CS6840 - Advanced Complexity Theory

Meetings  

• Theme 1 : Derandomization - 5 meetings

  • Meeting 01 : Wed, Apr 20, 08:00 am-08:50 am

  The BPP vs P problem. Pseudorandom generators. Nisan-Wigderson construction of PRGs (overview).

  References	[MB Notes]
  Exercises
  Reading

  The BPP vs P problem. Pseudorandom generators. Nisan-Wigderson construction of PRGs (overview).
  

  References:

  [MB Notes]

  • Meeting 02 : Fri, Apr 22, 12:00 pm-12:50 pm

  Proof of N-W constructions: Hybrid argument.

  References	[MBNotes]
  Exercises
  Reading

  Proof of N-W constructions: Hybrid argument.
  

  References:

  [MBNotes]

  • Meeting 03 : Mon, Apr 25, 10:00 am-10:50 am

  To Be Announced

  References
  Exercises
  Reading

  To Be Announced
  

  References:

  None

  • Meeting 04 : Tue, Apr 26, 09:00 am-09:50 am

  To Be Announced

  References
  Exercises
  Reading

  To Be Announced
  

  References:

  None

  • Meeting 05 : Wed, Apr 27, 08:00 am-08:50 am

  To Be Announced

  References
  Exercises
  Reading

  To Be Announced
  

  References:

  None

• Theme 2 : Complexity of Counting - 7 meetings

  • Meeting 06 : Mon, Jan 11, 10:00 am-10:50 am

  Administrative announcements. Complexity of coutning. Toda's theorem.

  References
  Exercises
  Reading

  Administrative announcements. Complexity of coutning. Toda's theorem.
  

  References:

  None

  • Meeting 07 : Tue, Jan 12, 09:00 am-09:50 am

  Isolating satisfying assignments: Valiant Vazirani Theorem. Proof. Isolating matchings: Mulmuley-Vazirani-Vazirani theorem (without proof).

  References	[MB Notes 1]
  Exercises
  Reading	Read Chapter 16 in [MB Notes 1]

  Isolating satisfying assignments: Valiant Vazirani Theorem. Proof. Isolating matchings: Mulmuley-Vazirani-Vazirani theorem (without proof).
  

  References:	[MB Notes 1]
  Reading:	Read Chapter 16 in [MB Notes 1]

  • Meeting 08 : Wed, Jan 13, 08:00 am-08:50 am

  Valiant Vazirani Theorem.

  References
  Exercises
  Reading

  Valiant Vazirani Theorem.
  

  References:

  None

  • Meeting 09 : Fri, Jan 15, 12:00 pm-12:50 pm

  Proof of Valiant - Vazirani (contd). The class Parity-P.

  References
  Exercises
  Reading

  Proof of Valiant - Vazirani (contd). The class Parity-P.
  

  References:

  None

  • Meeting 10 : Mon, Jan 18, 10:00 am-10:50 am

  Properties of parity-P: Closure under complementation, and Turing reductions.

  References
  Exercises
  Reading

  Properties of parity-P: Closure under complementation, and Turing reductions.
  

  References:

  None

  • Meeting 11 : Tue, Jan 19, 09:00 am-09:50 am

  BP-operator. Probability amplification. BP operator with quantifiers.

  References
  Exercises
  Reading

  BP-operator. Probability amplification. BP operator with quantifiers.
  

  References:

  None

  • Meeting 12 : Wed, Jan 20, 08:00 am-08:50 am

  Proof of Toda's theorem.

  References
  Exercises
  Reading

  Proof of Toda's theorem.
  

  References:

  None

• Theme 3 : Proof Systems - 25 meetings

  • Meeting 13 : Fri, Jan 22, 12:00 pm-12:50 pm

  Proof of isolation lemma. Introduction to proof systems. NP as a prover-verifier system. Generalization to a probabilistic verifier. Definition of interactive proof systems. The class IP.

  References	[MB Notes 1]
  Exercises
  Reading	Read the <a href="https://www.cs.princeton.edu/courses/archive/spring07/cos522/BabaiEmail.pdf"> article </a> by Babai for an account of development of interactive proof systems.

  Proof of isolation lemma. Introduction to proof systems. NP as a prover-verifier system. Generalization to a probabilistic verifier. Definition of interactive proof systems. The class IP.
  

