Mathematics Commons^™

Fibrations, Unique Path Lifting, And Continuous Monodromy, Hanspeter Fischer, Jacob D. Garcia

Mathematics Sciences: Faculty Publications

Given a path-connected space X and H≤π1(X,x0), there is essentially only one construction of a map pH:(X˜H,x˜0)→(X,x0) with connected and locally path-connected domain that can possibly have the following two properties: (pH)#π1(X˜H,x˜0)=H and pH has the unique lifting property. X˜H consists of equivalence classes of paths starting at x0, appropriately topologized, and pH is the endpoint projection. For pH to have these two properties, T1 fibers are necessary and unique path lifting is sufficient. However, pH always admits the standard lifts of paths. We show that pH has unique path lifting if it has continuous (standard) monodromies toward a T1 …

Analysis In Metric Spaces, Mario Bonk, Luca Capogna, Piotr Hajłasz, Nageswari Shanmugalingam, Jeremy Tyson

Mathematics Sciences: Faculty Publications

No abstract provided.

Neural Networks For Threshold Dynamics Reconstruction, Elisa Negrini, Almanzo Jiahe Gao, Abigail Bowering, Wei Zhu, Luca Capogna

Mathematics Sciences: Faculty Publications

We introduce two convolutional neural network (CNN) architec- tures, inspired by the Merriman-Bence-Osher (MBO) algorithm and by cellular automatons, to model and learn threshold dynamics for front evolution from video data. The first model, termed the (single-dynamics) MBO network, learns a specific kernel and threshold for each input video without adapting to new dynamics, while the second, a meta-learning MBO network, generalizes across diverse threshold dynamics by adapting its parameters per input. Both models are evaluated on synthetic and real-world videos (ice melting and fire front propagation), with performance metrics indicating effective reconstruction and extrapolation of evolving boundaries, even under …

A Filtration On The Cohomology Rings Of Regular Nilpotent Hessenberg Varieties, Megumi Harada, Tatsuya Horiguchi, Satoshi Murai, Martha Precup, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Let n be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in GL(n, C) / B such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in GL(n- 1 , C) / B, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce …

Graphs Admitting Only Constant Splines, Katie Anders, Alissa S. Crans, Briana Foster-Greenwood, Blake Mellor, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

We study generalized graph splines, introduced by Gilbert, Tymoczko, and Viel (2016). For a large class of rings, we characterize the graphs that only admit constant splines. To do this, we prove that if a graph has a particular type of cutset (e.g., a bridge), then the space of splines naturally decomposes as a certain direct sum of submodules. As an application, we use these results to describe splines on a triangulation studied by Zhou and Lai, but over a different ring than they used.

Sleep Stage Prediction With Raw Acceleration And Photoplethysmography Heart Rate Data Derived From A Consumer Wearable Device, Olivia Walch, Yitong Huang, Daniel Forger, Cathy Goldstein

Mathematics Sciences: Faculty Publications

Wearable, multisensor, consumer devices that estimate sleep are now commonplace, but the algorithms used by these devices to score sleep are not open source, and the raw sensor data is rarely accessible for external use. As a result, these devices are limited in their usefulness for clinical and research applications, despite holding much promise. We used a mobile application of our own creation to collect raw acceleration data and heart rate from the Apple Watch worn by participants undergoing polysomnography, as well as during the ambulatory period preceding in lab testing. Using this data, we compared the contributions of multiple …

Polymer Stress Growth In Viscoelastic Fluids In Oscillating Extensional Flows With Applications To Micro-Organism Locomotion, Becca Thomases, Robert D. Guy

Mathematics Sciences: Faculty Publications

Simulations of undulatory swimming in viscoelastic fluids with large amplitude gaits show concentration of polymer elastic stress at the tips of the swimmers. We use a series of related theoretical investigations to probe the origin of these concentrated stresses. First the polymer stress is computed analytically at a given oscillating extensional stagnation point in a viscoelastic fluid. The analysis identifies a Deborah number (De) dependent Weissenberg number (Wi) transition below which the stress is linear in Wi, and above which the stress grows exponentially in Wi. Next, stress and velocity are found from numerical simulations in an oscillating 4-roll mill …

Loose Legendrian And Pseudo-Legendrian Knots In 3-Manifolds, Patricia Cahn, Vladimir Chernov

