Barwick

18.915. The Kan seminar.

Autumn 2012.

• Time: Tuesday & Thursday, 1-2:30
• Place: 2-131
• This is a literature seminar for graduate students and very advanced undergraduates. Our focus will be on reading and discussing a range of classic papers in algebraic topology.
• Recommended papers
• Speaker list:
• 11 September: Akhil Matthew. G. Segal, Categories and cohomology theories.
• 13 September: Alisa Knizel. E. H. Brown, Cohomology theories.
• 18 September: No class.
• 20 September: Jay Shah. J. F. Adams and M. A. Atiyah, K-theory and the Hopf invariant.
• 20 September: Denis Nardin. J. F Adams, On the structure and applications of the Steenrod algebra.
• 25 September: Danny Shi. C. T. C. Wall, Finiteness conditions for CW complexes.
• 27 September: Lukas Brantner. P. J. Hilton, On the homotopy groups of the union of spheres.
• 2 October: John Berman. J. C. Moore, Semi-simplicial complexes and Postnikov systems.
• 2 October: Teng Fei. J. W. Milnor, On manifolds homeomorphic to the 7-sphere.
• 4 October: Michael Jemison. M. A. Kervaire, A manifold which does not admit any differentiable structure.
• 9 October: Columbus Day holiday.
• 11 October: No class.
• 16 October: Akhil Matthew. W. Browder, Homotopy type of differentiable manifolds.
• 18 October: Teng Fei. D. G. Quillen, Rational homotopy theory.
• 23 October: Jay Shah. J. P. May and R. W. Thomason, The uniqueness of infinite loop space machines.
• 25 October: Denis Nardin. G. Nishida, The nilpotency of elements of the stable homotopy groups of spheres.
• 30 October: Danny Shi. E. Dror (Farjoun), Acyclic spaces.
• 1 November: Michael Jemison. M. F. Atiyah and G. Segal, Equivariant K-theory and completion.
• 6 November: John Berman. G. Segal, Classifying spaces and spectral sequences.
• 8 November: Lukas Brantner. D. G. Quillen, On the formal group laws of unoriented and complex cobordism theory.
• 13 November: Alisa Knizel. D. G. Quillen, On the formal group laws of unoriented and complex cobordism theory, cont.
• 15 November: No class.
• 20 November: Denis Nardin. A. G. Bousfield, Localization of spectra with respect to homology.
• 22 November: Thanksgiving holiday.
• 27 November: Teng Fei. P. Deligne, P. Griffiths, Morgan, D. Sullivan, Real homotopy theory.
• 27 November: Alisa Knizel. D. G. Quillen, The Adams conjecture.
• 29 November: Jay Shah. S. P. Novikov, Methods of algebraic topology from the point of view of cobordism theory.
• 4 December: John Berman. I. M. James, Reduced product spaces, On the suspension triad, On the suspension triad of a sphere, On the suspension sequence.
• 4 December: Michael Jemison. D. G. Quillen, On the cohomology and K-theory of the general linear group over a finite field.
• 6 December: Danny Shi. J. F. Adams, Vector fields on spheres.
• 11 December: Lukas Brantner. D. G. Quillen, Higher algebraic K-theory I.
• 11 December: Akhil Mathew. E. S. Devinatz, M. J. Hopkins, and J. Smith, Nilpotence and stable homotopy theory, I.

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18.917. Algebraic K-theory.

Spring 2012.

• Outline
• This is a course on the algebraic K-theory of higher categories, culminating in the proof of the so-called "Theorem of the Heart."
• Notes from selected lectures:
• 1. Euler characteristics and K-theory.
• 2. Waldhausen categories, and a delooping of the universal property of the K.
• 3. From relative categories to quasicategories.
• 4. Fibrations.
• 5. Limits and colimits in ∞-categories.
• 6. Pairs of ∞-categories and Waldhausen ∞-categories.
• 7. Virtual Waldhausen ∞-categories I.
• 8. Filtered objects.
• 9. Distributive virtual Waldhausen ∞-categories.
• 10. Additive theories.
• 11. Goodwillie additivization.
• 12. Excision and Waldhausen's fibration theorem.
• 13. Labeled Waldhausen ∞-categories.
• 14. Localization.
• [ lectures 15-16: no notes ]
• 17. Exact ∞-categories.
• [ lectures 17-21: no notes ]
• 22. t-structures and K-theory.

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18.024. Calculus with theory II.

Spring 2012.

• Syllabus
• The focus of this course is the rigorous development of the concepts of multivariable calculus, including: linear algebra, total derivatives, partial derivatives, integration, multilinear algebra, and differential forms. This course is a continuation of 18.014; it covers essentially the same material as 18.02, but it does so from a conceptual and rigorous point of view, emphasizing careful reasoning and proofs.
• Problem sets:
• Problem set I.
• Problem set II.
• Problem set III.
• Problem set IV.
• Problem set V.
• Problem set VI.
• Problem set VII.
• Problem set VIII.
• Problem set IX.

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18.014. Calculus with theory I.

Autumn 2011.

• Syllabus
• The focus of this course is the rigorous development of the concepts of single-variable calculus, including: the axioms for the real number system; the Riemann integral; limits; continuity; differentiation; the fundamental theorems of calculus; Taylor's theorem; series and power series. This course covers essentially the same material as 18.01, but it does so from a conceptual and rigorous point of view, emphasizing careful reasoning and proofs.
• Problem sets:
• Problem set I.
• Problem set II.
• Problem set III.
• Problem set IV.
• Problem set V.
• Problem set VI.
• Problem set VII.
• Problem set VIII.
• Problem set IX.
• Problem set X.

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18.100C. Analysis I.

Spring 2011.

• Syllabus
• The goal of this course is to equip undergraduates with a concentration in mathematics or a closely related discipline with the basic tools of real analysis and to develop a facility at reading, writing, and communicating mathematics.
• Problem sets:
• Problem set I.
• Problem set II.
• Problem set III.
• Problem set IV.
• Problem set V.
• Problem set VI.
• Problem set VII.
• Problem set VIII.
• Problem set IX.
• Problem set X.

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18.02. Multivariable calculus (recitations).

Autumn 2010.

• Course website