Nielsen equivalence in Coxeter groups; and a geometric approach to group equivariant machine learning

In this thesis we study two main threads. In Part I, we initiate the study of Nielsen equivalence in Coxeter groups—the classification of finite generating sets up to a natural action of the automorphism group of a free group. We explore different Nielsen equivalence invariants and adapt the method of Lustig and Moriah [79] to the Coxeter case. We also adapt the completion sequences of Dani and Levcovitz [31] to give a method of testing when generating sets of right-angled Coxeter groups are Nielsen equivalent. Coxeter systems have a distinguished set of elements, called the reflections, from which generating sets can be drawn. We study generating sets of reflections separately. In this case, the natural notion of equivalence is generated by partial conjugations of one generator by another. This arises naturally for Weyl groups in the context of cluster algebras via quiver mutations [6]. We study this mutation equivalence for Weyl groups, and reflection equivalence for arbitrary Coxeter systems. In the latter case we leverage hyperplane arrangements in the Davis complex associated to a Coxeter system to give geometric criteria from when a set of reflections generates and a test for when generating sets of reflections are reflection equivalent. In Part II, we discuss the other main topic of the thesis is group equivariant machine learning, based on joint work with Aslan and Platt [3]. We propose a novel approach to defining machine learning algorithms for problems which are equivariant with respect to some discrete group action. Our approach involves pre-processing the input data from a learning algorithm by projecting it onto a fundamental domain for the group action. We give explicit and efficient algorithms for computing this projection. We test our approach on three example learning problems, and demonstrate improvements in accuracy over other methods in the literature

At the Interface of Algebra and Statistics

This thesis takes inspiration from quantum physics to investigate mathematical structure that lies at the interface of algebra and statistics. The starting point is a passage from classical probability theory to quantum probability theory. The quantum version of a probability distribution is a density operator, the quantum version of marginalizing is an operation called the partial trace, and the quantum version of a marginal probability distribution is a reduced density operator. Every joint probability distribution on a finite set can be modeled as a rank one density operator. By applying the partial trace, we obtain reduced density operators whose diagonals recover classical marginal probabilities. In general, these reduced densities will have rank higher than one, and their eigenvalues and eigenvectors will contain extra information that encodes subsystem interactions governed by statistics. We decode this information—and show it is akin to conditional probability—and then investigate the extent to which the eigenvectors capture concepts inherent in the original joint distribution. The theory is then illustrated with an experiment. In particular, we show how to reconstruct a joint probability distribution on a set of data by glueing together the spectral information of reduced densities operating on small subsystems. The algorithm naturally leads to a tensor network model, which we test on the even-parity dataset. Turning to a more theoretical application, we also discuss a preliminary framework for modeling entailment and concept hierarchy in natural language—namely, by representing expressions in the language as densities. Finally, initial inspiration for this thesis comes from formal concept analysis, which finds many striking parallels with the linear algebra. The parallels are not coincidental, and a common blueprint is found in category theory. We close with an exposition on free (co)completions and how the free-forgetful adjunctions in which they arise strongly suggest that in certain categorical contexts, the fixed points of a morphism with its adjoint encode interesting information

Stratified noncommutative geometry

We introduce a theory of stratifications of noncommutative stacks (i.e. presentable stable $\infty$-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as $E_n$-monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction, which is closely related to reflection functors and Verdier duality. Our main application is to equivariant stable homotopy theory: for any compact Lie group $G$, we give a symmetric monoidal stratification of genuine $G$-spectra, that expresses them in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory: the adelic stratification of the $\infty$-category of spectra.Comment: Added material on: reflection functors; Verdier duality; t-structures; alignment ("noncommutative general position"); the pullback and refinement operations; central co/augmented idempotents; non-presentable stratifications; categorical fixedpoints; gluing functors for $G$ nonabelian; naive $G$-spectra. (A version with improved formatting is available at https://etale.site/writing/strat.pdf.

Homotopy theory with bornological coarse spaces

We propose an axiomatic characterization of coarse homology theories defined on the category of bornological coarse spaces. We construct a category of motivic coarse spectra. Our focus is the classification of coarse homology theories and the construction of examples. We show that if a transformation between coarse homology theories induces an equivalence on all discrete bornological coarse spaces, then it is an equivalence on bornological coarse spaces of finite asymptotic dimension. The example of coarse K-homology will be discussed in detail.Comment: 220 pages (complete revision

Internal mathematics for stochastic calculus: a tripos-theoretic approach

We approach the problem of understanding the logical aspects of stochastic calculus through topos theoretic methods. In particular, we construct a tripos which encodes a higher-order logic tailor-made for a specific probability space, which we call Scott tripos. Some internal features and constructions of the associated topos are discussed. Furthermore, we study a family of adjoint modal operators arising from a filtration on a probability space. We explore whether these are related to modal operators in process logics (CTL*, PDL) and we give a negative answer

On the Semantics of Petri Nets: Processes, Unfoldings and Infinite Computations.

PhD Thesi