This talk is a introduction to the Drinfeld Double Construction, a way to make a new Hopf algebra from a given Hopf algebra so that the new algebra has a solution to the Yang-Baxter equation. This construction of Drinfeld is deeply related to the structure of quantum link invariants. Zoom invitation will be sent five minutes before 3PM

Consider a rational curve, described by a map f : P^1 \to P^n. The Shapiro--Shapiro conjecture says the following: if all the inflection points of the curve (roots of the Wronskian of f) are real, then the curve itself is defined by real polynomials, up to change of coordinates. Equivalently, certain real Schubert varieties in the Grassmannian intersect transversely — a fact with broad combinatorial and topological consequences. The conjecture, made in the 90s, was proven by Mukhin--Tarasov--Varchenko in '05/'09 using methods from quantum mechanics. I will present a generalization of the Shapiro--Shapiro conjecture, joint with Kevin Purbhoo, where we allow the Wronskian to have complex conjugate pairs of roots. We decompose the real Schubert cell according to the number of such roots and, under a certain orientation, we prove that the topological degree of the restricted Wronski map is given by a symmetric group character. In the case where all the roots are real, this implies that the restricted Wronski map is a topologically trivial covering map; in particular, this gives a new proof of the Shapiro-Shapiro conjecture.

The space of binary forms of degree d has a natural stratification given by the factorization pattern of a form, which is indexed by the partitions of $d$. For instance, those binary forms that are $d$-th powers of a linear form trace out a rational normal curve. Those that factor as $a^{(d-1)} b$, with $a,b$ linear forms, describe the tangent developable of the rational normal curve, etc. It is an interesting open problem to describe the defining equations of the closures of the factorization strata, as well as their higher syzygy modules. I will survey some of the known results and recent work on this problem, based on a beautiful interaction between geometry and the representation theory of $\textrm{SL}_2$.