CS6842 - Algorithmic Algebra

Meetings  

• Theme 1 : Introduction, Motivation and Basics - 3 meetings

  • Meeting 01 : Mon, Aug 03, 09:00 am-09:50 am

  Administrative Announcements. Motivation for the course. Algebraic Formulations for "Seemingly" Combinatorial Problems. <br> Example 1 : Graph Relabelling - Isomorphism Problem. Automorphism Group. Permutation groups.

  References
  Exercises
  Reading

  Administrative Announcements. Motivation for the course. Algebraic Formulations for "Seemingly" Combinatorial Problems.
  Example 1 : Graph Relabelling - Isomorphism Problem. Automorphism Group. Permutation groups.
  

  References

  :

  None

  • Meeting 02 : Tue, Aug 04, 08:00 am-08:50 am

  Example 2 : Primality testing Problem. Fermat's Little Theorem. Agrawal-Biswas formulation in terms of identity testing of polynomials in Rings.

  References	<a href="http://www.cse.iitk.ac.in/users/manindra/algebra/identity.pdf">Agrawal-Biswas Formulation</a> (Lemma 3.1).
  Exercises
  Reading

  Example 2 : Primality testing Problem. Fermat's Little Theorem. Agrawal-Biswas formulation in terms of identity testing of polynomials in Rings.
  

  References

  :

  Agrawal-Biswas Formulation (Lemma 3.1).

  • Meeting 03 : Wed, Aug 05, 12:00pm-12:50pm

  Example 3 : Perfect Matching Problem. Tutte Matrix. Multivariate identity testing problem.

  References
  Exercises
  Reading

  Example 3 : Perfect Matching Problem. Tutte Matrix. Multivariate identity testing problem.
  

  References

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  None

• Theme 2 : Algorithms for Groups & Applications - 22 meetings

  • Meeting 04 : Fri, Aug 07, 11:00 am-11:50 am

  Isomorphism and Automorphism Problems. Graph Rigidity problem. Presentation of Groups. Operations, Cyclic groups, Generators. Subgroup, Coset structure, Lagrange's theorem. Size of the Generating Set. GraphAuto Problem.

  References
  Exercises
  Reading

  Isomorphism and Automorphism Problems. Graph Rigidity problem. Presentation of Groups. Operations, Cyclic groups, Generators. Subgroup, Coset structure, Lagrange's theorem. Size of the Generating Set. GraphAuto Problem.
  

  References

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  None

  • Meeting 05 : Mon, Aug 10, 09:00 am-09:50 am

  Reduction from GI to GA. <br> Is GI relatively easier problem? Group Actions. Orbits, Orbit-Stabilizer Lemma.

  References
  Exercises
  Reading

  Reduction from GI to GA.
  Is GI relatively easier problem? Group Actions. Orbits, Orbit-Stabilizer Lemma.
  

  References

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  None

  • Meeting 06 : Tue, Aug 11, 08:00 am-08:50 am

  Coloured-GI Problem. Reduction to GI. Gadget construction. Computing the isomorphism using coloured GI and hence queries to GI.

  References
  Exercises	Use coloring idea to Reduce GR to GI.
  Reading

  Coloured-GI Problem. Reduction to GI. Gadget construction. Computing the isomorphism using coloured GI and hence queries to GI.
  

  References	:	None
  Exercises	:	Use coloring idea to Reduce GR to GI.

  • Meeting 07 : Wed, Aug 12, 12:00pm-12:50pm

  Tower of subgroups, coset representatives, strong generating set. Unique product representation. Computing the coset representatives using queries to GI. Using O(n^4) GI queries to solve GA. An idea for #GI using GI.

  References
  Exercises
  Reading

  Tower of subgroups, coset representatives, strong generating set. Unique product representation. Computing the coset representatives using queries to GI. Using O(n^4) GI queries to solve GA. An idea for #GI using GI.
  

  References

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  None

  • Meeting 08 : Fri, Aug 14, 11:00 am-11:50 am

  Abstracting out some computational problems in permutation groups.<br> 1) Given the generating set, find the size of the group.<br> 2) Given the generating set of a subgroup H of G, and an element in G, test for membership in H. Reduction from Problem 2 to 1. A recursive approach to problem 1 using Orbit (size) Computation.<br> 3) Orbit computation Problem.<br> 4) Set stabilizer problem. and Point Stabilizer problem.<br> Action of a group on itself. Action of S_n on E(X). Reduction from GA to Set Stabilizer Problem. Algorithm for computing orbits.