  References:	[MB Notes 1]
  Reading:	Read the article by Babai for an account of development of interactive proof systems.

  • Meeting 14 : Mon, Jan 25, 10:00 am-10:50 am

  No class (due to Shastra)

  References
  Exercises
  Reading

  No class (due to Shastra)
  

  References:

  None

  • Meeting 15 : Tue, Jan 26, 09:00 am-09:50 am

  No lecture (republic day)

  References
  Exercises
  Reading

  No lecture (republic day)
  

  References:

  None

  • Meeting 16 : Wed, Jan 27, 08:00 am-08:50 am

  No Lecture (instructor out of town)

  References
  Exercises
  Reading

  No Lecture (instructor out of town)
  

  References:

  None

  • Meeting 17 : Fri, Jan 29, 12:00 pm-12:50 pm

  No Lecture.

  References
  Exercises
  Reading

  No Lecture.
  

  References:

  None

  • Meeting 18 : Tue, Feb 02, 09:00 am-09:50 am

  Definition of interactive proof systems. A protocol for graph non-isomorphism.

  References	[MB Notes 1]
  Exercises
  Reading

  Definition of interactive proof systems. A protocol for graph non-isomorphism.
  

  References:

  [MB Notes 1]

  • Meeting 19 : Wed, Feb 03, 08:00 am-08:50 am

  IP is contained in PSPACE.

  References	The proof is from [Kozen2] chapter 17. Also see [MBnotes].
  Exercises
  Reading

  IP is contained in PSPACE.
  

  References:

  The proof is from [Kozen2] chapter 17. Also see [MBnotes].

  • Meeting 20 : Fri, Feb 05, 12:00 pm-12:50 pm

  Shamir's theorem: IP=PSPACE. Arithemtization of QBF.

  References	[MBnotes1]. Also see [Kozen2]
  Exercises
  Reading

  Shamir's theorem: IP=PSPACE. Arithemtization of QBF.
  

  References:

  [MBnotes1]. Also see [Kozen2]

  • Meeting 21 : Mon, Feb 08, 10:00 am-10:50 am

  Shamir's theorem (contd)

  References
  Exercises
  Reading

  Shamir's theorem (contd)
  

  References:

  None

  • Meeting 22 : Tue, Feb 09, 09:00 am-09:50 am

  Public coin protocols: Arthur vs Merlin. The class AM. GraphNonIso is in AM: a public coin protocol for graph non-isomorphism.

  References	[MBNotes1] and [AB]
  Exercises
  Reading

  Public coin protocols: Arthur vs Merlin. The class AM. GraphNonIso is in AM: a public coin protocol for graph non-isomorphism.
  

  References:

  [MBNotes1] and [AB]

  • Meeting 23 : Wed, Feb 10, 08:00 am-08:50 am

  Public coin protocol for Graph non isomorphism (contd). Public vs private coin (without proof). GI is NP complete implies PH collapses.

  References	[AB] For the proofs of public vs private coin proof systems, and constant round vs two round AM, see <a href="http://courses.cs.washington.edu/courses/cse532/04sp/"> lecture notes</a> (Lectures 9 and 10) by Paul Beame.
  Exercises
  Reading

  Public coin protocol for Graph non isomorphism (contd). Public vs private coin (without proof). GI is NP complete implies PH collapses.
  

  References:

  [AB] For the proofs of public vs private coin proof systems, and constant round vs two round AM, see lecture notes (Lectures 9 and 10) by Paul Beame.

  • Meeting 24 : Fri, Feb 12, 12:00 pm-12:50 pm

  GI is not NP complete unless PH collapses (contd). More on proof systems. Zero knowledge. Probabilistically checkable proofs.

  References	[AB] for GI. [MBNotes2] for PCP.
  Exercises
  Reading	See<a href="http://courses.cs.washington.edu/courses/cse533/05au/pcp-history.pdf"> History of PCP </a> a chronological note on the developments that lead to the PCP theorem.

  GI is not NP complete unless PH collapses (contd). More on proof systems. Zero knowledge. Probabilistically checkable proofs.
  

  References:	[AB] for GI. [MBNotes2] for PCP.
  Reading:	See History of PCP a chronological note on the developments that lead to the PCP theorem.

  • Meeting 25 : Mon, Feb 15, 10:00 am-10:50 am

  Probablistically checkable Proofs. Definition. Relation to hardness of approximation.

