À¶Ý®ÊÓƵ

Kaleb Ruscitti, University of À¶Ý®ÊÓƵ

Embedding a family of moduli spaces of SL(2,C) bundles into projective spaces

The moduli space of polystable degree-0 SL(2,C) bundles on a compact connected Riemann surface of genus g>=2 is a Kähler manifold, and an open subset of the moduli space of semi-stable bundles, which is a projective variety of dimension 3g-3. Biswas and Hurtubise constructed a toric degeneration of this moduli space, meaning a family of moduli spaces over C whose fiber over 0 is a toric variety. The toric variety has a moduli interpretation as a space of framed parabolic bundles.

In this talk, I will describe the family and then describe how one can embed the entire family into P^N x C. This is the key step in a current project I am working on, about relating different geometric quantizations of the moduli space of SL(2,C) bundles.

MC 5403

Paul Cusson, University of À¶Ý®ÊÓƵ

Vector bundles over a complex torus

We will cover basic results about the complex geometry of a complex torus X, followed by a discussion of holomorphic vector bundles over X. An immediate result due to Hodge theory is the existence of complex bundles that don't admit holomorphic structures when the complex dimension of X is at least 2. We will thus focus on bundles whose Chern classes lie in the diagonal of the Hodge diamond and ask which ones can be holomorphic.

MC 5403

Joey Lakerdas-Gayle, University of À¶Ý®ÊÓƵ

Effective Algebra 5

We will continue learning about computable Abelian groups following the monograph by Downey and Melnikov.

MC 5417

Rachael Alvir, University of À¶Ý®ÊÓƵ

More Torsion-Free Abelian Groups

We will continue learning about torsion-free Abelian Groups following the Monograph by Downey and Melnikov.

MC 5417

Sean Lee, University of À¶Ý®ÊÓƵ

Dynamics of the Rado graph II

We continue our discussion about the topological dynamics of the automorphism group of the Rado graph.

MC 5417

Faisal Romshoo, University of À¶Ý®ÊÓƵ

Constructing calibrated submanifolds through evolution equations

I will talk about how we can construct examples of calibrated submanifolds using the techniques of evolution equations. We will begin by defining the ideas involved in coming up with these evolution equations and then look at some of the examples of calibrated submanifolds that are constructed this way, following arXiv:math/0008021, arXiv:math/0008155, arXiv:math/0010036 and arXiv:math/0401123.

MC 5403

Xuemiao Chen, University of À¶Ý®ÊÓƵ

On the space of lines

I will make a story about the space of oriented lines in the three dimensional Euclidean space.

MC 5403

Jiahao Hu, Yau Mathematical Sciences Center, Tsinghua University

Homotopy theoretical holomorphic invariants of complex manifolds

In this talk, I will begin by presenting a method for extracting new holomorphic invariants of a complex manifold from its de Rham algebra of complex-valued differential forms. These invariants can be seen as refined versions of the complexified homotopy groups. I will then explore their potential connections with Hermitian geometry. Specifically: (1) the non-abelian part (refined fundamental group) should be related to a generalization of Higgs bundle; (2) the abelian part (refined higher homotopy) may provide new tools for studying the geometry of holomorphic mappings.

MC 5417

Sean Lee, University of À¶Ý®ÊÓƵ

Topological dynamics of the Rado graph

We introduce some concepts from topological dynamics, in particular the universal minimal flow, with the goal of showing that the universal minimal flow of the automorphism group of the Rado graph is the space of linear orders of the Rado graph.

MC 5417

Yash Singh, University of À¶Ý®ÊÓƵ

Vector Bundles on Toric Stacks

We give moduli interpretations of toric vector bundles and generalize this approach to a classification of bundles on arbitrary toric stacks.

MC 5403