### About

IWoAT Summer School 2023 is a six-day event which will take place at Yanqi Lake Beijing Institute of Mathematical Sciences and Applications in Beijing, China from August 14 to August 19, 2023. The topic of the summer school is “Operads, spectra, and multiplicative structures.”

The summer school is jointly sponsored by Yanqi Lake Beijing Institute of Mathematical Sciences and Applications and Morningside Center of Mathematics, Chinese Academy of Sciences.### Group Photo

### Syllabus and Schedule

#### Syllabus

**Notes**will be updated as they become available.

All lectures including evening sessions will take place in

**Lecture Room A6-1**.

We will also stream the lectures on Zoom (up to speakers' permission). ZOOM: 388 528 9728; Password: please email the organizers for the password.

#### Mon. Aug. 14, Day 1: History; higher homotopies the simple old-fashioned way

**[Notes]**9:30–10:30,

**Peter May**, Lecture 1-1:

*Overview*

**[Notes]**11:00–12:00,

**Meng Guo**, Lecture 1-2:

*$A_\infty$-spaces, classifying spaces, structures on classifying spaces*

**[Notes]**13:30–14:30,

**Yunze Lu**, Lecture 1-3:

*Adjunctions, $(\Sigma_n,\Omega_n)$, monads, and Beck’s monadicity theorem*

**[Notes]**15:00–16:00,

**Shangjie Zhang**, Lecture 1-4:

*Operads, monads, and their algebras; little $n$-cubes*

**[Notes]**19:00–20:00,

**Weinan Lin**, Lecture 1-5:

*James construction and the Hilton–Milnor theorem*

#### Tue. Aug. 15, Day 2: The recognition principle; multiplicative structures

**[Notes]**9:30–10:30,

**Shangjie Zhang**, Lecture 2-1:

*$E_n$-spaces, $E_\infty$-spaces and the recognition principle*

**[Notes]**11:00–12:00,

**Foling Zou**, Lecture 2-2:

*The approximation theorem: history and outline of the proof*

**[Notes]**13:30–14:30,

**Yu Zhang**, Lecture 2-3:

*Operad pairs; examples; the Steiner and linear isometries operads*

**[Notes]**15:00–16:00,

**Meng Guo**, Lecture 2-4:

*Monad pairs, $E_\infty$-ring spaces and $E_\infty$-ring spectra*

**[Notes]**19:00–20:00,

**Weinan Lin**, Lecture 2-5:

*$H_∗(CX)$ and $H_∗(\Omega_n\Sigma_nX)$ as functors of $H_∗(X)$*

#### Wed. Aug. 16, Day 3: Categorical multiplicative structure

**[Notes]**9:30–10:30,

**Hana Jia Kong**, Lecture 3-1:

*Symmetric monoidal and bimonoidal categories, permutative and bipermutative categories, endomorphism operad pairs, and strictification*

**[Notes]**11:00–12:00,

**Yu Zhang**, Lecture 3-2:

*From symmetric bimonoidal categories to $E_\infty$-ring spectra*

**[Notes]**13:30–14:30,

**Ningchuan Zhang**, Lecture 3-3:

*The Barratt–Priddy–Quillen theorem and algebraic K-theory*

**[Notes]**15:00–16:00,

**Peter May**, Lecture 3-4:

*Overview of equivariant generalizations*

19:00–20:00,

**Guchuan Li**, Lecture 3-5:

*The Goerss–Hopkins recognition of $E_\infty$-ring spectra*

#### Thu. Aug. 17, Day 4: Equivariant spaces and spectra

**[Notes]**9:30–10:30,

**Zhipeng Duan**, Lecture 4-1:

*$G$-spaces, $G$-CW complexes, $G$-Postnikov towers, fixed point and orbit adjunctions, homotopy groups and the Whitehead theorem*

**[Notes]**11:00–12:00,

**Zhipeng Duan**, Lecture 4-2:

*Equivariant stable homotopy theory, $G$-prespectra, $G$-spectra, the $(\Sigma_\infty,\Omega_\infty)$-adjunction, and Lewis’s theorem*

#### Fri. Aug. 18, Day 5: Equivariant recognition

9:30–10:30,**Guchuan Li**, Lecture 5-1:

*The additive equivariant recognition principle for $G$-categories*

11:00–12:00,

**Peter May**, Lecture 5-2:

*The multiplicative recognition principle for $G$-categories; the equivariant Barratt–Priddy–Quillen theorem and algebraic K-theory*

**[Notes]**13:30–14:30,

**Ningchuan Zhang**, Lecture 5-3:

*Orbit categories and the equivalence of homotopy categories, coefficient systems, Bredon cohomology of $G$-spaces and axioms*

15:00–16:00,

**Yunze Lu**, Lecture 5-4:

*Mackey functors and $RO(G)$-graded cohomology theory*

19:00–20:00,

**Guchuan Li**, Lecture 5-5:

*The Atiyah–Segal completion theorem, the Segal conjecture, and equivariant cobordism; the evenness conjecture*

#### Sat. Aug. 19, Day 6: Orbital presheaves and questions

9:30–10:30,**Hana Jia Kong**, Lecture 6-1:

*The homotopical Beck monadicity theorem*

11:00–12:00,

**Hana Jia Kong**, Lecture 6-2:

*The general theory of composite adjunctions*

13:30–14:30,

**Peter May**, Lecture 6-3:

*The recognition principle for orbital presheaves; examples: Eilenberg–Mac Lane $G$-spectra, unit $G$-spectra, and Picard spectra*

15:00–16:00,

**Peter May**, Lecture 6-4:

*Ending questions and speculations: homological and motivic applications of the general theory? equivariant and motivic chromatic theory?*

### Lecturers

### Application

Due to space restrictions, we kindly ask you to apply. Depending on your career status, we may require a registration fee. Some financial support is available, including support for lodging, traveling, as well as registration fee waiver. The registration is now closed. We have sent out replies to all applicants; please check your spam folder if you do not see the email.

### Organizers