AiM

Advances in Mathematics (AiM)

2025-2026

Spectral sequences

• McCleary. A user's guide to spectral sequences. 2nd ed., Cambridge uni., 2001.

• Hatcher. Spectral sequences. Cambridge uni., 2002.

• Fomenko-Fuchs - Homotopical topology. 2nd ed., Springer, 2016.

Lecture notes: SS01 - SS02 - SS03 - S-SS01 - 

See page E_0, E_1

Modular representation theory

• Benson. Representations and cohomology I. Cambridge uni., 1995.

• Benson. Representations and cohomology II. Cambridge uni., 1991.

• Thevenaz. G-algebras and modular representation theory. Lecture note, 1995.

• Some open problems: See here - connection to tensor triangulated category and here - 

Representation theory

• Serre. Linear representations of finite groups. Springer, 1977.

• Adams. Lecture on Lie groups. Uni. of Chicago, 1983.

• Benjamin. Representation theory of finite groups. Springer, 2012.

Algebraic topology (More Concise)

• May. More concise algebraic topology. Uni. of Chicago, 2012.

• May. A concise course in algebraic topology. Uni. of Chicago, 1999.

Algebraic topology

• Hatcher. Algebraic topology. Cambridge uni. , 2002.

• May. A concise course in algebraic topology. Uni. of Chicago, 1999.

• Fomenko-Fuchs - Homotopical topology. 2nd ed., Springer, 2016.

• John M. Lee. Introduction to topological manifolds. 2nd ed., Springer, 2011.

Homological algebra

• Cartan-Eilenberg. Homological algebra. Princeton uni., 1956.

• Weibel. An introduction to homological algebra. Cambridge uni., 1994. 

2024-2025

Applied & theoretical statistics

• James-Witten-Hastie-Tibshirani. An introduction to statistical learning - with applications in R. 2nd ed., Springer, 2021. 

Homological algebra

• Hilton-Stammbach. A course in homological algebra. 2nd ed., Springer, 1997.

• Weibel. An introduction to homological algebra. Cambridge uni., 1994. 

Lecture notes: HA01 - HA02 - HA03 - HA04 - HA05 - HA06 - HA07

Category theory

• Emily Riehl. Category Theory in Context. Dover, 2016. 

• Steve Awodey. Category Theory. 2nd ed., Oxford uni. , 2010.

Algebraic topology

• Hatcher. Algebraic topology. Cambridge uni. , 2002. 

• John M. Lee. Introduction to topological manifolds. 2nd ed., Springer, 2011.

Lecture notes: Fundamental groups - Homology - Cohomology

2023-2024

Differential topology

• John M. Lee, Introduction to smooth manifolds. 2nd ed., Springer, 2013. 

• Bott & Loring. Differential Forms in algebraic topology. Springer, 1982.

Differential geometry

• Barrett O’Neill. Elementary differential geometry. 2nd ed., Elsevier, 2006. 

• Andrew Pressley. Elementary differential geometry. 2nd ed., Springer, 2010. 

Learning roadmap

Possible coming topics: 

• Algebra: Module theory, Commutative algebra, Homological algebra, Representation theory, Modular representation theory, Lie theory, Spectral sequences, Hochschild homology, Sheaf theory, Algebraic K-theory

• Topology: General topology, Algebraic topology, Homotopy theory, Equivariant homotopy theory, (Rational homotopy theory), Stable homotopy theory, Equivariant stable homotopy theory

• Geometry: Differential geometry, Riemannian geometry, Cobordism theory, Morse theory, Algebraic geometry, Symplectic geometry

• Category: Category theory, Triangulated category, Higher category theory, Higher topos theory

Roadmap of learning tools

Aim: To grasp the basic established knowledge

Workshops/ Talkshows

2024

• Workshop: Linear algebra, real analysis, and topology

2025

• Workshop: Homological algebra

• Talk show: VD & MS (RUPP)

2026

• Workshop (offline): Theory - Differential geometry

• Workshop (online): Application - Data science

• Talk show: VD & .... (RUPP)

Learning Basic 

SR: CT - HA - MRT - AG (Wed)

SS: CT - HA - RT - (Sat)

KC: RT - MRT I+II (Wed)

CC: MT - GT - HA - HH (Loday) (Wed)

_Thesis: A study on Hochschild homology (Loday - Ch 1-4)

MT: GT - AT - CT - HA - AT (Hatcher) - KT - (rest for seminar) - SS (All are Hatcher) (Sat)

_Thesis: Spectral sequences and computations

SK: MT - GT - AT (Hatcher) - KT - HA - SS (All are Hatcher) (Sat)

_Thesis: A study on basic homotopy theory (Hatcher)

*A course on SS will be done in a group discussion among all the interested students

CC = Chantong CHEA - KC = Kimsan CHEA - MT = Makara THA - MR = Mean ROUS - SR = Sararn RIN - SS2 = Sovanpiseth SOEURN - SK = Sreykeopichmesa  KIM