Texas Algebraic Geometry Symposium

Texas A&M University 30 September – 2 October 2022.

College of Arts 
& Sciences

Winner of the poster contest: Byengsu Yu.
Runner up:     Josué Tonelli Cueto.


Desmond Colesabstract Tropicalization and Berkovich Analytification of Spherical Varieties
Byeongsu Yu abstract When are Zd-graded modules over affine semigroup rings Cohen–Macaulay?
Jordy Lopez GarciaabstractExtending Irreducibility of Bloch Varieties
Josué Tonelli Cuetoabstract Kushnirenko's fewnomials, the number of real zeros and condition number
Naufil SakranabstractUnipotent Wilf Conjecture
Thomas YahlabstractComputing Galois groups of Fano problems
C.J. Bottabstract SchubertIdeals.m2 : A Software Package for the Schubert Calculus of Flag Varieties
Javier González AnayaabstractThe geography of negative curves
Layla Sorkatti Nilpotent Symplectic Aternating Algebras


Desmond Coles  Tropicalization and Berkovich Analytification of Spherical Varieties
Abstract: Tropicalization is the process by which algebraic varieties are assigned a "combinatorial shadow". This poster reviews the notion of tropicalization of a toric variety and recent work on extending this to spherical varieties. It also presents recent work on how one can construct a deformation retraction from the Berkovich analytification of a spherical variety to its tropicalization.
Josué Tonelli Cueto  Kushnirenko's fewnomials, the number of real zeros and condition number
Abstract: Unlike complex zeros, we can bound the number of real zeros of a polynomial system only in terms of the number of variables and the number of monomials in the system, independently of the degree of the polynomials. However, as of today, it is open whether or not—even in the bivariate case—this bound is polynomial in the number of monomials. Known as Kushnirenko Hypothesis III, this is one of the biggest open problems in real algebraic geometry. In this poster, we present the probabilistic counterpart of Kushnirenko Hypothesis III and show how this might lead to a new approach toward the resolution of this open problem. Moreover, we show a new separation between real and complex zeros: we demonstrate that well-conditioned real polynomial systems cannot have many zeros. As a consequence, we obtain new probabilistic bounds for the number of real zeros of a random fewnomial system, i.e. a random polynomial system with few monomials. This also paves the way for a new family of numerical algorithms for solving real polynomial systems.
This is joint work with Elias Tsigaridas.
Byeongsu Yu   When are Zd-graded modules over affine semigroup rings Cohen–Macaulay?
Abstract: We give a new combinatorial criterion for Zd-graded modules of affine semigroup rings to be Cohen-Macaulay, by computing the homology of finitely many polyhedral complexes. This provides a common generalization of well-known criteria for affine semigroup rings and monomial ideals in polynomial rings. This is joint work with Laura Matusevich.
Jordy Lopez Garcia Extending Irreducibility of Bloch Varieties
Abstract: Bloch varieties arise from spectral problems on discrete periodic graphs. Upon extending these graphs, we are able to use techniques from discrete geometry and Floquet theory to investigate the irreducibility of their varieties. We present criteria to obtain irreducibility of Bloch varieties for infinite families of discrete periodic operators. This is joint work with Matthew Faust.
Thomas Yahl Computing Galois groups of Fano problems
Abstract: A Fano problem consists of enumerating linear spaces of a fixed dimension on a variety, generalizing the classical problem of 27 lines on a cubic surface. Those Fano problems with finitely many linear spaces have an associated Galois group that acts on these linear spaces and controls the complexity of computing them in coordinates via radicals. Galois groups of Fano problems were first studied by Jordan, who considered the Galois group of the problem of 27 lines on a cubic surface. Recently, Hashimoto and Kadets nearly classified all Galois groups of Fano problems by determining them in a special case and by showing that all other Fano problems have Galois group containing the alternating group. We use computational tools to prove that several Fano problems of moderate size have Galois group equal to the symmetric group, each of which were previously unknown.
Naufil Sakran Unipotent Wilf Conjecture
Abstract: The Wilf Conjecture is a longstanding conjecture regarding complement finite submonoids of the monoid of natural numbers N. There have been several attempts to generalize the conjecture for higher dimensions. We have generalized the conjecture for unipotent linear algebraic groups. We prove our conjecture for certain subfamilies (thick and thin) of the unipotent groups. The relation to algebraic geometry is that these objects have connection with the genus of smooth curves. For example, if we have a pair of rational points on a smooth curve X, the Weierstrass semigroup forms a complement finite submonoid of N2 and the cardinality of the complement is dependent on the genus of the curve.
C.J. Bott  SchubertIdeals.m2 : A Software Package for the Schubert Calculus of Flag Varieties
Abstract: Flag varieties form a fascinating class of algebraic manifolds, with important examples being projective spaces, Grassmannians, Lagrangian Grassmannians, and the full flag manifolds of classical Lie types. We present a Macaulay2 package that does Schubert calculus for flag varieties, i.e. computes intersections of their Schubert subvarieties. In the zero-dimensional case, we use cohomology calculations to count the number of points of intersection. In general, given a list of Schubert varieties in some flag variety, we compute the ideal of the intersection in terms of local coordinates.
Javier González Anaya The geography of negative curves
Abstract: The problem of determining the Mori Dream Space (MDS) property for blowups of weighted projective planes (WPP) at a general point has received renewed interest because of the essential role it plays in Castravet and Tevelev's proof that \bar M_{0,n} is not a MDS. Such a blow-up is a MDS if and only if it contains a non-exceptional negative curve and another curve disjoint from it. From a toric perspective, a WPP is defined by a rational plane triangle. We consider a parameter space of triangles and see how the negative curves and the MDS property vary within it. Using this approach we are able to recover and expand most known results in the area, including examples that do not contain a non-exceptional negative curve. This is Joint work with Jose Luis Gonzalez and Kalle Karu.
Swetank Mohan Heart failure prediction using machine learning model
Abstract: Heart failure is a complex clinical syndrome and not a disease. It prevents the heart from fulfilling the circulatory demands of the body since it impairs the ability of the ventricle to fill or eject blood. The symptoms include breathlessness, ankle swelling, and fatigue, which are often accompanied by signs of structural and/or functional cardiac or non-cardiac abnormalities, such as elevated jugular venous pressure, pulmonary crackles, and peripheral edema.
abstract Abstract:
Instructions You must register for TAGS, and contact Frank Sottile sottile@tamu.edu to let him know that you intend to present a poster at TAGS, as well as providing a title and abstract. Presenters are responsible for printing their own posters, and these should be 36 inches by 48 inches (no larger!).