2015-07-15

Complex and Real Rank of Homogeneous PolynomialsThis is an introductory talk of polynomial Waring problem. We will define "rank" of a homogeneous polynomial and introduce the Sylvester original idea to compute rank of binary forms. Then we will formulate his idea as modern terms of commutative algebra so called "APOLARITY LEMMA". Finally some open problems and recent progress will be annonced.

Real Waring Rank of Ternary Polynomials

If we can express the given homogeneous polynomial as the power sum of linear forms, the smallest number of such linear forms is called the waring rank of the polynomial. For the complex case, the general rank(constant on the Zariski open set) is well-known by Alexander and Hirschowitz. But for the real case, the typical rank(constant on the Euclidean open set) is not much known except for the binary case. For the real binary forms, the maximum typical rank is attained when the polynomial has distinct real roots only. From this fact, for the ternary case or the cases with more variables, we expect that hyperbolic polynomials, generalization of the binary forms with distinct real roots, has a maximum(at least big) typical real rank. In this talk, I will introduce some known facts about real rank of ternary forms, and what is expected and what can be done so far.

Parametrizations of Discriminantal Strata, and More.

This talk will treat two issues which correspond to two related mini-symposiums. For the first 15 minutes I will introduce some basics of Euclidean distance degree using beginner-adaptable examples. Afterwards I will discuss about discriminantal strata in the binary case, which generalize the classical discriminant. Roughly, this objects can be seen as the projective dual of the spaces of polynomials having some “types” of singularities. I will introduce parametrizations of this spaces and what they imply.

The Miracles of Tropical Spectral Theory

In this talk, I will introduce the notion of a tropical eigenpair of a tensor. I will discuss surprising theorems and distinguishing features of the tensor theory as contrasted with the matrix theory. No specialized knowledge of either tropical geometry or the theory of tensors will be required to understand the talk.

2015-07-16

Constructing Multistationary Biochemical Reaction Network by Real Algebraic GeometryIn this talk, we introduce the basic ideas and tools in computational real algebraic geometry by constructing multistationaries for dual phosphorylation-dephosphorylation cycle. We see an open problem (in real algebraic geometry) related to the multistationarity of biochemical reaction network is to check the emptiness of semi-algebraic sets.

Using Bertini.m2 in Numerical Algebraic Geometry

In this talk, we will discuss the basics of numerical algebraic geometry and an interface to the numerical algebraic geometry software Bertini.
In particular, the trace test’s recent generalization in “Numerical irreducible decomposition of multiprojective varieties” (joint work with Jon Hauenstein) and its application in “Critical points via monodromy and local methods” (joint work with Abraham Martin del Campo) will be presented.
The Bertini.m2 package is joint work with Dan Bates, Elizabeth Gross, and Anton Leykin.

Exponential Varieties

Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, generalizing those of toric varieties and their moment maps. Another special class, including Gaussian graphical models, are varieties of inverses of symmetric matrices satisfying linear constraints. We present a general theory of exponential varieties, with focus on those defined by hyperbolic polynomials. This is joint work with Mateusz Michałek, Caroline Uhler, and Piotr Zwiernik.

Cubic equations and relations for projective schemes

First, I'd like to explain quadratic equations and their higher syzygies with well known theorems. For cubic generators, we consider the generic initial ideal theory with respect to graded reverse lexicographic order which looks to be good for understanding similar questions about higher linear syzygies of cubic equations.