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The motivation for this paper is a better understanding of the basic building blocks of FJRW theory. The basics of FJRW theory will be briefly outlined, but the majority of the paper will deal with certain multivariate polynomials which are the most fundamental building blocks in FJRW theory. We will first describe what is already known about these polynomials and then discuss several properties we proved as well as conjectures we disproved. We also introduce a new conjecture suggested by computer calculations performed as part of our investigation.

Landau-Ginzburg Mirror Symmetry provides for the construction of two algebraic objects, called the A- and B-models. Special cases of these models–constructed using invertible polynomials and abelian symmetry groups–are well understood. In this thesis, we consider generalizations of the B-model, and specifically address the associativity of the multiplication in these models. We also prove an explicit B-model isomorphism for a class of polynomials in three variables.

Thesis: M.S., Massachusetts Institute of Technology, Sloan School of Management, 1980 / Bibliography: leaf 53. / by Jai Yong Oum. / M.S. / M.S. Massachusetts Institute of Technology, Sloan School of Management

Thesis (M.Sc. (Mathematics)) -- University of Limpopo, 2022 / Our discussion starts with the study of convergence and clustering of filters initiated in pointfree setting by Hong, and then characterize compact and almost compact frames in terms of these filters. We consider the strict extension and show that tQL is a zerodimensional compact frame, where Q denotes the set of filters in L. Furthermore, we study the notion of general filters introduced by Banaschewski and characterize compact frames and almost compact frames using them. For filter selections, we consider F−compact and strongly F−compact frames and show that lax retracts of strongly F−compact frames are also strongly F−compact. We study further the ideals Rs(L) and RK(L) of the ring of realvalued continuous functions on L, RL. We show that Rs(L) and RK(L) are improper ideals of RL if and only if L is compact. We consider also fixed ideals of RL and showthat L is compact if and only if every ideal of RL is fixed if and only if every maximalideal of RL is fixed. Of interest, we consider the class of isocompact locales, which is larger that the class of compact frames. We show that isocompactness is preserved by nearly perfect localic surjections. We study perfect compactifications and show that the Stone-Cˇech compactifications and Freudenthal compactifications of rim-compact frames are perfect. We close the discussion with a small section on Z−closed frames and show that a basically disconnected compact frame is Z−closed.

Thesis advisor: Jan Engelbrecht / Along with the proliferation of digital data collected using sensor technologies and a boost of computing power, Deep Learning (DL) based approaches have drawn enormous attention in the past decade due to their impressive performance in extracting complex relations from raw data and representing valuable information. At the same time, though, rooted in its notorious black-box nature, the appreciation of DL has been highly debated due to the lack of interpretability. On the one hand, DL only utilizes statistical features contained in raw data while ignoring human knowledge of the underlying system, which results in both data inefficiency and trust issues; on the other hand, a trained DL model does not provide to researchers any extra insight about the underlying system beyond its output, which, however, is the essence of most fields of science, e.g. physics and economics. The interpretability issue, in fact, has been naturally addressed in physics research. Conventional physics theories develop models of matter to describe experimentally observed phenomena. Tasks in DL, instead, can be considered as developing models of information to match with collected datasets. Motivated by techniques and perspectives in conventional physics, this thesis addresses the issue of interpretability in general information modeling. This thesis endeavors to address the two drawbacks of DL approaches mentioned above. Firstly, instead of relying on an intuition-driven construction of model structures, a problem-oriented perspective is applied to incorporate knowledge into modeling practice, where interesting mathematical properties emerge naturally which cast constraints on modeling. Secondly, given a trained model, various methods could be applied to extract further insights about the underlying system, which is achieved either based on a simplified function approximation of the complex neural network model, or through analyzing the model itself as an effective representation of the system. These two pathways are termed as guided model design (GuiMoD) and secondary measurements, respectively, which, together, present a comprehensive framework to investigate the general field of interpretability in modern Deep Learning practice. Remarkably, during the study of GuiMoD, a novel scheme emerges for the modeling practice in statistical learning: Algebraic Learning (AgLr). Instead of being restricted to the discussion of any specific model structure or dataset, AgLr starts from idiosyncrasies of a learning task itself and studies the structure of a legitimate model class in general. This novel modeling scheme demonstrates the noteworthy value of abstract algebra for general artificial intelligence, which has been overlooked in recent progress, and could shed further light on interpretable information modeling by offering practical insights from a formal yet useful perspective. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

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