Seminars

currently(Fall 2018) i am organizing homotopy theory and $K$-theory seminar

homotopy theory and k-theory reading seminar（Spring 2019)

This semester's homotopy theory and $K$-theory seminar we will be focusing on $\infty$-categories and simplicial homotopy theory.
Meeting time and location:

Each Friday 10:30am at 4N30

Schedules

Week 1 (Feb 1)
(Darrick) Simplicial sets, geometric realization and simplicial homotopy. Here are two other introductions to simplicial sets: https://arxiv.org/abs/0809.4221 and http://www.math.jhu.edu/~eriehl/ssets.pdf
Week 2 (FEB 8)
(Darrick) Anodyne extensions, function complexes and simplicial homotopy groups (1.4 - 1.7 in the book)
Week 3 (Feb 15)
(Darrick) Simplicial homotopy groups and minimal fibrations.
Week 4 (Feb 22)
(Prof. Block) Intro to model categories
Week 5 (Mar 1)
(Prof. Block) Model categories continued
Week 6 (Mar 7) Spring break, no seminar
Week 7 (Mar 14)
(Andy) Motivations for higher categories especially $(\infty,1)$-categories from both categorical and homotopical/topological points of view

references

homotopy theory and k-theory reading seminar（Fall 2018)

Together with Thomas and Jingye, I am running the Homotopy theory and $K$-theory learning seminar. Our plan is to start from Lecture notes on K-theory and Categorical homotopy theory, we hope to get a solid understanding higher algebraic K-theory and categorical homotopy theory by the end of this semester, and probably learn some $\infty$-categories theory.
Meeting time and location:

Each Wednesday 3:00pm at 4N49 and Friday 1:30pm at 4C6

references

Preliminary readings
Main references
On higher categories
  • Towards higher categories
  • Higer topos theory by Jacob Lurie

    • On Algebraic K-theory
    On Topological K-theory

    Schedules

    Week 1 (Sep 14)
  • #1. Introduction to algebraic $K$-theory (Thomas)

    • Week 2 (SEP 21) at 4C6
  • #2. Categories and functors (Jingye)

    • Week 3 (Sep 26)
  • (Sep 26) #3. 1) More categories (Jingye). 2) Trasnformations and equivalences (Perry) notes references: Ch. 4 of Rognes, and Ch.4 of Part III Category theory notes
  • (Sep 28)#4. Homotopy theory (Tianyue)

    • Week 4
  • (Oct 3)#5. Universal properties (Perry) notes

    • Week 5
  • (Oct 10)#6. Homotopy colimits and simplicial method (Thomas), references: first chapter of Dugger's notes for homotopy colimits, and Friedman's lovely intro to simplicial sets which I recommend everyone read
  • (Oct 12)#7 Simplicial homotopy, bisimplicial sets, and Dold-Kan correspondance (Andy) notes references: sec. 6.3-6.6 in Rognes, sec. 8.3-8.4 of Weibel's Introduction to homological algebra.

    • Week 6
  • (Oct 17)#8 Homotopy theory over categories and Quillen's theorem A and theorem B (Thomas) references: Rognes ch7, and Quillen's Higher algebraic K-theory I.
  • (Oct 19)#9 1) Intro to higher categories (Thomas). 2) $K_0$ in various setting (rings, symmetric monoids, exact categories, and Waldhausen categories) (Perry) Notes references: Ch. I of The K-Book.

    • Week 7
  • (Oct 24)#10 $K_1$, $K_2$ for rings and fields, Waldhausen caregories (Perry). Notes
  • (Oct 26) Update: we rescheduled this talk to Oct.29 (Monday). #11 Quillen's higher algebraic K-theory: plus construction and $Q$-construction (Andy) Notes References: Quillen's Higher algebraic K-theory I and Sec. 2.1-2.5 of Homotopical algebra by Yuri Berest and Sasha Patotski

    • Week 8
  • (Oct 31) no seminar today
  • (Nov 2)#12 $S^{-1}S$ construction, $K$-group of symmetric monoidal categories, and "+=$Q$" theorem (Andy) Notes

    • Week 9
  • (Nov 7)
    • #13 Sketch of the proof of "$S^{-1}=Q$", delooping in $K$-theory, and higher topological $K$-theory (Andy) notes
  • (Nov 9) no seminar

    • Week 10
  • (Nov 12) #14Higher Waldhausen $K$-theory (Perry) notes
  • (Nov 14) #15 Higher Waldhausen $K$-theory continued, introduction to delooping $K$-theory, and the iterated $S_{\bullet}$-construction (Perry) notes

    • Week 11
  • (Nov 19) #16 Spectra and cohomology theories (Thomas)
  • (Nov 21)non seminar

    • Week 12
  • (Nov 28) #17 Operads, $A_{\infty}$ spaces, $E_{\infty}$ spaces, recognition principles, and their connections to $E_{\infty}$-ring spectra (Andy) notes References: 1. What is an operad? by Jim Stacheff. 2. The geometry of iterated loop spaces by Peter May. 3.
  • (Nov 30) #18 (Thomas) "all concepts are Kan extensions." This talk will unify some discussions about homotopy theory and category theory we have had throughout the semester, and will include a brief discussion of derived functors. The reference is Emily Riehl's "Categorical Homotopy Theory" chapters 1-2.

    • Possible furture topics
  • Topological Hochschild homology and Topological cyclic homology
  • Motivic homotopy theory
  • Noncommutative geometry and index theory
  • Higher category theory