Nathaniel Bottman

400C Kaprielian Hall
Department of Mathematics, USC
3620 S Vermont Ave
Los Angeles, CA 90089

bottman at usc dot edu


Research
Upcoming travel
Teaching
Miscellaneous

Welcome! I am an Assistant Professor (postdoctoral position) at the University of Southern California. As of January 2021, I will have a W2 position as a Research Group Leader at the Max Planck Institute for Mathematics in Bonn.

From 2016 to 2019 I was an NSF Postdoctoral Fellow, with joint appointment at the Institute for Advanced Study and Princeton University. From 2015 to 2016 I was a Postdoctoral Researcher at Northeastern University. I got my PhD from MIT in 2015, advised by Katrin Wehrheim.

I study symplectic geometry, with connections to algebraic geometry and combinatorics. My research aims to relate Fukaya categories of different symplectic manifolds.

From 2019 through 2022, I am supported by NSF Standard Grant DMS-1906220.

Research (click text for abstracts; ** = undergraduate coauthor)

12 (in progress): A cellular decomposition of the Fulton–MacPherson operad from symplectic geometry. Nathaniel Bottman.

We equip the spaces in the Fulton–MacPherson operad with cellular decompositions that are compatible with the operad maps. Our construction is motivated by the author's construction of the symplectic \((A_\infty,2)\)-category.

11: A comparison of group testing architectures for COVID-19 testing. Nathaniel Bottman, Yaim Cooper, and Felix Janda. Preprint.

An important component of every country's COVID-19 response is fast and efficient testing &emdash; to identify and isolate cases, as well as for early detection of local hotspots. For many countries, producing a sufficient number of tests has been a serious limiting factor in their efforts to control COVID-19 infections. Group testing is a well-established mathematical tool, which can provide a serious and rapid improvement to this situation. In this note, we compare several well-established group testing schemes in the context of qPCR testing for COVID-19. We include example calculations, where we indicate which testing architectures yield the greatest efficiency gains in various settings. We find that for identification of individuals with COVID-19, array testing is usually the best choice, while for estimation of COVID-19 prevalence rates in the total population, Gibbs-Gower testing usually provides the most accurate estimates given a fixed and relatively small number of tests. This note is intended as a helpful handbook for labs implementing group testing methods.

10: The 2-associahedra are Eulerian (2019). Nathaniel Bottman, Dylan Mavrides**. 10pp; submitted to the Electronic Journal of Combinatorics.

We show that the 2-associahedra are Eulerian lattices, by exploiting their recursive structure.

9: A compactification of the moduli space of marked vertical lines in \(\mathbb{C}^2\) (2019). Nathaniel Bottman, Alexei Oblomkov. 37pp.

For \(r \geq 1\) and \(\mathbf{n} \in \mathbb{Z}_{\geq0}^r\setminus\{\mathbf{0}\}\), we construct a proper complex variety \(\overline{2M}_{\mathbf{n}}\). \(\overline{2M}_{\mathbf{n}}\) is locally toric, and it is equipped with a forgetful map \(\overline{2M}_{\mathbf{n}} \to \overline M_{0,r+1}\). This space is a compactification of \(2M_{\mathbf{n}}\), the configuration space of marked vertical lines in \(\mathbb{C}^2\) up to translations and dilations. In the appendices, we give several examples and show how the stratification of \(\overline{2M}_{\mathbf{n}}\) can be used to recursively compute its virtual Poincaré polynomial.

8: \((A_\infty,2)\)-categories and relative \(2\)-operads (2018). Nathaniel Bottman, Shachar Carmeli. 13pp; submitted to Higher Structures.

