Mathematical Simulations in Topology and Their Role in Mathematics Education

This thesis presents and discusses several software projects related to the learning of mathematics in general and topological concepts in particular, collecting the results from several publications in this field. It approaches mathematics education by construction of mathematical learning environments, which can be used for the learning of mathematics, as well as by contributing insights gained during the development and use of these learning environments. It should be noted that the presented software environments were not built for the use in schools or other settings, but to provide proofs of concepts and to act as a basis for research into mathematics and its education and communication. The first developed and analyzed environment is Ariadne, a software for the interactive visualization of dots, paths, and homotopies of paths. Ariadne is used as an example of a “mathematical simulation”, capable of supporting argumentation in a way that may be characterized as proving. The software was extended from two to three dimensions, making possible the investigation of two-dimensional manifolds, such as the torus or the sphere, using virtual reality. Another extension, KnotPortal, enables the exploration of three-dimensional manifolds represented as branched covers of knots, after an idea by Bill Thurston to portray these branched covers of knots as knotted portals between worlds. This software was the motivation for and was used in an investigation into embodied mathematics learning, as this virtual reality environment challenges users to determine the structure of the covering by moving their body. Also presented are some unpublished projects that were not completed during the doctorate. This includes work on concept images in topology as well as software for various purposes. One such software was intended for the construction of closed orientable surfaces, while another was focused on the interactive visualization of the uniformization theorem. The thesis concludes with a meta-discussion on the role of design in mathematics education research. While design plays an important role in mathematics education, designing seems to not to be recognized as research in itself, but only as part of theory building or, in most cases, an empirical study. The presented argumentation challenges this view and points out the dangers and obstacles involved

Energy and Charge Transfer in Organic Materials and Its Spectroscopic Signature: An Ab Initio Approach

Energy and charge transfer processes in organic materials have received a tremendous amount of attention in recent years, due to their impact on functionality within a wide range of applications. One prominent example is the field of organic photovoltaics (OPVs), where significant improvements in power conversion efficiency and durability have been achieved over the last decade. Another example is organic scintillators, which have seen a renewed interest due to the constrained supply of helium–3 gas, as well as their ability to discriminate between types of ionizing radiation. Advancement in the design of organic photovoltaic and luminescent materials can be facilitated by molecular level insights into the processes of energy transfer, gained through both experimental observations and theoretical and computational modeling. Thus, this thesis utilizes computational techniques to investigate excited states, and their spectroscopic signatures, in molecular systems that are experimentally relevant for OPVs and organic scintillators. In Chapter II of this thesis, a computational protocol based on density functional theory (DFT) is presented for calculating the dependence of the vibrational frequency of a carbonyl reporter mode on the electronic state of the molecular system, in the context of charge transfer (CT) in organic molecules. This protocol was utilized to study a system consisting of a phenyl–C61–butyric acid methyl ester electron acceptor with a N,N–dimethylaniline donor, in which small frequency shifts of less than 4 cm−1 were observed between the ground state and the CT excited state. A Stark tuning rate of 0.768 cm−1/(MV/cm) was calculated between the vibrational frequency and the electric field. In Chapter III of this thesis, the CT process in a carotenoid–porphyrin–C60 molecular triad was investigated in its two primary conformations (bent/linear) with an explicit tetrahydrofuran solvent via molecular dynamics. Vibrational frequency distributions were calculated for the amide I mode and found to be sensitive to the three electronic states relevant to CT: the Pi–Pi* excited state, the porphyrin-to-C60 CT state, and the carotenoid-to-C60 charge-separated state, with shifts as large as 40–60 cm−1 observed between the CT1 and CT2 states. Rate constants between these states were calculated with a hierarchy of approximations based on the linearized semiclassical method. The CT process was determined to occur via a two-step mechanism, Pi–Pi* -> CT1 -> CT2, where the second step is mediated by the bent-to-linear conformation change. In Chapter IV of this thesis, the role of intersystem crossing (ISC) from S1 to Tn in the pulse-shape discrimination (PSD) ability of single-crystal trans–stilbene was investigated. Time-dependent DFT was used with the newly developed OT– SRSH–PCM method to calculate the excited states, and an equilibrium Fermi’s golden rule approach was employed to calculate transition rate constants. The ISC rates were found to be too slow to compete with prompt fluorescence, and thus do not significantly impact the PSD ability. Deuteration of trans–stilbene was found to have a retarding effect on the ISC rates, with rate constants reduced by as much as 30%. Finally, in Chapter V of this thesis, a novel compute-to-learn pedagogy is presented, in which students design and develop interactive demonstrations of physical chemistry concepts in a peer-led studio environment. The rationale behind the pedagogy and improvements made over the course of three iterations are discussed, as well as an initial assessment of the pedagogy conducted via end-of-semester interviews.PHDChemistryUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147569/1/klwill_1.pd

Verifying safety and persistence in hybrid systems using flowpipes and continuous invariants

