Qualifying Exams

Algebra

General Algebra
  1. How does one make a subposet into a poset? (Bergman)
  2. State and prove the Fundamental Theorem of Finite Distributive Lattices. (Klass)
  3. Define and state things about posets. (Klass)
  4. Draw a non-lattice with 5 elements. Draw a lattice with 5 elements. Draw a Boolean lattice on 3 elements. (Lam)
  5. Give a ring $R$ and a free module $M$ that has a basis of $n$ elements for every integer $n\ge 1$. Can $R$ be choesen commutative unital?
  6. State the fundamental theorem of Galois theory.
Combinatorics
  1. State and prove Ramsey's Theorem, Hall's Theorem, Speiner's Theorem, Erdös-Ko-Rado Theorem, and the Erdös lower bound on Ramsey. (Karp)
  2. State Lovas local theorem and Alan's Theorem. (Karp) What is $R_1(n)$? (Karp)
  3. When is the Speiner bound tight? (Karp)
  4. What is the net work flow problem? (Sinclair)
  5. What is an algorithm to solve the problem? How do we know it terminates and what is a bound on the running time? (Sinclair)
  6. What is an algorithm with a better bound? (Sinclair)
  7. How can we use the algorithm to find a minimum cut? (Sinclair)
  8. What is a randomized algorithm for finding a minimal cut? (Sinclair)
  9. What is a bound on the error probability? (Sinclair)
  10. What does this tell us about how many minimal cuts there can be in a 1-graph? (Sinclair)
  11. What is an Eulerian poset? What is graded? What is a rank funciton? What is the length of a chain? What is $\mu$ of an interval? Why is it called Eulerian? (Sinclair)
  12. Consider monotonic paths from $(0,0)$ to $(n,n)$ consisting of unit steps either $+(1,0)$ or $+(0,1)$. $\alpha\ge\gamma$ if $\alpha$ is never below $\gamma$. Define a hill to be a $+(0,1)$ step followed by a $+(1,0)$ step. Define a valley to be a $+(1,0)$ step followed by a $+(0,1)$ step. Given $\alpha\ge\beta$, define hills of $\alpha$ and valleys of $\beta$ as good points. Define valleys of $\alpha$ and hills of $\beta$ as bad points. Show the number of good points is always greater than the number of bad points. (Sinclair)
  13. Talk about the Incidence Algebra on a poset. (Klass)
  14. If we are to implement the Mobius inversion on the poset, do we need the functions in the Incidence Algebra to take values in a field? Does a ring suffice? (Bergman)
  15. When is a function in the Incidence Algebra invertible? Prove it. (Bergman)
  16. Talk about the Mobius function for the product of two posets. Use it to describe the Mobius function on B_n, the Boolean poset of size n. (Klass)
  17. Prove that a finite meet-semilattice with 1 is a lattice. (Bergman)
  18. Is an infinite meet-semilattice with 1 necessarily a lattice? If not, find a counterexample. (Bergman)
  19. Prove that in a finite poset with a unique maximal element, that element is 1; find a counterexample in the infinite case. (Bergman)
  20. Consider the matroid of the hyperplane arrangement of the root system $B_n$. Draw this for $n=2$, and compute the characteristic polynomial. Is this a graphical matroid? Derive the characteristic polynomial for general $n$. How many regions does the hyperplane arrangement have? (Ardila (SFSU))
  21. State the finite field method for the characteristic polynomial of a hyperplane arrangement. Sketch a proof. (Ardila)
  22. Define Cohen-Macaulay posets. Show that the lattice of flats of a matroid is Cohen-Macaulay. What is the relationship between Cohen-Macaulay posets and Cohen-Macaulay rings? (Sturmfels)
  23. What is the Lagrange inversion formula? (Serganova)
  24. How do you use this formula? (Serganova)
  25. What is the proof? (Sturmfels)
  26. What is a Schur polynomial? (Serganova)
  27. How do you show that it is symmetric? (Serganova)
  28. What is the definition of the Schur polynomial using determinants? (Serganova)
  29. Why is the Schur polynomial important? (Serganova) [Hint]
    Inner products.
  30. Can you use Schur polynomials in computational biology? (Sturmfels)
Commutative Algebra
  1. What is Spec $\mathbb{C}[[x]]$? [Hint]
    $\mathbb{C}[[x]]$ is a DVR.
  2. State the Noether normalization theorem.
  3. Let $A$ be a commutative ring, $M$ a finitely generated $A$-module, and $x_1,\ldots,x_n$ elements of $M$ which generate $M/mM$ for every maximal ideal $m$ of $A$. Show that they generate $M$. (Vojta)
  4. What is $\mathbb{Z}^*_{q-1}$?
  5. Give an example of a commutative ring with unity which has prime ideals which are not maximal.
  6. Give two examples of UFD's which are not PID's.
  7. Suppose $p(x)\in F[x]$ where $F$ is a field. Let $p(a)=0$ for $a\in E$, an extension of $F$. Show that $p(x) = (x-a)q(x)$ for some $q(x)\in F[x]$.
