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Author: lenis2000

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

@@ -81,7 +81,7 @@ <h2 id="basics-section">Basics (common for complex and real parts)</h2>
   <section aria-labelledby="complex-analysis-section">
     <h2 id="complex-analysis-section">Complex Analysis</h2>
     <ol>
-      <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>
+      <li>Series of functions, power series, and power series of elementary functions, uniform convergence, Weierstrass <math role="math"><mi>M</mi></math> test. Formula for the radius of convergence of a power series.</li>
       <li>Analytic and harmonic functions, Cauchy-Riemann equations.</li>
       <li>Power series and Laurent series.</li>
       <li>Elementary conformal mappings, fractional linear mappings. The Cayley transform.</li>
@@ -96,21 +96,21 @@ <h2 id="complex-analysis-section">Complex Analysis</h2>
   <section aria-labelledby="real-analysis-section">
     <h2 id="real-analysis-section">Real Analysis</h2>
     <ol>
-      <li><math><mi>σ</mi></math>-algebras of sets.</li>
+      <li><math role="math"><mi>σ</mi></math>-algebras of sets.</li>
       <li>Lebesgue measures and abstract measures, signed measures. Lebesgue-Stieltjes measures on the real line and their correspondence with increasing, right continuous functions.</li>
       <li>Measurable functions. Approximation by simple functions. Riemann and Lebesgue integrals.</li>
       <li>Monotone convergence and dominated convergence theorems, Fatou's lemma.</li>
       <li>Product spaces and product measure, Fubini-Tonelli theorems.</li>
       <li>Absolute continuity of measures, Radon-Nikodym theorem. Lebesgue-Radon-Nikodym decomposition.</li>
       <li>Hardy-Littlewood maximal function: the maximal inequality for the Hardy-Littlewood maximal function and the Vitali covering lemma. The Lebesgue differentiation theorem.</li>
-      <li>Absolute continuity of functions, differentiation and the Fundamental Theorem of Calculus for absolutely continuous functions. The correspondence between absolutely continuous functions on <math><mi>ℝ</mi></math> and measures on <math><mi>ℝ</mi></math> which are absolutely continuous with respect to the Lebesgue measure. Bounded variation functions and their correspondence with complex measures on <math><mi>ℝ</mi></math>.</li>
+      <li>Absolute continuity of functions, differentiation and the Fundamental Theorem of Calculus for absolutely continuous functions. The correspondence between absolutely continuous functions on <math role="math"><mi mathvariant="double-struck">R</mi></math> and measures on <math role="math"><mi mathvariant="double-struck">R</mi></math> which are absolutely continuous with respect to the Lebesgue measure. Bounded variation functions and their correspondence with complex measures on <math role="math"><mi mathvariant="double-struck">R</mi></math>.</li>
       <li>Hölder's inequality, Jensen's inequality.</li>
-      <li><math><msup><mi>L</mi><mi>p</mi></msup></math> spaces, completeness. Approximation of <math><msup><mi>L</mi><mi>p</mi></msup></math>-functions on <math><msup><mi>ℝ</mi><mi>d</mi></msup></math> by compactly supported continuous functions.</li>
-      <li>Hilbert space, projection theorem, Riesz representation theorem, orthonormal sets, <math><msup><mi>L</mi><mn>2</mn></msup></math> spaces.</li>
-      <li><math><msup><mi>L</mi><mi>p</mi></msup></math>-<math><msup><mi>L</mi><msup><mi>p</mi><mo>′</mo></msup></msup></math> duality when <math><mrow><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><msup><mi>p</mi><mo>′</mo></msup></mfrac><mo>=</mo><mn>1</mn></mrow></math>.</li>
+      <li><math role="math"><msup><mi>L</mi><mi>p</mi></msup></math> spaces, completeness. Approximation of <math role="math"><msup><mi>L</mi><mi>p</mi></msup></math>-functions on <math role="math"><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></math> by compactly supported continuous functions.</li>
