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Featuring research from the 2017 research symposium of the Association for Women in Mathematics, this volume presents recent findings in pure mathematics and a range of advances and novel applications in fields such as engineering, biology, and medicine. Featured topics include geometric group theory, generalized iterated wreath products of cyclic groups and symmetric groups, Conway-Coxeter friezes and mutation, and classroom experiments in teaching collegiate mathematics. A review of DNA topology and a computational study of learning-induced sequence reactivation during sharp-wave ripples are also included in this volume. Numerous illustrations and tables convey key results throughout the book.

This volume highlights research from women working in academia, industry, and government. It is a helpful resource for researchers and graduate students interested in an overview of the latest research in mathematics.



Searching for Hyperbolicity

This paper is an expanded version of a talk given at the AWM Research Symposium 2017. It is intended as a gentle introduction to geometric group theory with a focus on the notion of hyperbolicity, a theme that has inspired the field from its inception to current-day research. The last section includes a discussion of some current approaches to extending techniques from hyperbolic groups to more general classes of groups.
Ruth Charney

Generalized Iterated Wreath Products of Cyclic Groups and Rooted Trees Correspondence

Consider the generalized iterated wreath product \(\mathbb {Z}_{r_1}\wr \mathbb {Z}_{r_2}\wr \ldots \wr \mathbb {Z}_{r_k}\) where \(r_i \in \mathbb {N}\). We prove that the irreducible representations for this class of groups are indexed by a certain type of rooted trees. This provides a Bratteli diagram for the generalized iterated wreath product, a simple recursion formula for the number of irreducible representations, and a strategy to calculate the dimension of each irreducible representation. We calculate explicitly fast Fourier transforms (FFT) for this class of groups, giving the literature’s fastest FFT upper bound estimate.
Mee Seong Im, Angela Wu

Generalized Iterated Wreath Products of Symmetric Groups and Generalized Rooted Trees Correspondence

Consider the generalized iterated wreath product \(S_{r_1}\wr \ldots \wr S_{r_k}\) of symmetric groups. We give a complete description of the traversal for the generalized iterated wreath product. We also prove an existence of a bijection between the equivalence classes of ordinary irreducible representations of the generalized iterated wreath product and orbits of labels on certain rooted trees. We find a recursion for the number of these labels and the degrees of irreducible representations of the generalized iterated wreath product. Finally, we give rough upper bound estimates for fast Fourier transforms.
Mee Seong Im, Angela Wu

Conway–Coxeter Friezes and Mutation: A Survey

In this survey chapter, we explain the intricate links between Conway–Coxeter friezes and cluster combinatorics. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster-tilting object in a cluster category of Dynkin type A in the sense of Caldero and Chapoton.
Karin Baur, Eleonore Faber, Sira Gratz, Khrystyna Serhiyenko, Gordana Todorov

Orbit Decompositions of Unipotent Elements in the Generalized Symmetric Spaces of

In this chapter, we determine the orbits of the fixed-point group on the unipotent elements in the generalized symmetric space for each involution of \({\mathrm{SL}}_2(\mathbb {F}_q)\) with \({\mathrm{char}}\left (\mathbb {F}_q \right ) \neq 2\). We discuss how the generalized symmetric spaces can be decomposed into semisimple elements and unipotent elements, and why this decomposition allows the orbits of the fixed-point group on the entire generalized symmetric space to be more easily classified. We conclude by providing a description of and a count for the orbits of the fixed-point group on the unipotent elements in the generalized symmetric space for each involution of \({\mathrm{SL}}_2(\mathbb {F}_q)\).
Catherine Buell, Vicky Klima, Jennifer Schaefer, Carmen Wright, Ellen Ziliak

A Characterization of the U( Ω, m) Sets of a Hyperelliptic Curve as Ω and m Vary

In this chapter, we consider a certain distinguished set U( Ω, m) ⊆{1, 2, …, 2g + 1, } that can be attached to a marked hyperelliptic curve of genus g equipped with a small period matrix Ω for its polarized Jacobian. We show that as Ω and the marking m vary, this set ranges over all possibilities prescribed by an argument of Poor.
Christelle Vincent