  8. Show that an element $a$ is irreducible if and only if $(a)$ is maximal, where $a\in R$ and $R$ is a PID.
  9. Let $R$ be a finite commutative ring free of zero divisors. Show that $R$ has a unit, show that each non-zero element has an inverse. Is the result still true if $R$ is infinite?
  10. Is there a general class of rings in which maximal ideals and prime ideals are the same?
  11. Let $P$ be a prime ideal in $R$ such that $R/P$ is a finite ring. Show that $P$ is maximal if $R$ is a commutative ring with unit.
  12. Show that in a commutative ring with unit every maximal ideal must be prime.
  13. Describe the ring of endomorphisms of the integers.
  14. If $R$ is a UFD, prove that $R[X]$ is a UFD.
  15. Let $R$ be a commutative ring with unity $1\not=0$, and let $S$ be a multiplicative subset of $R$ not containing $0$. Consider the set of all ideals $A$ which do not intersetct $S$. Show that a maximal element in this set must be a prime ideal.
  16. Let $R$ be the ring of real quaternions. Does $R[X]$ satisfy the division algorithm property?
  17. Distinguish algebraically between $\operatorname{GL}(3,\mathbb{R})$ and $\operatorname{GL}(2,\mathbb{R})$, not using topology.
  18. What is a Dedekind ring?
  19. Prove that $k[X,Y]$ is not Dedekind.
  20. Let $\Omega\subseteq \mathbb{C}$ be a domain, and $\mathscr{O} (\Omega)$, $\mathscr{O}_F(\Omega)$ the ring of homomorphic functions and the subring of functions with finitely many zeroes. Is $\mathscr{O} (\Omega)$ or $\mathscr{O}_F(\Omega)$ a UFD? What are the primes in these rings?
  21. Can you give an example of a ring $R$ which is not Cohen-Macauley? (Ogus)
  22. Can you give an example of a ring $R$ which is Cohen-Macauley but not Gorenstein? (Ogus)
  23. For a dimension zero Gorensteins ring, what can you say about the $R$-module ${\rm Hom}_k(R,k)$? ${\rm Hom}_R(k,{\rm Hom}_k(R,k))$? (Ogus)
  24. Given $R$ a domain, Noetherian, dimension 1, what can we say about $\tilde{R}$, its integral closure? (Lenstra)
  25. Let's prove that in the above case, $\tilde{R}$ is Noetherian. (Lenstra)
  26. Define "Hilbert function". (Sturmfels)
  27. What conditions on a graded ring $S=\oplus_{d=0}^{\infty} S_d$ will assure the agreement of the Hilbert function with a polynomiial? (Sturmfels)
  28. What sorts of invariants appear in the Hilbert polynomial? (Sturmfels)
  29. Given an ideal $I$, how would one compute its Hilbert function? (Sturmfels)
  30. Define Gröbner basis, term order and initial ideal. (Sturmfels)
  31. How would you calculate $\operatorname{lt}(I)$ where $\operatorname{lt}(I) = \langle \operatorname{lt}(f) \,|\, f\in I\rangle$ and $\operatorname{lt}(f)$ is the sum of terms of highest degree of $f$? (Sturmfels)
  32. Give an example of a tegrm order that refines the partial order by degree. (Casson)
  33. Given an example of an ideal $I = \langle f_1,\ldots,f_t\rangle$ such that $\operatorname{lt}(I)\not=\langle \operatorname{lt} (f_1),\ldots,\operatorname{lt}(f_t)\rangle$. (Sturmfels)
  34. Find the ideal $I(X)$ for $X$ the twisted cubic in $\mathbf{A}^3$. Show that two generators suffice. Find a term order such that these two generators are not a Gröbner basis for $I(X)$ but all three are.
  35. Show that $\langle x^2-yw, xz-y^2, xy-zw\rangle$ generate the ideal of $\overline{X}$, the closure of $X$ in $\mathbf{P}^3$. Compute the Hilbert polynomial of $X$.
  36. Tell me about integral extensions. (Hartshorne)
  37. What is a Dedekind domain? An example of a domain which is noetherian, integrally closed, and not one-dimensional. An example of a domain which is integrally closed, one-dimensional, and not noetherian. (Hartshorne)
  38. Suppose that $I\subseteq J$ are ideals of $A$, $B/A$ is an extension of rings such that $IB=JB$; does it follow that $I=J$? If not, can you give a counterexample? Is there some hypothesis that makes it work? (Hartshorne)
  39. Can you give an example of a surjective morphism of rings which is not finite? (Hartshorne)
  40. State Nakayama's Lemma. Give an example of a ring and a nonzero ideal that satisfy the hypothesis. (Bergman)
  41. Prove, possibly using Nakayama's Lemma, that if $M$ is an $n\times n$ matrix over a local ring, with coefficients in the maximal ideal $I$, then $I+M$ is invertible.
  42. Show that a PID has dimension $0$ or $1$.
  43. Let $A$ be a noetherian valuation ring which is not a field. Show that $A$ is a DVR.
  44. Let $I_1,\cdots,I_r$ be ideals of a commutative ring $A$ such that $I_i+I_j=(1)$ for all $i\not=j$. Show that $$\prod_{i=1}^r I_i = \bigcap_{i=1}^r I_i.$$
  45. Let $M$ be an $A$-module, $S\subset A$ a multiplicative subset of $A$. Do we have $$S^{-1}\operatorname{Ann}(M) = \operatorname{Ann} S^{-1}(M)?$$ If not, give a counterexample, and if yes prove it. What if $M$ is finitely generated?
