Student Geometry/Topology Seminar

Logistics

For Fall 2024, this seminar meets weekly on Wednesdays from 4-5:30pm in MSB 1147. The seminar is co-organized by Trevor Oliveira-Smith and Alex Simons. If you have any questions you can reach out via email at [email protected] and [email protected]

For the purposes of advertising the talks each week and sending out general information about topology/geometry at Davis, we will use the following listserv: [email protected]. You can email [email protected] if you would like to be added to this listserv, or you can go to lists.ucdavis.edu and search "geotopgrads" to manually add yourself.

Description

The purpose is to encourage students studying geometry/topology and related areas to engage in beneficial interactions and increase knowledge of important and basic ideas in topology and geometry, in addition to learning about currently active areas of investigation.

Talks will conceivably cover a broad spectrum, but in general, we envision not-too-technical talks emphasizing ideas. Talks are by students for students, but everyone is welcome. We also encourage speakers to take 2-3 seminars to delve into a topic in depth.

If you would like to give a talk, but on a very introductory level, you may want to consider the pure/applied grad seminar. That seminar has a general audience, while the audience for this seminar can be assumed to have familiarity with some algebraic topology and manifold theory. Practice qualifying exam talks are typically given in the pure/applied grad seminar.

Schedule Information

If you would like to give a talk, please email the seminar organizers (see Logistics above). The tentative schedule is available here:

Sept 27: Trevor Oliveira-Smith "Knots in 3-manifolds"

Oct 2: Trevor Oliveira-Smith "Knots, 3-manifolds, and 4-manifolds"

Oct 9: Bryce Thompson "Introduction to symplectic manifolds"

Oct 16: no seminar (conflicts with department Fall Welcome Event)

Oct 23: Daniela Cortes Rodriguez "Introduction to Weinstein Structures"

Oct 30: Zach Ibarra "Gromov-Witten Invariants, part i"

Nov 6: Zach Ibarra "Gromov-Witten Invariants, part ii"

Nov 13: Alex Simons "Introduction to contact topology"

Nov 20: Alex Simons "Legendrian knot theory"

Nov 27: no seminar (early Thanksgiving break)

Dec 4: Annette Belleman

Possible Seminar Topics

Anybody is welcome to talk about mathematics related to topology. If you would like to talk about something but don't have a topic, here are some suggestions.

The list below may be helpful in your search for a topic. Some items are specific talk ideas while others are keywords to look up and investigate, e.g. using MathSciNet. There are varying levels of expected difficulty, although each topic can really be taken in deep directions. Please email the organizers with any suggestions for this list.

For the more intrepid, you can find topics by looking through the arXiv using categories. Relevant categories include: Geometric Topology | Algebraic Topology | Differential Geometry | Group Theory. These categories will show you the most recent submissions so you can see what people are working on.

In addition, you can ask more experienced students and learn what they find interesting. The organizers are also happy to lend a hand in preparing a talk.

Beginner

[Note that some of these topics may be suitable for the pure/applied grad seminar. Also, one idea is to give a "basic" talk for the pure/applied grad seminar and a more advanced talk for the Student Topology Seminar.]?

• Map colorings and the five color theorem
• Different proofs of Euler's formula: V - E + F =2
• Gauss-Bonnet formula - concept, applications, generalizations, etc.
• Brouwer fixed point theorem -- different proofs, e.g. using Hex or Sperner's Lemma, differential topology (Sard's thm or Stoke's thm)
• Conway's ZIP proof of the classification of compact surfaces
• Dehn's solution to Hilbert's 3rd problem
• Take some family of knots and explain some interesting problems (solved and unsolved) about them. For example, alternating knots, 2-bridge knots, arborescent knots, Montesinos knots.
• Basic properties of knot genus and some results and conjectures
• The lens spaces L(5,1) and L(5,2) are not homeomorphic (a la Alexander)
• Wild/pathological objects
• Horned spheres and crumbled cubes
• Wild arcs -- Fox-Artin arcs
• The Hopf Fibration
• Whitehead manifold
• Lamplighter groups
• Heegaard splittings
• Legendrian knots (definition and classical invariants)
• Find an unsolved problem in Adams' The Knot Book and explain work that's been done on it

Intermediate

• Trisections
• Symplectic and/or contact manifolds
• "Unusual" uses of Sard's theorem
• e.g. in Gauss-Gersten-Stallings proof of the fundamental thm of algebra
• Four color theorem
• Explain main ideas of proof, e.g. "discharging"
• Relation to Lie algebras
• Yang-Baxter equation
• Use Conway notation to classify Euclidean 2-orbifolds/wallpaper groups
• Circle packings and Andreev's theorem
• Lickorish Twist theorem
• Isometric embeddings of the hyperbolic plane into R^3
• Topological proofs of Kneser's Conjecture and/or Grusko's Theorem
• Rubinstein-Thompson 3-sphere recognition algorithm
• Essential surfaces in knot complements take only finitely many boundary slopes
• Classification of Lens Spaces
• An Overview of the Eight 3-dimensional Geometries
• Interesting unknot diagrams and the unknotting problem
• Brown's algorithm for the BNS invariant of two generator one relator groups
• Finite type invariants of knots and 3-manifolds
• Dehn's problems and word-hyperbolic groups
• Recognizing hyperbolic 3-manifolds via SnapPea
• The curve complex and distances of Heegaard splittings
• Pick a topic from Lickorish's or Rolfsen's book on knot theory and discuss related solved and unsolved problems.
• Survey of results on knotted and linked spheres in higher dimensions
• A sample of results:
• No smooth circle knots in R^4 - it would be interesting to just explain the real proof of this
• Zeeman - PL knotting cannot occur in codimension > 2, e.g. no PL knotted S^3 in S^6
• The question of whether PL knotting occurs in codimension 1 is called the PL Schoenflies problem
• Haefliger - smooth knotting can occur in codimensions > 2, e.g. smoothly knotted S^3 in S^6
• Stallings - two S^50's link in dimensions 101, 100, 99, 98, but unlink in 97, 96, and then link again in 95, 94, ..., ??? , ..., 52.
• 4-dimensional Schoenflies problem
• Quantum algorithms and the Jones polynomial
• Equivariant versions of loop and sphere theorems

Advanced

• Double suspension of the Mazur homology 3-sphere is the 5-sphere
• Suggested reference: Daverman, Robert J. Decompositions of manifolds. Pure and Applied Mathematics, 124. Academic Press, Inc., Orlando, FL, 1986. xii+317 pp. ISBN: 0-12-204220-4
• Milnor's exotic 7-spheres
• Suggested reference: Milnor, J., Differential topology. 1964, Lectures on Modern Mathematics, Vol. II pp. 165--183 Wiley, New York (expository article)
• Thurston's compactification of Teichmuller space
• Smales's proof of higher-dimensional Poincare Conjecture
• Perelman's proof of the 3D Poincare Conjecture via Ricci flow
• Suggested reference:Morgan, John W., Recent progress on the Poincare conjecture and the classification of 3-manifolds. Bulletin Amer. Math. Soc. 42 (2005) no. 1, 57-78
• Heegaard Floer homology
• Casson invariant
• Volume Conjectures
• Is Weeks manifold the smallest closed hyperbolic 3-manifold?
• Jones polynomial and volume
• Khovanov Homology and volume