DEPARTMENT OF MATHEMATICS

FALL 2022

Math 520 Topics in Mathematical Physics

I will begin by giving an overview of the area (more on that below), but after that my goal for this course is that the students learn by preparing presentations and by actively querying one another.   The general area was determined in coversation with some students, but if you have a strong desire for a nearby topic we may be able to accommodate.  Rules of the game:  student presentations must provide context and must contain concrete examples to illustrate the main principles (speakers will not be allowed to "not have time" to get to the examples!).

The subjects will be cluster varieties, knots, and their relation to physics.

• Some promising models of string theory reduce in four dimensions to N=1 or N=2 supersymmetric theories.  These theories have "BPS states" with enhanced symmetries, whose enumeration and interaction can often be described mathematically, and often in more ways than one, leading to dualities.  Notable BPS states are "branes."  Categories of branes and their moduli of objects play a key role in elucidating the mathematical structure from physical theoreis.  For us, the mathematical stand-in for brane categories will be categories of constructible sheaves with singular support defined by a Legendrian variety --- for example, the boundary of a knot conormal.  These categories are invariants of the Legendrian up to Legendrian isotopy. 
• Foundational work on the above mathematics was provided by Kashiwara, Schapira and Guillermou.  We will be more concerned with various applications, where moduli of objects can be identified with various cluster varieties, especially those of Fock and Goncharov.

Here are some topics that students might explore.  A main goal of the course will be to thread them together.

• DT invariants of quivers (mathematically and as an effective theory of intersecting branes -- various papers, e.g. Kontsevich-Soibelman [COHA] and Cordoba-Cecotii-Vafa, others )
• Cluster varieties and their quantization, especially those of Fock-Goncharov:  framed local systems on surfaces.  (Goncharov-Shen survery paper)
• Cluster varieties and "curves on surfaces."  (Shende-Treumann-Williams)
• Clusters and braids (Casals-Gorsky-Gorsky-Shen-Simental)
• The chromatic Lagrangian inside the cluster variety of framed local systems on surfaces.  (Dijkgraaf-Gabella-Goncharov and other references)
• HOMFLY-PT invariants of knots as augmentations, and wavefunctions (Aganagic-Ekholm-Ng-Vafa)
• Skeins and branes (Ekholm-Shende) 
• Goncharov's representation of the cluster groupoid (Goncharov).

MATH 515-1 Topics in Geometry and Topology

Winter 2022

Math 445 Differential Geometry

• Prerequites: Manifolds, Tangent Bundle, Riemannian manifolds.  Some basics on pdes on R^n would be helpful.
• Breakdown of Class time (roughly):  There is more below than there are weeks, so in the end we will take a vote on which extra topics to cover.  I dont promise to get to Atiayah-Singer if we cover extras from previous topics.
• Prelimaries: ~ 1-3 weeks.  Topics will include Lie groups, principal bundles, vector bundles, connections/curvature, laplacians on manifolds, heat flows, brief discussion of other pdes.  How detailed and how much will depend on class familiarity.
• Hodge Theorem: ~ 3 Weeks.  Topics include basic functional analysis (closed operators on Hilbert spaces), Laplacians on Vector bundles, elliptic estimates, applications to eigenvalue behavior, deRham cohomology, proof of Hodge theorem.
• Chern-Weil Theory:  ~2 Weeks.  Topics include characteristic classes (a geometric viewpoint, topological viewpoint may be stated but not proved), invariant polynomials, construction of representatives given a connection on a bundle.  Extra possible topic (~1-2 week) : we can cover Quillen's chern-weil theory for superconnections, which is particularly useful on cotangent bundles of manifolds.  This has a nice application for understanding Thom Classes and generalized Atiyah-Singer, which we would also at least state in this case.
• Atiyah-Singer: ~ 3 weeks.  Topics include spin bundles, clifford actions, spin representation, dirac operators, index for spin manifolds and clifford modules.  If desired we can state Atiyah-Singer for general operators, but will only prove for clifford modules (time permitting).  

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

• Basic topics:  What is a supermanifold, mappings between them, vector bundles and forms on them, supersymplectic manifolds.  Emphasis on examples.
• Chern-Gauss-Bonet by supersymmetric methods:  This might belong in the basic topic section, as mainly it is an excuse to explore the basics in an interesting setup.  We follow the setup of Berwick to give an instructive proof of the Chern-Gauss-Bonet theorem in the super context which is fairly short.
• Supersymplectic Manifolds and their semiclassical analysis:  Unlike the real case Supersymplectic forms recover geometry and curvature and encode these into the associated flows.  Additionally, supersymmetry allows an even Hamiltonian vector field to be written as the square of an odd vector field, which is not possible in the real case.  The effect of this is that the semiclassical analysis in the supersymmetric case can see not just the top order part of an operator, but often lower order parts as well.  We will slowly explore this, again with an emphasis on examples.

Fall 2021

440-1,2,3 GEOMETRY/TOPOLOGY

Instructor: Elton Hsu
Meeting Time: MWF at 10a

Fall (Introduction to differentiable manifolds):  

• Differentiable manifolds; implicit function theorem and Sard’s theorem;tangent vectors and tensors; vector fields and flows; integral manifolds and Frobenius's theorem; differential forms, orientation and integration; Riemannian metrics, theLevi-Civita connection, geodesics, and the exponential map.
  