We define the notion of a 2-operad relative to an operad, and prove that the 2-associahedra form a relative 2-operad over the associahedra. Using this structure, we define the notions of an \((A_\infty,2)\)-category and \((A_\infty,2)\)-category in spaces and in chain complexes over a ring. Finally, we show that for any continuous map \(A\to X\), we can associate an \((A_\infty,2)\)-space \(\theta(A\to X)\), which specializes to \(\theta(\text{pt}\to X) = \Omega^2X\) and \(\theta(A \to \text{pt}) = \Omega A\times\Omega A\).

7: Explicit constructions of quilts associated to symplectic reductions (2018). Nathaniel Bottman. Accepted (2019), Kyoto Journal of Mathematics; 9pp.

If \(G\) is a Lie group acting in a Hamiltonian fashion on a symplectic manifold \(M\), we may form the symplectic quotient \(M/\!/G\). Associated to this situation is a Lagrangian correspondence \(\Lambda_G\) from \(M/\!/G\) to \(M\). In this short paper, we construct in two related examples quilts with seam condition given by such a correspondence \(\Lambda_G\), in the case of \(S^1\) acting on \(\mathbb{CP}^2\) with symplectic quotient \(\mathbb{CP}^2/\!/S^1 = \mathbb{CP}^1\). First, we study the quilted strips that would, if not for figure eight bubbling, identify the Floer chain group \(CF(\gamma,S^1_{\text{Cl}})\) and \(CF(\mathbb{RP}^2,T^2_{\text{Cl}})\), where \(\gamma\) is the connected double-cover of \(\mathbb{RP}^1\). Second, we produce a figure eight bubble that was predicted by Akveld–Cannas da Silva–Wehrheim. The figure eight bubbles we construct in this paper are the first concrete examples of this bubbling phenomenon, which is of key importance to functoriality for the Fukaya category.

6: Moduli spaces of witch curves topologically realize the 2-associahedra. Nathaniel Bottman. Accepted (2018), Journal of Symplectic Geometry; 21pp.

For \(r \geq 1\) and \(\mathbf{n} \in \mathbb{Z}_{\geq0}^r\), I construct the compactified moduli space \(\overline{2\mathcal{M}}_{\mathbf{n}}\) of witch curves of type \(\mathbf{n}\). These are the domain moduli spaces for witch balls, analogous to the domain moduli spaces \(\overline{\mathcal{M}}_r\) for pseudoholomorphic polygons. I equip \(\overline{2\mathcal{M}}_{\mathbf{n}}\) with a stratification by the 2-associahedron \(W_{\mathbf{n}}\), and prove that \(\overline{2\mathcal{M}}_{\mathbf{n}}\) is compact, second-countable, and metrizable. In addition, I show that the forgetful map \(\overline{2\mathcal{M}}_{\mathbf{n}} \to \overline{\mathcal{M}}_r\) to the moduli space of stable disk trees is continuous and respects the stratifications.

5: Pseudoholomorphic quilts with figure eight singularity. Nathaniel Bottman. To appear in Journal of Symplectic Geometry 18 (2020), no. 1; 34pp.

I show that the novel figure eight singularity in a pseudoholomorphic quilt can be continuously removed when composition of Lagrangian correspondences is cleanly immersed. The proof of this result requires a collection of width-independent elliptic estimates that allow for non-standard complex structures on the domain.

4: 2-associahedra. Nathaniel Bottman. Algebraic & Geometric Topology 19 (2019), no. 2, 743–806.

For any \(r\geq 1\) and \(\mathbf{n} \in \mathbb{Z}_{\geq0}^r\setminus\{\mathbf0\}\) I construct a poset \(W_{\mathbf{n}}\) called a 2-associahedron. The 2-associahedra arose in symplectic geometry, where they are expected to control maps between Fukaya categories of different symplectic manifolds. I prove that the completion of \(W_{\mathbf{n}}\) is an abstract polytope of dimension \(|\mathbf{n}|+r-3\). There are forgetful maps \(W_{\mathbf{n}}\to K_r\), where \(K_r\) is the \((r−2)\)-dimensional associahedron, and the 2-associahedra specialize to the associahedra (in two ways) and to the multiplihedra. In an appendix, I work out the 2- and 3-dimensional 2-associahedra in detail.

