About
This is a graduate-level seminar on topology and related fields. Topics are selected from common interests of the participants.
Here are some general suggestions on how to give a good talk.
Talks in Fall 2024 (mostly Mondays 4:20 in M5024)
Sep 23, '24,
Yifan Wu,
When is $K(n)^*(BG)$ concentrated in even degrees?
Being concentrated in even degrees is a nice property for cohomology rings. For example, it will lead to the collapses of spectral sequences in certain
cases. In the first part of the talk, I will introduce the notion of good groups $G$, which satisfy the property as stated in the title. Then I will try
to generalize these ideas from $K(n)$ to other field spectra.
Main reference: Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel,
Generalized group characters and
complex oriented cohomology theories
Notes
Sep 30, '24,
Chunshuang Yin,
An introduction to toric varieties
Toric varieties are a class of algebraic varieties. They can be seen as algebraic torus with added limit points such that the torus is dense within them,
and their action is extendable throughout the whole variety. Toric varieties have wide-ranging applications across many areas of mathematics.
Due to their explicit combinatorial descriptions and rich structures, they serve as the "geometric stage" for various mathematical theories and some areas of theoretical physics.
In this talk, I will start by discussing three equivalent descriptions of affine toric to introduce a construction method for general toric varieties.
Main reference: David A. Cox, John B. Little and Henry K. Schenck,
Toric Varieties
Slides
Oct 14, '24,
Zeyang Ding,
The space of persistence diagrams on $n$ points coarsely embeds into Hilbert space
In this talk, we prove that the space of persistence diagrams on $n$ points (with the bottleneck or a Wasserstein distance) coarsely embeds into Hilbert space by showing it is of asymptotic dimension $2n$.
Such an embedding enables utilization of Hilbert space techniques on the space of persistence diagrams. We also prove that when the number of points is not bounded, the corresponding spaces of persistence
diagrams do not have finite asymptotic dimension. Furthermore, in the case of the bottleneck distance, the corresponding space does not coarsely embed into Hilbert space.
Slides
Oct 28, '24,
Yingming Zhang,
Period Polynomials and the Eichler-Shimura isomorphism
Period polynomials and their coefficients have significant impact on the theory of modular forms. The coefficients of period polynomials
are certain special values of L-functions.And one can show that there is an isomorphism between two copies of space of cusp forms and the
cohomology group of period polynomials,i.e. the Eichler-Shimura isomorphism. I will introduce the basic properties of the theory of the
holomorphic and meromorphic modular forms and describe an example of the cohomology group.
Nov 4, '24,
Qingrui Qu,
Perspectives from differential and algebraic geometry on ruled surfaces I
In this presentation, we will begin by reviewing some basic concepts of classical differential geometry like curvature and torsion within the context of three-dimensional Euclidean space.
Then we will briefly recall the structure theorems pertaining to curves and surfaces. Following this foundation, we will concentrate on the properties and classifications of ruled surfaces and developable surfaces.
We will then proceed to demonstrate several explicit examples that are naturally derived from discriminant varieties of polynomial equations, highlighting some intriguing generalizations along the way.
Nov 11, '24,
Qingrui Qu,
Perspectives from differential and algebraic geometry on ruled surfaces II
Building upon last week's introduction to ruled surfaces, this talk focuses on discriminant surfaces arising from characteristic polynomials of 3-order matrices with special
symmetries. Our investigation is motivated by condensed matter physics, particularly the study of Hamiltonians exhibiting Parity-Time symmetry and Minkowski-like pseudo-Hermitian symmetry. We will
present several explicit examples of such discriminant surfaces and explore their geometric properties. Notably, we will showcase numerous counterexamples of non-ruled discriminant surfaces, leading to
an intriguing question: what conditions ensure that a discriminant surface is ruled?