  46. Let $M$ be an $A$-module, $m$ a maximal ideal of $A$. Prove that $M/mM\cong M_m/mM_m$.
  47. Let $M$ be a finitely generated $A$-module. If $M=mM$ for every maximal ideal ${\bf m}$ of $A$ show that $M=0$.
  48. Let $M$ be a finitely generated $A$-module. Suppose that $x_1,\ldots,x_n$ generate $M_m$ for every maximal ideal $m$ of $A$. Show they generate $M$.
  49. Let $B\supset A$ be commutative rings, with $B$ integral over $A$. Let $x\in A$, $x\in B^*$. Show that $x\in A^*$.
  50. Prove that any ideal in a Dedekind ring is generated by at most two elements.
  51. Show that $\mathbb{Z}_p$ is a DVR.
  52. Show that an Artinian ring has only finitely many maximal ideals.
  53. Let $A\supset B$ be rings with $B$ integral over $A$. Let $Q$ be a prime ideal of $B$, and let $P=Q\cap A$. Show that $Q$ is maximal in $B$ if an only if $P$ is maximal in $A$.
  54. Show that if $f\colon A\to A$ is a surjective endomorphism of a noetherian ring $A$, then $f$ is an automorphism.
  55. Give an example of an ideal in $k[x,y,z]$ whose associated primes are $(x,y)$ and $(x,y,z)$. (Eisenbud)
  56. Give an example of a surface that has only one singular point but is not normal. (Eisenbud)
  57. In the example $k[x,y]/(y^2 - x^3) = R$, prove that it's not integrally closed. State Serre's Criterion. Write down all the primes of $R$. Can you see them in the picture of $R$? What parts of Serre's criterion does $R$ satisfy? What parts does it not satisfy? Prove that $R_{(x,y)}$ is not a DVR. (Eisenbud)
  58. How do you decide if a given polynomial belongs to an ideal $I$? (Sturmfels)
  59. State the main theorem of Elimination Theory. Why is it called this? (Sturmfels)
  60. Let's talk about Noether normalization and flatness. Let $f$ be a homogeneous polynomial in $R = k[x,y,z]$ where $k$ is an infinite field. Show that there exists homogeneous linear forms $s$ and $t$ in $R$ such that $R/(f)$ is finite and flat as a $k[s,t]$-module. (Eisenbud) What if $k$ is finite? (Olsson)
  61. Give an example of an injective ring homomorphism which is not flat. (Eisenbud)
  62. Consider the product of two generic linear forms: $(ax+b)(cx+d) = acx^2+(ad+bc)x+bd$. Let $I = (ac,ad+bc+bd)$ be the ideal generated by the coefficients of this product. Compute a primary decomposition of $I$. (Sturmfels)
  63. Try computing a Gröbner basis for $I$ (as above) and finding a primary decomposition of the initial ideal $I$. What does this primary decomposition of $I$ mean? What are the associated primes? What is the Krull dimension? Geometrically, what does this variety look like? (Strumfel)
  64. Given a rank $k$ submodule of $\mathbb{Z}^n$, when is its image in $\mathbb{F}_p^n$ a rank $k$ subspace? Characterize the situation in terms of flatness condition on an appropriate family. (Eisenbud)
  65. State as many forms of Hensel's lemma as you know. Give an example demonstrating that the completeness hypothesis can't be dropped. Give an example of a ring which is Henselian but not complete. How would one go about making Hensel's lemma constructive? (Eisenbud)
  66. Compute a primary decomposition of $(x+y,x-y)$ over $\mathbb{Z}[x,y]$. (Sturmfels)
  67. State the Nullstellensatz. (Sturmfels)
  68. Do you know a ring which isn't Jacobsen? (Sturmfels)
  69. Prove that the more geometric versions of the Nullstellensatz follow from the general one. (Sturmfels)
  70. If $f\in R$, what familiar ring is $R[y]/(fy-1)$? (Eisenbud)
  71. Let $I=(x^2-y^2,x^3-y^3)$ . Find a primary decomposition of $I$, compute a Gröbner basis, compute the Hilbert function. Is this ideal Cohen Macaulay? (Sturmfels)
Algebraic Geometry
  1. State and prove the Riemann-Hurwitz formula. [Hint]
    Consider the conormal sequence.
  2. What are the involutions of an elliptic curve over $\mathbb{C}$? What quotient arises from this involution? What are the fixed points of this involution? Show this quotient is $\hat{\mathbb{C}}$. (McMullen)
  3. State and prove Riemann-Hurwitz. Given a nonconstant map between curves over $k$, is there an associated map on differentials? A resulting exact sequence? Is the right exact sequence short exact in this case? (Ogus)
  4. Calculate the Picard group of $k[t^2,t^3]\subset k[t]$. (Ogus)
  5. Give an example of a projective curve that is not rational.
  6. Prove that $\mathbf{P}^1\times \mathbf{P}^1$ is a projective variety. Find the explicit equation of the image of the Segre embedding of $\mathbf{P}^1\times\mathbf{P}^1\subset \mathbf{P}^3$.
