Title: Extended Somos and Gale-Robinson sequences, dual numbers, and cluster superalgebras

Date: 04/18/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

In 1980, Michael Somos invented integer sequences that have later been popularized and generalized (among others) by David Gale. A certain mystery around this class of sequences is probably due to their relation with a wealth of different topics, such as: elliptic curves, continued fractions, and more recently with cluster algebras and integrable systems.
I will describe a way to extend Somos-4 and Somos-5 and more general Gale-Robinson sequences, and construct a great number of new integer sequences that also look quite mysterious. The construction is based on the notion of 'cluster superalgebra' (which can be used as a machine to produce integer sequences).
Most of the talk will be accessible to non-experts in any of the above mentioned subjects.

I will begin by defining and giving examples of combinatorial species. I will then explain how they are related to generating functions and how to view some common operations on generating functions in this context. Time permitting I will talk about how combining combinatorial species with the idea of a monoidal category leads to a generalization of Hopf algebras.

Title: Constructing Sard-Smale Fundamental Classes

Date: 04/20/2017

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

The moduli spaces in gauge theory usually arise as generic fibers of a universal moduli space, and invariants are constructed using cobordisms between generic fibers.
I will describe a topological setting that, in important cases, produces a fundamental class on all fibers, and gives an alternative perspective on the resulting invariants.
This is joint work with E. Ionel.

Title: New developments in the theory of smooth actions.

Date: 04/20/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In resent years several new advances in the theory of lattice actions have been made. In this talk I will present some of the key ingredients to these advances. I plan to keep the talk at an elementary level so only some basic notions of measure theory and differentiation on manifolds should be needed.

Title: Numerical methods for energy-based models and its applicability to mixtures of isotropic and nematic flows with anchoring and stretching effects

Date: 04/21/2017

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

The study of interfacial dynamics between two different components has become the key role to understand the behavior
of many interesting systems. Indeed, two-phase flows composed of fluids exhibiting different microscopic structures
are an important class of engineering materials. The dynamics of these flows are determined by the coupling among
three different length scales: microscopic inside each component, mesoscopic interfacial morphology and macroscopic
hydrodynamics. Moreover, in the case of complex fluids composed by the mixture between isotropic (newtonian fluid)
and nematic (liquid crystal) flows, its interfaces exhibit novel dynamics due anchoring effects of the liquid crystal
molecules on the interface.
In this talk I will introduce a PDE system to model mixtures composed by isotropic fluids and nematic liquid
crystals, taking into account viscous, mixing, nematic, stretching and anchoring effects and reformulating the corre-
sponding stress tensors in order to derive a dissipative energy law. Then, I will present new linear unconditionally
energy-stable splitting schemes that allows us to split the computation of the three pairs of unknowns (velocity- pres-
sure, phase field-chemical potential and director vector-equilibrium) in three different steps. The fact of being able
to decouple the computations in different linear sub-steps maintaining the discrete energy law is crucial to carry out
relevant numerical experiments under a feasible computational cost and assuring the accuracy of the computed results.
Finally, I will present several numerical simulations in order to show the efficiency of the proposed numerical
schemes, the influence of the shape of the nematic molecules (stretching effects) in the dynamics and the importance
of the interfacial interactions (anchoring effects) in the equilibrium configurations achieved by the system.
This contribution is based on joint work with Francisco Guill´ en-Gonzal´ ez (Universidad de Sevilla, Spain) and Mar´ıa
´
Angeles Rodr´ıguez-Bellido (Universidad de Sevilla, Spain)