Mathematics Sciences: Faculty Publications

We prove a complete classification theorem for loose Legendrian knots in an oriented 3-manifold, generalizing results of Dymara and DingGeiges. Our approach is to classify knots in a 3-manifold M that are transverse to a nowhere-zero vector field V up to the corresponding isotopy relation. Such knots are called V -transverse. A framed isotopy class is simple if any two V -transverse knots in that class which are homotopic through V -transverse immersions are V -transverse isotopic. We show that all knot types in M are simple if any one of the following three conditions hold: 1. M is closed, …

Convergent Solutions Of Stokes Oldroyd-B Boundary Value Problems Using The Immersed Boundary Smooth Extension (Ibse) Method, David B. Stein, Robert D. Guy, Becca Thomases

Mathematics Sciences: Faculty Publications

The Immersed Boundary (IB) method has been widely used to solve fluid-structure interaction problems, including those where the structure interacts with polymeric fluids. In this paper, we examine the convergence of one such scheme for a well known two-dimensional benchmark flow for the Oldroyd-B constitutive model, and we show that the traditional IB-based scheme fails to adequately capture the polymeric stress near to embedded boundaries. We analyze the reason for such failure, and we argue that this feature is not specific to the case study chosen, but a general feature of such methods due to lack of convergence in velocity …

The Transition Matrix Between The Specht And Web Bases Is Unipotent With Additional Vanishing Entries, Heather M. Russell, Julianna S. Tymoczko

Mathematics Sciences: Faculty Publications

We compare two important bases of an irreducible representation of the symmetric group: the web basis and the Specht basis. The web basis has its roots in the Temperley-Lieb algebra and knot-theoretic considerations. The Specht basis is a classic algebraic and combinatorial construction of symmetric group representations which arises in this context through the geometry of varieties called Springer fibers. We describe a graph that encapsulates combinatorial relations between each of these bases, prove that there is a unique way (up to scaling) to map the Specht basis into the web representation, and use this to recover a result of …

Orientation Dependent Elastic Stress Concentration At Tips Of Slender Objects Translating In Viscoelastic Fluids, Chuanbin Li, Becca Thomases, Robert D. Guy

Mathematics Sciences: Faculty Publications

Elastic stress concentration at the tips of long slender objects moving in viscoelastic fluids has been observed in numerical simulations, but despite the prevalence of flagellated motion in complex fluids in many biological functions, the physics of stress accumulation near the tips has not been analyzed. Here, we theoretically investigate elastic stress development at the tips of slender objects by computing the leading-order viscoelastic correction to the equilibrium viscous flow around long cylinders, using the weak-coupling limit. In this limit, nonlinearities in the fluid are retained, allowing us to study the biologically relevant parameter regime of high Weissenberg number. We …

Proper Orthogonal Decomposition (Pod) Of The Flow Dynamics For A Viscoelastic Fluid In A Four-Roll Mill Geometry At The Stokes Limit, Paloma Gutierrez-Castillo, Becca Thomases

Mathematics Sciences: Faculty Publications

Numerical simulations of viscoelastic fluids in the Stokes limit with a four-roll mill background force were performed at a range of Weissenberg number (non-dimensional relaxation time). For small Weissenberg number the flow is steady and symmetric but upon increasing the Weissenberg number (corresponding to increased elasticity or flow memory time), the flow becomes unstable leading to a variety of temporal evolutions to different periodic and aperiodic solutions. These dynamics were analyzed using a Proper Orthogonal Decomposition (POD) that extracted elastic modes in terms of their contribution to the energy of the system. The temporal behavior of the system, captured by …

Springer Fibers And Schubert Points, Martha Precup, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis for the cohomology of the flag variety. This paper relates these two varieties combinatorially. We prove that the Betti numbers of the Springer fiber associated to a partition with at most three rows or two columns are equal to the Betti numbers of a specific union of Schubert varieties.

Conformality And Q-Harmonicity In Sub-Riemannian Manifolds, Luca Capogna, Giovanna Citti, Enrico Le Donne, Alessandro Ottazzi

Mathematics Sciences: Faculty Publications

We establish regularity of conformal maps between sub-Riemannian manifolds from regularity of Q-harmonic functions, and in particular we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth in all contact sub-Riemannian manifolds. Together with the recent results in [15], our work yields a new proof of the smoothness of boundary extensions of biholomorphims between strictly pseudoconvex smooth domains [29].

