A Unified Approach to Unimodality of Gaussian Polynomials

In 2013, Pak and Panova proved the strict unimodality property of $q$-binomial coefficients $\binom{\ell+m}{m}_q$ (as polynomials in $q$) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients. They showed it to be true for all $\ell,m\geq 8$ and a few other cases. We propose a different approach to this problem based on computer algebra, where we establish a closed form for the coefficients of these polynomials and then use cylindrical algebraic decomposition to identify exactly the range of coefficients where strict unimodality holds. This strategy allows us to tackle generalizations of the problem, e.g., to show unimodality with larger gaps or unimodality of related sequences. In particular, we present proofs of two additional cases of a conjecture by Stanley and Zanello.Comment: Supplementary material at https://wongey.github.io/unimodalit

Induced log-concavity of equivariant matroid invariants

Inspired by the notion of equivariant log-concavity, we introduce the concept of induced log-concavity for a sequence of representations of a finite group. For an equivariant matroid equipped with a symmetric group action or a finite general linear group action, we transform the problem of proving the induced log-concavity of matroid invariants to that of proving the Schur positivity of symmetric functions. We prove the induced log-concavity of the equivariant Kazhdan-Lusztig polynomials of $q$-niform matroids equipped with the action of a finite general linear group, as well as that of the equivariant Kazhdan-Lusztig polynomials of uniform matroids equipped with the action of a symmetric group. As a consequence of the former, we obtain the log-concavity of Kazhdan-Lusztig polynomials of $q$-niform matroids, thus providing further positive evidence for Elias, Proudfoot and Wakefield's log-concavity conjecture on the matroid Kazhdan-Lusztig polynomials. From the latter we obtain the log-concavity of Kazhdan-Lusztig polynomials of uniform matroids, which was recently proved by Xie and Zhang by using a computer algebra approach. We also establish the induced log-concavity of the equivariant characteristic polynomials and the equivariant inverse Kazhdan-Lusztig polynomials for $q$-niform matroids and uniform matroids.Comment: 36 page

The topology of the external activity complex of a matroid

We prove that the external activity complex $\textrm{Act}_<(M)$ of a matroid is shellable. In fact, we show that every linear extension of LasVergnas's external/internal order $<_{ext/int}$ on $M$ provides a shelling of $\textrm{Act}_<(M)$. We also show that every linear extension of LasVergnas's internal order $<_{int}$ on $M$ provides a shelling of the independence complex $IN(M)$. As a corollary, $\textrm{Act}_<(M)$ and $M$ have the same $h$-vector. We prove that, after removing its cone points, the external activity complex is contractible if $M$ contains $U_{3,1}$ as a minor, and a sphere otherwise.Comment: Comments are welcom

Learning, Arts, and the Brain: The Dana Consortium Report on Arts and Cognition

Reports findings from multiple neuroscientific studies on the impact of arts training on the enhancement of other cognitive capacities, such as reading acquisition, sequence learning, geometrical reasoning, and memory

Chaos in Ecology: The Topological Entropy of a Tritrophic Food Chain Model

An ecosystem is a web of complex interactions among species. With the purpose of understanding this complexity, it is necessary to study basic food chain dynamics with preys, predators and superpredators interactions. Although there is an elegant interpretation of ecological models in terms of chaos theory, the complex behavior of chaotic food chain systems is not completely understood. In the present work we study a specific food chain model from the literature. Using results from symbolic dynamics, we characterize the topological entropy of a family of logistic-like Poincaré return maps that replicates salient aspects of the dynamics of the model. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. This work is still another illustration of the role that the theory of dynamical systems can play in the study of chaotic dynamics in life sciences