Научно-исследовательский семинар "Гомотопическая топология"
- The seminar is intended to introduce the subject area to the students, and to offer them an opportunity to prepare and give a talk.
- Successful participants improve their presentation skills and prepare for participation in research projects in the subject area.
- Axioms for generalized (co)homology.
- Cofibration sequences for spaces. Omega-spectra and cohomology theories
- Fibration sequences for spaces
- First applications: the Dold–Thom theorem, representing rational homotopy classes by manifolds.
- Brown’s representability theorem for cohomology.
- Basic K-theory.
- Complex Bott periodicity; extending the complex K-theory to a cohomology theory.
- Applications of K-theory: the Hopf invariant 1 and almost complex structures on spheres.
- Spectra and stable homotopy category. Homotopy groups of spectra.
- Thom spectra and cobordism. The Pontrjagin–Thom theorem.
- Calculation of π∗(MO) and π∗ (MSO) ⊗ ℚ.
- Whitehead’s theorem for spectra.
- Spectra can be desuspended.
- Fibration and cofibration sequences for spectra.
- Duality for spectra. The Alexander duality.
- The Thom isomorphism for generalized cohomology and the Atiyah duality.
- The topological Riemann – Roch theorem and applications. Schwarzenberger’s conditions on the Chern numbers of complex vector bundles on ℂℙ^n.
- Cumulative gradecumulative grade is proportional to number of tasks solved
- Final exam
- Allen Hatcher. (2002). Algebraic topology. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.87FE219C
- Atiyah, M. F., & Anderson, D. W. (2018). K-theory. Boca Raton: CRC Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1728843