Session on Computational Algebra and Applications of Algebra:
Friday, August 9
14.30-15.00
Juan Sabia (Universidad de Buenos Aires, Argentina)
Hilbert's Nullstellensatz, sparse polynomials and degree bounds
Hilbert's Nullstellensatz states that a family $F$ of polynomials has no common zeroes in an algebraically closed field if and only if 1 is in the ideal $F$ generates. In this talk, we present new upper bounds for the degrees of the polynomials appearing in the writing of 1 as a linear combination of the polynomials in $F$ in terms of mixed volumes of convex sets associated with the supports of $F$. Moreover, for an ideal $I$, we give an upper bound for the Noether exponent of $I$ (a notion extremely related to Hilbert's Nullstellensatz) also in terms of mixed volumes of convex sets associated with the supports of a family of generators of $I$. Our bounds are the first one to distinguish the different supports of the arbitrary polynomials involved and can be considerably smaller than previously known ones. This is a joint work with María Isabel Herrero and Gabriela Jeronimo (Universidad de Buenos Aires and CONICET, Argentina).
15.00- 15.30
Laura Matusevich (Texas A&M University, USA)
Computations for monomial ideals in semigroup rings
We extend the notion of standard pairs, originally introduced for monomial ideals in polynomial rings by Sturmfels, Trung and Vogel, to the context of monomial ideals in semigroup rings. In this more general setting, subtle behaviors have to be taken into account, especially in the non-normal case. We discuss how to compute standard pairs, and how to use standard pairs to compute irreducible decompositions.
This is joint work with Byeongsu Yu.
15.40-16.10
Luis David García Puente (Sam Houston State University, USA)
Absolute concentration robustness: Algebra and Geometry foundations
Molecules inside the cell undergo various transformations known as chemical reactions. Chemical Reaction Network Theory correlates qualitative properties of ordinary differential equations corresponding to a reaction network to the network structure. During the last several decades, there has been a lot of impetus in analyzing the long-term dynamics, the existence, uniqueness, multiplicity and stability of fixed points and several other concepts associated to these networks. Particular attention has been paid to the connection between mass-action kinetics of biochemical reaction networks and toric varieties. This algebro-geometric perspective has provided strong insights into the subject.
In this talk, we will focus on the notion of absolute concentration robustness (ACR). In 2010, Shinar and Feinberg introduced this notion in order to study the question of how do cells maintain homeostasis in fluctuating environments? A biochemical system exhibits ACR in some species A if for every positive steady state, regardless of initial conditions, the value of A is the same.
The main goal of this talk will be to establish the relation of ACR to various algebraic objects (ideals and varieties) associated to a reaction network. We will also focus on the problem of deciding whether a network has ACR. We will discuss previous approaches to the problem together with their pitfalls and also introduce our own procedures based on comprehensive Groebner bases and numerical algebraic geometry.
This talk will not assume prior knowledge into the subject. In a broad sense, it will showcase a particular application of advanced computational algebra techniques in the biochemical sciences. This is joint work currently in progress with Elizabeth Gross, Heather Harrington, Matthew Johnson, Nicolette Meshkat, and Anne Shiu.
16.10-16.40
Jack Jeffries (CIMAT, México)
Neural rings
Neural rings and ideals, as introduced by Curto, Itskov, Veliz-Cuba, and Youngs, are a useful algebraic tool for organizing the combinatorial information of neural codes in the brain. In this talk, we will discuss this construction, and survey some of the results on this topic. In the end, we will focus on the notion of polarization of neural rings developed in joint work with Sema Güntürkün and Jeffrey Sun.
17.10-17.40
Rubí Pantaleón Mondragón (CIMAT, México)
An algorithm to detect foliations inside a Hilbert scheme
In 2001, Campillo and Olivares proved that a foliation of degree $d>1$ on the complex projective plane, is uniquely determined by its singular subscheme. This subscheme corresponds to a point in the Hilbert scheme of $d^2+d+1$ points on the projective plane. However, there are elements in the Hilbert scheme that do not come from foliations.
In this presentation, I will talk about a method to detect points in the Hilbert scheme arising from foliations. This method not only detect the foliations, in the affirmative case, it builds them too, which we will see it with some examples.
17.40-18.10
Carlos Valencia (CINVESTAV, México)
$k$-stable and $k$-recurrent ideals of a matrix
In this talk we introduce $k$-stable and $k$-recurrent ideals of a matrix.
Their monomial standard correspond to $k$-stable and $k$-recurrent configurations, which generalizes superstable and recurrent configurations in the chip-firing game.
The concept of $k$-stability of a configuration was inspired on the concept of the $k$-skeleta of the $G$-parking function ideal and the concept of $k$-recurrent configuration is dual to $k$-stability.
18.20-18.50
Víctor Castellanos Vargas (UJAT, México)
The Koszul complex of vector fields and the calculus of the Poincaré-Hopf index
In this conference we will to compute the Poincaré-Hopf index of real analytic vector fields at isolated singularities using the Eisenbud-Levine algebraic formula and we generalize this formula using the Koszul complex, $K$, and the annihilator of the $H_k(K)$ as a $H_0(K)$ module.