Winter (Introduction to algebraic topology):

• The fundamental group of a space; covering spaces; the Van-Kampen theorem; singular homology; homotopy invariance, excision; homological algebra; degree and CW homology.
  

Spring (Cohomology):

• Singular cohomology; the cup product; de Rham cohomology; sheaf cohomology; the de Rham theorem; orientability and Poincaré duality.

Math 470-1: algebra

Instructor: Paul Goerss

Math 483-1: algebraic geometry

Instructor: Nir Avni

Math 514-1: topics in geometry

SPRING 2021

Math 514-2: topics in geometry

• Physics terminology:  We'll construct nonperturbative \phi^4 field theory based on 2 and 3 dimensional manifolds. 
• Math terminology: we'll prove existence of certain (apriori singular) linear evolution equations and measures on some infinite dimensional spaces. 

• linear pde's on finite dimensional spaces,
• function spaces and elliptic regularity through besov,
• gaussian measures on finite and infinite dimensional Hilbert spaces,
• Hamiltonian quantization (differentiation), Lagrangian quantization (integration),
• field quantization, renormalization (a purely nonperturbative point of view),
• linear pde's on infinite dimensional spaces.

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:

• mixing properties of dynamical systems, shift spaces and subshifts (including shifts of finite type and sofic shifts, one sided vs. two sided), conjugacy of systems, automorphisms of shift systems, embeddings and factors, topological entropy, shift equivalence, and substitutions. 

MATH 511-1: Topics in ANALYSIS 

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

• No textbook required. I will very loosely follow Wolff’s “Lectures in harmonic analysis” and Javier Duoandikoetxea's "Fourier Analysis".

• Real analysis is highly recommended, as is some exposure to undergraduate Fourier analysis.

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

• Homework: TBA. Depends on whether there is a grader
• The Evaluation Component – Grades: There will be no in-class tests, but there will be guided projects in addition to the homework.
• Hybrid Format: As of December 1, 2020, this class is scheduled to be taught in a hybrid format, meaning there will be some in-person class meetings. Math 483 Algebraic Geometry met in the Fall quarter on Fridays, and was on-line the rest of the time. This may be a model for Math 440. Learning and teaching are highly interactive and if we can hold meetings it will add considerably to the experience. However, the final format of the class is not yet set; the final determination will depend on how enrollments proceed, the policies and procedures set by the University, and the course of COVID in the community
  • 1. Remote learning is possible, and even on-campus students should not attend a class if they are not well or not comfortable
  • 2. In the first week of the quarter all classes are only on-line. We will decide on format for later classes at that time
  • 3. All classes will be synchronously on-line, will be recorded, and will be available for later review. Canvas: All class materials, including the syllabus, more detailed descriptions of class topics, all assignments including homework and the projects, further policies and procedures, all grades, and the class Zoom link will all be available through Canvas. Zoom: All classes will be accessible virtually, over Zoom, and recorded. Attendance is restricted to registered students, and the link will be available only through Canvas. If, at any time, the internet connection fails and there is still substantial class time remaining, the instructor will try to reestablish connection for up to ten (10) minutes. If there is still no connection after that time, class will be deemed cancelled and the material will be rescheduled.

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:

• (a) Introduce a new problem/topic, cover what is known/not known
• (b) Explain one fundamental tool related to that area in detail
• (c) If not last week of classes, go back to (a) and repeat.

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.

• Pre-requisites: Math maturity should suffice. 450-1,2,3 (probability) is desirable so is 410-2 (functional). Classes: Classes are online, synchronous. They will be recorded and posted on Canvas. References: Tons. They will be shared in class as we go.
• Assessment : There will be no exams. Assessment will be based on 6 problem sets - one for each topic. If you want credit for this class you will need to submit all assignments. Another important info: Any student with a disability requesting accommodations is required to register with Services for Students with Disabilities (ssd@northwestern.edu; 847-467-5530) and provide an accommodation notification from SSD to his/her professor, preferably within the first two weeks of class. All information will remain confidential.
  
  

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 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.

• https://sites.math.northwestern.edu/~pgoerss/math483/syl483.pdf

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:

•  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

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: 

• Illusie
  • http://www.numdam.org/article/ASENS_1979_4_12_4_501_0.pdf
• Bhatt
  • https://arxiv.org/abs/1204.6560 
  • https://arxiv.org/abs/1204.6560
• Drinfeld
  • https://arxiv.org/abs/2005.04746
  • https://arxiv.org/abs/1810.11853
• Kaledin
  • https://arxiv.org/abs/1708.05660
  •  https://arxiv.org/abs/1708.05660
• Notes of Beilinson-Drinfeld seminar from 2019
• YouTube talks
  • https://www.youtube.com/watch?v=3wCk3qsFWLA 
  • https://www.youtube.com/watch?v=VW5NqoHi0Co
  • https://www.youtube.com/watch?v=mTLgU3JkZR0

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