MATH 583, FALL 2021
Course Title: K-Theory
Instructor: Nigel Higson
Meeting Times: Tuesdays and Thursdays, 1:35-2:50. This class will be held online, via Zoom. Contact me (at mylastname at psu.edu) for connection coordinates. Or, if you’re officially registered in the class, wait for my email.
Office Hours: By appointment, via Zoom. Contact me, as above.
Overview and Goals: The goal of the course is to become familiar with topological K-theory, as developed by Atiyah and Hirzebruch, and study some of its classic applications. High points will be the Bott periodicity theorem, and the application of topological K-theory to the Hopf invariant one problem. At the end of the semester we shall examine the extension of topological K-theory to C*-algebras, and use it to study noncommutative tori. I also plan to hold some optional extra sessions to present (what I understand about) connections between K-theory and condensed matter physics.
Prerequisites: Basic familiarity with general topology, linear algebra and functional analysis (roughly as in the first year graduate courses). From time to time further background material will be assumed (for example basic aspects of manifold theory and Banach spaces).
Texts: I recommend Atiyah's book K-Theory, which is widely available, and Hatcher's Vector Bundles and K-Theory. Neither of these texts cover K-theory for C*-algebras, which we shall examine towards the end of the course. For that I will provide other references.
Evaluation: There will be occasional homework assignments, which I hope will give you an opportunity to absorb the class material more deeply. You are asked to hand in a portion of the homework, although you may hand in more, if you wish. You are welcome to work in teams. There will be no exams.
COVID-19 Statement. It may be possible to switch to in-person classes later in the semester. If that happens, then everyone will be expected to follow all university and government guidelines regarding disease prevention measures, and this page will be updated with further information. In the mean time, be sensible and take care.
Academic Integrity Statement: Academic integrity is the pursuit of scholarly activity in an open, honest and responsible manner. Academic integrity is a basic guiding principle for all academic activity at The Pennsylvania State University, and all members of the University community are expected to act in accordance with this principle. Consistent with this expectation, the University’s Code of Conduct states that all students should act with personal integrity, respect other students’ dignity, rights and property, and help create and maintain an environment in which all can succeed through the fruits of their efforts. Academic integrity includes a commitment by all members of the University community not to engage in or tolerate acts of falsification, misrepresentation or deception. Such acts of dishonesty violate the fundamental ethical principles of the University community and compromise the worth of work completed by others.
Disability Accommodation Statement: Penn State welcomes students with disabilities into the University’s educational programs. Every Penn State campus has an office for students with disabilities. Student Disability Resources (SDR) website provides contact information for every Penn State campus. For further information, please visit the Student Disability Resources website. In order to receive consideration for reasonable accommodations, you must contact the appropriate disability services office at the campus where you are officially enrolled, participate in an intake interview, and provide documentation: See these documentation guidelines. If the documentation supports your request for reasonable accommodations, your campus disability services office will provide you with an accommodation letter. Please share this letter with your instructors and discuss the accommodations with them as early as possible. You must follow this process for every semester that you request accommodations.
Counselling and Psychological Services Statement: Many students at Penn State face personal challenges or have psychological needs that may interfere with their academic progress, social development, or emotional wellbeing. The university offers a variety of confidential services to help you through difficult times, including individual and group counseling, crisis intervention, consultations, online chats, and mental health screenings. These services are provided by staff who welcome all students and embrace a philosophy respectful of clients’ cultural and religious backgrounds, and sensitive to differences in race, ability, gender identity and sexual orientation. Go to Counseling and Psychological Services at University Park (CAPS) or call 814-863-0395, or call the Penn State Crisis Line (24 hours/7 days/week) at 877-229-6400 or reach the Crisis Text Line (24 hours/7 days/week) by texting LIONS to 741741.
Educational Equity: Consistent with University Policy AD29, students who believe they have experienced or observed a hate crime, an act of intolerance, discrimination, or harassment that occurs at Penn State are urged to report these incidents as outlined on the University’s Report Bias webpage.
