I am a Tenure-Track Assistant Professor (CV) at Swarthmore College, in the Philadelphia area, working in the Department of Mathematics and Statistics. My wife teaches at Swarthmore too. Prior to this position I was tenure-track at University of Georgia, and before that an NSF Postdoctoral Fellow at University of California, Berkeley, sponsored by Bernd Sturmfels. I finished my Ph.D. in 2011 at Brown University under the supervision of Dan Abramovich. My undergraduate studies were at University of Washington, working mostly with Jim Morrow, Chuck Doran, Henry Cohn, and Jerry Seidler. Before that, I was home schooled by my hippie parents. If my last name seems familiar in math, there's a good chance you know my brother.

Swarthmore College
Dept of Mathematics and Statistics
Swarthmore, PA 19081
Office: 140 Science Center
Phone: 610-690-6851
Email: ngiansi1@swarthmore.edu
News:
  • An interview with me by Joseph Rickert, discussing topological data analysis, is posted on the RStudio blog (a community for the statistical programming language R).
  • Will be speaking on the author Bely and his mathematician father, Bugaev, at the annual Association for Slavic, East European, and Eurasian Studies conference in Boston in December, 2018.
  • NSF grant proposal approved, funding included for a Math and Law summer high school outreach program for disadvantaged and underrepresented students that will take place in 2019 and 2021.
  • Joshua Mundinger (undergraduate mentee, Swat '18) awarded an NSF graduate fellowship to attend U. Chicago for his PhD.
  • Chul Moon (graduate mentee, UGA '18) completed his statistics PhD in TDA and accepted a tenure-track offer at Southern Methodist University.

Research Overview:

Algebraic geometry is rooted in a classical question: how do we solve a system of polynomial equations? From this perspective, it is a non-linear extension of linear algebra, so there is no surprise that it applies to areas as diverse as physics, engineering, biology, and statistics. Throughout the 20th century, however, algebraic geometry was an engine of modernity. The language of abstraction championed by Grothendieck opened the door to what is presently one of the most active areas of algebraic geometry: moduli spaces. A recurring theme in my research is the particular moduli space M_{0,n}, a compact manifold parameterizing configurations of n points on the Riemann sphere. This has been a remarkably fecund testing ground; many phenomena found in more complicated moduli spaces are manifest here in a combinatorial and concrete manner.

Tropical geometry and matroids have become another branch of my research, primarily the development of modern Grothendieck-style algebro-geometric foundations of these subjects. Tropical geometry, viewed as a combinatorial approach to algebraic geometry, was developed mostly in the past ten years, yet it has already yielded striking applications in variety of subjects. This has led me to the study of idempotent algebras, and in particular to an idempotent module-theoretic framework for matroids.

Topological Data Analysis (TDA) is a collection of tools and ideas aimed at using computational algebraic topology to quantify the higher-dimensional "shape" of data. Much recent work in the field has been in finding optimal ways to integrate TDA methods with machine learning algorithms, in essence allowing classification and prediction based on geometric, not just statistical, structure in data.  My work here has primarily been focused on this TDA-ML interface.

Voting theory offers an interesting combination of pure mathematics and data science.  Axiomatic voting preference models lead to precise predicted behavior that can be measured against actual voting data.  I am combining computational geometry and phylogenetic tools with empirical legal studies methodology to study Supreme Court judicial voting data in order to gain insight into the divisions and alignments of the Court.  I am also interested in exploring more general interactions between mathematics and the legal system, such as Bayesian statistics in evidence weighting.  

I am also interested, more as a casual curiosity, in interactions between mathematics and literature, for instance of the kind summarized here.

Papers:
(Blue author name denotes undergraduate coauthor.)

Algebraic Geometry:

Chow quotients of Grassmannians by diagonal subtori
with Xian Wu.
In preparation.

Equations for point configurations to lie on a rational normal curve
with Alessio Caminata, Han-Bom Moon, and Luca Schaffler.

Advances in Mathematics 340 (2018), 653-683.

Modular interpretation of a non-reductive Chow quotient,
with Patricio Gallardo.

Proceedings of the Edinburgh Mathematical Society 61 no. 2 (2018), 457-477
.

A simplicial approach to the effective cone of \bar{M}_{0,n},
with Brent Doran and Dave Jensen.

International Mathematics Research Notices no. 2 (2017), 529-565
.

Projective linear configurations via non-reductive actions
with Brent Doran.

Preprint on arXiv.

    The dual complex of \bar{M}_{0,n} via phylogenetics
      Archiv der Mathematik 106 no. 6 (2016), 525-529.

        Factorization of point configurations, cyclic covers and conformal blocks,
        with Michele Bolognesi.

          Journal of the European Mathematical Society 17 (2015), 2453-2471. 

          with W.D. Gillam.
          Turkish Journal of Mathematics 38 (2014), 625-648. 

          with Dave Jensen and Han-Bom Moon.
          Advances in Mathematics 248 (2013), 242-278.

          Journal of Algebraic Geometry 22 (2013), 773-793.

          with Angela Gibney.
          Advances in Mathematics 231 (2012), 798-814.

          with Matthew Simpson.
          International Mathematics Research Notices no. 14 (2011), 3315-3334.

          Tropical Geometry/Matroids:

          Grassmannians and Dressians in tropical geometry
          with J.H. Giansiracusa.
          In preparation.

          Projective hypersurfaces in tropical scheme theory
          with J.H. Giansiracusa.
          In preparation.

          with Jacob Manaker.
          Preprint on arXiv.

          with Joshua Mundinger and Colin Crowley.
          Preprint on arXiv.

          with J.H. Giansiracusa.
          Preprint on arXiv.

          with J.H. Giansiracusa.
          Manuscripta Mathematica 156 no. 1 (2018), 187-213.

          with J.H. Giansiracusa.
          Duke Mathematics Journal 165 no. 18 (2016), 3379-3433.

          Topological Data Analysis/Machine Learning:

          Machine learning with Morse slanting
          with Caleb Ho, Joe Jackson, Ryan Shi, and Sam Sokota.
          In preparation.

          Finding signal in persistent homology noise
          with Chul Moon and Nicole Lazar.
          In preparation.

          with Bob Giansiracusa and Chul Moon.
          Under revision at International Journal of Computational Geometry & Applications.

          with Chul Moon and Nicole Lazar.
          Journal of Computational and Graphical Statistics 27 no. 3 (2018), 576-586.

          Math and Law:

          An empirical investigation of an asymmetric voting model
          with Cameron Ricciardi.
          In preparation.

          with Cameron Ricciardi.
          Mathematical Social Sciences 98 (2019), 1-9.

          Under revision at Journal of Empirical Legal Studies.

          with Cameron Ricciardi.
          Preprint on arXiv.

          with Cameron Ricciardi.
          American Mathematical Monthly 125 no. 10 (2018), 867-877.

          Journal of Humanistic Mathematics 6 no. 2 (2016), 207-224.

          Misc:

          with Anastasia Vasilyeva.
          Preprint on arXiv.

          with Anastasia Vasilyeva.
          The Mathematical Intelligencer 40 no. 3 (2018), 2-11.

          The Phoenix (op-ed in Swarthmore student newspaper, response to John Fan).

          with P. Kishor and O. Seneviratne.

            Accepted in, but not presented at, ACMGIS 2009. arXiv.CS/1304.5755