Nov 18, '24,
Wenhui Yang,
A brief introduction to Hitchin's equations
Nov 25, '24,
Yunhao Sun,
Around algebraic cycles
The history of algebraic cycles can be traced back to the nineteenth century. However, it was not until Grothendieck proposed his conjectures that the significance of this concept was
fully recognized, leading to a flourishing of research in the field. Algebraic cycles are deeply interconnected with number theory, mathematical physics, differential geometry,
and other areas. A fundamental object within the study of algebraic cycles is the Chow group. It is now understood that Chow groups form part of a cohomology theory known as
motivic cohomology. Motivic cohomology bears a similar relationship to algebraic $K$-theory as singular cohomology does to topological $K$-theory. We aspire for an abelian category of
mixed motives such that motivic cohomology groups serve as extension groups for these mixed motives. On the other hand, studying general properties of Chow groups and performing concrete
calculations remain exceedingly challenging. In this talk, we will introduce some basic properties and perspectives regarding Chow groups. Due to technical complexity, a limited number of
concrete examples will be provided.
Slides
Dec 2, '24,
Zhou Fang,
A brief introduction to intersection homolgy
In recent years, intersection homology has become an indispensable tool for studying the topology of singular spaces.
While the main results of usual homology theories often fail for singular spaces, intersection homology effectively recovers these properties, bridging this critical gap.
In this talk, I will present the foundational concepts of GM intersection homology (including simplicial intersection homology, PL intersection homology, and singular intersection homology) and some examples.
Finally, I may explore the topic of non-GM (Goresky-MacPherson) intersection homology.
Notes
Dec 9, '24,
Qianyi Zhang,
Gluing method—An introduction to locally free sheaves
Mordell-Weil theorem states that for an abelian variety $A$ over a number field $K$, $K$-rational points of $A$ is a finitely generated group.
Except the way using Weil height as a quadratic form satisfying Northcott property to prove the case for dimension one, i.e. elliptic curve,
we can prove it in the language of algebraic geometry by a technical method of relative effective Cartier divisor.
Effective Cartier divisors “equal” invertible ideal sheaves in some sense. And invertible sheaf is the locally free sheaf of rank $1$.
In this talk, I will introduce locally free sheaves, the natural one-to-one correspondence between locally free sheaves and vector bundles, and give some examples of invertible sheaves on projective line.
Dec 16, '24 (zoom 587 160 6136),
Xiansheng Li (University of Copenhagen),
What is ${\rm D}(\mathbb{S})$?
In this talk, I will not provide a definitive answer to the question posed in the title. Instead, I will
outline a general framework for "deriving" a stable $\infty$-category in a manner relative to a well-behaved
homology theory.
This construction can be understood as:
- An $\infty$-categorical deformation, and
- A suitable context for the Goerss–Hopkins theory.
A key example of this framework is the (connective) synthetic spectra $\rm Syn$, defined relative to an Adams-type
spectrum $E$. This construction provides a way to view $\rm Syn$ as a "derived $\infty$-category of spectra,"
capturing $\infty$-categorical structures that reflect the properties of $E$-based Adams spectral sequences.
Slides
Dec 23, '24,
Siheng Yi,
TBA
TBA
Talks in Spring 2024 (mostly Tuesdays 4:20 in M5024)
Mar 12, '24,
Qingrui Qu,
An introduction to singularity theory, Milnor's fibration theorem and Brieskorn's construction of exotic spheres
Through this lecture, I will provide an overview of the singularity theory of complex analytic functions and polynomials, appreciating its wide-ranging
applications and contributions to various branches of mathematics. I will begin by introducing fundamental concepts and definitions in this field, then proceed
to discuss the invariants and classification of simple singular points, highlighting their significance. Furthermore, I will illustrate the intriguing
relationship between the ADE classification and other mathematical objects such as finite subgroups of the orthogonal group or the intersection form of a fiber
manifold. This connection sheds light on the broader mathematical landscape and enriches our understanding of singularity theory. Moving forward, I will
introduce the Milnor fibration of a singular point and describe its topological structure by Morse theory. Finally, I will provide a brief introduction to the
groundbreaking work of Milnor and Kervaire on the discovery and classification of exotic diffeomorphism structures on 7-dimensional spheres. Additionally, I
will touch upon Brieskorn's explicit construction, which involves a family of polynomial singularities that gives rise to all exotic spheres. This fascinating
exploration demonstrates the profound impact of singularity theory in the realm of differential topology.
