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‎graduate/docs/analysis-general-exam-syllabus.html‎

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<section aria-labelledby="basics-section">
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<h2 id="basics-section">Basics (common for complex and real parts)</h2>
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<ol class="topic-list">
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<ol>
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<li>Elementary set operations, countable and uncountable sets.</li>
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<li>Open, closed, compact, and connected sets on the line and in Euclidean space.</li>
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<li>Completeness, infima and suprema, limit points, lim inf, lim sup.</li>
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<section aria-labelledby="complex-analysis-section">
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<h2 id="complex-analysis-section">Complex Analysis</h2>
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<ol>
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<li>Series of functions, power series, and power series of elementary functions, uniform convergence, Weierstrass <math><mi>M</mi></math> test. Formula for the radius of convergence of a power series.</li>
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<li>Analytic and harmonic functions, Cauchy-Riemann equations.</li>
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<li>Power series and Laurent series.</li>
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<section aria-labelledby="real-analysis-section">
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<h2 id="real-analysis-section">Real Analysis</h2>
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<ol>
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<li><math><mi>σ</mi></math>-algebras of sets.</li>
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<li>Lebesgue measures and abstract measures, signed measures. Lebesgue-Stieltjes measures on the real line and their correspondence with increasing, right continuous functions.</li>
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<li>Measurable functions. Approximation by simple functions. Riemann and Lebesgue integrals.</li>
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</section>
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<section aria-labelledby="references-section">
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<h3 id="references-section">References</h3>
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<ul>
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<li>R. G. Bartle, <i>Elements of Real Analysis</i></li>

‎graduate/docs/topology-general-exam-syllabus.html‎

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<section aria-labelledby="differential-topology-section">
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<h2 id="differential-topology-section">I. Differential Topology</h2>
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<li>Multivariable calculus basics: definition of a smooth map <math><mrow><mi>f</mi><mo>:</mo><msup><mi>ℝ</mi><mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi><mi>m</mi></msup></mrow></math>, the inverse and implicit function theorems.</li>
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<li>Manifolds and smooth maps; submanifolds. Examples: 2-dimensional surfaces; the sphere <math><msup><mi>S</mi><mi>n</mi></msup></math>; the real projective space <math><msup><mi>ℝP</mi><mi>n</mi></msup></math>; examples of Lie groups: classical matrix groups.</li>
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<h2 id="algebraic-topology-section">II. Algebraic Topology</h2>
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<li>Basic properties of singular homology: functoriality, homotopy invariance, long exact sequence of a pair, excision, and the Meyer–Vietoris sequence.</li>
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<li>The homology groups of spheres, and the degree of a map between spheres. Classic applications, such as Brouwer fixed point theorem, but proved with homology.</li>
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<li><i>Homology Theory</i> by J. W. Vick (2nd edition).</li>
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<h4>References for general topology background material:</h4>
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<li>An outline summary of basic point set topology, by J.P. May, at<br>
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<a href="http://www.math.uchicago.edu/~may/MISC/Topology.pdf" target="_blank" rel="noopener noreferrer">http://www.math.uchicago.edu/~may/MISC/Topology.pdf</a></li>
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<h4>References for the Jordan-Alexander Complement Theorem:</h4>
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<h4>References for general topology background material:</h4>
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<li><i>Topology</i> by J. R. Munkres.</li>
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<li>An outline summary of basic point set topology, by J.P. May, at<br>
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<a href="http://www.math.uchicago.edu/~may/MISC/Topology.pdf" target="_blank" rel="noopener noreferrer">http://www.math.uchicago.edu/~may/MISC/Topology.pdf</a></li>
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</ul>
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<h4>References for the Jordan-Alexander Complement Theorem:</h4>
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<ul>
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<li>A Dold, A simple proof of the Jordan-Alexander Complement Theorem, <i>Amer. Math. Monthly</i> <strong>100</strong> (1993), 856–857.</li>
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<li>Both Hatcher and Vick prove special cases including the Jordan Curve Theorem.</li>
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</ul>
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</article>

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