  References
  Exercises
  Reading

  Probablistically checkable Proofs. Definition. Relation to hardness of approximation.
  

  References:

  None

  • Meeting 26 : Tue, Feb 16, 09:00 am-09:50 am

  PCPs and hardness of approximation(contd)

  References
  Exercises
  Reading

  PCPs and hardness of approximation(contd)
  

  References:

  None

  • Meeting 27 : Wed, Feb 17, 08:00 am-08:50 am

  Approximation algorithm and Gap problems. Weak PCP theorem. PCP for LIN.

  References	[MBNotes2] and <a href="http://www.cse.cuhk.edu.hk/~andrejb/csci5170/s10/notes/10L12.pdf"> Lecture notes </a> by Andrej Bogdanov
  Exercises
  Reading

  Approximation algorithm and Gap problems. Weak PCP theorem. PCP for LIN.
  Scribe : Vijayaraghunathan  (notes being written up)

  Scribe	:
  Key	:

  References:

  [MBNotes2] and Lecture notes by Andrej Bogdanov

  • Meeting 28 : Fri, Feb 19, 12:00 pm-12:50 pm

  PCP for LIN. Failed attempts. Need for linearity test. Notion of distance for Boolean functions.

  References	[MBnotes 2] and Notes by Bogdanov.
  Exercises
  Reading

  PCP for LIN. Failed attempts. Need for linearity test. Notion of distance for Boolean functions.
  Scribe : A Sundar.   (rough draft; better one coming soon)

  Scribe	:
  Key	:

  References:

  [MBnotes 2] and Notes by Bogdanov.

  • Meeting 29 : Mon, Feb 22, 10:00 am-10:50 am

  Fourier coefficients of Boolean functions. BLR Linearity test. Proof of correctness.

  References
  Exercises
  Reading

  Fourier coefficients of Boolean functions. BLR Linearity test. Proof of correctness.
  Scribe : A Sundar.  (notes being written up)

  Scribe	:
  Key	:

  References:

  None

  • Meeting 30 : Tue, Feb 23, 09:00 am-09:50 am

  Completing PCP for LIN. Proof of Weak PCP theorem.

  References	<a href="http://www.cse.cuhk.edu.hk/~andrejb/csci5170/s10/notes/10L12.pdf"> Notes </a> by Bogdanov.
  Exercises
  Reading

  Completing PCP for LIN. Proof of Weak PCP theorem.
  Scribe : Santhoshini  (notes being written up)

  Scribe	:
  Key	:

  References:

  Notes by Bogdanov.

  • Meeting 31 : Wed, Feb 24, 08:00 am-08:50 am

  Weak PCP theorem (contd). Dinur's proof of PCP theorem. Introduction and overview.

  References
  Exercises
  Reading

  Weak PCP theorem (contd). Dinur's proof of PCP theorem. Introduction and overview.
  Scribe : Santhoshini  (notes being written up)

  Scribe	:
  Key	:

  References:

  None

  • Meeting 32 : Thu, Feb 25, 04:00 pm-05:45 pm

  Dinur's proof of PCP theorem Overview (again). Step 1: Query reduction. Expander graphs. Definition os edge and spectral expansion. Relation between two. Rayleigh quotient characterization of spectral expansion.

  References	Bogdanov's notes and [MBNotes2]. For portion on expander graphs see <a href="http://www.tcs.tifr.res.in/~prahladh/teaching/07autumn/lectures/lec678.pdf"> Lecture notes </a> by Prahladh Harsha
  Exercises
  Reading

  Dinur's proof of PCP theorem Overview (again). Step 1: Query reduction. Expander graphs. Definition os edge and spectral expansion. Relation between two. Rayleigh quotient characterization of spectral expansion.
  Scribe : Subhadra  (notes being written up)

  Scribe	:
  Key	:

  References:

  Bogdanov's notes and [MBNotes2]. For portion on expander graphs see Lecture notes by Prahladh Harsha

  • Meeting 33 : Fri, Feb 26, 12:00 pm-12:50 pm

  Relation between spectral and edge expansion (proof of one direction).

