MaPhyAG Junior

MaPhyAG Junior Seminar

Alejandro Calleja

Configuration spaces of orbits and their application to character varieties

Thursday 4th December 12h Room TBA & Youtube Channel

The configuration space of orbits is a generalization of the well know configuration space of points but for algebraic varieties that are acted by some algebraic reductive group. The interesting fact about these spaces is that their geometry can be studied by relating it to the geometry of the usual configuration spaces of points. In this talk, we will use this approach to compute their E-polynomial in terms of the ones of the algebraic variety and of the algebraic group. We will also show how these spaces can be used to study character varieties of knots, focusing on the case of torus knots.

1st MaPhyAG Junior: 

Mini-course (3h):

 "Mirror-Symmetry and Enumerative Geometry"  by Carlos García Ordoñez 

Thursday 25th September 16h Room 117 & Google Meets

Friday 26th September 12h to 14:30h Room 117 & Google Meets

On the first day of the minicourse, I will briefly introduce the work done with my tutor Marina Logares Jiménez in enumerative geometry and mirror symmetry. I will then discuss enumerative geometry in general and specialise in the enumerative geometry of linear subspaces using Grassmannians and their cohomology. We will end this section by working out a specific example, computing the number of planes in a degree 2 hypersurface of Pˆ4

The second day, we will turn to physics and describe mirror symmetry in Calabi-Yau Manifolds. Mirror symmetry states that for any compact Calabi-Yau manifold modelling the 6 extra dimensions appearing in string Theory, we should have a ˋpartner´ Calabi-Yau Manifolds which produce the same physical output. It will be divided into two sections with a small break between them. 

In the first one, we will explore an enumerative geometry problem, which will lead to Gromov-Witten invariants, allowing us to connect to physics and define the A-model correlation functions. 

In the second section, we will use variations of Hodge structures to explore another alternative to correlation functions, corresponding to the B-model. After doing so, we will properly state the mathematical definición of Mirror Symmetry using its cohomological interpretation.

"Enumerative Geometry and Mirror Symmetry" by Carlos García Ordoñez