A First Step Toward Higher Order Chain Rules in Abelian Functor Calculus

One of the fundamental tools of undergraduate calculus is the chain rule. The notion of higher order directional derivatives was developed by Huang, Marcantognini, Young in Huang et al. (Math. Intell. 28(2):61–69, 2006), along with a corresponding higher order chain rule. When Johnson and McCarthy established abelian functor calculus, they proved a chain rule for functors that is analogous to the directional derivative chain rule when n = 1. In joint work with Bauer, Johnson, and Riehl, we defined an analogue of the iterated directional derivative and provided an inductive proof of the analogue to the chain rule of Huang et al.
This paper consists of the initial investigation of the chain rule found in Bauer et al., which involves a concrete computation of the case when n = 2. We describe how to obtain the second higher order directional derivative chain rule for functors of abelian categories. This proof is fundamentally different in spirit from the proof given in Bauer et al. as it relies only on properties of cross effects and the linearization of functors.
Christina Osborne, Amelia Tebbe

DNA Topology Review

DNA holds the instructions for an organism’s development, reproduction, and, ultimately, death. It encodes much of the information a cell needs to survive and reproduce. It is important for inheritance and coding for proteins, and contains the genetic instruction guide for life and its processes. But also, DNA of an organism has a complex and interesting topology. For information retrieval and cell viability, some geometric and topological features of DNA must be introduced, and others quickly removed. Proteins perform these amazing feats of topology at the molecular level; thus, the description and quantization of these protein actions require the language and computational machinery of topology. The use of tangle algebra to model the biological processes that give rise to knotting in DNA provides an excellent example of the application of topological algebra to biology. The tangle algebra approach to knotting in DNA began with the study of the site-specific recombinase Tn3 resolvase. This chapter is a summary of some basic knot theory and biology. We then describe the tangle model developed by Ernst and Sumners using the Tn3 resolvase as an example. We conclude with applications of the tangle model to other biological problems.
Garrett Jones, Candice Reneé Price

Structural Identifiability Analysis of a Labeled Oral Minimal Model for Quantifying Hepatic Insulin Resistance

Insulin resistance (IR) is associated with aging, trauma, and many diseases including obesity, type 2 diabetes, polycystic ovarian syndrome, and sepsis. Determining tissue-specificity of IR in a given individual or disease state may have important implications for clinical care and requires detailed assessment of glucose–insulin dynamics. Previous work introduced a differential-equations-based model to interpret data collected under a stable isotope-based oral glucose tolerance test designed to differentiate the dynamics of exogenous and endogenous glucose. We investigated the structural identifiability of this model using the Taylor expansion method. We found that the model is structurally unidentifiable due to parameters involving the rate of appearance of exogenous glucose and the volume of distribution of the compartment that cannot be separately identified. Our analysis informs a two-step approach to model implementation that overcomes limitations in identifiability and provides a reliable methodology to estimate parameters used to quantify tissue-specific IR. This work contributes to an improved understanding of methods designed to investigate tissue-specific IR.
Jacqueline L. Simens, Melanie Cree-Green, Bryan C. Bergman, Kristen J. Nadeau, Cecilia Diniz Behn

Spike-Field Coherence and Firing Rate Profiles of CA1 Interneurons During an Associative Memory Task

Flexible, dynamic activity in the brain is essential to information processing. Neurons in the hippocampus are capable of conveying information about the continually evolving world through changes in their spiking activity. This information can be expressed through changes in firing rate and through the reorganization of spike timing in unique rhythmic profiles. Locally projecting interneurons of the hippocampus are in an ideal position to coordinate task-relevant changes in the spiking activity of the network, as their inhibitory influence allows them to constrain communication between neurons to rhythmic, optimal windows and facilitates selective responses to afferent input. During a context-guided odor–reward association task, interneurons and principal cells in the CA1 subregion of the rat hippocampus demonstrate distinct oscillatory profiles that correspond to correct and incorrect performance, despite similar firing rates during correct and incorrect trials (Rangel et al., eLife 5:e09849, 2016). Principal cells additionally contained information in their firing rates about task dimensions, reflective of highly selective responses to features such as single positions and odors. It remains to be determined whether interneurons also contain information about task dimensions in their firing rates. To address this question, we evaluated the information content for task dimensions in the firing rates of inhibitory neurons. Interneurons contained low, but significant information for task dimensions in their firing rates, with increases in information over the course of a trial that reflected the evolving availability of task dimensions. These results suggest that interneurons are capable of manifesting distinct rhythmic profiles and changes in firing rate that reflect task-relevant processing.
Pamela D. Rivière, Lara M. Rangel