  References	See <a href="http://www.tcs.tifr.res.in/~prahladh/teaching/07autumn/lectures/lec678.pdf"> Lecture notes </a> by Prahladh Harsha.
  Exercises
  Reading

  Relation between spectral and edge expansion (proof of one direction).
  Scribe : Shijin  (notes being written up)

  Scribe	:
  Key	:

  References:

  See Lecture notes by Prahladh Harsha.

  • Meeting 34 : Mon, Feb 29, 10:00 am-10:50 am

  Randomk walks on expanders. Expander mixing. Step 2 in Dinur's proof: Degree reduction.

  References	See Lecture notes by Prahladh Harsha.
  Exercises
  Reading

  Randomk walks on expanders. Expander mixing. Step 2 in Dinur's proof: Degree reduction.
  Scribe : Shijin  (notes being written up)

  Scribe	:
  Key	:

  References:

  See Lecture notes by Prahladh Harsha.

  • Meeting 35 : Tue, Mar 01, 09:00 am-09:50 am

  Expanderization. Random walk in the expanderized constraint graph.

  References
  Exercises
  Reading

  Expanderization. Random walk in the expanderized constraint graph.
  Scribe : Shiva Krishna  (notes being written up)

  Scribe	:
  Key	:

  References:

  None

  • Meeting 36 : Wed, Mar 02, 08:00 am-08:50 am

  Gap amplification: Overview. The verifier GP-VER.

  References	See <a href="http://www.tcs.tifr.res.in/~prahladh/teaching/07autumn/lectures/lec678.pdf"> Lecture notes </a> by Prahladh Harsha.
  Exercises
  Reading

  Gap amplification: Overview. The verifier GP-VER.
  Scribe : Shiva Krishna  (notes being written up)

  Scribe	:
  Key	:

  References:

  See Lecture notes by Prahladh Harsha.

  • Meeting 37 : Fri, Mar 04, 12:00 pm-12:50 pm

  Gap Amplification: analysis of the ther verifier GP-VER with Lazy random walks

  References
  Exercises
  Reading

  Gap Amplification: analysis of the ther verifier GP-VER with Lazy random walks
  Scribe : Kavin Kumar  (notes being written up)

  Scribe	:
  Key	:

  References:

  None

• Theme 4 : Circuit Complexity - 25 meetings

  • Meeting 38 : Tue, Mar 08, 09:00 am-09:50 am

  Lecture Cancelled.

  References
  Exercises
  Reading

  Lecture Cancelled.
  

  References:

  None

  • Meeting 39 : Fri, Mar 11, 12:00 pm-12:50 pm

  Alphabet reduction (Sketch of the proof). Introduction to Boolean Circuits.

  References	For Boolean circuits, see <a href="http://www.cse.iitm.ac.in/~jayalal/teaching/CS6840/2012/lecture34.pdf"> Notes </a> from an earlier offering of the course.
  Exercises
  Reading

  Alphabet reduction (Sketch of the proof). Introduction to Boolean Circuits.
  

  References:

  For Boolean circuits, see Notes from an earlier offering of the course.

  • Meeting 40 : Mon, Mar 14, 10:00 am-10:50 am

  Post's characterization of complete basis. Complexity measures for Boolean Circuits: Size, Depth, Width.

  References	<a href="http://www.cse.iitm.ac.in/~jayalal/teaching/CS6840/2012/lecture34.pdf"> Notes </a> from an earlier offering of the course.
  Exercises
  Reading

  Post's characterization of complete basis. Complexity measures for Boolean Circuits: Size, Depth, Width.
  

  References:

  Notes from an earlier offering of the course.

  • Meeting 41 : Tue, Mar 15, 09:00 am-09:50 am

  Shannon's Lower boound. Lupanov's upper bound.

  References	<a href="http://www.cse.iitm.ac.in/~jayalal/teaching/CS6840/2012/lecture35.pdf"> Notes </a> from an earlier offering of the course.
  Exercises
  Reading

  Shannon's Lower boound. Lupanov's upper bound.
  

  References:

  Notes from an earlier offering of the course.

  • Meeting 42 : Wed, Mar 16, 08:00 am-08:50 am

  Lupanov's upper bound (contnued). Definitiong of PSIZE. Non uniform complexity classes. P/poly = PSIZE. Karp Lipton theorem.

