Held semester 2, 2018

##### References:

[D-K] J.F Davis and P.Kirk, *Lecture notes in algebraic topology*, Graduate Studies in Mathematics, 35. American Mathematical Society, Providence, RI, 2001. http://www.indiana.edu/~lniat/book.pdf

[H] A. Hatcher, *Algebraic topology,* Cambridge University Press 2002 https://www.math.cornell.edu/~hatcher/AT/ATpage.html

[H2] A. Hatcher, *Chapter 5 – Spectral Sequences, *Still in progress, http://pi.math.cornell.edu/~hatcher/AT/SSpage.html

[H3] A. Hatcher, *Vector Bundles and K-Theory*, Still in progress, http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html

[Hu] D. Husemoller, *Fibre bundles*, Third edition, GTM 20, Springer-Verlag, New York, 1994

[M] J. P. May, *A concise course in algebraic topology, *Chicago Lectures in Mathematics, The University of Chicago Press, 1999

[M-S] J. Milnor and Stasheff, *Characteristic classes, *Ann. of Math. Studies, No. 76, Princeton University Press, Princeton, N. J., University of Tokyo Press, Tokyo, 1974

[S] R. M. Switzer, *Algebraic topology – homotopy and homology, *Classics in Mathematics, reprint of the 1975 original, Springer-Verlag, 2002

#### Lecture 1: Introduction to cohomology I

Dualise for cohomology; examples of cohomology theories: singular, de Rham

Speaker: Ethan Armitage

Tutor: Marcy Robertson

Reference: [H, Ch 3.1] and [D-K, Ch 2.6]

**Time: 12:30 – 1:30pm, Friday 27/07 – Note the change of time
**Notes

#### Lectures 2 and 3: Introduction to cohomology II

Mayer-Vietoris for cohomology; universal coefficient theorem; examples

Speaker: Michelle Strumila

Tutor: Diarmuid Crowley

Reference: [H, Ch 3.2], [D-K Ch 2.6]

Time: 1-3pm, 02/08 (week 2)

#### Lecture 4: Cup products I

Eilenberg-Zilber theorem; Alexander Whitney map

Speaker: Patrick Elliot

Tutor: Xing Gu

Reference: [D-K]

Time: 1-2pm, 09/08 (week 3)

#### Research talk 1: Infinity modular operads and the Teichmuller tower

The goal of Grothendieck-Teichmuller theory is to study the absolute galois group of the rational numbers via faithful actions on certain geometric objects. In this talk we define the Teichmuller tower and outline its use in studying the absolute galois group of the rationals. We then give the definition of an infinity modular operad and explain their potential for modelling the Teichmuller tower and end with an explicit example.

Speaker: Ethan Armitage

Time: 2-3pm, 09/08 (week 3)

#### Lecture 5: Cup products II

Acyclic Models, Kunneth formula

Speaker: Adam Wood

Tutor: Xing Gu

Time: 1-2pm, 16/08 (week 4)

Reference: [S, 13.24 – 13.28]

Notes

#### Exercise session (Lecture 6)

Tutor: Xing Gu

Time: 2-3pm, 16/08 (week 4)

Notes

#### Lecture 7: Poincare-Lefschetz duality I

Statement and proof

Speaker: Finn McGlade

Tutor: Huijun Yang

Time: 1-2pm, 23/08 (week 5)

Reference: [M, Ch 20], [H]

Notes

#### Research talk 2: Infinity cyclic operads and asteroidal sets

Operads are a multi-input version of categories, while dagger categories are ones where the morphisms have no sense of direction. This talk will begin with an introduction to cyclic operads, which extend categories in both these directions. We will then explore asteroidal sets, the cyclic analogue of dendroidal sets, in order to work with infinity cyclic operads.

Speaker: Michelle Strumila

Time: 2-3pm, 23/08 (week 5)

Notes

#### Lecture 8: Poincare-Lefschetz duality II

Completion of proof and examples

Speaker: Tom Dove

Tutor: Huijun Yang

Time: 1-2pm, 30/08 (week 6)

Reference: [M, Ch 21], [H]

#### Lecture 9: Serre spectral sequence I

Summary statement, proof, and examples

Speaker: Songqi Han

Tutor: TriThang Tran

Reference: [H2]

Time: 2-3pm, 30/08 (week 6)

Notes

#### Lecture 10: Serre spectral sequence II

Speaker: Songqi Han

Tutor: TriThang Tran

Reference: [H2]

Time: 1-2pm, 6/09 (week 7)

Notes are as above

#### Research talk 3

Speaker: Patrick Elliott

Time: 2-3pm, 6/09 (week 7)

#### Lecture 11: The cohomology ring H*(CPn)

Computation, relation to intersection of cycles and Poincar ́e duality

Speaker: Blake Sims

Tutor: Iva Halacheva

Reference:

Time: 1-2pm, 13/09 (week 8)

#### Lecture 12: Vector bundles and classifying spaces

Pull-back, universal bundles and Grassmannians

Speaker: Tom Dove

Tutor: Diarmuid Crowley

Reference: [M-S, Sections 2 and 5]

Time: 2-3pm, 13/09 (week 8)

Notes

#### Lecture 13: Classifying spaces and Characteristic classes

Definition of universal principal G-bundles and characteristic classes

Speaker: Blake Sims

Tutor: Marcy Robertson

Reference: [Hu, 4.1-4.7]

Time: 1-2pm, 20/09 (week 9)

#### Research talk 4

Speaker: Csaba Nagy

Time: 2-3pm, 20/09 (week 9)

Notes

#### Mid-semester Break – no seminar

#### Lecture 14: The Euler class and the Thom isomorphism

Definitions, sketches of proofs, examples

Speaker: Marcy Robertson

Reference: [M-S, Sections 9 and 10]

Time: 1-2pm, 4/10 (week 10)

#### Lecture 15: Poincare-Hopf theorem

Statement, proof, applications

Speaker: Blake Sims

Tutor: Diarmuid Crowley

Reference: [M, Ch. 5]

Time: 2-3pm, 4/10 (week 10)

#### Lecture 16: Chern classes

Definition and basic properties

Speaker: Ethan Armitage

Tutor: Diarmuid Crowley

Reference: [M-S, Ch. 14, pp. 155-173]

Time: 1-2pm, 11/10 (week 11)

#### Research talk 5 – Smooth parametrised Whitehead torsion and the A-theory presheaf

Speaker: Songqi Han

Time: 2-3pm, 11/10 (week 11)

Notes

#### Lecture 17: An Introduction to Topological K-Theory

Definitions, basic properties, and the statement of Bott periodicity

Speaker: Tom Dove

Tutor: Iva Halaceva

Reference: [H3, 2.1 and 2.2]

Time: 1-2pm, 18/10 (week 12)

#### Lecture 18: Chern character

Definition, properties, complex bundles over spheres, division algebras

Speaker: Huijun Yang

Reference: [Hu, 20.9]

Time: 2-3pm, 18/10 (week 12)