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Articles 1 - 17 of 17
Full-Text Articles in Geometry and Topology
A Discrete Morse Approach For Computing Homotopy Types: An Exploration Of The Morse, Generalized Morse, Matching, And Independence Complexes, Connor Donovan
A Discrete Morse Approach For Computing Homotopy Types: An Exploration Of The Morse, Generalized Morse, Matching, And Independence Complexes, Connor Donovan
Mathematics Honors Papers
In this thesis, we study possible homotopy types of four families of simplicial complexes–the Morse complex, the generalized Morse complex, the matching complex, and the independence complex–using discrete Morse theory. Given a simplicial complex, K, we can construct its Morse complex from all possible discrete gradient vector fields on K. A similar construction will allow us to build the generalized Morse complex while considering edges and vertices will allow us to construct the matching complex and independence complex. In Chapter 3, we use the Cluster Lemma and the notion of star clusters to apply matchings to families of Morse, generalized …
The Dope Distance Is Sic: A Stable, Informative, And Computable Metric On Ordered Merge Trees, Jose Arbelo, Antonio Delgado, Charley Kirk, Zach Schlamowitz
The Dope Distance Is Sic: A Stable, Informative, And Computable Metric On Ordered Merge Trees, Jose Arbelo, Antonio Delgado, Charley Kirk, Zach Schlamowitz
Mathematics Summer Fellows
When analyzing time series data, it is often of interest to categorize them based on how different they are. We define a new dissimilarity measure between time series: Dynamic Ordered Persistence Editing (DOPE). DOPE satisfies metric properties, is stable to noise, is as informative as alternative approaches, and efficiently computable. Satisfying these properties simultaneously makes DOPE of interest to both theoreticians and data scientists alike.
Towards The Homotopy Type Of The Morse Complex, Connor Donovan
Towards The Homotopy Type Of The Morse Complex, Connor Donovan
Mathematics Summer Fellows
Mathematicians have long been interested in studying the properties of simplicial complexes. In 1998, Robin Forman developed gradient vector fields as a tool to study these complexes. Having gradient vector fields to study these simplicial complexes, in 2005, Chari and Joswig discovered the Morse complex, a complex consisting of all gradient vector fields on a fixed complex. Although the Morse complex has been studied since 2005, there is little information regarding its homotopy type for different simplicial complexes. Pursuing our curiosity of the topic, we extend a result by Ayala et. al., stating that the pure Morse complex of a …
Geometry Aided Sonification, Michael Tecce
Geometry Aided Sonification, Michael Tecce
Computer Science Summer Fellows
Sonification is the process of deriving an audio representation of a time series which conveys important information about that time series. Otology and vision science have established that humans process audio information more quickly than visual information, and sonification can convey data to the visually impaired. In our work, we implement pipelines using Python/Numpy, and we handle both ordinary 1D time series and multivariate time series. For 1D time series, we find that using data to modulate the pitch or timing of preselected sounds (such as sine waves) simply and effectively captures repeating patterns and anomalies/outliers within the data. To …
Merge Trees In Discrete Morse Theory, Benjamin Johnson
Merge Trees In Discrete Morse Theory, Benjamin Johnson
Mathematics Summer Fellows
The field of topological data analysis seeks to use techniques in topology to study large data sets. The hope is that rather than single quantities that summarize the data, such as mean or standard deviation, information about the data can be learned by studying the overall ``shape” of the data. One way to summarize this data is through a merge tree. Merge trees can be thought of as keeping track of certain clusters of data and determining when they merge together. In this paper, we will study merge trees induced by a discrete Morse function on a tree. Under a …
Connectedness- Its Evolution And Applications, Nicholas A. Scoville
Connectedness- Its Evolution And Applications, Nicholas A. Scoville
Topology
No abstract provided.
Persistence Equivalence Of Discrete Morse Functions On Trees, Yuqing Liu
Persistence Equivalence Of Discrete Morse Functions On Trees, Yuqing Liu
Mathematics Summer Fellows
We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if they induce the same persistence diagram. We compare this notion of equivalence to other notions of equivalent discrete Morse functions. We then compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree. We conclude with an example illustrating our construction.
Nearness Without Distance, Nicholas A. Scoville
From Sets To Metric Spaces To Topological Spaces, Nicholas A. Scoville
From Sets To Metric Spaces To Topological Spaces, Nicholas A. Scoville
Topology
No abstract provided.
The Closure Operation As The Foundation Of Topology, Nicholas A. Scoville
The Closure Operation As The Foundation Of Topology, Nicholas A. Scoville
Topology
No abstract provided.
A Compact Introduction To A Generalized Extreme Value Theorem, Nicholas A. Scoville
A Compact Introduction To A Generalized Extreme Value Theorem, Nicholas A. Scoville
Topology
In a short paper published just one year prior to his thesis, Maurice Frechet gives a simple generalization one what we might today call the Extreme value theorem. This generalization is a simple matter of coming up with ``the right" definitions in order to make this work. In this mini PSP, we work through Frechet's entire 1.5 page paper to give an extreme value theorem in more general topological spaces, ones which, to use Frechet's newly coined term, are compact.
The Failure Of The Euclidean Parallel Postulate And Distance In Hyperbolic Geometry, Jerry Lodder
The Failure Of The Euclidean Parallel Postulate And Distance In Hyperbolic Geometry, Jerry Lodder
Geometry
No abstract provided.
The Cantor Set Before Cantor, Nicholas A. Scoville
The Cantor Set Before Cantor, Nicholas A. Scoville
Topology
A special construction used in both analysis and topology today is known as the Cantor set. Cantor used this set in a paper in the 1880s. Yet it appeared as early as 1875 in a paper by the Irish mathematician Henry John Stephen Smith (1826 - 1883). Smith, who is best known for the Smith normal form of a matrix, was a professor at Oxford who made great contributions in matrix theory and number theory. In this project, we will explore parts of a paper he wrote titled On the Integration of Discontinuous Functions.
Topology From Analysis, Nicholas A. Scoville
Topology From Analysis, Nicholas A. Scoville
Topology
Topology is often described as having no notion of distance, but a notion of nearness. How can such a thing be possible? Isn't this just a distinction without a difference? In this project, we will discover the notion of nearness without distance by studying the work of Georg Cantor and a problem he was investigating involving Fourier series. We will see that it is the relationship of points to each other, and not their distances per se, that is a proper view. We will see the roots of topology organically springing from analysis.
Connecting Connectedness, Nicholas A. Scoville
The Exigency Of The Euclidean Parallel Postulate And The Pythagorean Theorem, Jerry Lodder
The Exigency Of The Euclidean Parallel Postulate And The Pythagorean Theorem, Jerry Lodder
Geometry
No abstract provided.
Discrete Morse Functions, Vector Fields, And Homological Sequences On Trees, Ian B. Rand
Discrete Morse Functions, Vector Fields, And Homological Sequences On Trees, Ian B. Rand
Mathematics Summer Fellows
The goal of this project is to construct a discrete Morse function which induces both a unique gradient vector field and homological sequence on a given tree. After reviewing the basics of discrete Morse theory, we will show that the two standard notions of equivalence of discrete Morse functions, Forman and homological equivalence, are independent of one another. We then show through a constructive algorithm the existence of a discrete Morse function on a tree inducing a desired gradient vector field and homological sequence. After proving that our algorithm is correct, we give an example to illustrate its use.