  7. How do you use Hurwitz's formula to calculate the geneus of a give curve? (Coleman)
  8. What can you say about curves over perfect fields? (Coleman)
  9. Define the degree of a projective variety. Show that a hypersurface of degree $d$ in $\mathbf{P}^n$ ofhas degree $d$. What does the constant term of the Hilbert polynomial represent? (Sturmfels)
  10. What does the degree of a hypersurface have to do with the line bundles on $\mathbf{P}^1$? (Ogus)
  11. Let $X$ be the twisted cubic in $\mathbf{P}^3$, is $X$ a set-theoretical intersection of two surfaces in $\mathbf{P}^3$? (Ogus)
  12. Define separated morphisms. Give an example of a non-seperated morphism. What about qusi-seperated morphisms? What are the good properties of separated morphisms? (Ogus)
  13. Let $g,h:Z\to X$ be two morphisms of schemes over $Y$, via $f:X\to Y$. If $g$ and $h$ agree on a dense open subset of $Z$, what can be said if $f$ is separated? What if $Z$ is reduced?
  14. Define differentials. Are differentials quasicoherent? (Ogus)
  15. What does the going up theorem mean in algebraic geometry? (Hartshorne)
  16. What can you say about the dimension of the image of a map from $\mathbf{P}^n$ to $\mathbf{P}^m$? (Hartshorne)
  17. What is the genus of a curve? Does the genus of a curve depend on the embedding? (Hartshorne)
  18. When is a canonical divisor very ample? (Wodzicki)
  19. State Riemann-Roch. (Wodzicki)
  20. Compute the dimension of the space of holomorphic differentials on a Riemann surface of genus $g$. (Wodzicki)
  21. State Abel's theorem. (Wodzicki)
  22. What is the significance of the Jacobian? What kind of map is the Abel-Jacobi map? What is it in the case of genus 1? (Wodzicki)
  23. What is the connection between $H^1$ and line bundles? (Wodzicki)
  24. What is a scheme? How can you tell if a scheme is affine? Can you weaken the Noetherian hypothesis in Serre's criterion for affineness? Prove that if $X$ is a Noetherian scheme such that $H^1(X,\mathscr{I}) = 0$ for all coherent sheaves of ideals $\mathscr{I}$ then $X$ is affine. Can you give an example where the theorem is false if we drop the quasi-compactness assumption?(Ogus)
  25. What can you say about curves of genus 0? Prove that such a curve can always be embedded as a line or a quadric in $\mathbf{P}^2$. If the base field is finite, can the latter occur?(Ogus)
  26. Calculate $H^0(\mathbf{P}^1,\Omega_{\mathbf{P}^1})$. (Poonen)
  27. If $f(x,y)$ and $g(x,y)$ are two polynomials such that the curves they define have inifinitely many points in common, is it true that they have a common factor?
  28. Give two criteria for a curve to be nonsingular (over an algebraically closed field.) (Ogus)
  29. What is a normal domain? How is this related to regular local rings? (Ogus)
  30. Find the singularities of the curve in $\mathbf{P}^2$ defined by the equation $X^3+y^3+z^3 = 3cxyz$. (Ogus)
  31. Describe Weil divisors and Cartier divisors on curves. How do you get a Weil divisor from an element $f\in K(X)^*$ in the canonical isomorphism?(Ogus)
  32. What is the degree of a divisor? (Ogus)
  33. Does there exist a variety $V$ with Pic$(V) =\mathbb{Z}/3$? (Poonen)
  34. Is the complement of a hypersurface in $\mathbf{P}^2$ affine? (Poonen)
  35. Define the geometric genus. (Poonen)
  36. What might be the geometric genus of a singular curve? (Poonen)
  37. Find the arithmetic genus of $y^3 = x^2z$. (Frenkel)
  38. Define sheaf cohomology. What's a right derived functor? (Olsson)
  39. Let $E$ be the curve in $\mathbf{P}^2$ defined by $y^2 = x^3-1$. Compute the cohomology of the structure sheaf $\mathscr{O}_E$. (Olsson)
  40. Define projective morphisms and what are they good for? What's a morphism that is not projective? (Eisenbud)
  41. Define Cartier and Weil divisor and relate them to each other. Do you know a Weil divisor which is not Cartier? Compute the Picard group and the class group of the cone over a conic. (Eisenbud)
  42. What can you say about curves of degree $4$ in $\mathbf{P}^3$? What if they are contained in a plane? What if they are singular? (Eisenbud)
  43. Let $X$ be a quartic surface in $\mathbf{P}^3$. Does $X$ contain a curve with negative self intersection, i.e. can the normal bundle to the curve have negative degree? (Eisenbud)
Representation Theory
  1. Prove Engel's theorem. (Serganova)
  2. Prove Lie's theorem. What would happen if the hypothesis was not that $g$ is solvable but that $g \ne [g,g]$? (Serganova)
  3. Why is $g = [g,g]$ for $g$ semisimple? (Weinstein)
  4. What is the exponential map, and what is it good for? (Serganova)
  5. Classify the real connected abelian Lie groups. (Serganova)
  6. Prove that a Lie group homomorphism $\phi\colon H \to G$ for $H$ connected is determined by the derivative at the identity. (Serganova)
