Linear Algebra
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Analysis
Finite, countable and uncountable sets---metric spaces---compact sets---perfect sets---connected sets---convergent and Cauchy sequences ---power series---absolute convergence---rearrangements---limits--- continuity and compactness---connectedness---discontinuities---differentiablity---Riemann integration ---sequences and series of functions---equicontinuous family of functions and the Stone-Weierstrass theorem.
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Probability and Statistics I
Prerequisite: Real Analysis
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Probability and Statistics II
Prerequisite: Probability and Statistics I
Probability and Statistics III
Prerequisite: Probability and Statistics I and II
Programming Techniques
Introduction to basic programming principles using Python, including object-oriented design, big-oh notation, sorting and search algorithms, elementary data structures (lists, heaps, binary trees).
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Measure Theoretic Probability
Measure and integration: sigma fields and monotone class theorem, probability measures, statement of Caratheodory extension theorem, measurable functions, integration, Fatou, MCT, DCT, product spaces, Fubini. (about 1/2 time to be spent) Probability: 1-1 correspondence between distribution functions and probabilities on R, independence, Borel-Cantelli, Weak and Strong laws in the i.i.d. case, Kolmogorov 0-1 law, various modes of convergence, characterstic functions, uniqueness/inversion/Levy continuity theorems, CLT for the iid case with finite variance. (about 1/2 time to be spent)
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Discrete mathematics
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Differential Equations
Solution of First-order ODE's, Linear ODE's, Especially Second Order with Constant Coefficients; Undetermined Coefficients and Variation of Parameters; Sinusoidal and Exponential Signals: Oscillations, Damping, Resonance; Fourier Series, Periodic Solutions; Delta Functions, Convolution, and Laplace Transform Methods; Matrix and First-order Linear Systems: Eigenvalues and Eigenvectors;
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Algorithms
A quick revision of sorting, searching, selection and Big Oh; Divide and Conquer; Dynamic Programming; Graphs, BFS, DFS, connectivity; Algorithms on Matrices; Combinatorial Optimization --- Linear Programming, Simplex, Duality, Primal Dual Algorithms (shortest paths, max flow, matching).Suggested Textbooks
Economics
This course covers material useful for an understanding of both theoretical and empirical finance. It is not intended as a comprehensive survey of economics. The approach is analytical (as befits a graduate math course) and stresses understanding of concepts. Topics from both micro and macro economics are covered.Suggested Textbooks
Stochastic Processes I
Computational methods
Simulation techiques
Pseudo random numbers, Linear congruential generator, Mersenne twister RNG, Simulation of random variables, illustrations, Monte Carlo integration, Simulation of Random walk and approximations to diffusion processes, Applications to credit risk, complex derivatives pricing and portfolio optimization.Econometrics
Prerequisite: Probability and Statistics I and II
Regression and Classification
Prerequisite: Probability and Statistics I and II
Finance
Prerequisite: Real Analysis, Probability and Statistics I
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Mathematical Finance
Prerequisite: Probability and Statistics I,II and Finance
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Financial risk management
Prerequisite: Finance and Mathematical Finance
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Applied Statistics
Correlations, significance, regression (logistic, linear, time series), testing of hypotheses, clustering techniques, factor analysis, principal component analysis, distributions (parametric and non parametric).Advanced Algorithms
This course is about techniques for dealing with algorithmically hard problems. NP completeness; Approximation Algorithms (including LP rounding and primal dual algorithms); Randomized algorithms; fixed parameter algorithms; branch and bound, local search; heuristicsSuggested Textbooks
Algorithms on Strings, trees and sequences
This is a course on topics in algorithms oriented towards applications in Biology. String Matching and variations; Suffix trees and its uses; Inexact matching; Sequence Alignment; Sequence databases and their usesSuggested Textbook
Data Mining and Machine Learning
Association rules, frequent itemsets; Finding high-correlation with low-support; Classifiers -- Bayesian, Nearest Neighbour; Decision Trees; Clustering techniques; Vector space (TF-IDF) model; Stop words and stemming; Supervised learning : Bayesian Networks, Support Vector Machines; Semisupervised learning: Expectation maximization; Web search: HITS and PageRank;Suggested Textbooks
Cryptography
Elementary number theory --- Pseudo-random bit generation --- elementary cryptosystems --- number theoretic algorithms (RSA) --- symmetric key cryptosystems - DESIDEA, AES, --- authentication --- digital signatures, electronic commerce (anonymous cash, micropayments), key management--- PGP --- zero-knowledge protocols --- fairness.Suggested Textbooks
Finite automata---regular languages---pumping lemma--- stack automata---context free languages---applications to compilers---Turing machines---universal Turing machines--- halting problem---non deterministic Turing machines--- complexity classes---P v/s NP.
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Programming Language Concepts
Imperative programming---Scope rules---Object oriented-programming--- Java---Shell programming---PERL--- Functional programming---Logic programming---Query Language for databases.
Laboratory: Programming assignments in Java, PERL and SQL.
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