Slides
Mar 19, '24,
Jiacheng Liang,
Barr-Beck Theorem, Morita theory and Brauer groups in $\infty$-categories
In this talk, we will introduce the $\infty$-categorical version of the Barr-Beck theorem, Morita theory and Brauer groups. And we will see that the Brauer
group Br(S) of the sphere spectrum is zero by a spectral sequence using etale cohomology from Antieau–Gepner's work in 2012.
Slides
Mar 26, '24,
Yunsheng Li,
The Teichmüller TQFT and the Quantum Modularity Conjecture
The Quantum Modularity Conjecture is a refined form of the Volume Conjecture, one of the main problems of quantum topology. In recent studies, it is
discovered that the Quantum Modularity Conjecture enjoys rich connections to the Teichmüller TQFT, a TQFT promoted from the quantum Teichmüller theory. In
this talk, I will briefly introduce the construction of Teichmüller TQFT and its connections to the Quantum Modularity Conjecture.
References:
-
Li Yunsheng, On the Quantum
Modularity Conjecture for knots
-
Eiichi Piguet, Teichmüller TQFT calculations for infinite families of knots
Apr 2, '24,
Yunsheng Li,
The Teichmüller TQFT and the Quantum Modularity Conjecture, II
Apr 9, '24,
Yifan Wu,
Pipe Rings and Pipe Formal Groups
In this talk, we review some basic facts about formal schemes and formal groups, then we generalize these ideas to a more general setting called Pipe rings
and Pipe formal groups. These objects are useful to describe homotopy groups of spectra localized at Morava K theories and their moduli interpretations.
Slides
Notes
Apr 16, '24,
Qingrui Qu,
From Adams's $v_1$-periodicity to Andrews's motivic $w_1$-periodicity in homotopy groups
The purpose of this talk is to introduce Michael Andrews's work on discovering exotic periodicity in the homotopy groups of the motivic sphere spectrum and the
identification of six infinite families of elements. Firstly, we will provide a brief overview of the chromatic phenomenon in classical homotopy theory and Adams's
8-periodic isomorphism, expressed by a Massey product in the Adams spectral sequence. Then we will delve into motivic homotopy theory and explore the
philosophy behind Haynes Miller's square, which serves as an analogy and connection between the motivic Adams spectral sequence and the classical Adams–Novikov
spectral sequence. This perspective allows us to consider multiplication by the non-nilpotent element $\eta$ as a $w_0$-periodic map, similar to the role
played by $2$ for the $v_0$-periodic map. After defining a Thom reduction map to compare the $E_2$-pages of two spectral sequences, we can reduce the
existence of motivic $w_1^4$-periodic maps to the existence of classical $v_1^4$-periodic maps, as initially found by Adams in the early 1960s.
Notes
Apr 23, '24,
Jiacheng Liang,
Picard $\infty$-groupoids, Picard groups of $\mathbb{E}_\infty$–rings and generalized Thom spectra
In this talk, we will introduce the Picard $\infty$-groupoids and calculate Picard groups of several $\mathbb{E}_\infty$ rings,
such as K-theory spectra and topological modular forms.
Besides, we also introduce how to generalize the Thom spectrum functor into any presentably symmetric monoidal $\infty$-category,
by using the universal property of Picard $\infty$-groupoids.
