Topics in topology

2023/2024
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
core mandatory
Group:
M3
ECTS:
6
Language:
slovenian, english
Lecturer (contact person):
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
1
Tutorial
2
Lab
0
Content (Syllabus outline)

The lecturer chooses one or more important topics such as:
Knots and links. Basic invariants. Knots and surfaces, the Seifert form. Connected sum and decomposition into prime knots. Polynomial invariants; Alexander’s and Jones’ polynomials. Khovanov homology. Braid groups. Lattice homology.
3- and 4-manifolds. Constructions: handle decomposition, surgery, Heegaard splitting and covering spaces. Casson’s and Rohlin’s invariants. Fundamental group and its representations. The basics of Seiberg-Witten theory and Heegaard-Floer homology.
The topology of smooth manifolds. Morse theory. Differential forms and de Rham cohomology. Hodge theory. Contact and symplectic structures on manifolds.
Discrete Morse theory. Simplicial complexes and maps: abstract simplicial complexes, geometric realization, subdivision, simplicial approximation. Discrete Morse functions, gradient vector fields, the Morse chain complex and Morse homology, discrete Morse inequalities. Computational algorithms and implementation.
Topological robotics. Configuration space of a robot, motion and navigation plans. The concept of topological complexity of motion planning and navigation; upper and lower bounds on topological complexity.
Persistent homology. Filtrations on spaces, persistence complex and its homology; stability theorems.
Topological methods in group theory. Finiteness properties of groups. Cohomology of infinite groups. Homotopy theory of groups. PL Morse theory.
Characteristic classes of vector bundles. Cohomology ring of a smooth manifold and vector bundles over smooth manifolds. Stiefel-Whitney characteristic classes. Orientability and Thom’s theorem. Complexification; Chern and Pontryagin characteristic classes.

Readings
  • R. Gompf in A. Stipsicz: 4-manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society, 1999.
  • L. Kauffman: On knots, Annals of Mathematics Studies 115, Princeton University Press, 1987.
  • D. Rolfsen: Knots and links, Mathematics lecture series 7, Publish or Perish, 1990.
  • N. Saveliev: Lectures on the topology of 3-manifolds, An introduction to the Casson invariant, De Gruyter, 2012.
  • M. Hirsch: Differential topology, Graduate Texts in Mathematics 33, Springer, 1976.
  • H. Edelsbrunner, J. Harer: Computational Topology: an Introduction, American Mathematical Society, 2010.
  • R. Ghrist: Applied algebraic topology and sensor networks, https://www2.math.upenn.edu/~ghrist/preprints/ATSN.pdf
  • N. Scoville: Discrete Morse Theory, Student Mathematical Library 90, American Mathematical Society, 2019.
  • M. Farber: Invitation to topological robotics, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), 2008.
  • R. Geoghegan, Topological methods in group theory, Graduate Texts in Mathematics 243, Springer, 2008.
  • M. Bestvina, PL Morse theory, Math. Commun. 13 (2008), št. 2, 149–162.
  • J. Milnor, J. Stasheff: Characteristic classes, Annals of mathematics studies 76, Princeton University Press, 1974.
  • D. Husemoller, Fibre bundles, Graduate Texts in Mathematics 20, Springer, 1994.
  • L. Nicolaescu, Lectures on the geometry of manifolds, World Scientific Publishing Co., 2021.
Objectives and competences

The objectives and competences coincide with those of the study program.

Intended learning outcomes

Students get acquainted with one or more important or advanced topics in topology to the extent of being able to be introduced to research problems.

Learning and teaching methods

Lectures, discussion, exercises, homework assignments.

Assessment

Theoretical knowldege exam (oral or written), exercise-based exame (written), homework assignments.

Lecturer's references

Petar Pavešić:
GOVC, Dejan, MARZANTOWICZ, Wacław, PAVEŠIĆ, Petar. Estimates of covering type and the number of vertices of minimal triangulations. Discrete & computational geometry. Jan. 2020, vol. 63, iss. 1, str. 31-48. [COBISS-SI-ID 18627417]
PAVEŠIĆ, Petar. Topological complexity of a map. Homology, homotopy, and applications. Jan. 2019, vol. 21, no. 2, str. 107-130. [COBISS-SI-ID 18590297]
PAVEŠIĆ, Petar. Triangulations with few vertices of manifolds with non-free fundamental group. Proceedings. Section A, Mathematics. Dec. 2019, vol. 149, iss. 6, str. 1453-1463. [COBISS-SI-ID 18671705]
Jaka Smrekar:
SMREKAR, Jaka. Rational Reidemeister trace of an outer automorphism of finite order. Journal of Pure and Applied Algebra. July 2023, vol. 227, iss. 7, art. 107350 (13 str.). ISSN 0022-4049. [COBISS-SI-ID 141555715]
SMREKAR, Jaka. Gottlieb’s theorem for a periodic equivalence. Journal of fixed point theory and its applications. Dec. 2022, vol. 24, iss. 4, art. 73 (15 str.). ISSN 1661-7738. [COBISS-SI-ID 125772547]
FORSTNERIČ, Franc, SMREKAR, Jaka, SUKHOV, Alexandre. On the Hodge conjecture for q-complete manifolds. Geometry & topology. 2016, vol. 20, no. 1, str. 353-388. ISSN 1465-3060. [COBISS-SI-ID 17622361]
Sašo Strle:
RUBERMAN, Daniel, SLAPAR, Marko, STRLE, Sašo. On the Thom conjecture in CP^3. International mathematics research notices, ISSN 1687-0247, 2021, vol. 2022, 1 spletni vir (1 datoteka pdf (14 str.)). [COBISS-SI-ID 89922563]
LEVINE, Adam Simon, RUBERMAN, Daniel, STRLE, Sašo, GESSEL, Ira M. Nonorientable surfaces in homology cobordisms. Geometry & topology, ISSN 1465-3060, 2015, vol. 19, no. 1, str. 439-494. [COBISS-SI-ID 17557337]
OWENS, Brendan, STRLE, Sašo. Dehn surgeries and negative-definite four-manifolds. Selecta mathematica, New series, ISSN 1022-1824, 2012, vol. 18, iss. 4, str. 839-854. [COBISS-SI-ID 16808025]