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spring 2022

Math 516-1: topics in topology

Instructor: Gus Shrader
Meeting Time: MWF 11a

Description:

This course will be an introduction to geometric representation theory, with the aim of providing an overview of some of the fundamental results and techniques in the field. Material we will cover includes:

  1. Geometry of the flag variety of a complex semisimple Lie group
  2. The Borel-Weil-Bott theorem; proof of the Weyl character formula bylocalization in equivariant K-theory
  3. D-modules on the flag variety and the Beilinson-Bernstein theorem
  4. Convolution algebrasTime permitting, we may also explore more specialized topics according to participants' interests.

 

Winter 2022

Math 445 Differential Geometry

Instructor: Aaron Naber
Time: MWF 12pm
Main Topics List:  Prelimaries , Hodge Theorem for deRham Cohomology, Chern-Weil Theory, Atiyah-Singer Index theorem.

483-2 Algebraic geometry

Instructor: Yuchen Liu
Meeting Time: MWF at 2p

This course is an introduction to the modern language of algebraic geometry and its basic technical tools: schemes and sheaves. This language pervades a great deal of modern mathematics, and is now fundamental to algebraic number theory, geometric representation theory, and some aspects of homotopy theory as well as K-theory and algebraic geometry. We will largely follow Robin Hartshorne's book (Chapters 2 and 3) and Ravi Vakil's notes.

Math 514 class Topics in Geometry

Instructor: Aaron Naber
Meeting Time: MWF 3pm
Class Breakdown:  There will be one main topic covered twice per week and several smaller topics covered once per week by guest speakers.
Main Topic:  Supermanifolds and their analysis.  A basic overview of what they are and how they work with emphasis on examples.  One practical goal is to see how this structure can be used to build very accurate approximations of heat kernels (much more accurate than classical approximations, which is what interests me).  Another practical goal is to build a supersymmetric semiclassical analysis in the sense of Weinstein (this is ambitious as this does not exist in the literature and involves quite a few subtle new points.  As for the heat kernels, the answer seems to give much more information than the classical case).  We will cover the classical case first of course.
Smaller Topics:  Optimal Transport on Manifolds, lower sectional curvature and its implications, analysis on ginzburg-landau equations.
Main Topic Breakdown:

 

Fall 2021

440-1,2,3 GEOMETRY/TOPOLOGY

Instructor: Elton Hsu
Meeting Time: MWF at 10a

Fall (Introduction to differentiable manifolds):  

Winter (Introduction to algebraic topology):

Spring (Cohomology):

Math 470-1: algebra

Instructor: Paul Goerss

Meeting Time: MWF at Noon
Click here for Syllabus Link

 

Math 483-1: algebraic geometry

Instructor: Nir Avni

Meeting Time: MWF at 2p
The Fall quarter will be an introduction to algebraic geometry. We'll spend half of the course on curves (highlights: Bezout's theorem, abelian integrals, Riemann--Roch) and the other half on higher dimensional varieties (highlights: Lefschetz principle, dimension, tangent cone).

Math 514-1: topics in geometry

Instructor: Ezra Getzler
Meeting Time: MWF at 11a
FQ21 Math 14-1 Syllabus: https://canvas.northwestern.edu/courses/152234/pages/syllabus

 

Math 518-1: topics in number. theory

Instructor: Ilya Kkayutin
Meeting Time: MWF at 10a

This is a basic notions class about homogenous dynamics and measure rigidity. There will be no prerequisites in dynamics or algebraic groups. 
In many arithmetic questions we have an algebraic group G acting transitively on a space X, and we want to understand the orbits of the action of a subgroup H<G on X. Dynamical methods describe the behavior of a typical orbit, but for arithmetic applications we need to understand a specific orbit. The magic of rigidity in homogeneous dynamics is that in favorable situations there is an explicit algebraic description of the asymptotic properties of all orbits. After a survey of homogeneous spaces and elements of ergodic theory, this course covers some of the methods used in the proof of Ratner’s measure rigidity results for unipotent flows. Particular attention is given to selected applications in number theory.
 

SPRING 2021

Math 514-2: topics in geometry

Instructor: Aaron Naber
Meeting Time: MWF at Noon
Remote
Title:  Analysis on finite and infinite dimensional spaces
Main Topic:  
A classic result proved by Glimme/Jaffe, we will give a new proof which takes just a few pages.  Much of the class will be background working up to understanding all the words used above and the classical tools needed: 
Secondary topic: Anthony McCormick will lecture on the Ginsburg Landau  pde, beginning with the basics and working up to some open problems in the field.
Possible other topics:  Time allowing, the second part of the course may include several other topics of a few lectures each, including:  Man-Chun, uniformization in kahler geometry; Neumayer, Gromov's torus rigidity for nonnegative scalar curvature; Hupp, Simon’s construction of minimal hyper surfaces with irregular singular sets.

 

winter 2021

Math 430-2: Dynamical Systems 430-2  

Instructor: Bryna Kra
Meeting Time: MWF at 1p
Remote

Although this is the second quarter in a series, the course will be self contained and you can register for it without having taken the first quarter. The course will cover Topological and Symbolic Dynamics. 