  References
  Exercises
  Reading

  Abstracting out some computational problems in permutation groups.
  1) Given the generating set, find the size of the group.
  2) Given the generating set of a subgroup H of G, and an element in G, test for membership in H. Reduction from Problem 2 to 1. A recursive approach to problem 1 using Orbit (size) Computation.
  3) Orbit computation Problem.
  4) Set stabilizer problem. and Point Stabilizer problem.
  Action of a group on itself. Action of S_n on E(X). Reduction from GA to Set Stabilizer Problem. Algorithm for computing orbits.
  

  References

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  None

  • Meeting 09 : Mon, Aug 17, 09:00 am-09:50 am

  Schreier Generators. An algorithm for computing strong generating set with respect to the tower of stabilizer subgroups.

  References
  Exercises
  Reading

  Schreier Generators. An algorithm for computing strong generating set with respect to the tower of stabilizer subgroups.
  

  References

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  None

  • Meeting 10 : Tue, Aug 18, 08:00 am-08:50 am

  Reduction Step. Pointwise Stabilizer problem.

  References
  Exercises
  Reading

  Reduction Step. Pointwise Stabilizer problem.
  

  References

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  None

  • Meeting 11 : Wed, Aug 19, 12:00pm-12:50pm

  Finding the Coset representatives. Final algorithm for point-wise stabilizer problem. Membership Testing in Permutation Groups given by generators. Group Intersection Problem. Naive attempts. From Set-Stabilizer Problem to Group Intersection Problem.

  References
  Exercises
  Reading

  Finding the Coset representatives. Final algorithm for point-wise stabilizer problem. Membership Testing in Permutation Groups given by generators. Group Intersection Problem. Naive attempts. From Set-Stabilizer Problem to Group Intersection Problem.
  

  References

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  None

  • Meeting 12 : Fri, Aug 21, 11:00 am-11:50 am

  From Group Intersection Problem to Set-Stabilizer Problem. Jerrum's Filter. Linear sized generating sets.

  References
  Exercises
  Reading

  From Group Intersection Problem to Set-Stabilizer Problem. Jerrum's Filter. Linear sized generating sets.
  

  References

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  None

  • Meeting 13 : Mon, Aug 24, 09:00 am-09:50 am

  Normal Subgroups. Normalizer and Centralizers. Algorithmic problems in normal subgroups. Group intersection problem for special case : when G normalizes H in S_n.

  References
  Exercises
  Reading

  Normal Subgroups. Normalizer and Centralizers. Algorithmic problems in normal subgroups. Group intersection problem for special case : when G normalizes H in S_n.
  

  References

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  None

  • Meeting 14 : Tue, Aug 25, 08:00 am-08:50 am

  Details of the group intersection tower.<br> From Special Set-Stabilizer to Point-Stabilizer on an extended action. GI with bound on the color class size is in P.

  References
  Exercises	We left the class to think about the normalizer computation algorithm and switched the special case of set stabilizer problem.
  Reading

  Details of the group intersection tower.
  From Special Set-Stabilizer to Point-Stabilizer on an extended action. GI with bound on the color class size is in P.
  

  References	:	None
  Exercises	:	We left the class to think about the normalizer computation algorithm and switched the special case of set stabilizer problem.

  • Meeting 15 : Wed, Aug 26, 12:00pm-12:50pm

  Normailizer Computation. Non-transitivity. Subnormality. Testing sub-normality. Another Special Case of Group Intersection Problem - G is subnormal in <G,H>.

  References
  Exercises
  Reading

  Normailizer Computation. Non-transitivity. Subnormality. Testing sub-normality. Another Special Case of Group Intersection Problem - G is subnormal in .
  

  References

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  None

  • Meeting 16 : Mon, Aug 31, 09:00 am-09:50 am

  Divide and Conquer Methods. Transitive actions. Blocks. Primitive actions. Examples. Algebraic Characterization.

  References
  Exercises
  Reading

  Divide and Conquer Methods. Transitive actions. Blocks. Primitive actions. Examples. Algebraic Characterization.
  

  References

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  None

  • Meeting 17 : Tue, Sep 01, 08:00 am-08:50 am

  Proof of the characterization.

  References
  Exercises
  Reading

  Proof of the characterization.
  

  References

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  None

  • Meeting 18 : Wed, Sep 02, 12:00pm-12:50pm

  Computing the blocks of a transitive action.