  7. Give an example of a Lie group G where the exponential map is not surjective. (Weinstein)
  8. Given the standard representation of $sl_n({\mathbb C})$ identify the simple roots and explain the correlation between the height of the root and the corresponding "location" in the matrix. (Frenkel)
  9. Decompose $\operatorname{Sym}_n(V)\otimes \operatorname{Sym}_m(V)$ where $V$ is the 2-dimensional irreducible representation of $sl_2({\mathbb C})$. (Frenkel)
  10. Do the calculation above using a character formula. (Reshetikhin)
  11. State and explain the Harish-Chandra isomorphism. (Wodzicki)
  12. Explain how to write down the Weyl group of $SL_n$ using generators and relations. (Frenkel)
  13. What is a Verma module? (Reshetikhin)
  14. When is a Verma module finite-dimensional? (Wodzicki)
  15. What is the exponential map for $sl_2({\mathbb C})$? What is it a map from and to? Is it a homomorphism, is it surjective? What proofs of the Weyl character formula do you know? (Reshetikhin)
  16. What is Weyl's Integration formula? How do you use it to prove Weyl character formula? (Reshetikhin)
  17. What is the dimension of $E_8$? (Borcherds)
  18. Decompose $E_8$ as a representation of $E_7$. (Borcherds)
  19. Given a point in a semisimple Lie-algebra, how can we tell whether it lies in a Cartan subalgebra? (Knutsen)
  20. What is the relation between the Lie groups $SU(2)$ and $SO(3,\mathbb{R})$? Prove that the center of $SU(2)$ is $\mathbb{Z}/2\mathbb{Z}$ using Schur's lemma. (Weinstein)
  21. Under what condition on $G$ is every discrete normal subgroup of a Lie group $G$ contained in the center of $G$? Prove this. (Reshetikhin)
  22. List all the irreducible complex representations of the Lie group $SO(3,\mathbb{R})$. (Knutsen)
  23. Decompose the square of the adjoint representation of $sl(3)$ into irreducibles. (Hint: Weyl Character Formula.) (Haiman)
  24. State a theorem that explains the basic relationship between Lie algebras and Lie groups. Say some words about the proof. (Haiman)
  25. What is the relationship between Lie groups and Lie algebras? How do you show the existence of a Lie group with a given finite dimensional Lie algebra? (Reshetikhin)
  26. How many Lie groups are there with Lie algebra $sl_n$? (Serganova)
  27. What can you tell me about Bruhat decomposition and Bruhat cells? (Reshetikhin)
  28. What does the Borel-Weil theorem say? How does $G$ act on sections of the relevant line bundle? How do vectors in the dual of the irreducible representation with highest weight lambda give sections of this bundle? (Serganova)
  29. Suppose a finite group $G$ has only 1-dimensional representations. Is $G$ necessarily abelian (over $\mathbb{C}$, over $\mathbb{R}$)? (Serganova)
  30. What can you say about the multiplicities of irreducibles in an induced representation? (Reshetikhin)
  31. How does the representation of $S_4$ induced from the trivial representation on $S_2 \times S_2$ decompose into irreducibles, without using characters? (Serganova)

Mathematical Analysis

Banach spaces and Spectral Theory
  1. What is a nuclear operator? (Coleman)
  2. Give an example of an integral operator which is nuclear. (Coleman)
  3. What can you say about the specturm of a nuclear operator? (Coleman) Could it be the empty set? (Arveson)
  4. Give an example of an operator on a real Banach space with no specturm. (Arveson)
  5. Does the sum of the elements of the spectrum of a nuclear operator converge? (Coleman)
  6. What is a trace class operator? (Coleman)
  7. What is a Hilbert-Schmidt operator? Can you give an example over $\mathcal{ L}^2$ of the unit interval? (Coleman)
  8. Can $[0,1]$ be the spectrum of a compact operator? ( Arveson)
  9. What is the spectrum of $M_{e^{2\pi it}}$? How could you know that it is invertible? What is the inverse? (Arveson)
  10. If $T$ is an operator on a Banach space, what is $\cos^2T + \sin^2T$? (Arveson)
  11. What is $\cos T$ (Arveson)
  12. If $f$ is an entire function, what is $fT$? ( Arveson)
  13. List the properties of the functional calculus. ( Arveson)
  14. Consider $\mathcal{ C}\bigl([0,1], \mathbb{R}\bigr)$. Is there a natural topology on this space? (Arveson)
  15. Let $S=\{f\in \mathcal{ C}[0,1]\,|\, |f(x)-f(y)|\leq |x-y|\}$. What properties does it have (e.g. closed, complete, bounded compact)? (Arveson)
  16. Let $S_0=\{f\in S\,|\, f(0)=0\}$. What properties does it have (e.g. closed, complete, bounded, compact) ? (Arveson)
  17. What is the Riesz theory of compact operators?
  18. What is a Fredholm operator? Can any Fredholm operator be written as the sum of an invertible operator with a compact operator? What is the Fredholm index? What are its properties? How can you obtain an isomorphism between the abstract index group and the integers?
