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Geometric interpretation of valuated term (pre)orders

Valuated term orders are studied for the purposes of Gröbner theory over fields with valuation. The points of a usual tropical variety correspond to certain valuated terms preorders. Generalizing both of these, the set of all ``well-behaved'' valuated term preorders is canonically in bijection with the points of a space introduced in our previous [Read More...]

Presenter: Kalina Mincheva, Tulane University
Authors: Netanel Friedenberg, Kalina Mincheva
Symposium Year: 2023
Session: Tropical Geometry [Organized by Josephine Yu and Abeer Al Ahmadieh]
Presentation Time: October 1, 2023; 9:45 am

The rank 2 symmetric determinantal variety

We discuss a notion of tropical rank for symmetric matrices, and describe its motivation from and differences to the usual notion of the tropical rank of a matrix. We will show how to establish a bijection between the tropical determinantal variety of symmetric tropical rank 2 matrices and with a particular type of phylogenetic trees. We use this bijection [Read More...]

Presenter: May Cai, Georgia Institute of Technology
Authors: May Cai, Kisun Lee, Josephine Yu
Symposium Year: 2023
Session: Tropical Geometry [Organized by Josephine Yu and Abeer Al Ahmadieh]
Presentation Time: October 1, 2023; 10:10 am

Putting the volume back into the volume polynomial

Associated to any divisor in the Chow ring of a simplicial tropical fan, we discuss the construction of a family of polytopal complexes, called normal complexes, which we propose as an analogue of the well-studied notion of normal polytopes from the setting of complete fans. The talk will describe certain closed convex polyhedral cones of divisors for which [Read More...]

Presenter: Anastasia Nathanson, University of Minnesota
Authors: Anastasia Nathanson, Dustin Ross
Symposium Year: 2023
Session: Tropical Geometry [Organized by Josephine Yu and Abeer Al Ahmadieh]
Presentation Time: October 1, 2023; 10:35 am

Matroids as positive vector bundles

What are some properties that one may wish to have in a theory of tropical vector bundles on tropical manifolds? We discuss some possibilities from the perspective of matroids and their tautological [Read More...]

Presenter: Christopher Eur, Harvard University
Symposium Year: 2023
Session: Tropical Geometry [Organized by Josephine Yu and Abeer Al Ahmadieh]
Presentation Time: October 1, 2023; 11:00 am

Community Discussion

A discussion on current developments and open problems in the [Read More...]

Presenter: Abeer Al Ahmadieh, Georgia Institute of Technology
Symposium Year: 2023
Session: Tropical Geometry [Organized by Josephine Yu and Abeer Al Ahmadieh]
Presentation Time: October 2, 2023; 8:30 am

The topology of the moduli spaces of tropical unramified p-covers

Let $G$ be a cyclic group of prime order $p$. The moduli space $\mathcal{M}_{g, G}$ parametrizes smooth curves of genus $g$ with degree $p$ étale coverings with a $G$-action. This space is an étale covering of the moduli space $\mathcal{M}_g$ of curves of degree $p^{2g} - 1$. We study its tropicalization $M^\mathrm{trop}_{g, G}$ via the admissible covering [Read More...]

Presenter: Claudia Yun, University of Michigan
Authors: Yassine El Maazouz, Paul Alexander Helminck, Felix Röhrle, Pedro Souza and Claudia He Yun
Symposium Year: 2023
Session: Tropical Geometry [Organized by Josephine Yu and Abeer Al Ahmadieh]
Presentation Time: October 2, 2023; 8:55 am

The Tropicalization the image of the space of positive semidefinite matrices under the principal minor map.

The principal minor map takes an $n$ by $n$ square matrix with entries in a real field $\mathcal{R}$ and maps it to the $2^n$-vector of its principal minors. In this talk, I will describe the tropicalization of the image of the cone of positive semidefinite matrices under this map using a combinatorial characterization of the support of multiaffine stable [Read More...]

Presenter: Abeer Al Ahmadieh, Georgia Institute of Technology
Authors: Abeer Al Ahmadieh Felipe Rincon, Cynthia Vinzant and Josephine Yu.
Symposium Year: 2023
Session: Tropical Geometry [Organized by Josephine Yu and Abeer Al Ahmadieh]
Presentation Time: October 2, 2023; 9:20 am

Tropical Logistic Regression Model on Space of Phylogenetic Trees

Classification of gene trees is an important task both in analysis of multi-locus phylogenetic data, and assessment of the convergence of Markov Chain Monte Carlo (MCMC) analyses used in Bayesian phylogenetic tree reconstruction. The logistic regression model is one of the most popular classification models in statistical learning, thanks to its [Read More...]

Presenter: Ruriko Yoshida, Naval Postgraduate School
Authors: Georgios Aliatimis, Ruriko Yoshida, Burak Boyaci, and James A. Grant
Symposium Year: 2023
Session: Tropical Geometry [Organized by Josephine Yu and Abeer Al Ahmadieh]
Presentation Time: October 2, 2023; 9:45 am

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