  References
  Exercises
  Reading

  Computing the blocks of a transitive action.
  

  References

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  None

  • Meeting 19 : Fri, Sep 04, 11:00 am-11:50 am

  Trivalent Graph Isomorphism. Overview and basics.

  References
  Exercises
  Reading

  Trivalent Graph Isomorphism. Overview and basics.
  

  References

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  None

  • Meeting 20 : Sat, Sep 05, 02:00 pm-03:30 pm

  Tutte's theorem. Reduction from Trivalent GI to Restricted SetStab.

  References
  Exercises
  Reading

  Tutte's theorem. Reduction from Trivalent GI to Restricted SetStab.
  

  References

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  None

  • Meeting 21 : Tue, Sep 08, 08:00 am-08:50 am

  Structure of 2-groups. Sylow's theorem. Structure of groups for degree d graphs. The overview of the approach. Blocks and Structure Forest.

  References
  Exercises
  Reading

  Structure of 2-groups. Sylow's theorem. Structure of groups for degree d graphs. The overview of the approach. Blocks and Structure Forest.
  

  References

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  None

  • Meeting 22 : Wed, Sep 09, 06:00 am-06:00 am

  Solving the Restricted Set Stabilizer. divide and conquer approach. Babai-Luks-Palfy bound.

  References
  Exercises
  Reading

  Solving the Restricted Set Stabilizer. divide and conquer approach. Babai-Luks-Palfy bound.
  

  References

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  None

  • Meeting 23 : Fri, Sep 11, 11:00 am-11:50 am

  Coloring and color refinements. Stable coloring. Properties of stable coloring. Individualization of vertices.

  References
  Exercises
  Reading

  Coloring and color refinements. Stable coloring. Properties of stable coloring. Individualization of vertices.
  

  References

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  None

  • Meeting 24 : Mon, Sep 14, 09:00 am-09:50 am

  Approaching the general graph isomorphism problem using color refinements and vertex individualization. Notion of color valence. Runtime of the final algorithm assuming nice individualization sequences.

  References
  Exercises
  Reading

  Approaching the general graph isomorphism problem using color refinements and vertex individualization. Notion of color valence. Runtime of the final algorithm assuming nice individualization sequences.
  

  References

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  None

  • Meeting 25 : Mon, Sep 14, 02:30 pm-03:30 pm

  Greedy Algorithm for Computing the individualization sequence to reach bounded colour valence. Solving the instances with Bounded color valence using bounded degree algorithm.

  References
  Exercises
  Reading

  Greedy Algorithm for Computing the individualization sequence to reach bounded colour valence. Solving the instances with Bounded color valence using bounded degree algorithm.
  

  References

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  None

• Theme 3 : Algorithms for Polynomials & Applications - 32 meetings

  • Meeting 26 : Tue, Sep 15, 08:00 am-08:50 am

  Solving Polynomial Equations. Ideals, Examples. Lessons from simple case of linear equations. Rings, Division Rings, Fields, Polynomial Rings. Formal definition of ideals and properties.

  References
  Exercises
  Reading

  Solving Polynomial Equations. Ideals, Examples. Lessons from simple case of linear equations. Rings, Division Rings, Fields, Polynomial Rings. Formal definition of ideals and properties.
  

  References

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  None

  • Meeting 27 : Wed, Sep 16, 06:00 am-06:00 am

  Algorithmic problems to address. Ideal Membership problem. Ideals and Varieties. Examples, basic observations.

  References
  Exercises
  Reading

  Algorithmic problems to address. Ideal Membership problem. Ideals and Varieties. Examples, basic observations.
  

  References

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  None

  • Meeting 28 : Fri, Sep 18, 11:00 am-11:50 am

  Noetherian Rings. Ascending chain termination. Hilbert Basis theorem.

  References
  Exercises
  Reading

  Noetherian Rings. Ascending chain termination. Hilbert Basis theorem.
  

  References

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  None

  • Meeting 29 : Mon, Sep 21, 09:00 am-09:50 am

  Simple cases first. Linear equations. Division algorithm. Univariate case. Division algorithm. Uniqueness. Gcd of polynomials. Proof that F[x] is a PID.

  References
  Exercises
  Reading

  Simple cases first. Linear equations. Division algorithm. Univariate case. Division algorithm. Uniqueness. Gcd of polynomials. Proof that F[x] is a PID.
  

