Student Geometry/Topology Seminar

For Spring 2026, this seminar meets weekly on Mondays from 4-5:00pm 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 tdoliveirasmith@ucdavis.edu and asimons@ucdavis.edu

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: geotopgrads@ucdavis.edu. You can email asimons@ucdavis.edu 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.

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. Furthermore, this seminar is meant to prepare graduate students at Davis to give research and job talks at other universities by providing opportunities to practice giving talks. We especially encourage you to give a talk if you haven't had much practice presenting math. (People on hiring committees have repeatedly said that it is obvious when an applicant has had lots of practice giving math talks, and when they haven't. Get your practice in now so you are a seasoned presenter when you go on the job market!)

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. Practice qualifying exam talks are typically given in the Student Run Research Seminar. However, the Student Run GT Seminar is a great place for practice conference talks or if you are invited to speak at another institution's seminar.

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

• Apr 6: Trevor Oliveira-Smith: “Leveraging Kirby Diagrams”
• Apr 13: Stephen Hutchins: “Summarizing a Proof About Diffeomorphisms of the 2-Sphere”
• Apr 20: Evan Ortiz: “Curvature on Principal Bundles and Characteristic Classes”
• Apr 27: Can Gormez: “Contact Projective Structures ”
• May 4: Annette Belleman
• May 11: Cindy Zhang
• May 18: Daniela Cortes Rodriguez
• May 25: NO SEMINAR–HOLIDAY
• Jun 1: Soyeon Kim

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.

• 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

• 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

• Choose an unsolved problem on the Kirby problem list and explain the most recent progress on that problem.
• 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

• Jan 5: Trevor Oliveira-Smith: “Extending Singular Fibrations of 3-manifolds”
• Jan 12: Trevor Oliveira-Smith: “Intro to Sutured Manifolds”
• Jan 19: MLK Day (no seminar)
• Jan 26: Zach Ibarra: “Characterizations of Connections over the Field of Algebraic Numbers”
• Feb 2: Trevor Oliveira-Smith: “Dehn Surgery and RBG-links”
• Feb 9: Alex Simons “Lagrangian Cobordisms and Functoriality of the Legendrian Contact DGA”
• Feb 16: President's Day (no seminar)
• Feb 23: Cindy Zhang: “The h-cobordism theorem”
• Mar 2: Ian Sullivan “Khovanov Homology's Kirby Object”
• Mar 9: CANCELLED due to sick speaker

• Sep 29: Alex Simons: “A hyperbolic structure on the figure eight knot complement”
• Oct 6: Trevor Oliveira-Smith: “An Intro to Morse Type Functions and Handle Decompositions”
• Oct 13: Cancelled for Indigenous People's Day!
• Oct 20: Cancelled (scheduled speaker was sick)
• Oct 27: Trevor Oliveira-Smith: “Slice vs. Ribbon”
• Nov 3: Bryce Thomson: “Lagrangian 2-knots are Trivial”
• Nov 10: Cancelled (scheduled speaker was sick)
• Nov 17: Cindy Zhang: “Exotic Surfaces in 4-Manifolds and Relative Corks”
• Nov 24: Ian Sullivan
• Dec 1: Zach Ibarra: “Introduction to Moduli Spaces”

• Apr 7: Bryce Thompson: “Classification of Lagrangian Fillings of the Legendrian Hopf Link”
• Apr 14: Josh Turner
• Apr 21: Evan Ortiz
• Apr 28: Evan Ortiz
• May 5: Ian Sullivan
• May 12: Soyeon Kim
• May 19: NO SEMINAR (too many folks at Georgia Topology Conference)
• May 26: NO SEMINAR (Memorial Day)
• June 2: Liza Jacoby

• Jan 8: Tonie Scroggin: “Understanding geometric objects through a combinatorial lens, part 1”
• Jan 22: Tonie Scroggin: “Understanding geometric objects through a combinatorial lens, part 2”
• Jan 29: Tonie Scroggin: “Understanding geometric objects through a combinatorial lens, part 3”
• Feb 5: Ian Sullivan: “Exotically knotted surfaces and their stabilizations through skein lasagna modules”, part 1
• Feb 12: Ian Sullivan: “Exotically knotted surfaces and their stabilizations through skein lasagna modules”, part 2
• Feb 19: Peyton Wood: “Defining n-distinguishing Quandles”
• Mar 12: Zach Ibarra

• 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 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”
• Dec 4: Annette Belleman “Cubical Homotopy Theory”