  References	<a href="http://www.cse.iitm.ac.in/~jayalal/teaching/CS6840/2012/lecture36.pdf"> Notes</a> from an earlier offering of the course.
  Exercises
  Reading

  Lupanov's upper bound (contnued). Definitiong of PSIZE. Non uniform complexity classes. P/poly = PSIZE. Karp Lipton theorem.
  

  References:

  Notes from an earlier offering of the course.

  • Meeting 43 : Fri, Mar 18, 12:00 pm-12:50 pm

  Karp-Lipton theorem: NP subseteq P/poly implies PH collapses. Proof.

  References	[MBNotes]
  Exercises
  Reading

  Karp-Lipton theorem: NP subseteq P/poly implies PH collapses. Proof.
  

  References:

  [MBNotes]

  • Meeting 44 : Mon, Mar 21, 10:00 am-10:50 am

  Notion of uniformity for circuits: P-uniform and los-space uniform circuits. Uniform circuit complexity classes. Circuit complexity classes: NCi. NC hierarchy. Sequentiol vs Parallel computation. Parity is in NC1. Reach is in NC2.

  References	[Vollmer]
  Exercises
  Reading

  Notion of uniformity for circuits: P-uniform and los-space uniform circuits. Uniform circuit complexity classes. Circuit complexity classes: NCi. NC hierarchy. Sequentiol vs Parallel computation. Parity is in NC1. Reach is in NC2.
  

  References:

  [Vollmer]

  • Meeting 45 : Tue, Mar 22, 09:00 am-09:50 am

  The NC Hierarchy. ADD is in NC1. Definition of AC. ADD is in AC0. Reach is in AC1. AC0 vs NC1.

  References	See [Vollmer] for a more detailed exposition.
  Exercises
  Reading

  The NC Hierarchy. ADD is in NC1. Definition of AC. ADD is in AC0. Reach is in AC1. AC0 vs NC1.
  

  References:

  See [Vollmer] for a more detailed exposition.

  • Meeting 46 : Wed, Mar 23, 08:00 am-08:50 am

  More functions inside NC1: ITerated-Addition (ITADD), Bit Count (BCOUNT), Majority (MAJ) and Threshold functions. Comparing functions inside NC1: constant depth reductions. Comparing functions MULT, ITADD, BCOUNT and MAJ.

  References	See [Vollmer] for a more detailed exposition.
  Exercises
  Reading

  More functions inside NC1: ITerated-Addition (ITADD), Bit Count (BCOUNT), Majority (MAJ) and Threshold functions. Comparing functions inside NC1: constant depth reductions. Comparing functions MULT, ITADD, BCOUNT and MAJ.
  

  References:

  See [Vollmer] for a more detailed exposition.

  • Meeting 47 : Mon, Mar 28, 10:00 am-10:50 am

  ITADD constand depth reduces to BCOUNT. ITADD, BCOUNT, MAJ, Th and MULT are constant depth equivalent. logITADD is in AC0.

  References	[Vollmer]
  Exercises
  Reading

  ITADD constand depth reduces to BCOUNT. ITADD, BCOUNT, MAJ, Th and MULT are constant depth equivalent. logITADD is in AC0.
  

  References:

  [Vollmer]

  • Meeting 48 : Tue, Mar 29, 09:00 am-09:50 am

  Threshold circuits. The TC hierarchy: TC^i. Symmetric functions. Parity vs AC0. Definition of the class ACC0. Formula vs NC1

  References	Your class notes. Also see [vollmer]
  Exercises
  Reading

  Threshold circuits. The TC hierarchy: TC^i. Symmetric functions. Parity vs AC0. Definition of the class ACC0. Formula vs NC1
  

  References:

  Your class notes. Also see [vollmer]

  • Meeting 49 : Wed, Mar 30, 08:00 am-08:50 am

  Formula = NC1. Branching programs.

  References	Your class notes.
  Exercises
  Reading

  Formula = NC1. Branching programs.
  

  References:

  Your class notes.

  • Meeting 50 : Fri, Apr 01, 12:00 pm-12:50 pm

  Barrington's theorem: NC1= BWBP. Proof.

  References	<a href="http://www.complexity.ethz.ch/education/Lectures/ComplexityHS10/ScriptChapterTwelve"> See</a> T. Holenstein's lecture notes. Also see, <a href="http://www.ccs.neu.edu/home/viola/classes/gems-08/lectures/le11.pdf"> Notes </a> by E. Viola.
  Exercises
  Reading

  Barrington's theorem: NC1= BWBP. Proof.
  