  19. Suppose you have an operator $x$ on a Hilbert space such that $x - x^2$ is compact. What can you tell me about it?
  20. In the previous question, you had a projection in the Calkin algebra, and you showed that it can be lifted to $B(H)$. Can you do the same for a unitary?
  21. What is the polar decomposition? What can you say about it?
$C^*$ and Von Neumann Algebras
  1. What are Fredholm operators?
  2. What do they have to do with $K$-theory for operator algebras?
  3. Could you give some examples of interesting $C^*$-algebras with nontrivial $K$-theory?
  4. How does one recognize a compact operator? Give examples.
  5. Prove that the Hilbert-Schmidt integral operators are compact.
  6. One usually calls a $C^*$-algebra separable if it is represented on a separable Hilbert space. What are the $C^*$-algebras that are in fact separable as topological spaces?
  7. State Kaplansky's Density Theorem. ( Jones)
  8. What is it good for? (e.g. in $L^\infty(S^1)$) ( Jones)
  9. Are the von Neumann algebras $l^\infty(\mathbb{Z})$ and $l^\infty(S^1)$ isomorphic? Can they be embedded in a $II_1$ factor? ( Jones)
  10. Define the index of a subfactor. ( Jones)
  11. What are all the hyperfinite subfactors of index $< 4$? ( Jones)
  12. Let $S$ be the unilateral shift. What is the commutant of $C^*(S^2)$?
  13. Do the Hilbert-Schmidt and trace class operators constitute $C^*$ algebras under the Hilbert-Schmidt and trace norms, respectively?
Complex Analysis
  1. Given a function continuous in a disk and analytic everwhere but at the center, prove that the function is analytic in the entire disk.
  2. Give a proof of Picard's theorem using, for example, the fact that the $j$ invariant of a modular curve uniformizes the $2,3,\infty$ hyperbolic triangle as the upper half-plane.
  3. Show that the mapping group of the torus is $SL(2,\mathbb{Z})$.
  4. Let $\Omega={\mathbb C}\backslash\left\{x\in \mathbb{R}\mid x<{1\over 4}\right\}$. Is there a conformal isomorphism $f\colon \Delta\to\Omega$, where \Delta is the open unit disk? (McMullen)
  5. Is there one with $f(0)=0$? (McMullen)
  6. How can we arrange for a unique $f$ with $f(0)=0$? (McMullen)
  7. What can you say about the coefficients $a_i$ of the power series expansion $f=\sum a_jz^j$? (McMullen)
  8. For $f$ with $f'(0)\in \mathbb{R}$, $f'(0)\geq 0$, what ring do the $a_i$ lie in? (McMullen)
  9. So to show that the $a_i$ lie in this ring, can we write down another function in terms of $f$ and --'s which maps $\Delta$ to $\Omega$? (McMullen)
  10. Calculate $a_i$. Now what ring do the $a_i$ lie in? (McMullen)
  11. How would you write down the power series for $\tan z$? (McMullen)
  12. What is its radius of convergence? (McMullen)
  13. Can you prove what the zeroes of $\cos z$ are? (McMullen)
  14. Why does the radius of convergence correspond this way? (McMullen)
  15. What is the area of a spherical triangle? Can you prove it? (McMullen)
  16. Same for hyperbolic triangle. (McMullen)
  17. Define a complex torus. (McMullen)
  18. What is the automorphism group of a complex torus? (McMullen)
  19. Show that if all the zeroes of a polynomial lie in a half-plane, then all zeroes of the derivative lie in the same half plane.
  20. What is the area of a spherical triangle?
  21. What is the automorphism group of a complex torus?
  22. If $f_n$ is a family of holomorphic functions such that $f_n\to F$ uniformly on compact subsets of some domain $\Omega$, what can you say about $f_n'$?
  23. Give an example of a sequence $f_n\to f$ where every $f_n$ is holomorphic and injective, and $f$ is not. Is this the most general such example?
  24. Why is there no conformal automorphism from the punctured disk to an annulus?
  25. Show that for a doubly periodic function f the number of zeroes of f and the number of poles of $f$ (counting with multiplicities) is equal.
  26. Suppose $f_i$ are harmonic functions on the unit disk $D$. Show that no linear combination of the $f_i$ can be negative on $\partial D$ and positive at some point in the interior of $D$.
  27. Find the poles and residues of $1/\sin(z)$.
  28. Give the formula for a conformal map from the unit disk to the inside of a polygon with angles $2\pi-\beta_i\pi$.
  29. Show that a continuous real-valued function $u$ on some region $\Omega$ which has the mean-value property is harmonic.
