Deformation Theory and Partition Lie Algebras

A theorem of Lurie and Pridham establishes a correspondence between formal moduli problems and differential graded Lie algebras in characteristic zero, thereby formalising a well-known principle in deformation theory. We introduce a variant of differential graded Lie algebras, called partition Lie algebras, in arbitrary characteristic. We then explicitly compute the homotopy groups of free algebras, which parametrise operations. Finally, we prove generalisations of the Lurie-Pridham correspondence classifying formal moduli problems via partition Lie algebras over an arbitrary field, as well as over a complete local base.Comment: 89 page

When Mommy or Daddy is Gay: Developing Constitutional Standards for Custody Decisions

Divorce can be one of the most traumatic and stressful experiences a person will undergo in his or her lifetime. When the trauma of divorce is intensified by a battle over custody of one\u27s children, the process becomes even more difficult. When someone involved in that process is at the same time dealing with issues of same-sex orientation, perhaps for the first time, it is easy to see why gay, lesbian, or bisexual individuals involved in a dissolution of marriage may experience extreme pressure, since the issue of their sexual orientation could become a critical issue in the court proceedings. When gay and lesbian parents are forced to make the choice between their children and their partners, or when courts make that choice for them by imposing draconian restrictions based on myths about how homosexuality impacts child-rearing, the impact of divorce is particularly heartbreaking

Simplified electrometer has excellent operating characteristics

Simplified and improved electrometer circuit provides high-input impedance, stability of gain and operating point, linear response, and low power requirements

Regional evolutionary distinctiveness and endangerment as a means of prioritizing protection of endangered species

Conservation is costly, and choices must be made about where to best allocate limited resources. I propose a regional evolutionary diversity and endangerment (RED-E) approach to prioritization of endangered species. It builds off of the evolutionary diversity and global endangerment (EDGE) approach, but will allow conservation agencies to focus their efforts on species in specific regions. I used the RED-E approach to prioritize mammal and bird species listed under the U.S. Endangered Species Act (ESA), as well as to make a ranking of species without ESA critical habitat (CH), as a practical application. Regional conservation approaches differ significantly from global approaches. The RED-E approach places a high significance on the level of endangerment of a species, but also allows for very distinct species to have increased prioritization on the RED-E list. Using the CH RED-E list, the U.S. government could begin focusing resources toward endangered and genetically diverse species

Symplectic Autoencoders for Model Reduction of Hamiltonian Systems

Many applications, such as optimization, uncertainty quantification and inverse problems, require repeatedly performing simulations of large-dimensional physical systems for different choices of parameters. This can be prohibitively expensive. In order to save computational cost, one can construct surrogate models by expressing the system in a low-dimensional basis, obtained from training data. This is referred to as model reduction. Past investigations have shown that, when performing model reduction of Hamiltonian systems, it is crucial to preserve the symplectic structure associated with the system in order to ensure long-term numerical stability. Up to this point structure-preserving reductions have largely been limited to linear transformations. We propose a new neural network architecture in the spirit of autoencoders, which are established tools for dimension reduction and feature extraction in data science, to obtain more general mappings. In order to train the network, a non-standard gradient descent approach is applied that leverages the differential-geometric structure emerging from the network design. The new architecture is shown to significantly outperform existing designs in accuracy