We describe a method for verifying the temporal property of persistence in non-linear hybrid systems. Given some system and an initial set of states, the method establishes that system trajectories always eventually evolve into some specified target subset of the states of one of the discrete modes of the system, and always remain within this target region. The method also computes a time-bound within which the target region is always reached. The approach combines flowpipe computation with deductive reasoning about invariants and is more general than each technique alone. We illustrate the method with a case study showing that potentially destructive stick-slip oscillations of an oil-well drill eventually die away for a certain choice of drill control parameters. The case study demonstrates how just using flowpipes or just reasoning about invariants alone can be insufficient and shows the richness of systems that one can handle with the proposed method, since the systems features modes with non-polynomial ODEs. We also propose an alternative method for proving persistence that relies solely on flowpipe computation

Some issues on the automatic computation of plane envelopes in interactive environments

This paper addresses some concerns, and describes some proposals, on the elusive concept of envelope of an algebraic family of varieties, and on its automatic computation. We describe how to use the recently developed Gröbner Cover algorithm to study envelopes of families of algebraic curves, and we give a protocol toward its implementation in dynamic geometry environments. The proposal is illustrated through some examples. A beta version of GeoGebra is used to highlight new envelope abilities in interactive environments, and limitations of our approach are discussed, since the computations are performed in an algebraically closed field

Principled and Efficient Motif Finding for Structure Learning of Lifted Graphical Models

Structure learning is a core problem in AI central to the fields of neuro-symbolic AI and statistical relational learning. It consists in automatically learning a logical theory from data. The basis for structure learning is mining repeating patterns in the data, known as structural motifs. Finding these patterns reduces the exponential search space and therefore guides the learning of formulas. Despite the importance of motif learning, it is still not well understood. We present the first principled approach for mining structural motifs in lifted graphical models, languages that blend first-order logic with probabilistic models, which uses a stochastic process to measure the similarity of entities in the data. Our first contribution is an algorithm, which depends on two intuitive hyperparameters: one controlling the uncertainty in the entity similarity measure, and one controlling the softness of the resulting rules. Our second contribution is a preprocessing step where we perform hierarchical clustering on the data to reduce the search space to the most relevant data. Our third contribution is to introduce an O(n ln n) (in the size of the entities in the data) algorithm for clustering structurally-related data. We evaluate our approach using standard benchmarks and show that we outperform state-of-the-art structure learning approaches by up to 6% in terms of accuracy and up to 80% in terms of runtime.Comment: Submitted to AAAI23. 9 pages. Appendix include

Stochastic Self-Energy in a Self-Consistent Second-Order Green's Function Scheme

The description of electron correlation has been a critical problem among theoretical and computational chemistry researchers. To describe the physics of many new materials, this interaction is crucial. Typically, chemists and physicists have constructed approximations to classic electronic structure methods - wavefunction methods and density functional theory - to attempt to solve this problem. Both have had many successes in the computational chemistry field, but are hindered by either their computational cost or ability to rigorously describe the electron correlation. More recently, researchers have come back to investigating how Green's functions can be of use to study this correlation. This thesis focuses on the second-order Green's function which has shown successes in its moderate computational cost and ability to describe electron correlation. However, in solid-state chemistry where the systems to be studied are quite large, the method still must be implemented by other means. Instead of exploring this method deterministically, this thesis moves towards investigating the method via a stochastic method. While stochastic methods such as diagrammatic Monte Carlo are quite common in the physics community, they are far lesser explored among theoretical chemists. This work attempts to develop a computationally feasible way to implement diagrammatic Monte Carlo within the second-order Green's function scheme. This method is preferred to be implemented in a fully self-consistent matter. Thus, chapter 4 of this work will explore the algorithmics required to sample for the second-order self-energy component of the method. Further in chapter 5, the correct statistical exploration required to obtain important expectation values such as the one-body and two-body energies are discussed. It is found that due to the non-linearity of the data, non-parametric statistical resampling methods are required. In this work, it is shown that via a jackknife algorithm built into the second-order Green's function scheme, the stochastic error from a Monte Carlo evaluation of the self-energy can be controlled. To show the power of this analysis, in chapter 6 self-consistency is shown to be possible via calculations of a few model systems as well as larger more realistic chemical systems. Finally, chapter 7 of this thesis segues into a problem within the chemistry community. Coding skills are at the core of developing the methods described in this thesis; however, coding is not required in the chemistry curriculum in the United States. This chapter describes a curriculum that encouraged chemistry students to develop coding skills in a low stakes environment. While the overall hypothesis going into the study was that students who use coding to study a quantitative problem in chemistry will increase their understanding of that problem, the study left with further results. Via surveys, it was discovered that many chemistry students desire to learn coding and feel that developing the skill positively impacts their studies and future goals. Overall, this thesis has made great progress in developing a diagrammatic Monte Carlo technique that could be of interest to the chemistry community. The method has proven to be computationally feasibly and quantitatively correct in comparison to the analogous deterministic method. This work has formulated a scheme that can provide an accessible approach to solving other Green's function methods of interest and hopefully bring us closer to finding a method that is computationally approachable to quantitatively describing electronic correlation.PHDChemistryUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155152/1/bawinogr_1.pd