## Bo Lin (林博)

Postdoc
School of Mathematics
Georgia Institute of Technology

### Contact

Email: linbomath [at] gmail [dot] com
bo.lin [at] math [dot] gatech [dot] edu
Phone (office): +1 (404)894-9426
Office: Skiles 017
Mail:
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160 USA

## Publications

Tropical Fermat-Weber Points (joint with Ruriko Yoshida)
SIAM Journal on Discrete Mathematics, 2018, 32(2), 1229–1245.
arXiv version

In a metric space, the Fermat-Weber points of a sample are statistics to measure the central tendency of the sample and it is well known that the Fermat--Weber point of a sample is not necessarily unique in the metric space. We investigate the computation of Fermat--Weber points under the tropical metric on the quotient space $\mathbb{R}^{n} \!/ \mathbb{R} {1}$ with a fixed $n \in \mathbb{N}$, motivated by its application to the space of equidistant phylogenetic trees with $N$ leaves (in this case $n=\binom{N}{2}$) realized as the tropical linear space of all ultrametrics. We show that the set of all tropical Fermat--Weber points of a finite sample is always a classical convex polytope, and we present a combinatorial formula for a key value associated with this set. We identify conditions under which this set is a singleton. We apply numerical experiments to analyze the set of the tropical Fermat--Weber points within a space of phylogenetic trees. We discuss the issues in the computation of the tropical Fermat-Weber points.

Towards a Tropical Hodge Bundle (joint with Martin Ulirsch)
Combinatorial Algebraic Geometry, 353-368, Fields Institute Communications (2017), Volume 80, Springer 2017, Editors: Gregory G. Smith and Bernd Sturmfels.
arXiv version

The moduli space $M_g^{trop}$ of tropical curves of genus $g$ is a generalized cone complex that parametrizes metric vertex-weighted graphs of genus $g$. For each such graph $\Gamma$, the associated canonical linear system $\vert K_\Gamma\vert$ has the structure of a polyhedral complex. In this article we propose a tropical analogue of the Hodge bundle on $M_g^{trop}$ and study its basic combinatorial properties. Our construction is illustrated with explicit computations and examples.

Convexity in Tree Spaces (joint with Bernd Sturmfels, Xiaoxian Tang, and Ruriko Yoshida)
SIAM Journal on Discrete Mathematics, 2017, 31(3), 2015-2038.
arXiv version

We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all ultrametrics. The ${CAT}(0)$ metric of Billera--Holmes--Vogtman arises from the theory of orthant spaces. While its geodesics can be computed by the Owen--Provan algorithm, geodesic triangles are complicated. We show that the dimension of such a triangle can be arbitrarily high. Tropical convexity and the tropical metric exhibit properties that are desirable for geometric statistics, such as geodesics of small depth.

Computing Linear Systems on Metric Graphs
Journal of Symbolic Computation, 2018, 87(3), 54-67.
arXiv version

The linear system of a divisor $D$ on a metric graph has the structure of a cell complex. We introduce the anchor divisors and anchor cells in it – they serve as the landmarks for us to compute the $f$-vector of the complex and find all cells in the complex. A linear system can also be identified as a tropical convex hull of rational functions. We compute its extremal generators using the landmarks. We apply these methods to some examples – namely the canonical linear systems of some small trivalent graphs.

Almost-toric Hypersurfaces
Beiträge zur Algebra und Geometrie, 2016, 57(2), 343-359.
arXiv version

An almost-toric hypersurface is parameterized by monomials multiplied by polynomials in one extra variable. We determine the Newton polytope of such a hypersurface, and apply this to give an algorithm for computing the implicit polynomial equation.

Two-player incentive compatible outcome functions are affine maximizers
(joint with Ngoc Mai Tran)
Linear Algebra and its Applications, 2019, 578, 133-152.
arXiv version

In mechanism design, for a given type space, there may be incentive compatible outcome functions which are not affine maximizers. We prove that for two-player games on a discrete type space, any given outcome function can be turned into an affine maximizer through a nontrivial perturbation of the type space. Furthermore, our theorems are the strongest possible in this setup.

## Preprints

Optimizing coordinate choice for locomotion systems with toroidal shape spaces.
(joint with Baxi Chong et al.)

Geometric mechanics has been used to analyze and design motion of various locomotors. In this framework, the configuration space is decomposed into a shape space and a position space. The internal motion of the system is prescribed by a closed loop in the shape space, which causes net motion in the position space. If the shape space is a simply connected domain in Euclidean space, then with an optimal choice of the body frame, the displacement in the position space is reasonably approximated by the surface integral of the height function, a visualization tool provided by geometric mechanics. Recent work in geometric mechanics extended its scope to more complicated systems, such as the multi-legged system, where the shape space is the torus. However, to the best of our knowledge, the optimal choice of the body frame on the torus shape space was not explored. In this paper, we develop a method to optimally choose the body frame on the torus which results in good approximation of displacement by the integral of the height function. We apply our methods to the centipede locomotion system and observe quantitative agreement of our prediction and experimental results.

Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables. Special families of these polynomials have appeared in economics, discrete convex analysis, and combinatorics. Our theorems rely on an intrinsic characterization of regular mixed subdivisions of integral polytopes, and lead to many open problems of interest in discrete geometry.

We introduce a novel framework for the statistical analysis of phylogenetic trees: Palm tree space is constructed on principles of tropical algebraic geometry, and represents phylogenetic trees as a point in a space endowed with the tropical metric. We show that palm tree space possesses a variety of properties that allow for the definition of probability measures, and thus expectations, variances, and other fundamental statistical quantities. This provides a new, tropical basis for a statistical treatment of evolutionary biological processes represented by phylogenetic trees. In particular, we show that a geometric approach to phylogenetic tree space - first introduced by Billera, Holmes, and Vogtmann, which we reinterpret in this paper via tropical geometry - results in analytic, geometric, and topological characteristics that are desirable for probability, statistics, and increased computational efficiency.

We study the problem of optimal transport on phylogenetic tree space from the perspective of tropical geometry, and thus define the Wasserstein-p distances for probability measures in this continuous metric measure space setting. With respect to the tropical metric---a combinatorial metric on the space of phylogenetic trees---the cases of p=1,2 are treated in detail, which give an efficient way to compute geodesics and also provide theoretical foundations for geometric insight a statistical framework on tree space. We construct explicit algorithms for the computation of the tropical Wasserstein-1 and 2 distances, and prove their convergence. Our results provide the first study of the Wasserstein distances and optimal transport on sets of phylogenetic trees. Several numerical examples are provided.

## In Preparation

Finding Roots of the Tropical Rational Function by using Adversarial Example in Deep Neural Network
(joint with Rui Zhu).

Tropical Exponential Distribution
(joint with Ngoc Mai Tran).

Frechet Means and Stickiness in Palm Tree Space
(joint with Carlos Amendola , Stephan Huckemann, Anthea Monod and Ruriko Yoshida).

Last update: 03/03/2020.

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