The Impact Of College Athletic Success On Donations And Applicant Quality, Benjamin Baumer, Andrew Zimbalist

Mathematics Sciences: Faculty Publications

For the 65 colleges and universities that participate in the Power Five athletic conferences (Pac 12, Big 10, SEC, ACC, and Big 12), the football and men’s basketball teams are highly visible. While these programs generate tens of millions of dollars in revenue annually, very few of them turn an operating “profit.” Their existence is thus justified by the claim that athletic success leads to ancillary benefits for the academic institution, in terms of both quantity (e.g., more applications, donations, and state funding) and quality (e.g., stronger applicants, lower acceptance rates, higher yields). Previous studies provide only weak support for …

Convergence In A Disk Stacking Model On The Cylinder, Christophe Golé, Stéphane Douady

Mathematics Sciences: Faculty Publications

We study an iterative process modeling growth of phyllotactic patterns, wherein disks are added one by one on the surface of a cylinder, on top of an existing set of disks, as low as possible and without overlap. Numerical simulations show that the steady states of the system are spatially periodic, lattices-like structures called rhombic tilings. We present a rigorous analysis of the dynamics of all configurations starting with closed chains of 3 tangent, non-overlapping disks encircling the cylinder. We show that all these configurations indeed converge to rhombic tilings. Surprisingly, we show that convergence can occur in either …

Double Lie Algebroids And Representations Up To Homotopy, A. Gracia-Saz, M. Jotz Lean, K. C. H. Mackenzie, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

We show that a double Lie algebroid, together with a chosen decomposition, is equivalent to a pair of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the double Lie algebroids arising from the tangent bundle of a Lie algebroid and the cotangent bundle of a Lie bialgebroid.

Constant Symplectic 2-Groupoids, Rajan Amit Mehta, Xiang Tang

Mathematics Sciences: Faculty Publications

We propose a definition of symplectic 2-groupoid which includes integrations of Courant algebroids that have been recently constructed. We study in detail the simple but illustrative case of constant symplectic 2-groupoids. We show that the constant symplectic 2-groupoids are, up to equivalence, in one-to-one correspondence with a simple class of Courant algebroids that we call constant Courant algebroids. Furthermore, we find a correspondence between certain Dirac structures and Lagrangian sub-2-groupoids.

Germs Of Fibrations Of Spheres By Great Circles Always Extend To The Whole Sphere, Patricia Cahn, Herman Gluck, Haggai Nuchi

Mathematics Sciences: Faculty Publications

We prove that every germ of a smooth fibration of an odd-dimensional round sphere by great circles extends to such a fibration of the entire sphere, a result previously known only in dimension three.

Symplectic Structures On The Integration Of Exact Courant Algebroids, Rajan Amit Mehta, Xiang Tang

Mathematics Sciences: Faculty Publications

We construct an infinite-dimensional symplectic 2-groupoid as the integration of an exact Courant algebroid. We show that every integrable Dirac structure integrates to a “Lagrangian” sub-2-groupoid of this symplectic 2-groupoid. As a corollary, we recover a result of Bursztyn-Crainic-Weinstein-Zhu that every integrable Dirac structure integrates to a presymplectic groupoid.

A Variation On The Theme Of Nicomachus, Florian Luca, Geremías Polanco, Wadim Zudilin

Mathematics Sciences: Faculty Publications

In this paper, we prove some conjectures of K. Stolarsky concerning the first and third moments of the Beatty sequences with the golden section and its square.

Clustered Networks Protect Cooperation Against Catastrophic Collapse, Gwen Spencer

Mathematics Sciences: Faculty Publications

Assuming a society of conditional cooperators (or moody conditional cooperators), this computational study proposes a new perspective on the structural advantage of social network clustering. Previous work focused on how clustered structure might encourage initial outbreaks of cooperation or defend against invasion by a few defectors. Instead, we explore the ability of a societal structure to retain cooperative norms in the face of widespread disturbances. Such disturbances may abstractly describe hardships like famine and economic recession, or the random spatial placement of a substantial numbers of pure defectors (or round-1 defectors) among a spatially-structured population of players in a laboratory …

The Fivethirtyeight R Package: ‘Tame Data’ Principles For Introductory Statistics And Data Science Courses, Albert Y. Kim, Chester Ismay, Jennifer Chunn

Mathematics Sciences: Faculty Publications

As statistics and data science instructors, we often seek to use data in our courses that are rich, real, realistic, and relevant. To this end we created the fivethirtyeight R package of data and code behind the stories and interactives at the data journalism website FiveThirtyEight.com. After a discussion on the conflicting pedagogical goals of "minimizing prerequisites to research" (Cobb 2015) while at the same time presenting students with a realistic view of data as it exists "in the wild," we articulate how a desired balance between these two goals informed the design of the package. The details behind this …