  30. Suppose $f$ is an analytic map from the punctured disk to ${\mathbb C}$. Can you write a power series expansion for $f$? What general form does it have? (McMullen)
  31. What can you say about the growth of the $a_n$'s?
  32. Relate this to radii of convergence.
  33. If $f$ is bounded, what additional things can you say?
  34. How would you compute the integral $\int_0^{\infty}{x^{1\over 2}\over {1+x^2}}\,dx$?(McMullen)
  35. Is the top half of a disk conformally isomorphic to the whole disk? What is the isomoprhism? (McMullen)
  36. What is the argument principle? (Sarason)
  37. Why is it called the argument principle? (Sarason)
  38. Generalize the argument principle to a statement about an arbitrary continuous function $f$ from a domain to ${\mathbb C}$. You can assume $f$ has isolated zeroes. (Sarason)
  39. You don't want to calculate any integrals do you? (Sarason)
  40. How do you prove the uniqueness part of the Riemann Mapping Theorem? (Sarason)
  41. What are the conformal automorphisms of the disk? (Sarason)
  42. What are the conformal automorphisms of the upper half plane? (Sarason)
  43. What is the modular group? (Sarason)
  44. Let $f$ be holomorphic in $\Delta^{*}$, the punctured unit disk, and suppose that $|f(z)| \le {1 \over {\sqrt {|z|}}}$. Show that the singularity at $0$ is removable. (McMullen)
  45. What is the Riemann zeta function? (Poonen)
  46. State the Riemann hypothesis. (Poonen)
  47. What is analytic continuation? Why is it unique?
  48. Does every complex analytic function have a power series?
  49. What is a a holomorphic function? What have properties do they have? (Sarason)
  50. How can you prove they have a Taylor series expansion? (Sarason)
  51. What connection does complex analysis have to algebraic geometry? (Sarason)

Geometry and Topology

Algebraic Topology
  1. Compute the homotopy group $\pi_3(S^2)$.
  2. Which homotopy classes $\alpha\colon \mathbf{P}^2\to \mathbf{P}^2$ are there which is the identity on $\pi_1(\mathbf{P}^2)$, the fundamental group?
  3. What can you say about the homotopy type of the "dunce cap"?
  4. Tell us about the Van Kampen theorem. (Stallings)
  5. Can you use the Van Kampen theorem to compute the fundamental group of the Hawaiian earring? (Kirby)
  6. Is the fundamental group of the Hawaiian earring finite? Free? Countable or Uncountable? (Kirby)
  7. Why do you have the Fundamental Theorem of Algebra under algebraic topology in your syllabus? (Stallings)
  8. How do you know the fundamental group of $S^1$ is $\mathbb{Z}$? (Stallings)
  9. Find all $2$-fold coverings of the figure $8$.
  10. Find an example of: $H_p(X)=H_p(Y)$ for all $p$, but $X$ and $Y$ not homeomorphic. $\pi_p(X)=\pi_p(Y)$ for all $p$, but $H_*(X)\not=H_*(Y)$.
  11. The example to (2) above seems to contradict Whitehead's Theorem. Do you know why it doesn't contradict it?
  12. Compute $H_*(\mathbf{P}_\mathbb{C}^n)$ and $H^*(\mathbf{P}_\mathbb{C}^n)$.
  13. Define Chern classes and compute them for some examples like $\mathbf{P}_\mathbb{C}^n$.
  14. Does "Euler Class" classify all disk bundles over $S^2$?
  15. Does $C_1$, the first Chern class, classify all complex line bundles over $T^2$?
  16. Compute the intersection form from the framed link which represents the $4$-manifold.
  17. What is $\pi_2(S^2\vee S^2)$?
  18. Give an example of two spaces which are not homotopy equivalent, but have the same homology.
  19. What is $\pi_2$ of $S^2\vee S^1$?
  20. Calculate $\pi_1(X)$ where $X$ is the three manifold obtained from $T^2\times I$ by identifying the opposite faces by the glueing map $(1,0)\mapsto (2,1)$, $(0,1)\mapsto (1,1)$.
  21. Show that the free group on two generators contains the free group on $n$ generators with finite index.
  22. Show that every subgroup of a free group is free.
  23. Given an example of a pair $(X,A)$ such that $\pi_i(X,A)\ne \pi_i(X/A)$ for some $i$.
  24. Give an example of two spaces with the same cohomology groups but with a different ring structure.
  25. Show that a compact surface with sectional curvature positive everywhere is homeomorphic to $S^2$.
  26. Calculate the homology with coefficients in $\mathbb{Z}$ of the Lens space $L(a,b)$.
  27. Prove that for any orientable compact $3$-manifold $M$ with boundary $\partial M$ that half the first rational homology of $\partial M$ is killed by inclusion into $M$.
  28. Show that an element of $H_{n-1}$ for an orientable $n$-manifold is represented by a smoothly embedded $n-1$-manifold.
  29. If a simply-connected CW complex $\Sigma$ satisfies $H_2(\Sigma) = \mathbb{Z}\oplus\mathbb{Z}$ and $H_i(\Sigma) = 0$ for all $i\ne 2$, then show that $\Sigma$ is homotopy equivalent to $S^2\vee S^2$.
  30. Show that if $G$ is a finitely generated finitely presented group, then $G$ is the fundamental group of some compact $4$-manifold.