+      <li>Hilbert space, projection theorem, Riesz representation theorem, orthonormal sets, <math role="math"><msup><mi>L</mi><mn>2</mn></msup></math> spaces.</li>
+      <li><math role="math"><msup><mi>L</mi><mi>p</mi></msup></math>-<math role="math"><msup><mi>L</mi><msup><mi>p</mi><mo>′</mo></msup></msup></math> duality when <math role="math"><mrow><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><msup><mi>p</mi><mo>′</mo></msup></mfrac><mo>=</mo><mn>1</mn></mrow></math>.</li>
       <li>Elementary Fourier series, Riesz-Fischer and Parseval theorems. Riemann-Lebesgue theorem. Dirichlet kernel.</li>
-      <li>Fourier transforms in <math><msup><mi>ℝ</mi><mi>d</mi></msup></math>, Plancherel and Parseval's theorems, Fourier transforms of derivatives and translations. Riemann-Lebesgue lemma. Hausdorff-Young's inequality.</li>
-      <li>Convolutions: Fourier transforms of convolutions, approximations to the identity, approximation of functions on <math><msup><mi>ℝ</mi><mi>d</mi></msup></math> by compactly supported smooth functions. Young's inequality for convolution.</li>
+      <li>Fourier transforms in <math role="math"><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></math>, Plancherel and Parseval's theorems, Fourier transforms of derivatives and translations. Riemann-Lebesgue lemma. Hausdorff-Young's inequality.</li>
+      <li>Convolutions: Fourier transforms of convolutions, approximations to the identity, approximation of functions on <math role="math"><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></math> by compactly supported smooth functions. Young's inequality for convolution.</li>
     </ol>
   </section>
 

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

@@ -77,16 +77,16 @@ <h1 id="syllabus-title">Topology General Exam Syllabus</h1>
   <section aria-labelledby="differential-topology-section">
     <h2 id="differential-topology-section">I. Differential Topology</h2>
     <ol>
-      <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>
-      <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>
+      <li>Multivariable calculus basics: definition of a smooth map <math role="math"><mrow><mi>f</mi><mo>:</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>→</mo><msup><mi mathvariant="double-struck">R</mi><mi>m</mi></msup></mrow></math>, the inverse and implicit function theorems.</li>
+      <li>Manifolds and smooth maps; submanifolds. Examples: 2-dimensional surfaces; the sphere <math role="math"><msup><mi>S</mi><mi>n</mi></msup></math>; the real projective space <math role="math"><msup><mrow><mi mathvariant="double-struck">R</mi><mi>P</mi></mrow><mi>n</mi></mrow></msup></math>; examples of Lie groups: classical matrix groups.</li>
       <li>The differential of a smooth map, tangent vectors, and tangent spaces. The tangent bundle.</li>
       <li>Regular and critical values. Embeddings, immersions. Transversality.</li>
       <li>Sard's theorem.</li>
       <li>The embedding theorem: every closed manifold embeds in a Euclidean space.</li>
       <li>Orientability.</li>
       <li>Vector fields, the Euler characteristic.</li>
       <li>Invariants of manifolds and smooth maps: mod 2 degree of a map, the integer-valued degree of a map between oriented manifolds, intersection numbers, linking numbers.</li>
-      <li>Applications to compact manifolds: <math><mrow><mo>∂</mo><mi>M</mi></mrow></math> is not a retract of <math><mi>M</mi></math>, Brouwer Fixed Point Theorem, zeros of vector fields, etc.</li>
+      <li>Applications to compact manifolds: <math role="math"><mrow><mo>&PartialD;</mo><mi>M</mi></mrow></math> is not a retract of <math role="math"><mi>M</mi></math>, Brouwer Fixed Point Theorem, zeros of vector fields, etc.</li>
       <li>Vector bundles: tangent bundle, normal bundle, duals, tensor bundles. Structures on bundles including inner products, specifically Riemannian metrics.</li>
       <li>Differential forms: exterior algebra, exterior derivative.</li>
       <li>Integration of forms on oriented manifolds; Riemannian volume form; Stokes' theorem.</li>
@@ -99,15 +99,15 @@ <h2 id="algebraic-topology-section">II. Algebraic Topology</h2>