Slides
Apr 30, '24,
Yunhao Sun,
Motives, motivic cohomology, and motivic homotopy: A historical introduction
In this talk, we will overview some aspects of the theory of motives. There are some significant conjectures. Currently, there are still more conjectures than
theorems in the field of motives. We will also discuss motivic homotopy theory. It can be thought of as a non-abelian generalization of motives. The
philosophy of motives is to study universal invariants. We will provide some examples to illustrate it. (The last part of this report was delivered by handwritting.)
Slides
May 14, '24,
Peng Huang,
Functoriality of the Postnikov tower
In this talk, I will introduce the Moore tower, Whitehead tower and particular the Postnikov tower.
I will show how to construct a Postnikov tower of a 0-connected CW complex and discuss the functoriality of the Postnikov tower.
On the other hand, I will give an example to show that the Moore tower has no functoriality.
Notes
May 15, '24 (Wednesday in M4009)
- 10:00–10:30, Jiacheng Liang, Master's thesis defense: Elliptic cohomology theories and the $\sigma$-orientation
- 10:50–11:20, Zhonglin Wu, Master's thesis defense: Spectral sequence calculation for unstable $v_n$-periodic homotopy groups of spheres
- 3:00–3:45, Xuecai Ma, Doctoral thesis defense: Methods of spectral algebraic geometry in chromatic homotopy theory
- 4:30–5:00, Yunhao Sun, Master's thesis proposal: Algebraic cobordism and cycle class map
May 21, '24,
Zhou Fang, Wenhui Yang, Chenlu Huang,
Topology and geometry of singularities
In this talk, we will introduce topological classifications of Hamiltonians, especially concerning singularities in the relevant moduli spaces.
The talk consists of three parts:
Zhou will explain the motivations and background, together with a specific example of an eigenbundle in three-band systems;
Wenhui will then discuss eigenbundles in two-band Hermitian systems and non-Hermitian systems;
Chenlu will show the eigenbundle along some interesting loops in a moduli space for three-band non-Hermitian systems.
Slides
May 28, '24,
Ni An,
The colored Jones polynomial, Nahm sum and stability
In this talk, we will introduce Braid group and enhanced Yang-Baxter operator firstly. Next give the definition of colored Jones polynomial
from the represention of Braid group by using Yang-Baxter operator. After knowing the definition, we will show a state sum formula for colored Jones polynomial
which can be uesd to compute it conveniently. Then we will give the limit of colored Jones polynomial for alternating knot which is in form of Nahm sum.
Slides
Jun 4, '24,
Zhonglin Wu,
On "Secondary power operations and the Brown–Peterson spectrum at the prime 2"
Understanding whether an $E_{\infty}$ structure exists on the Brown-Peterson spectra can enhance our comprehension of many other spectra in chromatic homotopy theory. In 2018, Tyler Lawson disproved the existence of an $E_{\infty}$ structure on the mod 2 Brown-Peterson spectrum. However, there are interesting insights to be gained from the proof process, which can help us better understand how to use secondary operations.
The presentation is divided into two parts. The first part sets up the fundamental knowledge, focusing mainly on constructing secondary operations and providing a brief introduction to Dyer-Lashof operations. The second part demonstrates how to construct a specific secondary operation and use it to disprove the existence of an $E_{\infty}$ structure on the mod 2 Brown-Peterson spectrum.
Jun 14, '24,
Zhonglin Wu,
On "Secondary power operations and the Brown–Peterson spectrum at the prime 2", II
Understanding whether an $E_{\infty}$ structure exists on the Brown-Peterson spectra can enhance our comprehension of many other spectra in chromatic homotopy theory. In 2018, Tyler Lawson disproved the existence of an $E_{\infty}$ structure on the mod 2 Brown-Peterson spectrum. However, there are interesting insights to be gained from the proof process, which can help us better understand how to use secondary operations.
The presentation is divided into two parts. The first part sets up the fundamental knowledge, focusing mainly on constructing secondary operations and providing a brief introduction to Dyer-Lashof operations. The second part demonstrates how to construct a specific secondary operation and use it to disprove the existence of an $E_{\infty}$ structure on the mod 2 Brown-Peterson spectrum.