  References

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  None

  • Meeting 30 : Tue, Sep 22, 08:00 am-08:50 am

  Euclidean Algorithm and the correctness in terms of ideals generated. Solving the Ideal membership problem for uni-variate case. Towards generalizations.

  References
  Exercises
  Reading

  Euclidean Algorithm and the correctness in terms of ideals generated. Solving the Ideal membership problem for uni-variate case. Towards generalizations.
  

  References

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  None

  • Meeting 31 : Wed, Sep 23, 12:00pm-12:50pm

  Term orders. Well-ordering. Termination of the division algorithm. Example term orderings.

  References
  Exercises
  Reading

  Term orders. Well-ordering. Termination of the division algorithm. Example term orderings.
  

  References

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  None

  • Meeting 32 : Tue, Sep 29, 08:00 am-08:50 am

  Examples of term orderings. Division algorithm with multiple multivariate polynomials.

  References
  Exercises
  Reading

  Examples of term orderings. Division algorithm with multiple multivariate polynomials.
  

  References

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  None

  • Meeting 33 : Wed, Sep 30, 12:00pm-12:50pm

  Examples for remainder being non-zero depending on the order of the division. Definition of Grobner basis. Characterization of Grobner basis.

  References
  Exercises
  Reading

  Examples for remainder being non-zero depending on the order of the division. Definition of Grobner basis. Characterization of Grobner basis.
  

  References

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  None

  • Meeting 34 : Mon, Oct 05, 09:00 am-09:50 am

  Computing Grobner basis. Constructing violating examples for Grobner condition. Leading the intuition to S-polynomials and Buchberger's criterion. Proof of Buchberger's criterion - A special case first.

  References
  Exercises
  Reading

  Computing Grobner basis. Constructing violating examples for Grobner condition. Leading the intuition to S-polynomials and Buchberger's criterion. Proof of Buchberger's criterion - A special case first.
  

  References

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  None

  • Meeting 35 : Tue, Oct 06, 08:00 am-08:50 am

  Main lemma : any violating example can be expressed in terms of S-polynomials. General case of the criterion.

  References
  Exercises
  Reading

  Main lemma : any violating example can be expressed in terms of S-polynomials. General case of the criterion.
  

  References

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  None

  • Meeting 36 : Wed, Oct 07, 12:00pm-12:50pm

  Buchberger's algorithm. Termination and Correctness. Minimal and Reduced Grobner Basis.

  References
  Exercises
  Reading

  Buchberger's algorithm. Termination and Correctness. Minimal and Reduced Grobner Basis.
  

  References

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  None

  • Meeting 37 : Fri, Oct 09, 11:00 am-11:50 am

  Solving 3-coloring problem using Grobner basis. Formulation in terms of polynomials. Integer programming. A special case first. Reduction to the problem of testing membership in the image of a ring-homomorphism.

  References
  Exercises
  Reading

  Solving 3-coloring problem using Grobner basis. Formulation in terms of polynomials. Integer programming. A special case first. Reduction to the problem of testing membership in the image of a ring-homomorphism.
  

  References

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  None

  • Meeting 38 : Sat, Oct 10, 09:00 am-09:50 am

  A characterization of kernel in terms of elimination ideals. Elimination theory and Grobner basis. Examples.

  References
  Exercises
  Reading

  A characterization of kernel in terms of elimination ideals. Elimination theory and Grobner basis. Examples.
  

  References

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  None

  • Meeting 39 : Sat, Oct 10, 10:00 am-10:50 am

  Checking whether a polynomials is in the image of the homomorphism. Observations about Grobner basis generated by the Buchberger algorithm in this case. Using the term ordering to solve the optimization version of the problem.

  References	<a href="http://math.harvard.edu/~waffle/ufds2.pdf">Notes by Waffle.</a>
  Exercises
  Reading	<a href="http://homepage.math.uiowa.edu/~goodman/22m121.dir/2005/section6.6.pdf">Chapter of a Book</a>

  Checking whether a polynomials is in the image of the homomorphism. Observations about Grobner basis generated by the Buchberger algorithm in this case. Using the term ordering to solve the optimization version of the problem.
  

  References	:	Notes by Waffle.
  Reading	:	Chapter of a Book

  • Meeting 40 : Mon, Oct 12, 09:00 am-09:50 am

  Integral Domains, Units, Irreducibles, Primes, Primes are Irreducibles. Integral Domains with factorisation is a UFD if and only if all Irreducibles are primes.