3: Gromov compactness for squiggly strip shrinking in pseudoholomorphic quilts. Nathaniel Bottman, Katrin Wehrheim. Selecta Mathematica (2018) 24, pp. 3381–3443.

We establish a Gromov compactness theorem for strip shrinking in pseudoholomorphic quilts when composition of Lagrangian correspondences is immersed. In particular, we show that figure eight bubbling occurs in the limit, argue that this is a codimension-0 effect, and predict its algebraic consequences — geometric composition extends to a curved \(A_\infty\)-bifunctor, in particular the associated Floer complexes are isomorphic after a figure eight correction of the bounding cochain. An appendix with Felix Schmäschke provides examples of nontrivial figure eight bubbles.

2: Elliptic solutions of the defocusing NLS equation are stable. Nathaniel Bottman, Bernard Deconinck, Michael Nivala. J. Phys. A 44 (2011), no. 28, 24pp.

The stability of the stationary periodic solutions of the integrable (one-dimensional, cubic) defocusing nonlinear Schrodinger (NLS) equation is reasonably well understood, especially for solutions of small amplitude. In this paper, we exploit the integrability of the NLS equation to establish the spectral stability of all such stationary solutions, this time by explicitly computing the spectrum and the corresponding eigenfunctions associated with their linear stability problem. An additional argument using an appropriate Krein signature allows us to conclude the (nonlinear) orbital stability of all stationary solutions of the defocusing NLS equation with respect to so-called subharmonic perturbations: perturbations that have period equal to an integer multiple of the period of the amplitude of the solution. All results presented here are independent of the size of the amplitude of the solutions and apply equally to solutions with trivial and nontrivial phase profiles.

1: KdV cnoidal waves are spectrally stable. Nathaniel Bottman, Bernard Deconinck. Discrete Contin. Dyn. Syst. A 25 (2009), no. 4, 1163–1180.

Going back to considerations of Benjamin (1974), there has been significant interest in the question of stability for the stationary periodic solutions of the Korteweg-deVries equation, the so-called cnoidal waves. In this paper, we exploit the squared-eigenfunction connection between the linear stability problem and the Lax pair for the Korteweg-deVries equation to completely determine the spectrum of the linear stability problem for perturbations that are bounded on the real line. We find that this spectrum is confined to the imaginary axis, leading to the conclusion of spectral stability. An additional argument allows us to conclude the completeness of the associated eigenfunctions.

Upcoming talks/research visits (as of 2/20)

  • Topology seminar, Northeastern University, February 18, 2020
  • Topology seminar, Northwestern University, February 20, 2020
  • Southern California Algebraic Geometry Seminar, UC San Diego, February 29, 2020
  • Workshop "Recent developments in Lagrangian Floer theory," Simons Center, March 16–20, 2020
  • Workshop "Structural aspects of Fukaya categories," Harvard University, May 12–15, 2020
  • BIRS–CMO Workshop, "Locality and Functoriality in Symplectic Geometry", Casa Matemática Oaxaca, October 18—23, 2020.

    Teaching

  • Math 226g, multivariable calculus, Fall 2019, USC
  • Math 5122, graduate course on manifolds, Spring 2016, Northeastern
  • Math 2321, multivariable calculus, Fall 2015, Northeastern
  • Miscellaneous

  • In the note A new approach to Symp, I show that a linear \((A_\infty,2)\)-algebra \(A\) induces an \(A_\infty\)-structure on the bar complex \(TA[1]\).
  • Check out complex-2associahedra, a repository whose main point is a Python script that computes virtual Poincare polynomials of complex 2-associahedra. This is in support of my paper 9 with Alexei Oblomkov.
  • Aleksandar Subotic's thesis: A monoidal structure for the Fukaya category.