  References:

  See T. Holenstein's lecture notes. Also see, Notes by E. Viola.

  • Meeting 51 : Mon, Apr 04, 10:00 am-10:50 am

  Barrington's Theorem (contd).

  References
  Exercises
  Reading

  Barrington's Theorem (contd).
  

  References:

  None

  • Meeting 52 : Tue, Apr 05, 09:00 am-09:50 am

  Circuit lower bounds. Gate elimination method.

  References	[vollmer]
  Exercises
  Reading

  Circuit lower bounds. Gate elimination method.
  

  References:

  [vollmer]

  • Meeting 53 : Wed, Apr 06, 08:00 am-08:50 am

  Gate elimination method 2: lower bound for parity. Nechiporuk's formula size lower bound.

  References	Your class notes.
  Exercises
  Reading

  Gate elimination method 2: lower bound for parity. Nechiporuk's formula size lower bound.
  

  References:

  Your class notes.

  • Meeting 54 : Thu, Apr 07, 11:00 am-12:00 pm

  Nechiporuk's formula size lower bound.

  References	[Jukna]
  Exercises
  Reading

  Nechiporuk's formula size lower bound.
  

  References:

  [Jukna]

  • Meeting 55 : Fri, Apr 08, 10:00 am-11:00 am

  Effect of restrictions. Random restrictions on formulas: Subbatovskaya's theorem.

  References	[Jukna]
  Exercises
  Reading

  Effect of restrictions. Random restrictions on formulas: Subbatovskaya's theorem.
  

  References:

  [Jukna]

  • Meeting 56 : Mon, Apr 11, 10:00 am-10:50 am

  Andreev's cubic lower bound. Lower bounds against constant depth circuits. Hastad's switching lemma.

  References	[Jukna]
  Exercises
  Reading

  Andreev's cubic lower bound. Lower bounds against constant depth circuits. Hastad's switching lemma.
  

  References:

  [Jukna]

  • Meeting 57 : Tue, Apr 12, 09:00 am-09:50 am

  Proof of the switching lemma.

  References	[Jukna]
  Exercises
  Reading

  Proof of the switching lemma.
  

  References:

  [Jukna]

  • Meeting 58 : Tue, Apr 12, 12:00 pm-01:00 pm

  Parity not in AC0.

  References	[Jukna]
  Exercises
  Reading

  Parity not in AC0.
  

  References:

  [Jukna]

  • Meeting 59 : Wed, Apr 13, 08:00 am-08:50 am

  Polynomial method. Approximation of AC0 by polynomials.

  References	[MBNotes]
  Exercises
  Reading

  Polynomial method. Approximation of AC0 by polynomials.
  

  References:

  [MBNotes]

  • Meeting 60 : Mon, Apr 18, 10:00 am-10:50 am

  Parity cannot be approximated by low degree polynomials.

  References	[MBNotes]
  Exercises
  Reading

  Parity cannot be approximated by low degree polynomials.
  

  References:

  [MBNotes]

  • Meeting 61 : Tue, Apr 19, 09:00 am-09:50 am

  Lower bound for parity via polynomial method (contd)

  References
  Exercises
  Reading

  Lower bound for parity via polynomial method (contd)
  

  References:

  None

  • Meeting 62 : Thu, Apr 21, 04:00 pm-06:00 pm

  TBA

  References
  Exercises
  Reading

  TBA
  

  References:

  None

• Theme 5 : Evaluation Meetings - 3 meetings

  • Meeting 63 : Sat, Mar 12, 02:00 pm-03:30 pm

  Syllabus: Complexity of counting + Proof systems.

  References
  Exercises
  Reading

  Syllabus: Complexity of counting + Proof systems.
  

  References:

  None

  • Meeting 64 : Sat, Apr 23, 01:00 pm-01:50 pm

  Matching is in Quasi NC. (Sundar A)

  References	http://eccc.hpi-web.de/report/2015/177/download/
  Exercises
  Reading

  Matching is in Quasi NC. (Sundar A)
  

  References:

  http://eccc.hpi-web.de/report/2015/177/download/

  • Meeting 65 : Sat, Apr 23, 01:55 pm-02:45 pm

  TBA

  References
  Exercises
  Reading

  TBA
  

  References:

  None