Blank Slides
Slides with Notes
Jun 17, '24 (Monday in M616)
Jun 18, '24 (Tuesday in M5024)
Talks in Fall 2023 (Tuesdays 4:20 in M4009)
Oct 17, '23,
Yifan Wu,
Dyer–Lashof theories and algebras
In this lecture, we briefly introduce algebraic theories and operations. In particular, we specify these settings in homotopy theory and land on the so called Dyer–Lashof theories.
In the second part, we calculate Dyer–Lashof operations for some spectra, and discuss the relation between them and power operations.
Slides
Nov 7, '23,
Zhonglin Wu,
The motivic Adams spectral sequence
In this lecture, we briefly introduce some properties and examples of calculations of the motivic Admas spectral sequence and show how to use it to help us compute some differentials in the classical Admas spectral sequence.
Slides
Nov 28, '23,
Qingrui Qu,
On "The special fiber of the motivic deformation of the stable homotopy category is algebraic"
In this talk, I will introduce the main content of the paper ''The special fiber of the motivic deformation of the stable homotopy
category is algebraic'' by Bogdan Gheorghe, Guozhen Wang, and Zhouli Xu. They define a Chow-Novikov t-structure on the category of module
spectra over cofiber $S^{0,0}/\tau$, and establish that this category is equivalent to the derived category of $BP_*BP$-comodule category
as stable $\infty$-categories equipped with t-structures.
An exciting application is the isomorphism between the motivic Adams spectral sequence for $S^{0,0}/\tau$ and the algebraic Novikov spectral
sequence. This isomorphism enables Isaksen, Wang, and Xu to compute the stable homotopy groups of spheres up to the 90-stem.
Slides
Dec 5, '23 (4:20, Yidan 307),
Yunhao Sun,
On "The Chow $t$-structure on the $\infty$-category of motivic spectra"
Dec 6, '23 (4:20–5:10, M5024),
Wenbo Liao (Chinese University of Hong Kong),
Guts of sutured manifolds and knots
Given any sutured manifold, Gabai (1983) constructed a canonical way to find its certain subspace which we call guts.
Agol and Zhang (2022) extended this concept to certain homology classes and futhermore to knots.
In this talk, we introduce Gabai, Agol and Zhang's works and see the guts of knots defines a new invariant for knots
other than Heegaard Floer homology by computing guts of 2-bridge knots.
Dec 19, '23 (4:20, M1001),
Qingrui Qu,
Chromatic structures in motivic and equivariant stable homotopy categories
Dec 26, '23,
Jiacheng Liang,
An overview of $\infty$-categories and higher algebra
Jan 4, '24 (4:20–5:10, M5024),
Wenhui Yang,
Homotopy theory for directed graphs
Talks in Spring 2023 (Tuesdays 4:20 in M5024)
Mar 7, '23,
Yingxin Li,
The Goerss–Hopkins obstruction theory
Apr 11, '23,
Zhonglin Wu,
Multiplicative structures in the Adams spectral sequence
Apr 18, '23,
Xuecai Ma,
Formal moduli problems
Apr 25, '23,
Yuanding Li,
An overview of $v_n$-periodic homotopy theory
May 9, '23,
Yunhao Sun,
An overview of motivic homotopy theory
May 16, '23,
Hongxiang Zhao,
Explicit local class field theory à la Lubin and Tate with an application to algebraic topology
May 17, '23 (10–11, M4009),
Tongtong Liang,
Thesis defense: Methods of homotopy theory in algebraic geometry from the viewpoint of cohomology operations
May 23, '23,
Jiacheng Liang,
Thom spectra, infinite loop spaces, generalized cocycles, and the $\sigma$-orientation
May 30, '23,
Qingrui Qu,
Functor calculus and chromatic homotopy theory
Talks in Fall 2022 (Tuesdays 4:20–5:20 / 4:20–6:10 in M5024)
Sep 20, '22,
Jiacheng Liang,
Elliptic Curves and Abelian Varieties
Sep 27, '22,
Yifan Wu,
Introduction to bordism and Thom spectra
