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We use persistent homology in order to define a family of fractal dimensions, denoted \(\dim _{\mathrm {PH}}^i(\mu )\) for each homological dimension i ≥ 0, assigned to a probability measure μ on a metric space. The case of zero-dimensional homology (i = 0) relates to work by Steele (Ann Probab 16(4): 1767–1787, 1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if μ is supported on a compact subset of Euclidean space \(\mathbb {R}^m\) for m ≥ 2, then Steele’s work implies that \(\dim _{\mathrm {PH}}^0(\mu )=m\) if the absolutely continuous part of μ has positive mass, and otherwise \(\dim _{\mathrm {PH}}^0(\mu )

Motivated by hexagonal patterns with defects in natural and laboratory systems, we compare quantitative measures of order for nearly hexagonal, planar lattices. These include a spectral measure of order based on the Fourier transform, a geometric measure of order using the Delaunay triangulation, and topological measures of order introduced in this paper. The topological measures are based on a tool from topological data analysis called persistent homology. We contrast these measures of order by comparing their sensitivity to perturbations of Bravais lattices. We then study the imperfect hexagonal arrangements of nanodots produced by numerical simulations of partial differential equations that model the surface of a binary alloy undergoing erosion by a broad ion beam. These numerical experiments further distinguish the various measures of hexagonal order and highlight the role of various model parameters in the formation and elimination of defects. Finally, we quantify the dependence of order on prepatterning the surface to suggest experimental protocols that could lead to improved order in nanodot arrays.

While one-dimensional persistent homology can be an effective way to discriminate data, it has limitations. Multidimensional persistent homology is a technique amenable to data naturally described by more than a single parameter, and is able to encoding more robust information about the structure of the data. However, as indicated by Carlsson and Zomorodian (Discrete Comput Geom 42(1):71–271, 2009), no perfect higher-dimensional analogue of the one-dimensional persistence barcode exists for higher-dimensional filtrations. Xia and Wei (J Comput Chem 36:1502–1520, 2015) propose computing one-dimensional Betti number functions at various values of a second parameter and stacking these functions for each homological dimension. The aim of this visualization is to increase the discriminatory power of current one-dimensional persistence techniques, especially for datasets that have features more readily captured by a combination of two parameters. We apply this practical approach to three datasets, relating to (1) craniofacial shape and (2) Lissajous knots, both using parameters for scale and curvature; and (3) the Kuramoto–Sivashinsky partial differential equation, using parameters for both scale and time. This new approach is able to differentiate between topologically equivalent geometric objects and offers insight into the study of the Kuramoto–Sivashinsky partial differential equation and Lissajous knots. We were unable to obtain meaningful results, however, in our applications to the screening of anomalous facial structures, although our method seems sensitive enough to identify patients at severe risk of a sleep disorder associated closely with craniofacial structure. This approach, though still in its infancy, presents new insights and avenues for the analysis of data with complex structure.

To succeed in physical chemistry courses, students need competence with a significant amount of mathematics, including ordinary and partial differential equations and linear operators. This can present a barrier to student success because many of these topics are not taught in the traditional prerequisite calculus sequence. Through a collaboration of the Departments of Chemistry and Mathematics, Colorado State University has developed and implemented a two-semester sequence of courses, Applied Mathematics for Chemists, aimed specifically at providing students with the mathematical tools necessary for success in physical chemistry. This report summarizes one of the invited papers to the ConfChem online conference on Mathematics in Undergraduate Chemistry Instruction, held from October 23 to November 27, 2017, and hosted by the ACS DivCHED Committee on Computers in Chemical Education (CCCE).

Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs., Version 3 contains updated theoretical results supporting methodology; expanded discussion of related works; extended list of references; extended applications section; additional experimental results and new figures

Aberrant expression of the epithelial cell adhesion protein P-cadherin has been observed to be independently causative and preventative with respect to tumor progression, depending on tissue type. Oral epithelial tumors have been shown to exhibit a unique signature of P-cadherin expression in which P-cadherin expression increases in dysplasia and is subsequently lost during malignancy. These studies suggest that early increases in P-cadherin expression may be critical events in tumor progression. Since E- and N-cadherin have been shown to physically interact with and modulate function of Epidermal Growth Factor (EGF) and Fibroblast Growth Factor (FGF) respectively, we hypothesized that P-cadherin could promote tumor development via its own interaction with receptor tyrosine kinases in a ligand-dependent manner. To identify functional interactions between P-cadherin and receptor tyrosine kinases, we retrovirally transduced two independently isolated oral squamous carcinoma cell lines to overexpress P-cadherin. Transduced cells were treated independently with one of seven different growth factors to identify alterations in mitogen-activated protein kinase (MAPK) and phosphoinositide 3-kinase (PI3K) signaling pathways. For all experiments, protein was collected over the course of six hours at multiple timepoints to identify fluctuations in signaling. Although P-cadherin did not alter AKT phosphorylation, it was found to alter MAPK signaling in both a positive and negative manner. The most robust alterations in MAPK signaling occurred in response to activation of Insulin-like Growth Factor (IGF), Epithelial Growth Factor (EGF), and Keratinocyte Growth Factor (KGF). P-cadherin overexpression also induced phenotypic changes. Behavioral assays revealed that P-cadherin expression increased motility, but decreased expression of invasive proteases MMP-9 and MMP-14. P-cadherin cells grown in serum-free media for 48 hours exhibited increased cell number compared to control cells. A cell proliferation analysis (MTT assay) indicated lower rates of proliferation in P-cadherin -overexpressing cells compared to controls. This data suggests that the increased cell number of serum-starved P-cadherin cells was due to increased survival, rather than proliferation. These results demonstrate that P-cadherin overexpression can modulate ligand-dependent growth factor signaling and alter those cellular characteristics most commonly associated with tumor aggressiveness. Since P-cadherin expression has been shown to increase early on in the development of tumors, our research may provide a possible mechanism as to how this increase may play a role in tumor progression during dysplasia. Citation Format: James Mar, Adam Robinson, Rachel Neville, Desseree Lysne, Kathryn Lawson. P-cadherin modulates signaling of multiple growth factor receptors and cellular aggressiveness in oral carcinoma cells. [abstract]. In: Proceedings of the 105th Annual Meeting of the American Association for Cancer Research; 2014 Apr 5-9; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2014;74(19 Suppl):Abstract nr 4409. doi:10.1158/1538-7445.AM2014-4409