       <li>Basic properties of singular homology: functoriality, homotopy invariance, long exact sequence of a pair, excision, and the Meyer–Vietoris sequence.</li>
       <li>Homological algebra: chain complexes, maps, homotopies. The long exact homology sequence associated to a s.e.s. of chain complexes. The snake lemma. The 5–lemma.</li>
       <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>
-      <li>The Jordan–Alexander Complement Theorem: <math><mrow><msub><mi>H</mi><mo>*</mo></msub><mo>(</mo><msup><mi>ℝ</mi><mi>n</mi></msup><mo>−</mo><mi>A</mi><mo>)</mo><mo>≅</mo><msub><mi>H</mi><mo>*</mo></msub><mo>(</mo><msup><mi>ℝ</mi><mi>n</mi></msup><mo>−</mo><mi>B</mi><mo>)</mo></mrow></math> if <math><mi>A</mi></math> and <math><mi>B</mi></math> are homeomorphic closed subsets of <math><msup><mi>ℝ</mi><mi>n</mi></msup></math>. The Jordan Curve Theorem is a special case.</li>
+      <li>The Jordan–Alexander Complement Theorem: <math role="math"><mrow><msub><mi>H</mi><mo>*</mo></msub><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>−</mo><mi>A</mi><mo>)</mo><mo>≅</mo><msub><mi>H</mi><mo>*</mo></msub><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>−</mo><mi>B</mi><mo>)</mo></mrow></math> if <math role="math"><mi>A</mi></math> and <math role="math"><mi>B</mi></math> are homeomorphic closed subsets of <math role="math"><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></math>. The Jordan Curve Theorem is a special case.</li>
       <li>The homology of a C.W. complex: cellular chains. This includes delta complex homology as a special case. Examples: real and complex projective spaces, closed surfaces.</li>
       <li>Euler characteristic and its properties. Classic calculations: spheres, closed surfaces.</li>
       <li>Construction of the fundamental group as a homotopy functor of a space with basepoint.</li>
       <li>Covering spaces: definition and examples.</li>
-      <li>The lifting theorem: under appropriate point set conditions, a continuous <math><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow></math> lifts through a covering map <math><mrow><mover><mi>Y</mi><mo>~</mo></mover><mo>→</mo><mi>Y</mi></mrow></math> iff it does on the level of <math><msub><mi>π</mi><mn>1</mn></msub></math>.</li>
-      <li>Deck transformations, and the correspondence between subgroups of the fundamental groups and covering spaces. A variant: if <math><mover><mi>Y</mi><mo>~</mo></mover></math> is simply connected, there is a 1–1 correspondence between covering spaces <math><mrow><mover><mi>Y</mi><mo>~</mo></mover><mo>→</mo><mi>Y</mi></mrow></math> and free, proper group actions on <math><mover><mi>Y</mi><mo>~</mo></mover></math>.</li>
+      <li>The lifting theorem: under appropriate point set conditions, a continuous <math role="math"><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow></math> lifts through a covering map <math role="math"><mrow><mover><mi>Y</mi><mo>~</mo></mover><mo>→</mo><mi>Y</mi></mrow></math> iff it does on the level of <math role="math"><msub><mi>π</mi><mn>1</mn></msub></math>.</li>
+      <li>Deck transformations, and the correspondence between subgroups of the fundamental groups and covering spaces. A variant: if <math role="math"><mover><mi>Y</mi><mo>~</mo></mover></math> is simply connected, there is a 1–1 correspondence between covering spaces <math role="math"><mrow><mover><mi>Y</mi><mo>~</mo></mover><mo>→</mo><mi>Y</mi></mrow></math> and free, proper group actions on <math role="math"><mover><mi>Y</mi><mo>~</mo></mover></math>.</li>
       <li>Seifert–Van Kampen Theorem.</li>
-      <li><math><msub><mi>H</mi><mn>1</mn></msub></math> is the abelianization of <math><msub><mi>π</mi><mn>1</mn></msub></math>.</li>
+      <li><math role="math"><msub><mi>H</mi><mn>1</mn></msub></math> is the abelianization of <math role="math"><msub><mi>π</mi><mn>1</mn></msub></math>.</li>
       <li>Examples, including the fundamental group of spheres, projective space, surfaces, etc. Classic applications, e.g., to group theory.</li>
     </ol>
   </section>