Jo Nelson

Assistant Professor, Department of Mathematics at Rice University

My research is supported by an NSF CAREER grant
Email: jo.lastname [at] rice [dot] edu Office: 402 Hermann Brown Hall, x5209
Math Alliance Symplectic Zoominar
Rice AWM Chapter Western Hemisphere Virtual Symplectic Seminar
Advice for potential PhD advisees Obstruction Bundle Zooming, Fall 2020
Past Teaching Rice Geometry Seminar
Math 401: Differential Geometry of Curves and Surfaces Math 451/551: Differentiable Manifolds
Statement on Collegiality, Respect, and Sensitivity. I condemn racism and the systematic oppression of Black people. Please join me in continuing the important work that gained traction during the Academic Strike for Black Lives on Wednesday June 10, 2020 to actively promote diversity and inclusion in mathematics, STEM, and academia more broadly.

I would like to acknowledge that the land I occupy today has served as a site of Indigenous peoples, specifically the Atakapa-Ishak, Tāp Pīlam Coahuiltecan, the Sana band of the Tonkawa tribe, and Karankawa nations. Only the Atakapa-Ishak and Tāp Pīlam Coahuiltecan Nations have survived colonization.

UPDATES:


My 2024 IAS Symplectic Geometry Seminar on Algebraic Torsion of Concave Boundaries of Linear Plumbings


My 2023 IAS Members Colloquium on Floer theories and Reeb dynamics


My IAS Scholar Spotlight with fellow AMIAS trustees Michael Hanchard (Social Science) and Tracy Slatyer (Natural Sciences).


I was to be a mentor for at the SYNC workshop for early career researchers in Symplectic and Contact Topology August 8-12, 2022, but then I caught COVID.

My research group's Topology Seminar and Undergrad Colloquium on Symplectic Embeddings on the Virtual BeECH.

Slides of my talk Reflections on Cylindrical Contact Homology at the Symplectic Zoominar on Friday May 15.


For further distraction, check out Prof. Niles Johnson's visualization of the Hopf fibration.
Try to count all the J-holomorphic curves!
On 3/10/2020 we postponed our conference Dubrovnik 2020: Holomorphic curves and low-dimensional topology due to the ongoing COVID-19 pandemic.
My research involves the interactions between symplectic and contact topology and is supported by NSF grants DMS-1840723, DMS-2104411, and CAREER DMS-2142694. Presently, I am working on using geometric techniques to define and compute various flavors of contact homology and related spectral invariants. I'm interested in the relationships between symplectic and contact homology theories. I am also exploring applications to dynamics and symplectic embedding problems.

Congratulations to Dr. Leo Digiosia who defended and deposited his PhD thesis under my supervision in Summer 2022. He will be joining US Bank as a Quantitative Model Development Analyst in January 2023!

A video of my research lecture on Embedded contact homology of prequantization bundles at the Princeton/IAS Symplectic Seminar in November 2022.



Slides of my talk with Morgan Weiler on Embedded contact homology of prequantization bundles at the Western Hemisphere Virtual Symplectic Seminar on Friday May 1, 2020.





A video of my research lecture on ``Equivariant and Nonequivariant Contact Homology" at the 2019 IAS Symplectic Geometry/Dynamics Seminar

My Meet our Faculty profile from Fall 2018 by the Wiess School of Natural Sciences, Rice University.

My survey article From Dynamics to Contact and Symplectic Topology and Back, was published in the 2016 Institute for Advanced Study Newsletter.

Here are my 2019 Colloquium slides, Reeb dynamics and contact invariants.

Here are my research group's Undergraduate Colloquium on Symplectic Embeddings on the Virtual BeECH

Here are my Spring 2020 Undergraduate Colloquium slides, Contact encounters in the third dimension.

Videos of my expository lectures on the origins of cylindrical contact homology at the 2019 IMPA Winter school on Symplectic Topology.



Publications and arXiv preprints.

