Mathematics

Groupoids: symmetry beyond groups

Rui Fernandes (Department of Mathematics, University of Illinois)

Abstract: The notion of symmetry (of a dynamical system, of some geometry, of some algebraic system, etc.) is often taken as synonym of a group acting. In modern day mathematics, after the works of Charles Ehresmann and Alexander Grothendieck, we have come to realize that groupoids also play a fundamental role in describing symmetry. In this lecture I will discuss these ideas and how my research relates to them.

Rational Surfaces from an Algebraic perspective

Eliana Duarte (UIUC Math)

Abstract: It is common practice in Algebraic Geometry to study objects such as curves and surfaces by embedding them into some projective space. In this talk the objects of study are the blow-up of $\mathbb{P}^2$ and $\mathbb{P}^1\times\mathbb{P}^1$ at a generic set of points. I will explain how to obtain an embedding of the blow-up of $\mathbb{P}^2$ by using linear systems of hypersurfaces in $\mathbb{P}^2$ which vanish at the set of generic points. Using this embedding I will use commutative algebra to obtain the defining equations of the blow up of $\mathbb{P}^2$ at a generic set of points. The goal is to try to do the same process for blow-ups of generic sets of points in $\mathbb{P}^1\times\mathbb{P}^1$.

Forcing and Baire category: an introduction (Part VI, last)

Anton Bernshteyn (UIUC Math)

Abstract: In the last talk of this series we will see how the techniques of forcing are applied to derive properties of $M[G]$ (for instance, to show that $M[G]$ is a model of ZFC). In particular, we will give an overview of the proof that the Continuum Hypothesis is independent of ZFC.

Continued Fractions and Operator Algebras

Florin Boca (UIUC Math)