Current year: FALL 2020
Math 429 - Fourier Analysis
Instructor: Xiumin Du
Meeting Time: Asynchronous
Fourier analysis is a tool that takes a complicated function and breaks it into a sum of simple functions. This decomposition is a useful tool in a remarkable variety of problems, including partial differential equations, number theory, combinatorics, geometry, and applied math. We will rigorously study this decomposition, and look at many applications.
Textbook: We will use a textbook from the Princeton Lectures in Analysis sequence, written by Stein and Shakarchi: Fourier Analysis: An Introduction. We will cover Chapters 1-6.
Math 430-1: Dynamical systems
MATH 483 - Algebraic Geometry
Instructor: Paul Goerss
Meeting Time: MWF 4:10-5:00p
This is the first quarter of a two-quarter sequence on schemes and their coherent cohomology, following Hartshorne, with added material to raise the emphasis on the modern “functors of points” point of view. The fall quarter will develop schemes, quasicoherent sheaves, divisors, maps to projective space, and differentials. Topics that require cohomology, such as Serre duality and Reimann-Roch will be left of the winter quarter.
MATH 484 – Lie Theory
Instructor: Nir Avni
Meeting Time: MWF 6:30-7:20p
Math 484 will be about the classical groups and their representations. Topics I plan to cover are:
- The classical (compact) groups.
- Representations of SU_n.
- Representations of the symmetric groups.
- The relation between representations of SU_n and S_n: Schur--Weyl duality and symmetric functions.
- (time permitting) Representations of SL_n(F_p).
MATH 514 – Topics in Geometry
Instructor: Emmy Murphy
Meeting time: MWF 12:40-1:30p
The intent is to have a broad-focused “stroll” through 3-dimensional topology: not too many multiple-lecture proofs, but still getting into some deep and pretty mathematics. Lots of pictures. We’ll maybe do some student presentations depending on turnout. No pre-reqs beyond the standard 440-sequence.
We survey a number of results in knot theory and the topology of 3–manifolds. We’ll start from the classical theory: Alexander's theorem, \pi_1 of knot complements, the Alexander module, Dehn surgery, the Lickorish-Wallace theorem. We’ll discuss Papa’s theorems (but probably not prove them). The Jones polynomial and the Tait conjectures. Haken manifolds, irreducible surfaces, and Seifert fibered spaces. Uniqueness of prime decompositions, and the JSJ theorem. Other possible topics depending on time/interest: relationships with 4-manifolds, foliation theory, sutured hierarchies, Heegaard-Floer homology, eight model geometries and the classification of surface diffeomorphisms, Hatcher’s theorem.
MATH 517 – Topics in Algebra
Instructor: Boris Tsygan
Title: Differential calculus in positive characteristic.
Meeing time: MWF 10:20am
The course will be as concrete as possible. I will start with the Cartier isomorphism between differential forms and De Rham cohomology. I will use it to prove Deligne and Illusie’s theorem about degeneration of the Hodge to De Rham spectral sequence. Then I will define the ring of Witt vectors, following both the classical definition and Joyal’s interpretation via delta-rings. Then I will define De Rham-Witt complex and prove its main properties.
After that, I will discuss derived De Rham, crystalline, and prismatic cohomologies. I will also outline noncommutative versions of the above notions, such as Kaledin’s Cartier isomorphism, noncommutative Witt vectors, and degeneration theorem. I will say at least something about D-modules and deformations of symplectic manifolds in positive characteristics.
- Notes of Beilinson-Drinfeld seminar from 2019
- YouTube talks
MATH 521 – Topics in Representation Theory
Instructor: Ben Antieau
Meeting Time TBA
This will be a course on derived algebraic geometry from the perspective of animated (formerly known as simplicial) commutative rings. I will post weekly lecture notes to study asyncronously; remote course meetings will take the form of structured discussions about the material. These meetings are tentatively Mondays and Fridays at 9am and Tuesdays at 8pm, but times are subject to change depending on the needs of the participants. Our weekly schedule is below.
Week 1 Simplicial commutative rings
Week 2 Animated commutative rings
Week 3 The cotangent complex
Week 4 Derived de Rham cohomology
Week 5 Circle actions
Week 6 The HKR theorem
Week 7 The de Rham filtration on periodic cyclic homology
Week 8 Derived stacks
Week 9 Artin--Lurie representability
Week 10 The moduli stack of objects
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