Oct 11, '22,
Tongtong Liang,
Power Operations in Complex Cobordism and Quillen's Theorem on Formal Group Laws
Oct 18, '22,
Zhonglin Wu,
Introduction to Adams spectral sequence
Nov 15 & Nov 22, '22,
Xuecai Ma,
An overview of chromatic homotopy theory
Nov 29, '22,
Hongxiang Zhao,
Formal Group Laws and Formal Groups
Dec 6, '22,
Jiacheng Liang,
Sites, Sheaves, Formal Groups and Stacks
Dec 13, '22,
Tongtong Liang,
Operads and the Recognition Principle
Dec 20, '22,
Tongtong Liang,
An Introduction to Structured Ring Spectra
In this talk, I will introduce how Peter May developed the theory of E-infinity-ring spectra, especially the construction of the twisted half-smashed product. From this viewpoint, we will see how linear isometry operads parametrize internal smash products from external smash products. If time permits, I will give a gentle introduction to the EKMM framework of spectra.
Jan 5, '23,
Yingxin Li,
Elliptic homology theory and tmf
Talks in Summer 2022
Jul 25, '22 (2:30–3:30, M2002),
Xuecai Ma,
Methods of spectral algebraic geometry in chromatic homotopy theory
Jul 25, '22 (3:30–4:30, M2002),
Tongtong Liang,
Methods of homotopy theory in algebraic geometry from the viewpoint of cohomology operations
Jul 28, '22 (4–5, M714),
Kecheng Shi,
Reciprocity and isogeny-based cryptography
This informal talk consists of two relatively independent parts. In the first part, I will talk about my interest in reciprocity. Most of this part is based on Frank Calegari’s talk “30 years of modularity” at the ICM 2022. I will start with Diophantine equations, giving some simple examples of their relationship to other mathematical objects such as L-functions and modular forms, and explain some related progress in number theory. In the second part, I would like to introduce some of the mathematical background of isogeny-based cryptography, on which I intend to work in the future.
Slides
Frank Calegari, Reciprocity in the Langlands program since Fermat's last theorem
Luca De Feo, Mathematics of isogeny based cryptography
Talks in Spring 2021 (mostly Mondays 4:30–5:30 in Yidan 307)
Apr 19, '21,
Tongtong Liang,
Homotopy coherence problem and ∞-categories
In this talk, I will introduce homotopy coherence problems as the motivation
of ∞-categories and give a brief introduction to ∞-categories. Most of topological invariants are homotopy invariants and many important functors are
representable in the homotopy category of spaces, for example, ordinary cohomology, K-theory and cobordism theory (Brown representability theorem),
which shows the need for homotopy category of spaces. However, there must be
loss of information when passing to homotopy category. An important problem
to measure how much the data lose is: Can a homotopy commutative diagram
in the category of space be replaced by a strictly commutative diagram by a
natural transformation up to homotopy equivalence? Dywer, Kan and Smith
pointed out in 1989 that the necessary and sufficient condition is homotopy
coherence. However, it is very complicated to describe the phenomenon of homotopy coherence by ordinary categories. Therefore, we need a new framework
to do with homotopy coherence problem and ∞-categories are what we need.
Slides
Apr 26, '21,
Tongtong Liang,
Introduction to spectral sequences
May 10, '21,
Pengcheng Li,
Introduction to spectral sequences, II
May 17, '21, Huiyuan 3-518,
Siheng Yi,
Rational homotopy theory
May 24, '21,
Siheng Yi,
Rational homotopy theory (cont'd)
May 31, '21, 10-11 am,
Yifan Wu,
K-theory
Jun 7, '21, Huiyuan 3-518,
Tongtong Liang,
Steenrod operations