng in 1950, consist of spheres, Elementary Moore spaces and Elementary Chang complexes.
These are elementary homotopy types in homotopy theory. In this talk, I introduce some homotopy of An2-complexes, especially the homotopy
decomposition problems of them. These are joint works with Prof. Pan Jianzhong.
Oct 31, '22 (Monday, 16:20–17:20, Tencent Meeting 402-5413-1062),
Ruizhi Huang (Chinese Academy of Sciences),
Homotopy of Manifolds
Homotopy properties are basic global properties of geometric objects. From 2014, techniques in unstable homotopy theory are used to
study homotopy of manifolds effectively. In this talk, I will review the progress in this topic, especially on a series of recent
work of myself joint with Stephen Theriault.
Dec 12, '22 (Monday, 16:20–17:20, Tencent Meeting 402-5413-1062),
Xueqi Wang (Beijing International Studies University),
On the Classification of Certain 1-connected 7-manifolds
The classification of manifolds is a basic and important problem intopology. In this talk,
I will focus on 1-connected 7-manifolds with specialhomology groups or cohomology rings
and report some previous work along with my progress on their classification.
I will also present some interesting applications of these classification results in geometry.
Dec 19, '22 (Monday, 16:20–17:20, Tencent Meeting 402-5413-1062),
Yichen Tong (Kyoto University),
Rational Self-closeness Numbers of Mapping Spaces
For a closed simply-connected 2n-dimensional manifold M, it has been proved that the components
of the free mapping space from M to 2n-sphere have exactly two different rational homotopy types.
However, since this result is proved by algebraic models of components, we do not know whether other homotopy
invariants distinguish these two types or not. In this talk, we completely determine the self-closeness
numbers of rationalized components of the mapping space and prove that they do distinguish different
rational homotopy types. The methods also have potential to be extended on other mapping spaces.
Talks in 2021 (mostly Mondays 4:20–5:10, zoom: 846 6434 3534)
Oct 22, '21 (Colloquium, Friday, 9:00–10:00 am, zoom: 976 6122 0179, passcode: 645744),
Zhouli Xu (University of California, San Diego),
In and around stable homotopy groups of spheres
The computation of stable homotopy groups of spheres is one of the most fundamental problems in topology.
Despite its simple definition, it is notoriously hard to compute. It has connections to problems in many
areas of mathematics, such as the problem of smooth structures on spheres. In this talk, I will discuss
a recent breakthrough on this computation, which depends on motivic homotopy theory in a critical way.
This talk is based on several joint works with Bogdan Gheorghe, Dan Isaksen, and Guozhen Wang.
Oct 25, '21 (3:00–4:00 pm, Room 201,
SUSTech International Center for Mathematics),
Satoshi Nawata (Fudan University),
Recent development about BPS q-series
From physics perspective, Gukov(-Pei)-Putrov-Vafa have introduced a non-perturbative definition of complex
Chern-Simons invariants of a 3-manifold, which is called BPS q-series. Although its mathematical definition
is not given yet, it exhibit remarkable mathematical properties: (non-semisimple) TQFT structure, modularity,
cyclotomic expansions. In this talk, I will survey the recent development of the BPS q-series.
Nov 1, '21 (zoom: 937 2129 3631),
Ruizhi Huang
(Chinese Academy of Sciences),
Algebraic topology of 24 dimensional string manifolds
String can be viewed as higher version of spin, while the latter plays a fundamental role in Atiyah-Singer
index theory. People try to develop parallel theory for string, the full story of which is still mystery.
Geometry and topology of string manifolds then attract increasing attentions and interests, while the ones
of dimensional 24 are quite special among string manifolds.
In this talk, we will discuss the algebraic-topological aspect of 24-dimensional string manifolds,
following the work of Hirzebruch, Ochanine, Landweber-Stong, Mahowald-Hopkins, Chen-Han, and very recent
work of mine joint with Fei Han. We will talk about string cobordism, various index-theoretic genera and
some applications.
Nov 8, '21 (zoom),
Weiyan Chen
(Tsinghua University),
Choosing points on cubic plane curves
It is a classical topic to study structures of certain special points on complex smooth cubic plane curves,
for example, the 9 flex points and the 27 sextactic points. We consider the following topological question
asked by Farb: Is it true that the known algebraic structures give all the possible ways to continuously
choose n distinct points on every smooth cubic plane curve, for each given positive integer n? This work is
joint with Ishan Banerjee.
