Math 8230: Topics in Topology and Geometry

Email: akram.alishahi@uga.edu
Time and Place: Tuesday-Thursday: 11:10am-12:25pm at TBA
Course webpage:
Office hours: Please email me, and we will schedule a one on one Zoom meeting.

References:

• Expository papers:

  • [G] J. Greene, Heegaard Floer homology
  • [H] J. Hom, Lecture notes on Heegaard Floer homology
  • [L] R. Lipshitz, Heegaard Floer homologies
  • [M] C. Manolescu, An introduction to knot Floer homology
  • [OS-1] P. Ozsváth and Z. Szabó, An introduction to Heegaard Floer homology
  • [OS-2] P. Ozsváth and Z. Szabó, Lectures on Heegaard Floer homology
  • [OS-3] P. Ozsváth and Z. Szabó, Heegaard diagrams and holomorphic disks

• Research Papers

  • [OSz04a] Peter Ozsváth and Zoltán Szabó, Holomorphic disks and topological invariants for closed three-manifolds. Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. arXiv:math/0101206
  • [OSz04b] Peter Ozsváth and Zoltán Szabó, Holomorphic disks and three-manifold invariants: properties and applications. Ann. of Math. (2) 159 (2004), no. 3, 1159–1245. arXiv:math/0105202
  • [OSz04c] Peter Ozsváth and Zoltán Szabó, Holomorphic disks and knot invariants. Adv. Math. 186 (2004), no. 1, 58–116. arXiv:math/0209056
  • [OSz06] Peter Ozsváth and Zoltán Szabó, Holomorphic triangles and invariants for smooth four-manifolds. Adv. Math. 202 (2006), no. 2, 326–400. arXiv:math/0110169
  • [Per08] Timothy Perutz, Hamiltonian handleslides for Heegaard Floer homology. Proceedings of Gökova Geometry-Topology Conference 2007, 15–35, Gökova Geometry/Topology Conference (GGT), Gökova, 2008. arXiv:0801.0564

• Basic Morse theory, symplectic geometry and Fleor homology:

  • [Mi-1] Milnor, Morse theory
  • [Mi-2] Milnor, Lectures on the h-cobordism theorem
  • [Ca] A. Cannas da Silva. Lectures on Symplectic Geometry
  • [Mc] D. McDuff, Floer theory and low-dimensional topology
  • [AD] M. Audin and M. Damian, Morse theory and Floer homology
  • [Hu] M. Hutchings, Lecture notes on Morse homology (with an eye towards Floer theory and pseudoholomorphic curves)

• Low-dimensional topology:

  • [S] N. Saveliev, Lectures on the topology of 3-manifolds
  • [R] D. Rolfsen, *Knots and links
  • [GS] R. Gompf and A. Stipsicz, 4-manifolds and Kirby calculus
  • [L] R. Lickorish, An introduction to knot theory

Prerequisite: The course will assume a basic understanding of smooth manifolds (smooth maps, derivatives, differential forms) and algebraic topology (homology, cohomology).

Homework: Registered students are expectedly to regularly attend class and give a presentation on a topic related to the course. Please come talk to me to discuss the topic of your talk and to schedule a date.

• Suggested Topics:

  • [SW] S. Sarkar and J. Wang, An algorithm for computing some Heegaard Floer homologies, Ann. of Math., 171 (2010), 1213–1236, arXiv:math/0607777.
  • Grid homology from:
    (1) C. Manolescu and P. Ozsváth and S. Sarkar, A combinatorial description of knot Floer homology, Ann. of Math., 169 (2009), 633–660, arXiv:math/0607691.
    (2) P. Ozsváth and A. Stipsicz and Z. Szabó, Grid Homology for Knots and Links, Also available at https://web.math.princeton.edu/~petero/GridHomologyBook.pdf with comment: please go and buy a hard copy, too!
  • J. Hom, A survey on Heegaard Floer homology and concordanceJ. of Knot Theo. and Its Ram.(2) 26 (2017) arXiv:1512.00383
  • K. Honda and W. Kazez and G. Matić, On the contact class in Heegaard Floer homology, J. Differential Geom. (2) 83 (2009), 289-311, arXiv:math/0609734
  • Sutured Floer homology from:
    (1) [L] Lipshitz expository paper listed above
    (2) A. Juhász Holomorphic discs and sutured manifolds Algebr. Geom. Topol., (3) 6 (2006), 1429-1457, arXiv:math/0601443
    (3) A. JuhászKnot Floer homology and Seifert surfaces Algebr. Geom. Topol., (1) 8 (2008), 603-608 arXiv:math/0702514

Exam: This course does not have a final exam.

Academic honesty: As a University of Georgia student, you have agreed to abide by the Universitys academic honesty policy, A Culture of Honesty, and the Student Honor Code. All academic work must meet the standards described in A Culture of Honesty found at: www.uga.edu/honesty. Lack of knowledge of the academic honesty policy is not a reasonable explanation for a violation. Questions related to course assignments and the academic honesty policy should be directed to the instructor.

Tentetive Schedule: