Lectures

Following is an outline of lectures given along with references and links to additional reading. For abbreviations, [RP], [SY] and [CKW], check the references page.

Theme: Basics - arithmetic models of computation and arithmetic classes

Lec 1 (07 Jan, Wed) : Course information. A brief course overview. Central objects - multivariate polynomials. Computing polynomials diagrammatically, the arithmetic circuit representation. Circuits for computing some simple polynomials. Goals of this course.

^{Reading : Class notes, Chapter 1 of [RP] survey}

Lec 2 (09 Jan, Fri) : Course evaluation policy discussed. The interpolation technique, computing the elementary symmetric polynomial using small sized arithmetic circuits. Determinant polynomial. Algebraic formulation for bipartite perfect matching.

^{Reading : Class notes, Chapter 1 of [RP] survey. Section 7.2 of this notes.}

Lec 3 (12 Jan, Mon) : Two problems expressed via polynomials – primality checking and hamiltonian cycle. Naive algorithm for primality check is inefficient. Agrawal-Biswas theorem for primality checking. The polynomial identity testing question.

^{Reading : Lemma 3.1 of this paper.}

^{Additional References : A writeup on primality testing by Scott Aaronson.}

Lec 4 (14 Jan, Wed) : Algebraic circuit model. Size and depth. Number of monomials in an $n$ variate degree $d$ polynomial. Naive upper bounds on circuit size. Shannon’s counting lower bound and existence of polynomials requiring large circuit size. The circuit size lower bound question.

^{Reading : Class notes. Section 10.3.2 of this notes.}

Lec 5 (16 Jan, Fri) : Explicitness and polynomial families. Permanent and Determinant polynomial families. The Valiant’s P class (VP). Valiant’s conjecture. An arithmetic circuit computing permanent (Ryser’s construction). Universality of permanent - expressing formulas as “projections” of permanent of matrices. Characterisation of permanent via cycle covers.

^{Reading :}

Lec 6 (19 Jan, Mon) : Notion of reduction and completeness. The class Valiant’s NP (VNP). Permanent is in VNP. Towards showing Permanent is VNP-complete - cycle cover characterization of permanent. Rosette and details of the construction.

^{Reading :}

Lec 7 (21 Jan, Wed) : Determinant is universal. Introduction to algebraic algebraic branching program (ABP). Iterated matrix multiplication polynomial and equivalence to ABPs. Expressing determinant using iterated matrix multiplication. Determinants can compute ABPs. An ABP that computes determinant. Characterization of determinants using Closed walk (clow) sequence. Description of the ABP (Mahajan-Vinay algorithm).

^{Reading :}

Lec 8 (23 Jan, Fri) : More ABPs for determinant – Construction due to Ben-Or and Cleve. Construction using Newton-Girard identities.

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Lec () (26 Jan, Mon) : Holiday due to Republic day.

Theme: Structural results

Lec 9 (28 Jan, Wed) : Homogenization, Depth reduction for Formulas (construction due to Brent, Spira).

^{Reading :}

Lec 10 (30 Jan, Fri) : Handling divisions, depth reduction for circuits - Valiant-Skyum-Berkowitz-Rackoff construction.

^{Reading :}

Lec 11 (02 Feb, Mon) : Reduction to depth 4 - Agrawal-Vinay, Koiran, Tavenas.

^{Reading :}

Theme: Classical lower bounds

Lec 12 (04 Feb, Wed) : Lower bounds for circuits due to Baur and Strassen.

^{Reading :}

Lec 13 (06 Feb, Fri) : Lower bound for ABPs

^{Reading :}

Lec 14 (09 Feb, Mon) : Quadratic lower bound for Formulas due to Kalarkoti.

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Lec 15 (11 Feb, Wed) : Lower bounds for monotone circuits due to Jerrum-Snir.

^{Reading :}

Theme: Lower bounds for restricted settings

Lec 16 (13 Feb, Fri) : Natural lower bounds based on measure. Partial derivative matrix method. Lower bound for homogeneous circuits (Nisan-Wigderson).

^{Reading :}

Lec 17 (16 Feb, Mon) : Lower bounds against set multilinear ABPs and Read once ABPs.

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Lec 18 (18 Feb, Wed) : Relative rank measure. Raz’s lower bound for multilinear formulas.

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Lec 19 (20 Feb, Fri) : Constant depth arithmetic circuit lower bounds (LST) - 1

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Lec 20 (23 Feb, Mon) : Constant depth arithmetic circuit lower bounds (LST) - 2

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Lec 21 (25 Feb, Wed) : Constant depth arithmetic circuit lower bounds (LST) - 3

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Lec 22 (27 Feb, Fri) : Separating monotone circuits and monotone ABPs - 1

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Lec 23 (02 Mar, Mon) : Separating monotone circuits and monotone ABPs - 2

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Lec () (04 Mar, Wed) : Holiday due to Holi.

Theme: Polynomial Identity Testing and connections to circuit lower bounds

Lec 24 (06 Mar, Fri) : White-box, Black-box PIT. Hitting sets. Klivans-Spielman hitting sets for sparse polynomials.

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Lec 25 (09 Mar, Mon) : Raz-Shpilka polynomial time white-box PIT for ROABPs - 1

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Lec 26 (11 Mar, Wed) : Raz-Shpilka polynomial time white-box PIT for ROABPs - 2

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Lec 27 (13 Mar, Fri) : Forbes-Shpilka quasi-polynomial time black-box PIT for ROABPs - 1

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Lec 28 (16 Mar, Mon) : Forbes-Shpilka quasi-polynomial time black-box PIT for ROABPs - 2

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Lec 29 (18 Mar, Wed) : Hitting set for depth $3$ powering circuits using Shpilka-Volkovitch generators.

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Lec 30 (20 Mar, Fri) : Hitting sets to lower bounds. Combinatorial designs. Impagliazzo-Kabanets theorem - Hitting sets from lower bounds.

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Lec 31 (23 Mar, Mon) :

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Lec 32 (25 Mar, Wed) :

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Lec 33 (27 Mar, Fri) :

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Lec 34 (30 Mar, Mon) :

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Lec 35 (01 Apr, Wed) :

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Lec () (03 Apr, Fri) : Holiday due to Good Friday.

Lec 36 (06 Apr, Mon) :

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Lec 37 (08 Apr, Wed) :

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Theme: Introduction to Geometric Complexity Theory

Lec 38 (10 Apr, Fri) : Introduction and goals of the program.

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Lec 39 (13 Apr, Mon) :

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Lec 40 (15 Apr, Wed) : Border complexity and closure of algebraic classes

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Lec 41 (17 Apr, Fri) :

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Lec 42 (20 Apr, Mon) : Waring representation, apolarity and secant varieties

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Lec 43 (22 Apr, Wed) :

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