Nov 19 (Friday, 9:00–10:00 am, zoom),
Prasit Bhattacharya
(University of Notre Dame),
Equivariant Steenrod Operations
Classical Steenrod algebra is one of the most fundamental algebraic gadgets in stable homotopy theory.
It led to the theory of characteristic classes, which is key to some of the most celebrated applications
of homotopy theory to geometry. The G-equivariant Steenrod algebra is not known beyond the group of
order 2. In this talk, I will recall a geometric construction of the classical Steenrod algebra and
generalize it to construct G-equivariant Steenrod operations. Time permitting, I will discuss potential
applications to equivariant geometry.
Dec 3, '21 (Colloquium, Friday, 4:30–5:30 pm, Lecture Hall 1-111),
Don Zagier
(Max Planck Institute for Mathematics),
Topology and number theory
Number theory is the study of completely discrete things that can be
counted or numbered, while topology is the study of continuous things
and of properties that are not changed by deformation. So it is
perhaps surprising that questions of pure topology lead to surprisingly
subtle questions of number theory. In this talk I will sketch several
instances of this phenomenon, starting with Euler's discovery that a
discrete invariant (now called the Euler characteristic) of a
topological space can be computed by triangulating it any way, and then
move on to more sophisticated examples: the "Betti numbers" of a
manifold (roughly speaking, the number of k-dimensional "holes" for
each integer k up to the dimension of the manifold), the appearance of
Bernoulli numbers in connection with so-called exotic spheres in higher
dimensions, the appearance of Dedekind sums (another simple
number-theoretical object going back to the 19th century) in the study
of 4-dimensional manifolds, and most recently a whole series of
discoveries relating "quantum invariants" of knots to various deep
number-theoretical concepts (units, K-theory, Bloch groups, Habiro
ring,...). This last is ongoing research in collaboration with Stavros
Garoufalidis in SUSTech. The talk will be at a very general level and
will not require any prior knowledge. In particular, rough
explanations of all of the various technical words occurring above will
be given.
In the Math Center seminar talk that I will give on December 6, I will
tell more about one of the specific questions appearing above, namely,
the question of the possible Betti numbers that can occur for smooth
n-dimensional manifolds. The answer here turns out to involve
elementary but quite hard number theory, and is quite amusing. This
talk, too, is meant to be understandable by good undergraduates or by
mathematicians in all fields.
Dec 6, '21 (Monday, 3:00–4:00 pm,
SUSTech International Center for Mathematics),
Don Zagier
(Max Planck Institute for Mathematics),
What can the Betti numbers of a smooth manifold be?
Dec 6, '21 (Monday, 4:20–5:10 pm, zoom: 846 6434 3534),
Mayuko Yamashita
(Kyoto University),
On the absence of all heterotic global anomalies
In this talk, I will explain the work "Topological modular forms and the absence of all heterotic global
anomalies" (
https://arxiv.org/abs/2108.13542)
with Yuji Tachikawa from a mathematical point of view. That work is aimed at settling a physical problem to
show the vanishing of anomalies in heterotic string theories. We translate the problem into a mathematical
problem to show that a certain transformation of generalized cohomology theories from TMF (Topological
Modular Forms) to the Anderson dual to String bordism vanishes, and prove that it is indeed the case. Here,
the Anderson dual of a generalized homology theory plays the crucial role in the classification of anomalies
by a conjecture of Freed and Hopkins. I will also explain this point, as well as related works on the
Anderson duals.
Dec 13, '21 (Göttsche Workshop talk, 5:00–6:00 pm,
SUSTech International Center for Mathematics),
Don Zagier
(Max Planck Institute for Mathematics),
Mock modular forms and wall crossing
The theory of mock modular forms was created in the 2002 thesis of Sander Zwegers as a generalization of the
famous "mock theta functions" of Ramanujan, of which Ramanujan had given many examples but no intrinsic
characterization. One of the approaches given by Zwegers involves the Taylor expansion of meromorphic Jacobi
forms, which have a "wall-crossing" property as some contour of integration crosses a pole of the function, and
he noticed that the functions defined some years earlier in a paper by Lothar Göttsche and myself on Donaldson
invariants of 4-manifolds, which were also based on a wall-crossing property, were in fact examples of mock
modular forms before either the notion or the term existed. The same wall-crossing property occurred some years
later in mathematical physics in the study of the string theory of black holes, the explanation there too being
given by mock modular forms. In the lecture I will try to explain what mock modular forms are and how these
various properties arise.