| Desmond Coles | abstract |
Tropicalization and Berkovich Analytification of Spherical Varieties |
| Byeongsu Yu | abstract |
When are Zd-graded modules over affine semigroup rings Cohen–Macaulay? |
| Jordy Lopez Garcia | abstract | Extending Irreducibility of Bloch Varieties |
| Josué Tonelli Cueto | abstract |
Kushnirenko's fewnomials, the number of real zeros and condition number |
| Naufil Sakran | abstract | Unipotent Wilf Conjecture |
| Thomas Yahl | abstract | Computing Galois groups of Fano problems |
| C.J. Bott | abstract |
SchubertIdeals.m2 : A Software Package for the Schubert Calculus of Flag Varieties |
| Javier González Anaya | abstract | The 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.
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| abstract Abstract:
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