Colloquium

Speaker: Ilia Binder (University of Toronto)

"Computability in Geometric Function Theory and Complex Dynamics"

Time: 15:30 - 16:30

Room: MC107

In this talk, I will discuss recent advances in the computability of various objects arising in Complex Dynamics and the Theory of Univalent functions. After a brief introduction to the general Computability Theory, I will talk about the computation properties of polynomial Julia sets, conformal maps, and Harmonic measure. Based on joint work with M. Braverman (Princeton), A. Glucksam (Northwestern), C. Rojas ( Pontificia Universidad Catolica de Chile), and M. Yampolsky (University of Toronto).

Transformation Groups Seminar

Speaker: Larry So (Western)

"Polyhedral products and their cohomologies"

Time: 09:30 - 10:30

Room: MC 107

Polyhedral products are topological spaces formed by gluing products of spaces according to the inclusion relation of faces in the underlying simplicial complexes. Examples include Davis-Januszkiewicz spaces and moment-angle complexes. There is a beautiful interplay between their topology and the combinatorial information of simplicial complexes, which makes them important in toric topology, algebraic and combinatorial geometry, geometric group theory and homotopy theory. In this seminar I will talk about the definition of polyhedral products, their functorial properties and how to compute their cohomologies. If time permits, then I will mention my recent work that uses polyhedral products to study the realization problem of graded commutative rings.

Transformation Groups Seminar

Speaker: Steven Amelotte (Western)

"Cohomology operations for moment-angle complexes and minimal free resolutions of Stanley-Reisner rings"

Time: 09:30 - 11:00

Room: MC 107

After reviewing some results from last week's talk concerning moment-angle complexes $\mathcal{Z}_K$ and their cohomology rings, I will describe some further structure on $H^*(\mathcal{Z}_K)$ given by cohomology operations induced by the standard torus action. Under the identification of $H^*(\mathcal{Z}_K)$ with the Koszul homology of the Stanley-Reisner ring of $K$, these operations assemble to give an explicit differential on the minimal free resolution of the Stanley-Reisner ring. Using this topological interpretation of the minimal free resolution, we give simple algebraic and combinatorial characterizations of equivariant formality for torus actions on moment-angle complexes. This is joint work with Benjamin Briggs.

Analysis Seminar

Speaker: Blake J. Boudreaux (Western)

"Rational Convexity of Totally Real Sets"

Time: 14:30 - 15:30

Room: MC 107

A compact set $X\subset\mathbb C^n$ is said to be rationally convex if for every point $z\not\in X$ there is a polynomial $P$, depending on $z$, so that $P(z)=0$ but $P^{-1}(0)\cap X=\varnothing$. In view of the Oka-Weil theorem, any function holomorphic on a rationally convex compact $X$ can be approximated uniformly on $X$ by rational functions with poles off $X$. A totally real manifold $M$ is one whose tangent space has no complex structure, i.e., $J(T_pM)\cap T_pM=\{0\}$ for all $p\in M$. $$ $$ By a classical result of Duval-Sibony, a totally real manifold $M$ in $\mathbb{C}^n$ is rationally convex if and only if there exists a Kähler form $dd^c\varphi$ for which $M$ is isotropic. Under a mild technical assumption, we generalize this necessary and sufficient condition to the setting of totally real sets (zero loci of strictly plurisubharmonic functions).

Basic Notions Seminar

Speaker: Taylor Brysiewicz (Western)

"Basic Notions: Numerical Algebraic Geometry"

Time: 15:30 - 16:30

Room: MC 107

Numerical algebraic geometry is a computational framework for studying solution sets to polynomial equations (called varieties) using numerical algorithms. In contrast to symbolic methods (e.g. Grobner bases) which manipulate polynomials, numerical algorithms manipulate points on varieties. In this sense, numerical algebraic geometry may be thought of as the geometric side of computational algebraic geometry. In this talk, I will explain the basics ideas underlying the theory of numerical algebraic geometry.