How Often Does The Best Team Win? A Unified Approach To Understanding Randomness In North American Sport, Michael J. Lopez, Gregory J. Matthews, Benjamin Baumer

Mathematics Sciences: Faculty Publications

Statistical applications in sports have long centered on how to best separate signal (e.g. team talent) from random noise. However, most of this work has concentrated on a single sport, and the development of meaningful cross-sport comparisons has been impeded by the difficulty of translating luck from one sport to another. In this manuscript, we develop Bayesian state-space models using betting market data that can be uniformly applied across sporting organizations to better understand the role of randomness in game outcomes. These models can be used to extract estimates of team strength, the between-season, within-season, and game-to-game variability of team …

The Dihedral Genus Of A Knot, Patricia Cahn, Alexandra Kjuchukova

Mathematics Sciences: Faculty Publications

Let K ⊂ S³ be a Fox p-colored knot and assume K bounds a locally flat surface S ⊂ B⁴ over which the given p-coloring extends. This coloring of S induces a dihedral branched cover X → S⁴ . Its branching set is a closed surface embedded in S⁴ locally flatly away from one singularity whose link is K. When S is homotopy ribbon and X a definite four-manifold, a condition relating the signature of X and the Murasugi signature of K guarantees that S in fact realizes the four-genus of K. We exhibit an infinite …

Action Of The Symmetric Group On The Free Lanke: A Catalanke Theorem, Tamar Friedmann, Philip Hanlon, Richard P. Stanley, Michelle L. Wachs

Mathematics Sciences: Faculty Publications

We initiate a study of the representation of the symmetric group on the multilinear component of an n-ary generalization of the free Lie algebra, which we call a free LAnKe. Our central result is that the representation of the symmetric group S_2n−1 on the multilinear component of the free LAnKe with S_2n−1 generators is given by an irreducible representation whose dimension is the nth Catalan number. This leads to a more general result on eigenspaces of a certain linear operator. A decomposition, into irreducibles, of the representation of S_3n−2 on the multilinear component the free LAnKe …

Strong Comparison Principle For P-Harmonic Functions In Carnot-Caratheodory Spaces, Luca Capogna, Xiaodan Zhou

Mathematics Sciences: Faculty Publications

We extend Bony’s propagation of support argument to C1 solutions of the nonhomogeneous subelliptic p-Laplacian associated to a system of smooth vector fields satisfying Hörmander’s finite rank condition. As a consequence we prove a strong maximum principle and strong comparison principle that generalize results of Tolksdorf.

Mapping Class Group Orbits Of Curves With Self-Intersections, Patricia Cahn, Federica Fanoni, Bram Petri

Mathematics Sciences: Faculty Publications

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity.

We also consider the minimal genus of a subsurface that contains the curve. We determine the asymptotic number of orbits of curves with a fixed minimal genus and a bounded self-intersection number, as the complexity of the surface tends to infinity.

As a corollary of our methods, we obtain that most curves that are homotopic are also isotopic. …

Flagellar Swimming In Viscoelastic Fluids: Role Of Fluid Elastic Stress Revealed By Simulations Based On Experimental Data, Chuanbin Li, Boyang Qin, Arvind Gopinath, Paulo E. Arratia, Becca Thomases, Robert D. Guy

Mathematics Sciences: Faculty Publications

Many important biological functions depend on microorganisms' ability to move in viscoelastic fluids such as mucus and wet soil. The effects of fluid elasticity on motility remain poorly understood, partly because the swimmer strokes depend on the properties of the fluid medium, which obfuscates the mechanisms responsible for observed behavioural changes. In this study, we use experimental data on the gaits of Chlamydomonas reinhardtii swimming in Newtonian and viscoelastic fluids as inputs to numerical simulations that decouple the swimmer gait and fluid type in order to isolate the effect of fluid elasticity on swimming. In viscoelastic fluids, cells employing the …

Lessons From Between The White Lines For Isolated Data Scientists, Benjamin Baumer

Mathematics Sciences: Faculty Publications

Many current and future data scientists will be “isolated”—working alone or in small teams within a larger organization. This isolation brings certain challenges as well as freedoms. Drawing on my considerable experience both working in the professional sports industry and teaching in academia, I discuss troubled waters likely to be encountered by newly minted data scientists and offer advice about how to navigate them. Neither the issues raised nor the advice given are particular to sports and should be applicable to a wide range of knowledge domains.