CO 601Fundamentals of Enumerative Combinatorics0.50 units
Elementary counting, bijections. Sets, weights, decompositions. The ordinary generating series, sum, product, composition and differentiation lemmas; formal use in combinatorial identities. The algebra of formal power series. Strings on a finite alphabet. Unique factorization of strings. Pattern and noncommutative methods. The Maximal Decomposition Theorem. Rational and algebraic generating series, linear recurrence equations. Integer partitions. Functional equation. Ferrers graph, geometric decompositions, Euler's Pentagonal Number Theorem. Vector spaces over GF(q), linear transformations, finite forms of partition theorems, Gaussian coefficients. Inversions in permutations. The exponential generating series, rule of sum and product, composition and differentiation lemmas. Disjoint cycles and permutations, functions, set partitions, graphs. Lagrange's Implicit Function Theorem. Lattice paths, Wiener-Hopf Factorization. Lattice polygons, polyominos, trees and functions. Formal use in combinatorial identities. Enumeration under group action, Orbit-Stabilizer Theorem, Burnside's Lemma, Polya's Theorem. Necklaces, abstract trees and trees embedded in the plane.
Duality in linear optimization. Farkas' Lemma and the theorems of the alternative. Duality theorem for linear programming. Complementary Slackness theorem. Simplex Method. Polyhedra and elementary convex geometry.
Combinatorial Optimization (4 weeks):
Linear diophantine equations. Facets of polyhedra. Integrality of polyhedra. Shortest paths and optimal flows. Total Unimodularity. Konig's Theorem. Max. Flow-Min. Cut Theorem. (Hungarian) Bipartite matching algorithm.
Continuous Optimization (4 weeks):
Convex functions. Analytic characterizations of convexity. Existence and uniqueness of optimal solutions. Separating and supporting hyperplane theorems. Lagrangean Duality. Karush-Kuhn-Tucker Theorem. Conic Optimization problems. Ellipsoild Method.
The algebra of formal Laurent series. Multivariate ordinary generating functions and exponential generating functions. The Lagrange Implicit Function Theorem, the MacMahon Master Theorem. Enumeration of planar triangulations. The Transfer Matrix method. Sieve methods, Inclusion/Exclusion, Möbius inversion. Pólya Enumeration, Enumeration of Trees. Basic hypergeometric series, q-analogues, Rogers-Ramanujan identities. Asymptotic methods.
The ring of symmetric functions, standard bases, the Hall inner project. Young tableaux. The Robinson-Schensted-Knuth correspondence, the hood-length formula, the Jacobi-Trudi formula, the Pieri rule, the Littlewood-Richardson rule. Representation theory of the symmetric groups. Enumeration of plane partitions. Enumeration of maps on surfaces. Other topics.
Pairwise orthogonal latin squares. Transversal designs and finite planes. Balanced incomplete block designs, group divisible designs and pairwise balanced designs. Symmetric designs and Hadamard matrices. Recursive constructions. Wilson's fundamental construction.
Colouring: Brooks' Theorem and Vizing's Theorem. Flows: integer and group-valued flows, the flow polynomial, the 6-flow theorem. Extremal graph theory; Ramsey's theorem, Turan's theorem, Mader's theorem on graphs with no n-clique-minor. Probabilistic methods: Lower bounds for Ramsey numbers, graphs with large girth and chromatic number.
Automorphisms. Cayley graphs and their properties. Arc and distance transitive graphs. Generalised polygons. homomorphisms and covers. Adjacency and incidence matrices. Eigenvectors of graphs. Quotients. Interlacing. Strongly regular graphs. Line graphs and graphs with least eigenvalue -2. Expanders. Shannon capacity.
This is an introductory course on matroid theory, with particular emphasis on graphic matroids and on topics that are applicable to graph theory. The topics include: matroid intersection and partition, graphic matroids, matroid connectivity, regular matroids, and representable matroids. Applications include: matching, disjoint spanning trees, the 8-flow theorem for graphs, planarity testing, and recognizing totally unimodular matrices.
Characterizations of optimalsolutions and efficient algorithms for optimization problems over discrete structures. Topics include network flows, optimal matchings, T-joins and postman tours, matroid optimization.
Formulation of problems as integer linear programs. Solution by branch-and-bound and cutting plane algorithms. Introduction to the theory of valid inequalities and polyhedral combinatorics.