  References	<a href="http://math.harvard.edu/~waffle/ufds2.pdf">Notes by Waffle.</a>
  Exercises
  Reading	<a href="http://homepage.math.uiowa.edu/~goodman/22m121.dir/2005/section6.6.pdf">Chapter of a Book</a>

  Integral Domains, Units, Irreducibles, Primes, Primes are Irreducibles. Integral Domains with factorisation is a UFD if and only if all Irreducibles are primes.
  

  References	:	Notes by Waffle.
  Reading	:	Chapter of a Book

  • Meeting 41 : Tue, Oct 13, 08:00 am-08:50 am

  All Principal Ideal Domains are Unique Facotorization Domains. Z, F[x] are all UFDs. Towards proving : if R is a UFD, so is R[x]. Gauss's Lemma and its proof.

  References	<a href="http://math.harvard.edu/~waffle/ufds2.pdf">Notes by Waffle.</a>
  Exercises
  Reading	<a href="http://homepage.math.uiowa.edu/~goodman/22m121.dir/2005/section6.6.pdf">Chapter of a Book</a>

  All Principal Ideal Domains are Unique Facotorization Domains. Z, F[x] are all UFDs. Towards proving : if R is a UFD, so is R[x]. Gauss's Lemma and its proof.
  

  References	:	Notes by Waffle.
  Reading	:	Chapter of a Book

  • Meeting 42 : Wed, Oct 14, 12:00 pm-12:50 pm

  Content of a polynomial. Primitive polynomials. Characterization of non-constant irreducibles in R[x] using irreducibility in F[x] where F is the Field of fractions of R. Proving that a factorization exists. Proof that all non-constant irreducibles are primes in R[x]. Completing the proof of the theorem : F[x_1, x_2, ... x_n] is a UFD. Factorization Problem : Given a polynomial f, factorize it into irredcubiles. Given a polynomial can we test if it is irreducible?

  References	<a href="http://math.harvard.edu/~waffle/ufds2.pdf">Notes by Waffle.</a>
  Exercises
  Reading	<a href="http://homepage.math.uiowa.edu/~goodman/22m121.dir/2005/section6.6.pdf">Chapter of a Book</a>

  Content of a polynomial. Primitive polynomials. Characterization of non-constant irreducibles in R[x] using irreducibility in F[x] where F is the Field of fractions of R. Proving that a factorization exists. Proof that all non-constant irreducibles are primes in R[x]. Completing the proof of the theorem : F[x_1, x_2, ... x_n] is a UFD. Factorization Problem : Given a polynomial f, factorize it into irredcubiles. Given a polynomial can we test if it is irreducible?
  

  References	:	Notes by Waffle.
  Reading	:	Chapter of a Book

  • Meeting 43 : Fri, Oct 16, 11:00 am-11:50 am

  A characterization of irreducibility of polynomials. Quotient Ring is a field if and only if the polynomial is irreducible.

  References	<a href="http://sierra.nmsu.edu/morandi/OldWebPages/Math331Spring2003/Chapter5.pdf">Notes Online</a>
  Exercises
  Reading	<a href="http://homepage.math.uiowa.edu/~goodman/22m121.dir/2005/section6.6.pdf">Chapter of a Book</a>

  A characterization of irreducibility of polynomials. Quotient Ring is a field if and only if the polynomial is irreducible.
  

  References	:	Notes Online
  Reading	:	Chapter of a Book

  • Meeting 44 : Mon, Oct 19, 09:00 am-09:50 am

  Chinese remainder theorem for integers. Relatively prime ideals. Chinese reminder theorem for ideals.

  References	<a href="http://www3.ntu.edu.sg/home/Frederique/chap2.pdf">Section 2.3 of the notes</a>
  Exercises
  Reading

  Chinese remainder theorem for integers. Relatively prime ideals. Chinese reminder theorem for ideals.
  

  References

  :

  Section 2.3 of the notes

  • Meeting 45 : Mon, Oct 26, 09:00 am-09:50 am

  Lecture Cancelled. To be compensated.

  References
  Exercises
  Reading

  Lecture Cancelled. To be compensated.
  

  References

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  None

  • Meeting 46 : Tue, Oct 27, 08:00 am-08:50 am

  From Quotient Rings of Irreducibile polynomials to Field extensions. Viewing field extensions as vector spaces. Characteristic of a field. Sizes of fields. Constructing extensions and uniqueness of fields of a given size (up to isomorphism).

