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Austin Christian

UCLA Mathematics

Student seminar: Symplectic capacities and embedding problems (Fall 2020)

Location: Zoom     Time: Fridays, 3pm-4:30pm

Schedule of talks (abstracts at bottom)

Talks will be 60 minutes in length, followed by 15-30 minutes of discussion.

Date Title Reference(s) Notes
Talk 0
Austin Christian
Oct 2 Propaganda talk [MS17, Ch 12], [HZ12, Ch 2] PDF
Talk 1
Joseph Breen
Oct 9 The Hofer-Zehnder capacity I [MS17, Sec 12.3-5], [HZ12, Ch 1,3] PDF
Talk 2
Joseph Breen
Oct 16 The Hofer-Zehnder capacity II [MS17, Sec 12.3-5], [HZ12, Ch 1,3] PDF
Talk 3
Roman Krutowskiy
Oct 23 ECH capacities background [Hut11, Sec 2], [Hut02], [HT07, Sec 7], [HT13] PDF
Talk 4
Eilon Reisin-Tzur
Oct 30 The full ECH capacities [Hut11, Sec 3],[Hut12] PDF
Talk 5
Tianyu Yuan
Nov 6 The distinguished ECH capacities [Hut11, Secs 4 & 7] PDF/Video
Talk 6
Zachary Smith
Nov 13 ECH capacities and ellipsoid embeddings [McD11], [Hut11] PDF
Talk 7
Austin Christian
Nov 20 Embeddings of toric domains: concave into convex [CG19] PDF
Talk 8
Guoran Ye
Dec 4 Guth's higher-dimensional ellipsoid embeddings [Gut08] PDF
Talk 9
Morgan Weiler (Rice)
Dec 11 Infinite staircases of symplectic embeddings into Hirzebruch surfaces [BHMMMPW20]

References

[BHMMMPW20] Bertozzi, Holm, Maw, McDuff, Mwakyoma, Pires, & Weiler. Infinite staircases for Hirzebruch surfaces . arXiv preprint. arXiv:2010.08567 (2020)

[CCGFHR14] Choi, Cristofaro-Gardiner, Frenkel, Hutchings, & Ramos. Symplectic embeddings into four-dimensional concave toric domains . J. Topol. 7(4) (2014) 1054-1076.

[CHLS07] Cieliebak, Hofer, Latschev, & Schlenk. Quantitative Symplectic Geometry. Cambridge University Press (2007)

[CG19] Cristofaro-Gardiner. Symplectic embeddings from concave toric domains into convex ones . J. Diff. Geom. 112(2) (2019) 199-232.

[EH89] Ekeland & Hofer. Symplectic topology and Hamiltonian dynamics. Math. Z. 200(3) (1989) 355-378.

[EH90] Ekeland & Hofer. Symplectic topology and Hamiltonian dynamics II. Math. Z. 203(4) (1990) 553-567.

[FH94] Floer & Hofer. Symplectic homology I open cets in \( \mathbb{C}^n \). Math. Z. 215(1) (1994) 37-88.

[FHW94] Floer, Hofer, & Wysocki. Applications of symplectic homology I. Math. Z. 217(1) (1994) 577-606.

[Gut08] Guth. Symplectic embeddings of polydisks . Invent. Math. 172(3) (2008) 477-489.

[Hut02] Hutchings. An index inequality for embedded pseudoholomorphic curves in symplectizations . J. Eur. Math. Soc. (JEMS) 4(4) (2002) 313-361.

[Hut11] Hutchings. Quantitative embedded contact homology . J. Diff. Geom. 88(2) (2011) 231-266.

[Hut12] Hutchings. Lecture notes on ECH 8: The ellipsoid example . blog post. URL:https://floerhomology.wordpress.com/2012/07/14/lecture-notes-on-ech-8-the-ellipsoid-example/

[Hut16] Hutchings. Beyond ECH capacities . Geom. Topol. 20(2) (2016) 1085-1126.

[HT07] Hutchings & Taubes. Gluing pseudoholomorphic curves along branched covered cylinders I . J. Symplectic Geom. 5(1) (2007) 43-137.

[HT13] Hutchings & Taubes. Proof of the Arnold chord conjecture in three dimensions II . Geom. Topol. 17(5) (2013) 2601-2688.

[HZ90] Hofer & Zehnder. A new capacity for symplectic manifolds. Analysis et cetera. (1990) 405-428.

[HZ12] Hofer & Zehnder. Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser (2012)

[McD11] McDuff. The Hofer conjecture on embedding symplectic ellipsoids . J. Diff. Geom. 88(3) (2011) 519-532.

[MS17] McDuff & Salamon. Introduction to Symplectic Topology. Oxford University Press (2017)

[MS12] McDuff & Schlenk. The embedding capacity of 4-dimensional symplectic ellipsoids . Ann. Math. (2012) 1191-1282.

[Sie19] Siegel. Higher symplectic capacities . arXiv preprint. arXiv:1902.01490 (2019)

Abstracts

Talk 0. In this introductory talk, we will motivate the study of symplectic embedding problems and discuss some of the ways these problems have been studied. We will introduce the main star of this quarter's seminar, symplectic capacities, and outline the plan for the seminar.

Talks 1 and 2. In these two talks, we will provide the necessary symplectic geometry background for the quarter and define the Hofer-Zehnder capacity. We will use a variational technique to show that this is indeed a symplectic capacity and explore some applications.

Talk 3. In this talk, we'll provide enough background on embedded contact homology (ECH) so that the next few talks may construct the ECH capacities of a four-dimensional Liouville domain.

Talk 4. To a contact 3-manifold \( (Y,\lambda) \) we will associate a sequence of numbers known as its full ECH spectrum, measuring the amount of symplectic action needed to represent certain classes in the ECH. The full ECH capacities of a four-dimensional Liouville domain are then given by the full ECH spectrum of its boundary.

Talk 5. We define a modified version of the full ECH capacities of a Liouville domain, which we call the distinguished ECH capacities. We use ECH cobordisms to prove monotonicity for these capacities.

Talk 6. We present a result of McDuff, which shows that one open four-dimensional ellipsoid embeds symplectically into another if and only if the ECH capacities of the first are no larger than those of the second.

Talk 7. We present a class of symplectic embedding problems for which the ECH capacities give sharp obstructions: the problem of embedding a concave toric domain into a convex one.

Talk 8. In dimensions greater than four, we construct symplectic embeddings \( E(1,S,\ldots,S)\hookrightarrow E(R,R,\infty,\ldots,\infty) \) for a particular (finite) \( R \) and arbitrarily large \( S \). This shows that higher-dimensional embeddings of symplectic ellipsoids are allowed "an infinite amount of squeezing" in certain factors, unlike the four-dimensional case.

Talk 9. In 2012, McDuff and Schlenk analyzed symplectic embeddings of four-dimensional ellipsoids into 4-ball, and found the ellipsoid embedding function of the 4-ball contained a complex number-theoretic structure called an infinite staircase. Recently, Cristofaro-Gardiner--Holm--Mandini--Pires found a conjectural list of all rational convex toric domains in R^4 whose ellipsoid embedding function admits an infinite staircase. We will introduce recent work identifying six new infinite families of infinite staircases (which fall outside the scope of the conjecture of CGHMP), as well as extensive numerical evidence supporting the CGHMP conjecture in the case of one-point blowups of the complex projective plane. We will attempt to explain how we found these staircases and why we believe the CGHMP conjecture is provable in this case.