(with A. Marinković, A. Rechtman, L. Starkston, S. Tanny, L. Wang) Contact structures induced concavely on the on boundaries of linear plumbings, in preparation, 56 pages.

(with M. Weiler) Torus knotted Reeb dynamics and the Calabi invariant, 88 pages, arxiv:2310.18307, submitted.

(with M. Weiler) Torus knot filtered embedded contact homology of the tight contact 3-sphere, J. Topology 2024, (7) Issue 2, 74 pages.

(with L. Digiosia) A contact McKay correspondence for links of simple singularities, 60 pages, revisions in progress.

(with M. Weiler) Embedded contact homology of prequantization bundles, J. Symp. Geom. vol 21, issue 6, 2023, 1077-1189, 113 pages.

(with L. Digiosia, H. Ning, M. Weiler, Y. Yang)* Symplectic embeddings of four-dimensional polydisks into ellipsoids, J. Fix. Point Theory and Appl. 24, 69 (2022), 38 pages.

(with M. Hutchings) S^1-equivariant contact homology for hypertight contact forms, J. Topology 2022, (15) 1455--1539, 84 pages.

(with M. Hutchings) Axiomatic S^1 Morse-Bott theory, Algebraic & Geometric Topology 20 (2020) 1641-1690, 49 pages

Automatic transversality for contact homology II: filtrations and computations, Proceedings of the London Mathematical Society. Volume 120, Issue 6, 853-917, June 2020, 64 pages.

(with K. Christianson)* Symplectic embeddings of 4-dimensional polydisks into balls, Algebraic & Geometric Topology 18 (2018) 2151-2178, 27 pages.

(with L. Abbrescia, I. Huq-Kuruvilla, & N. Sultani)* Reeb Dynamics of the Link of the A_n Singularity, Involve Vol. 10 (2017), No. 3, 417-442, 25 pages.

(with M. Hutchings) Cylindrical contact homology for dynamically convex contact forms in three dimensions, J. Symp. Geom. 14, No, 4, 983-1012, 2016, 29 pages.

From Dynamics to Contact and Symplectic Topology and Back, The Institute for Advanced Study Newsletter, Summer 2016,

Automatic transversality for contact homology I: Regularity, Abh. Math. Semin. Univ. Hambg. 85 (2015), no. 2, 125-179, 54 pages.

* denotes a paper written with undergraduate research mentees.

My research involves the interactions between symplectic, contact and complex geometry and Reeb dynamics. I am working on foundational results in defining and computing contact invariants via pseudoholomorphic curves, often with Michael Hutchings and more recently finding applications to dynamics and low dimensional topology with others.

I obtained my PhD in 2013 under the supervision of Mohammed Abouzaid while at the University of Wisconsin - Madison. Before that I spent 2006-2007 as a foreign exchange student at the Technische Universität Berlin. I earned my bachelors degree in mathematics and minored in German at the University of Illinois at Urbana-Champaign. A long time ago I went to the Illinois Mathematics and Science Academy.

Among other things I enjoy the color purple, zombies, running, swimming, and traveling.

Previously, I was a trustee of the Association of Members at the Institutue for Advanced Study from 2019-2022. I was a Ritt Assistant professor at Columbia University (2017-2018), an NSF MSPRF postdoc at Barnard College, Columbia University (2013-2014, 2016-2017), a Member at the Institute for Advanced Study from 2014-2016, a Visitor at the Institute for Advanced Study from 2013-2014, and at the Simons Center for Geometry and Physics in the spring of 2014.

Here are some exciting workshops and conferences I was involved with in the (not so) recent past:
2017 Eilenberg Lectures by Paul Seidel
2016 Eilenberg Lectures by Denis Auroux

ALIAS: Augmentations and Legendrians at the IAS
Thursday and February 11-12, 2016.

Moduli Problems in Symplectic geometry
IHÉS Summer School, July 6-17, 2015.
Chair of the organizing committee.
Videos of lectures
Link to notes

Transversality in Contact Homology
AIM, December 8-12, 2014