Jo Nelson

Assistant Professor, Department of Mathematics at Rice University

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 Symplectic Group Action
Statement on Collegiality, Respect, and Sensitivity. I condemn racism and the systematic oppression of Black people. Please join me in continuing the important work initiated 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. A very comprehensive set of resources, compiled by Prof. Ila Varma.

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

Slides of my talk with Morgan Weiler on ECH of Prequantization Bundles at the Western Hemisphere Virtual Symplectic Seminar on Friday May 1.

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 to 2022 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 and DMS-2104411. Presently, I am working on using geometric techniques to define and compute various flavors of contact homology and related product structures. I'm interested in the relationships between symplectic and contact homology theories. I am also exploring applications to dynamics and symplectic embedding problems.

My research group consists of Dr. Morgan Weiler (RTG Lovett Instructor 2019-2021) and Leo Digiosia (PhD Candidate). Rice Undergraduates Haoming Ning and Yirong Yang joined us in summer and fall of 2020 to study symplectic embedding problems via the methods of Beyond ECH Capacities.

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 M. Hutchings) S^1-equivariant contact homology for dynamically convex contact forms, in preparation.

(with M. Weiler) Knot filtered embedded contact homology of trefoils in $S^3$, in preparation.

(with L. Digiosia, H. Ning, M. Weiler, Y. Yang)* Symplectic embeddings of four-dimensional polydisks into ellipsoids, 27 pages, arxiv:2010.06687, submitted.

(with M. Weiler) Embedded contact homology of prequantization bundles, 85 pages, arxiv:2007.13883, submitted.

(with M. Hutchings) S^1-equivariant contact homology for hypertight contact forms, 88 pages, arXiv:1906.03457, submitted.

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

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

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

(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.

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

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.

* denotes a paper written with undergraduate research mentees.

I am a trustee of the Association of Members at the Institutue for Advanced Study. Previously 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.

My research involves the interactions between symplectic, contact and complex geometry. I am working on foundational results in defining contact invariants via pseudoholomorphic curves, often with Michael Hutchings and more recently with Morgan Weiler. I've also started thinking about pair of pants products in various formulations of contact homology and relationships to loop spaces, in part with Umberto Hryniewicz.

With Prof. Danielle King (Industrial and Organizational Psychology) and Prof. Mikki Hebl (IO Psychology and Business Management) we are planning two studies to delineate forms of anti-racism in academic advising and to examine the effect of academic advisers’ practices of engaging in direct anti-racism, general inclusion practices or racism on BIPOC student psychological experiences and outcomes. Read more in the press release about our grant from the Race and Anti-Racism Research Fund at Rice University

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.

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