  References
  Exercises
  Reading

  From Quotient Rings of Irreducibile polynomials to Field extensions. Viewing field extensions as vector spaces. Characteristic of a field. Sizes of fields. Constructing extensions and uniqueness of fields of a given size (up to isomorphism).
  

  References

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  None

  • Meeting 47 : Wed, Oct 28, 12:00pm-12:50pm

  Feild Extensions and Quotient Rings of irreducible polynomials. Back to factorization problem. Square free factorization. Distinct degree factorization. A starting idea from Fermat's little theorem. Extracting product of linear factors.

  References
  Exercises
  Reading

  Feild Extensions and Quotient Rings of irreducible polynomials. Back to factorization problem. Square free factorization. Distinct degree factorization. A starting idea from Fermat's little theorem. Extracting product of linear factors.
  

  References

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  None

  • Meeting 48 : Thu, Oct 29, 05:00 pm-06:00 pm

  Generalizing Fermat's little theorem. Distinct Degree factorization Algorithm. Irreducibility Test.

  References
  Exercises
  Reading

  Generalizing Fermat's little theorem. Distinct Degree factorization Algorithm. Irreducibility Test.
  

  References

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  None

  • Meeting 49 : Fri, Oct 30, 11:00 am-11:50 am

  Equal Degree Factorization. Cantor-Zassenhaus Algorithm. Quadratic residues.

  References
  Exercises
  Reading

  Equal Degree Factorization. Cantor-Zassenhaus Algorithm. Quadratic residues.
  

  References

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  None

  • Meeting 50 : Mon, Nov 02, 09:00 am-09:50 am

  Berlekamp Subalgebra. Dimension of the subalgebra and number of factors. Factorization algorithm assuming basis for the subalgebra.

  References
  Exercises
  Reading

  Berlekamp Subalgebra. Dimension of the subalgebra and number of factors. Factorization algorithm assuming basis for the subalgebra.
  

  References

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  None

  • Meeting 51 : Tue, Nov 03, 08:00 am-08:50 am

  Finding a basis for the subalgebra. Reducing the dependence on Q. Outline of the Zassenhaus's idea.

  References
  Exercises
  Reading

  Finding a basis for the subalgebra. Reducing the dependence on Q. Outline of the Zassenhaus's idea.
  

  References

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  None

  • Meeting 52 : Wed, Nov 04, 12:00pm-12:50pm

  Hensel lifting (general ideal version). Polynomial version. Reducing bivariate factorization to univariate factorization.

  References
  Exercises
  Reading

  Hensel lifting (general ideal version). Polynomial version. Reducing bivariate factorization to univariate factorization.
  

  References

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  None

  • Meeting 53 : Thu, Nov 05, 02:15 pm-03:15 pm

  Details of Hensel lifting. Hensel lifting and factorization over integers.

  References
  Exercises
  Reading

  Details of Hensel lifting. Hensel lifting and factorization over integers.
  

  References

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  None

  • Meeting 54 : Fri, Nov 06, 11:00 am-11:50 am

  References
  Exercises
  Reading

  
  

  References

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  None

  • Meeting 55 : Mon, Nov 09, 09:00 am-09:50 am

  Implicit Representation of Polynomials: Arithmetic Circuits. Example: Elementary symmetric polynomials More on structure of arithmetic circuits. VSBR construction.

  References
  Exercises
  Reading

  Implicit Representation of Polynomials: Arithmetic Circuits. Example: Elementary symmetric polynomials More on structure of arithmetic circuits. VSBR construction.
  

  References

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  None

  • Meeting 56 : Tue, Nov 10, 08:00 am-08:50 am

  A Chasm at Depth 4.

  References
  Exercises
  Reading

  A Chasm at Depth 4.
  

  References

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  None

  • Meeting 57 : Fri, Nov 13, 11:00 am-11:50 am

  Lower Bounds vs Identity Testing.

  References
  Exercises
  Reading

  Lower Bounds vs Identity Testing.
  

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• Theme 4 : Student Presentations - 1 meetings

  • Meeting 58 : Sat, Nov 07, 10:00 am-05:00 pm

  Group I (Group Theoretic Algorithms)<br> <ul> <li>Topic 1 : </li> <li>Topic 2 : </li> <li>Topic 3 : </li> </ul>

  References
  Exercises
  Reading

  Group I (Group Theoretic Algorithms)
  

  • Topic 1 :
  • Topic 2 :
  • Topic 3 :

  
  

  References

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  None