Learning-Induced Sequence Reactivation During Sharp-Wave Ripples: A Computational Study

During sleep, memories formed during the day are consolidated in a dialogue between cortex and hippocampus. The reactivation of specific neural activity patterns—replay—during sleep has been observed in both structures and is hypothesized to represent a neuronal substrate of consolidation. In the hippocampus, replay happens during sharp-wave ripple complexes (SWR), when short bouts of excitatory activity in area CA3 induce high-frequency oscillations in area CA1. In particular, recordings of hippocampal cells which spike at a specific location (“place cells”) show that recently learned trajectories are reactivated during CA1 ripples in the following sleep period. Despite the importance of sleep replay, its underlying neural mechanisms are still poorly understood.
We used a previously developed model of sharp-wave ripples activity, to study the effects of learning-induced synaptic changes on spontaneous sequence reactivation during CA3 sharp waves. In this study, we implemented a paradigm including three epochs: Pre-sleep, learning, and Post-sleep activity. We first tested the effects of learning on the hippocampal network activity through changes in a minimal number of synapses connecting selected pyramidal cells. We then introduced an explicit trajectory-learning task to the learning portion of the paradigm, to obtain behavior-induced synaptic changes. Our analysis revealed that recently learned trajectories were reactivated during sleep more often than other trajectories in the training field. This study predicts that the gain of reactivation rate during sleep following vs sleep preceding learning for a trained sequence of pyramidal cells depends on Pre-sleep activation of the same sequence, and on the amount of trajectory repetitions included in the training phase.
Paola Malerba, Katya Tsimring, Maxim Bazhenov

A DG Method for the Simulation of CO2 Storage in Saline Aquifer

To simulate the process of CO2 injection into deep saline aquifers, we use the isothermal two-phase two-component model, which takes mass transfer into account. We develop a new discontinuous Galerkin method called the “partial upwind” method for space discretization, incorporated with the backward Euler scheme for time discretization and the Newton–Raphson method for linearization. Numerical simulations show that the new method is a promising candidate for the CO2 storage problem in both homogenous and heterogenous porous media and is more robust to the standard discontinuous Galerkin method for some subsurface fluid flow problems.
Beatrice Riviere, Xin Yang

Regularization Results for Inhomogeneous Ill-Posed Problems in Banach Space

We prove continuous dependence on modeling for the inhomogeneous ill-posed Cauchy problem in Banach space X, then use these results to obtain a regularization result. The particular problem we consider is given by \(\frac {\mathrm{d} u(t)}{\mathrm{d} t} = A u(t) + h(t), 0 \leq t <T, u(0) = \chi \), where − A generates a uniformly bounded holomorphic semigroup {ezA|Re(z) ≥ 0} and h : [0, T) → X. In the approximate problem, the operator A is replaced by the operator fβ(A), β > 0, which approximates A as β goes to 0. We use a logarithmic approximation introduced by Boussetila and Rebbani. Our results extend earlier work of the author together with Fury and Huddell on the homogeneous ill-posed problem.
Beth M. Campbell Hetrick

Research in Collegiate Mathematics Education

The chapter sketches some of the landscape of current research in undergraduate mathematics education and offers useful information for present and future faculty members. Six research projects related to the teaching and learning of post-secondary mathematics are summarized. Approaches in the research reported here include individual interviews, classroom observations, national survey, and in-depth study of a particular instance or case of learning. The collegiate mathematics topics at the heart of the respective studies range from calculus, combinatorics, linear algebra, and foundations of proof to the application of mathematics to teaching in the development of future teachers.
Shandy Hauk, Chris Rasmussen, Nicole Engelke Infante, Elise Lockwood, Michelle Zandieh, Stacy Brown, Yvonne Lai, Pao-sheng Hsu


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