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) : Need for explicitness and the need for polynomial families. Permanent and Determinant polynomial families. Valiant’s conjecture and need for a theory. Arithmetic formula model of computation. Size of a formula. The circuit versus formula model.

^{Reading : Class notes. Chapter 2 of [RP] survey.}

Lec () (19 Jan, Mon) : Class cancelled due to Institute day. Compensated on 24 Jan.

Lec 6 (21 Jan, Wed) : Bounds on the formula and circuit size of simple polynomials. Towards an intermediate model - the iterated matrix multiplication polynomial and algebraic branching programs (ABP). Some examples of ABPs computing polynomials.

^{Reading : Class notes. Chapter 2 of [RP] survey.}

Lec 7 (23 Jan, Fri) : Obtaining ABPs from arithmetic formulas and obtaining arithmetic circuits from ABPs with a constant size blowup. Correctness arguments and argument for size bounds.

^{Reading : Class notes. Chapter 2 of [RP] survey.}

Lec 8 (24 Jan, Sat) : (Compensatory class) Characterisation of permanent via cycle covers. Universality of permanent - expressing arithmetic formulas as “projections” of permanent of matrices. Classical complexity classes $\P$ and $\NP$. Polynomial classes VF for formulas and VBP for algebraic branching programs, class VP for circuits. VF $\subseteq$ VBP $\subseteq$ VP. Notion of polynomial being hard for a class.

^{Reading :}

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

Lec 9 (28 Jan, Wed) : Projection reductions. Expressing any ABP as a projection of iterated matrix multiplication. Iterated matrix multiplication polynomial is VBP-complete. Hardness and completeness. The classical complexity class NP and its analog Valiant’s NP (VNP).

^{Reading : Class notes. Chapter 3 (Section 1) of [RP] survey.}

Lec 10 (30 Jan, Fri) : An depth three arithmetic circuit computing permanent (Ryser’s construction) via inclusion exclusion argument. Permanent is in VNP.

^{Reading : Class notes.}

Lec 11 (02 Feb, Mon) : Matching polynomial is VNP-hard. Towards showing Permanent is VNP-complete. Class VNP stays the same with $g \in $ VP replaced by $g \in $ VF in the definition (statement only). Special case when auxillary variables occurs once in ABP of $g$. Rosette and glue gadget details of the construction.

^{Reading : Class notes, Section 3.3.4 of [RP] survey.}

Lec 12 (04 Feb, Wed) : Determinant is universal. Expressing determinant using iterated matrix multiplication. Determinants can compute ABPs. An ABP that computes determinant. Characterization of determinants using Closed walk (clow) sequence.

^{Reading :}

Lec 13 (06 Feb, Fri) : Description of the ABP (Mahajan-Vinay algorithm). More ABPs for determinant – Construction due to Ben-Or and Cleve. Construction using Newton-Girard identities.

^{Reading :}

Theme: Structural results

Lec 14 (09 Feb, Mon) : Homogenization, Depth reduction for arithmetic Formulas (construction due to Brent, Spira). Handling divisions, depth reduction for circuits - Valiant-Skyum-Berkowitz-Rackoff construction.

^{Reading :}

Lec 15 (11 Feb, Wed) : Reduction to depth 4 - Agrawal-Vinay, Koiran, Tavenas.

^{Reading :}

Theme: Classical lower bounds

Lec 16 (13 Feb, Fri) : Lower bounds for circuits due to Baur and Strassen. Lower bound for ABPs

^{Reading :}

Lec 17 (16 Feb, Mon) : Quadratic lower bound for Formulas due to Kalarkoti.Lower bounds for monotone circuits due to Jerrum-Snir.

^{Reading :}

Theme: Lower bounds for restricted settings

Lec 18 (18 Feb, Wed) : Natural lower bounds based on measure. Partial derivative matrix method. Lower bound for homogeneous circuits (Nisan-Wigderson).

^{Reading :}

Lec 19 (20 Feb, Fri) : Lower bounds against set multilinear ABPs and Read once ABPs.

^{Reading :}

Lec 20 (23 Feb, Mon) : Relative rank measure. Raz’s lower bound for multilinear formulas.

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Lec 21 (25 Feb, Wed) : Constant depth arithmetic circuit lower bounds (LST) - 1

^{Reading :}

Lec 22 (27 Feb, Fri) : Constant depth arithmetic circuit lower bounds (LST) - 2

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Lec 23 (02 Mar, Mon) : Constant depth arithmetic circuit lower bounds (LST) - 3

^{Reading :}

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

Lec 24 (06 Mar, Fri) : Separating monotone circuits and monotone ABPs - 1

^{Reading :}

Lec 25 (09 Mar, Mon) : Separating monotone circuits and monotone ABPs - 2

^{Reading :}

Theme: Polynomial Identity Testing and connections to circuit lower bounds

Lec 26 (11 Mar, Wed) : Polynomial Identity Testing and connections to circuit lower bounds

^{Reading :}

Lec 27 (13 Mar, Fri) : White-box, Black-box PIT. Hitting sets. Klivans-Spielman hitting sets for sparse polynomials.

^{Reading :}

Lec 28 (16 Mar, Mon) : Raz-Shpilka polynomial time white-box PIT for ROABPs - 1

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Lec 29 (18 Mar, Wed) : Raz-Shpilka polynomial time white-box PIT for ROABPs - 2

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Lec 30 (20 Mar, Fri) : Forbes-Shpilka quasi-polynomial time black-box PIT for ROABPs - 1

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Lec 31 (23 Mar, Mon) : Forbes-Shpilka quasi-polynomial time black-box PIT for ROABPs - 2

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Lec 32 (25 Mar, Wed) : Hitting set for depth $3$ powering circuits using Shpilka-Volkovitch generators.

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Lec 33 (27 Mar, Fri) : Hitting sets to lower bounds. Combinatorial designs. Impagliazzo-Kabanets theorem - Hitting sets from lower bounds.

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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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