Course Title: K-Theory
Instructor: Nigel Higson
Meeting Times: Tuesdays and Thursdays, 1:35-2:50. This class will be held online, via Zoom. Contact me (at mylastname at psu.edu) for connection coordinates. Or, if you’re officially registered in the class, wait for my email.
Office Hours: By appointment, via Zoom. Contact me, as above.
Overview and Goals: The goal of the course is to become familiar with topological K-theory, as developed by Atiyah and Hirzebruch, and study some of its classic applications. High points will be the Bott periodicity theorem, and the application of topological K-theory to the Hopf invariant one problem. At the end of the semester we shall examine the extension of topological K-theory to C*-algebras, and use it to study noncommutative tori. I also plan to hold some optional extra sessions to present (what I understand about) connections between K-theory and condensed matter physics.
Prerequisites: Basic familiarity with general topology, linear algebra and functional analysis (roughly as in the first year graduate courses). From time to time further background material will be assumed (for example basic aspects of manifold theory and Banach spaces).
Texts: I recommend Atiyah's book K-Theory, which is widely available, and Hatcher's Vector Bundles and K-Theory. Neither of these texts cover K-theory for C*-algebras, which we shall examine towards the end of the course. For that I will provide other references.
Evaluation: There will be occasional homework assignments, which I hope will give you an opportunity to absorb the class material more deeply. You are asked to hand in a portion of the homework, although you may hand in more, if you wish. You are welcome to work in teams. There will be no exams.
COVID-19 Statement. It may be possible to switch to in-person classes later in the semester. If that happens, then everyone will be expected to follow all university and government guidelines regarding disease prevention measures, and this page will be updated with further information. In the mean time, be sensible and take care.
Academic Integrity Statement: Academic integrity is the pursuit of scholarly activity in an open, honest and responsible manner. Academic integrity is a basic guiding principle for all academic activity at The Pennsylvania State University, and all members of the University community are expected to act in accordance with this principle. Consistent with this expectation, the University’s Code of Conduct states that all students should act with personal integrity, respect other students’ dignity, rights and property, and help create and maintain an environment in which all can succeed through the fruits of their efforts. Academic integrity includes a commitment by all members of the University community not to engage in or tolerate acts of falsification, misrepresentation or deception. Such acts of dishonesty violate the fundamental ethical principles of the University community and compromise the worth of work completed by others.
Disability Accommodation Statement: Penn State welcomes students with disabilities into the University’s educational programs. Every Penn State campus has an office for students with disabilities. Student Disability Resources (SDR) website provides contact information for every Penn State campus. For further information, please visit the Student Disability Resources website. In order to receive consideration for reasonable accommodations, you must contact the appropriate disability services office at the campus where you are officially enrolled, participate in an intake interview, and provide documentation: See these documentation guidelines. If the documentation supports your request for reasonable accommodations, your campus disability services office will provide you with an accommodation letter. Please share this letter with your instructors and discuss the accommodations with them as early as possible. You must follow this process for every semester that you request accommodations.
Counselling and Psychological Services Statement: Many students at Penn State face personal challenges or have psychological needs that may interfere with their academic progress, social development, or emotional wellbeing. The university offers a variety of confidential services to help you through difficult times, including individual and group counseling, crisis intervention, consultations, online chats, and mental health screenings. These services are provided by staff who welcome all students and embrace a philosophy respectful of clients’ cultural and religious backgrounds, and sensitive to differences in race, ability, gender identity and sexual orientation. Go to Counseling and Psychological Services at University Park (CAPS) or call 814-863-0395, or call the Penn State Crisis Line (24 hours/7 days/week) at 877-229-6400 or reach the Crisis Text Line (24 hours/7 days/week) by texting LIONS to 741741.
Educational Equity: Consistent with University Policy AD29, students who believe they have experienced or observed a hate crime, an act of intolerance, discrimination, or harassment that occurs at Penn State are urged to report these incidents as outlined on the University’s Report Bias webpage.