Basic Minicourses

The target audience for these minicourses is graduate and advanced undergraduate students of mathematics who are new to the field of Low-dimensional Topology.

Minicourse 1: Geometric group theory 

 Hyperbolic groups - Radhika Gupta

a) Morse lemma and invariance under quasi-isometry

b) Quasiconvex subgroups (buildup for Agol-Wise theorem)

Cubulation – Mahan Mj

a) Cat(0) cube complexes, hyperplanes, Gromov criterion in terms of flag complexes

b) Definition and examples of special complexes. Implications of specialness and relation to virtual betti number

Minicourse 2: Knot Theory and 4-manifolds

Knot theory - Shane D'Mello

a) Knots and knot groups 

b) Seifert surfaces and the Alexander polynomial

c) Linking numbers

4-manifolds - Marc Kegel

a) Handlebody description and handle slides, Heegaard splittings for 3-manifolds. 

b) Kirby calculus

c) Kirby’s theorem, Lickorish-Wallace theorem

Minicourse 3: Hyperbolic Geometry and 3-manifolds

 Hyperbolic geometry - Siddhartha Gadgil

a) 2-dimensional hyperbolic geometry, construction of lattices in PSl(2,R)

b) Preissman's theorem and necessity of atoroidal condition for hyperbolization in 3-dimension

c) Statement of hyperbolization

 3-manifolds - Siddhartha Gadgil

a) Loop theorem, sphere theorem, prime decomposition

b) JSJ decomposition 

c) Statement of Geometrisation