An introduction to the modern theory of convex programming, its extensions and applications. Structure of convex sets, separation and support, subgradient calculus for convex functions, Fenchel conjugacy and duality, Lagrange multipliers. Ellipsoid method for convex optimization.
A course on theory and solution algorithms for the minimization of a convex quadratic function subject to linear constraints. Karush-Kuhn-Tucker conditions, duality theory. Active set solution algorithms and their parametric extensions. Quadratic programmes as linear complementarity problems. Applications in portfolio optimization and structural analysis.
Numerical algorithms for nonlinear optimization. Newton, variable-metric, quasi-Newton and conjugate gradient methods. Obtaining derivatives. Convexity. Trust region methods. Constrained optimization including optimality conditions, sequential quadratic programming, interior point and active set strategies.
Optimization over convex sets described as the intersections of the set of symmetric, positive semidefinite matrices with affine spaces. Formulations of problems from combinatorial optimization, graph theory, number theory, probability and statistics, engineering design and control theory. Theoretical and practical consequences of these formulations. Duality theory and algorithms.
Review of basics of quantum information and computational complexity; Simple quantum algorithms; Quantum Fourier transform and Shor factoring algorithm: Amplitude amplification, Grover search algorithm and its optimality; Completely positive trace-preserving maps and Kraus representation; Non-locality and communication complexity; Physical realizations of quantum computation: requirements and examples; Quantum error-correction, including CSS codes, and elements of fault-tolerant computation; Quantum cryptography; Security proofs of quantum key distribution protocols; Quantum proof systems. Familiarity with theoretical computer science or quantum mechanics will also be an asset, though most students will not be familiar with both.
CO 685The Mathematics of Public-Key Cryptography0.50 units
An in-depth study of public-key cryptography, including: number-theoretic problems - prime generation, integer factorization, discrete logarithms; public-key encryption; digital signatures; key establishment; secret sharing; and security definitions and proofs.
A broad introduction to cryptography, highlighting the major developments of the past twenty years, including symmetric ciphers, hash functions and data integrity, public-key encryption and digital signatures, key establishment, and key management. Applications to internet security, computer security, communications security, and electronic commerce will be studied.
Methods of obtaining asymptotic estimates for sums arising in enumeration. Application to Bell numbers, the distribution of balls into cells, and random graphs. Methods for obtaining asymptotic estimates for the coefficients of generating functions. Application to random planar graph problems and to unimodality problems.
CO 738Probabilistic Methods in Discrete Mathematics0.50 units
The probabilitistic method proves the existence of a combinatorial structure by showing a random element in an appropriate probability space has the desired properties with positive probability. The course will introduce the basic techniques and give applications in graph theory, combinatorics, discrete optimization, theoretical CS. In particular, some of the following will be considered: linearity of expectation; alterations; the second moment method; bounding of large deviations; the local lemma; correlation inequalities; martingales and tight concentration; discrepancy; derandomization.
An introduction to the concepts of directed graphs, problems about directed circuits and directed cuts, and minimax equalities relating these and other graphical objects.
A general study of algorithms for the solutions of problems in graph theory. Included are algorithms for graph isomorphism, planarity, connectedness, chromatic number, spanning trees, circuits and many others.
CO 754Approximation Algorithms in Combinatorial Optimization0.50 units
The course studies some of the successful paradigms for designing approximation algorithms and for proving approximation guaranteees, such as the greedy method, formulating and solving linear programming relaxations, the primal-dual method, randomized rounding, semidefinite programming relaxations, approximation of metrics. Also, the PCP theorem, and its application to the hardness of approximating specific problems, e.g., set covering and shortest lattice vector.
Basic optimization: quadratic minimization subject to leanear equality constraints. Effecient portfolios: the efficient frontier, the capital market line, Sharpe ratios and threshold returns. Practical portfolio optimization: short sales restrictions target portfolios, transactions costs. Quadratic programming theory. Special purpose quadratic programming algorithms for portfolio optimization: today's large investment firms expect to solve problems with at least 1000 assets, transactions costs and various side constraints in just a few minutes of computation time. This requires very specialized QP algorithms. An overview of such algorithms will be presented with computational results from commercial problems. The efficient frontier, the capital market line, Sharpe ratios and threshold returns in practice.
Fundamental problems of elementary and algebraic number theory from an algorithmic and computational complexity point of view with emphasis placed on analysis of algorithms. Topics include basic arithmetic algorithms; computation over finite fields; primality testing; algorithms for integer factorization; algorithms in number fields.