I will discuss the question of when a contractible manifold fails to be a cork; this happens either if there is no exotic involution on its boundary, or it fails to be Stein. I will give examples of Stein non-corks, and non-Stein non-loose corks. I will discuss open questions of whether there are loose-corks that can not be corks, or if there are infinite order corks (joint with Danny Ruberman).

Title: Symplectic Geometry of the moduli space of Projective Structures on Riemann surfaces.

Date: 11/14/2017

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

The moduli space of quadratic differentials on Riemann surfaces can be viewed as the total space of the cotangent bundle to the moduli space of Riemann surfaces. By choosing a base projective connection which varies holomorphically in moduli, the moduli space of projective structures is identified with the moduli space of quadratic differentials. A projective connection defines, via the monodromy map, a representation of the fundamental group of the Riemann surface into PSL(2,C), i.e. a point in the character variety.
We study the symplectic geometry induced via these maps and show: The homological symplectic structure on the moduli space of quadratic differentials (defined explicitly in terms of Darboux coordinates which involve the double cover arising from a quadratic differential) is identified with the canonical symplectic structure on the cotangent bundle to the moduli space of Riemann surfaces. Choosing the base projective connection as Bergman, Schottky, and Wirtinger induces equivalent symplectic structures on the moduli space of projective connections. Finally we show that under the monodromy map with base Bergman projective connection, the homological symplectic structure induces the Goldman bracket on the character variety. A comparison with the work of Kawai is made. This is joint work with Marco Bertola and Dmitry Korotkin.

Speaker: Eliot Bongiovanni, Michigan State Universtiy

Title: Proving the Least-Area Tetrahedral Tile of Space

Date: 11/14/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

We prove the least-area, unit-volume, tetrahedral tile of Euclidean space, without the assumption that the tiling uses only orientation-preserving images of the tile. Using a graph-theoretical approach, we define a class of tetrahedra that potentially tile with less surface area than the orientation-preserving minimizer, the Sommerville No. 1. We find that without the assumption of orientation preservation, the winner remains the Sommerville No. 1.
The talk summarizes "The Least-Area Tetrahedral Tile of Space" by Eliot Bongiovanni, Alejandro Diaz, Arjun Kakkar, and Nat Sothanaphan, a product of the 2017 NSF Williams College SMALL REU. Preprint: https://arxiv.org/abs/1709.04139

Speaker: Vladimir Chernousov, University of Alberta

Title: Classification of torsors over Laurent polynomial rings

Date: 11/15/2017

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

We will talk about classification of torsors of reductive group schemes over Laurent polynomial rings and applications in infinite dimensional Lie algebras. Joint work with A. Pianzola and P. Gille.

In this talk after going through the statements of Hodge decomposition and sketch of proof, we will try to visit some applications of it in Riemannian Geometry and Complex Geometry.

Title: Cluster Algebras and Representation Theory Part I

Date: 11/16/2017

Time: 10:00 AM - 10:50 AM

Place: C304 Wells Hall

One of the main motivations for the study of cluster algebras is to provide an algebraic-geometric framework for studying problems arising from representation theory. In the first talk, we will focus on the cluster theory of configuration spaces of decorated flags. The tropical sets of the latter spaces parametrize top components of the affine Grassmannian convolution varieties. By the geometric Satake Correspondence, they parametrize bases in the tensor invariants of representations of the Langlands dual groups. If time permits, I will talk about their connections with Fock-Goncharov Duality Conjecture and Mirror Symmetry.

Title: Approximating Continuous Functions on Persistence Diagrams for Machine Learning Tasks

Date: 11/16/2017

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Many machine learning tasks can be boiled down to the following idea: Approximate a continuous function defined on a topological space (the ``ground truth'') given the function values (or approximate function values) on some subset of the points. This formulation has been well studied Euclidean data; however, more work is necessary to extend these ideas to arbitrary topological spaces.
In this talk, we focus on the task of classification and regression on the space of persistence diagrams endowed with the bottleneck distance, (D,d_B). These objects arise in the field of Topological Data Analysis as a signature which gives insight into the underlying structure of a data set. The issue is that the structure of (D,d_B) is not directly amenable to the application of existing machine learning theories. In order to properly create this theory, we will give a full characterization of compact sets in D; provide simple, exemplar functions for vectorization of persistence diagrams; and show that, in practice, this method is quite successful in classification and regression tasks on several data sets of interest.

Speaker: Gavish Nir, Technion Israel Institute of Technology

Title: A model of highly concentrated electrolyte solutions

Date: 11/17/2017

Time: 4:10 PM - 5:00 PM

Place: C100 Wells Hall

The Poisson-Nernst-Planck (PNP) theory is one of the most widely used analytical methods to describe electrokinetic phenomena for electrolytes. The model, however, considers isolated charges and thus is valid only for dilute ion concentrations. The key importance of concentrated electrolytes in applications has led to the development of a large family of generalized PNP models. However, the wide family of generalized PNP models fails to capture key phenomena recently observed in experiments and simulations, such as self-assembly, multiple-time relaxation, and under-screening in concentrated electrolytes.
In this talk, we derive a thermodynamically consistent mean-field model for concentrated solutions that goes beyond the PNP framework. The result is a modeling framework that contains the essential ingredients for describing electrolytes over the whole range of concentrations - from dilute electrolyte solutions to highly concentrated solution, such as ionic liquids. The model describes self-assembly, multiple-time relaxation, and under-screening, and reveals a mechanism of under-screening. Furthermore, the model predicts distinct transport properties which are not governed by Einstein-Stokes relations, but are rather effected by inter-diffusion and emergence of nano-structure.
Joint work with Doron Elad and Arik Yochelis.