  31. Show that a simply connected differentiable manifold is orientable.
  32. Classify $S^3$ bundles over $S^5$.
  33. Show that any two embeddings of a connected closed set $X$ in $S^2$ has homeomorphic complements $C_1$, $C_2$.
  34. Show that $\mathbf{P}_\mathbb{C}^2$ does not cover any manifold other than itself.
  35. Compute the homology of $\mathbf{P}^n$. (Stallings)
  36. Compute the homology of $\mathbf{P}^n$ with $\mathbb{Z}/2$. (Stallings)
  37. Compute the cohomology of $\mathbf{P}^n$. (Stallings)
  38. Use intersection theory to compute the cup structure of the cohomology of $\mathbf{P}^n$ with $\mathbb{Z}/2$ coefficients where $n$ is odd. (Stallings)
  39. What are all of the $n$-fold covers of the genus $2$ surface.
  40. What is an $H$-space? What special property does $\pi_1$ of an $H$-space have? Prove it. (Casson)
  41. Why can't $S^2$ be an $H$-space? (Stallings)
  42. What is the homology of $S^2\times S^2$? Cohomology? How is the cohomology related to the homology? What is the cup product structure? (Casson)
  43. Suppose that $X$ and $Y$ are simply-connected CW complexes which have the same homology groups. Do they necessarily have the same homotopy groups? Are they necessarily homotopy equivalent? (Givental)
  44. Why does Hurwitz's theorem fail for non-simply connected spaces? Give an example of a space $X$ where the action of $\pi_1(X)$ on the higher homotopy groups is not trivial. (Weinstein)
  45. What is the Thom class? Let $E$ be the universal bundle over $BU(n)$ and consider $F = \mathbf{P}(E \oplus \mathbb{C})$. What is the Thom class of the normal bundle to the zero section in $F$? (Givental)
  46. What is the homology of $\mathbb{R}^4 - S^1$? (Hutchings)
  47. How does Poincare duality show up in Morse theory? (Hutchings)
Differential Topology
  1. What is Sard's Theorem?
  2. Give an application of Sard's Theorem.
  3. Give a smooth map from $S^3$ to $S^3$. Can "most" points have an infinite number of preimages?
  4. Define the Lie bracket of two vector fields on a $\mathcal{C}^{\infty}$ manifold. (Casson)
  5. What does it mean to compose two vector fields? i.e. what does $XY-YX$ mean? (Casson)
  6. Define vector field in terms of the ring ${\cal F}$ of $\mathcal{C}^{\infty}$ functions $M\to\mathbb{R}$. What does it mean to compose two vector fields? (Casson) Is $XY$ necessarily a vector field? Why is $[X,Y]$ a vector field?
  7. When does a vector field determine a flow? (Casson)
  8. What does it mean for a vector field to have compact support? (Casson)
  9. Define flow. (Casson)
  10. In what sense do the diffeomorphisms in a flow vay "in a $\mathcal{C}^{\infty}$ fashion"? (Casson)
  11. Does a flow determine a vector field? (Casson)
  12. Give conditions on $M$ so that every vector field on $M$ determines a flow. (Casson)
  13. Relate "tangent vector to a curve at a point" to "point derivation". (Casson)
  14. Give an example of a vector field on a manifold that does not determine an everywhere-defined flow. (Casson)
  15. A knot is a $\mathcal{C}^{\infty}$ embedding $S^1\to \mathbb{R}^3$. Consider the following two statements about two knots $f,g\colon S^1\to \mathbb{R}^3$: 1) There is an isotopy between $f$ and $g$. 2) There is a diffeomorphism of $\mathbb{R}^3$ inducing a diffeomorphism $f(S^1)\to g(S^1)$. Relate these conditions. (Casson) Can you use any of this information to say something about classifying diffeomorphisms $\mathbb{R}^3\to\mathbb{R}^3$? [Hint]
    The trefoil knot cannot be deformed into its mirror image.
  16. How many components does $\operatorname{Diff}(\mathbb{R}^3,\mathbb{R}^3)$ have, and what is meant by this? (Casson)
  17. State a Lemma about a diffeomorphism $f\colon \mathbb{R}^3\to \mathbb{R}^3$ if $f(0)=0$; in particular, how may $f$ be rewritten? (Casson)
  18. Write down a path from $f$ to $\operatorname{Diff}|_0$, where $f(0)=0$ and $f\colon \mathbb{R}^3\to\mathbb{R}^3$ is a diffeomorphism. (Casson)
  19. Define a Morse function. Define index. (Casson)
  20. What is $h^{-1}(a,b)$ if $(a,b)$ does not contain any critical values? What if it contains exactly one critical value? (Casson)
  21. If a morse function on a manifold $M$ has exactly two critical points, what can you say about $M$? (Casson)
  22. What is the Frobenius Integrability Theorem? (Serganova)
  23. What is an integral submanifold? (Serganova)
  24. What is $[x,y]$? (Serganova)
  25. Can you give an example of a distribution which is not integrable? (Serganova)
  26. Explain how the characteristic classes of a vector bundle arise. What are all the characteristic classes of a vector bundle? (Wodzicki)
  27. What is the Thom isomorphism? (Wodzicki)
  28. What is the Thom class? What is it an obstruction to? (Wodzicki)
  29. Construct the Thom class explicitly for a trivial bundle. (Frenkel)
  30. Prove that the cohomology of a compact Lie group is that of its Lie algebra. (Wodzicki)
  31. How does one use Sard's theorem to prove the Whitney embedding theorem? (Harrision)