Symbolic dynamics is a key tool that was developed to study topological, smooth, and measurable dynamical systems. More recently, the methods and ideas have found applications in linear algebra, computer science, and particularly in data storage and transmission. The basic idea is to use an infinite sequence of symbols, each of which corresponds to some state of the system, to construct a symbolic model for a complicated system.  Then properties of this model can be used to understand properties of the system. 

This idea goes back to the work of Hadamard in the late 1890's, who used codings to study geodesics on surfaces of negative curvature, and was further developed by Artin, Morse, Hedlund, Birkhoff and many others. The first formal treatment of the subject is the fundamental paper of Morse and Hedlund that plays a key role in motivating the results under discussion today.  This paper from 1938 named the subject and introduced the general philosophy that still holds sway over the subject.

Topics to be covered include:

MATH 511-1: Topics in ANALYSIS 

Instructor: Xiumin Du
Meeting Time: MWF at 2p
Remote

Textbook
Prerequisite
Course Description
  • In this course, we'll cover some classic topics in harmonic analysis, including tempered distributions, Hilbert transform, Calderon-Zygmund singular integral operators, and Littlewood-Paley theory and multipliers. If time permits, we'll also briefly discuss the Fourier restriction problem and Kakeya conjecture.

Math 440-2: geometry and topology

Instructor: Paul Goerss
Meeting Time: MWF at 10am
Hybrid

This is the second quarter of a three-quarter course; the emphasis here will be on basic algebraic topology. The intent is to go from the definition of singular homology to Poincar´e Duality.

  1. Singular homology; homotopy invariance
  2. Basic homological algebra
  3. Local-to-global computations; Mayer-Vietoris
  4. CW complexes: CW-homology, simplicial homology
  5. More advanced homological algebra: Universal Coefficient Theorems, Eilenberg-Zilber
  6. Cohomology: cohomology rings, the cohomology of projective space
  7. Orientations
  8. Poincar´e Duality

Math 520-1: Topics in mathematical physics

Instructor: Antonio Auffinger
Meeting Time: MWF at Noon
Remote

This is a topics class that touches a few classical and recent areas of probability theory. I plan to cover six independent topics. They are listed below. Classes will proceed through the following routine:

Topics:
  1. KPZ universality (a) Particle systems, totally asymmetric simple exclusion process, RSK correspondence. (b) Young tableaux: limit shapes and fluctuations.
  2. Random metrics on Z d . (a) Subadditive Ergodic Theorem and Gromov-Hausdorff convergence. (b) Asymptotic direction of geodesics
  3. Brownian motion and PDE’s. (a) Dirichlet’s problem, Feynman-Kac formula. (b) Polar sets for Brownian motion, capacity, and existence of bounded harmonic functions on domains
  4. Random Graphs (a) Phase transitions for Erdos-Renyi. (b) Preferential attachment models
  5. Riemann Zeta Function and Log Correlated Fields (a) Branching Brownian motion, Gaussian Free Field, extremal processes. (b) Fyodorov-Hiary-Keating conjecture
  6. The de Brujin-Newman constant. (a) Relation to Lee-Yang’s Theorem. (b) Rodgers-Tao’s inequality.

There are positive and negative aspects in this plan. First, we will not be able to go deep in any of these areas. Each item could (and deserves to) be developed as a single course. I will try to mitigate this problem by providing enough references and a possible study plan for each one. So if you fall in love with a topic, you will get guidance and tons of stuff to read. At the same time, if you are not a fan of a particular subject, you can always turn off your camera and wait for the next one to see something new. 

Think this class as a tasting menu. Each dish will be small but unique with tons of flavor. They will also fit a broad narrative. Will this experience help you to become a better chef? Probably not. It should at least bring perspective and hopefully some things to think about.

FALL 2020

Math 429 - Fourier Analysis

Instructor: Xiumin Du
Meeting Time: Asynchronous
Remote

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

Instructor: Aaron Brown
Meeting Time: MWF 11:30
Remote Synchronous 
This course will be an introduction to dynamical systems.  Through a number of standard examples, I plan to introduce and study various definitions and phenomena occurring in the settings of topological, measure theoretic, symbolic, and smooth dynamical systems.  
Prerequisites for the course are undergrad background in real analysis, measure theory, and abstract algebra.  I will not assume any previous background in dynamical systems.  
I will primarily follow the text: Brin and Stuck, Introduction to dynamical systems.

 

MATH 483 - Algebraic Geometry

Instructor: Paul Goerss

Meeting Time: MWF 4:10-5:00p

Hybrid

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 6:30-7:2pm 

Remote

Math 484 will be about the classical groups and their representations. Topics I plan to cover are:

MATH 514 – Topics in Geometry

Instructor: Emmy Murphy

Meeting time: MWF 12:40-1:30p

Remote

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 

Remote

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.

 

Click Here to view FQ20 Math 517 Syllabi 

Sources include: 

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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