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Fantastic Topological Surfaces and How to Classify Them Khorben Boyer Western Oregon University, Kboyer12@Mail.Wou.Edu Western Oregon University Digital Commons@WOU Academic Excellence Showcase Proceedings Student Scholarship 2018-06-01 Fantastic Topological Surfaces and How to Classify Them Khorben Boyer Western Oregon University, [email protected] Follow this and additional works at: https://digitalcommons.wou.edu/aes Part of the Physical Sciences and Mathematics Commons Recommended Citation Boyer, Khorben, "Fantastic Topological Surfaces and How to Classify Them" (2018). Academic Excellence Showcase Proceedings. 101. https://digitalcommons.wou.edu/aes/101 This Presentation is brought to you for free and open access by the Student Scholarship at Digital Commons@WOU. It has been accepted for inclusion in Academic Excellence Showcase Proceedings by an authorized administrator of Digital Commons@WOU. For more information, please contact [email protected], [email protected], [email protected]. Introduction Overview of Topological Surfaces How to work with Surfaces Fantastic Topological Surfaces and How to Classify Them Khorben Boyer Western Oregon University May 31, 2018 Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them The field of Topology is concerned with the properties of space that are preserved under continuous deformations such as stretching, crumpling, and bending, but not tearing new holes or sewing up old ones. Think of it as the field of \rubber-sheet" geometry since the rigid shapes of the objects are not relevant. In particular, mathematicians often study mathematical structures in topology called surfaces. These surfaces come in a diverse array of variations and combinations. Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction Our context: Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Think of it as the field of \rubber-sheet" geometry since the rigid shapes of the objects are not relevant. In particular, mathematicians often study mathematical structures in topology called surfaces. These surfaces come in a diverse array of variations and combinations. Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction Our context: The field of Topology is concerned with the properties of space that are preserved under continuous deformations such as stretching, crumpling, and bending, but not tearing new holes or sewing up old ones. Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them In particular, mathematicians often study mathematical structures in topology called surfaces. These surfaces come in a diverse array of variations and combinations. Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction Our context: The field of Topology is concerned with the properties of space that are preserved under continuous deformations such as stretching, crumpling, and bending, but not tearing new holes or sewing up old ones. Think of it as the field of \rubber-sheet" geometry since the rigid shapes of the objects are not relevant. Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them These surfaces come in a diverse array of variations and combinations. Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction Our context: The field of Topology is concerned with the properties of space that are preserved under continuous deformations such as stretching, crumpling, and bending, but not tearing new holes or sewing up old ones. Think of it as the field of \rubber-sheet" geometry since the rigid shapes of the objects are not relevant. In particular, mathematicians often study mathematical structures in topology called surfaces. Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction Our context: The field of Topology is concerned with the properties of space that are preserved under continuous deformations such as stretching, crumpling, and bending, but not tearing new holes or sewing up old ones. Think of it as the field of \rubber-sheet" geometry since the rigid shapes of the objects are not relevant. In particular, mathematicians often study mathematical structures in topology called surfaces. These surfaces come in a diverse array of variations and combinations. Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Sphere, S2 Torus, T2 Klein Bottle, K 2-holed Torus, T2#T2 Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction Examples of Topological Surfaces: Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Torus, T2 Klein Bottle, K 2-holed Torus, T2#T2 Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction Examples of Topological Surfaces: Sphere, S2 Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Klein Bottle, K 2-holed Torus, T2#T2 Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction Examples of Topological Surfaces: Sphere, S2 Torus, T2 Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them 2-holed Torus, T2#T2 Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction Examples of Topological Surfaces: Sphere, S2 Torus, T2 Klein Bottle, K Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction Examples of Topological Surfaces: Sphere, S2 Torus, T2 Klein Bottle, K 2-holed Torus, T2#T2 Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Question: If two surfaces are not the same, how can we tell them apart? Question: How many categories of surfaces are there? Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Motivating Questions Question: When are two surfaces the same? Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Question: How many categories of surfaces are there? Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Motivating Questions Question: When are two surfaces the same? Question: If two surfaces are not the same, how can we tell them apart? Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Motivating Questions Question: When are two surfaces the same? Question: If two surfaces are not the same, how can we tell them apart? Question: How many categories of surfaces are there? Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Connected Sums Compact How to work with surfaces Locally Euclidean Polygonal Presentations Homeomorphism Klein Lemma Coffee and Donuts General Remarks The Classification Theorem Conclusion Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts Surfaces Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Connected Sums How to work with surfaces Locally Euclidean Polygonal Presentations Homeomorphism Klein Lemma Coffee and Donuts General Remarks The Classification Theorem Conclusion Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts Surfaces Compact Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Connected Sums How to work with surfaces Polygonal Presentations Homeomorphism Klein Lemma Coffee and Donuts General Remarks The Classification Theorem Conclusion Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts Surfaces Compact Locally Euclidean Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Connected Sums How to work with surfaces Polygonal Presentations Klein Lemma Coffee and Donuts General Remarks The Classification Theorem Conclusion Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts Surfaces Compact Locally Euclidean Homeomorphism Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Connected Sums How to work with surfaces Polygonal Presentations Klein Lemma General Remarks The Classification Theorem Conclusion Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts Surfaces
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