UCB博士资格考试试题

 

https://math.berkeley.edu/~myzhang/qual.html?tdsourcetag=s_pcqq_aiomsg

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<h3 style="text-align: center">Qualifying Exams</h3>
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The following is a collection of past qualifying exam questions at UC Berkeley, to serve as practices for graduate students preparing their exams.
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<h4 class="sectionTitle">Algebra</h4>
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<h5 class="sectionTitle">General Algebra</h5>
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<ol class="subject">
<li class="question">
How does one make a subposet into a poset? (Bergman)
</li>
<li class="question">
State and prove the Fundamental Theorem of Finite Distributive Lattices. (Klass)
</li>
<li class="question">
Define and state things about posets. (Klass)
</li>
<li class="question">
Draw a non-lattice with 5 elements. Draw a lattice with 5 elements. Draw a Boolean lattice on 3 elements. (Lam)
</li>
<li class="question">
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?
</li>
<li class="question">
State the fundamental theorem of Galois theory.
</li>
</ol>
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<h5 class="sectionTitle">Combinatorics</h5>
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<ol class="subject">
<li class="question">
State and prove Ramsey's Theorem, Hall's Theorem, Speiner's Theorem, Erd&oumls-Ko-Rado Theorem, and the Erd&oumls lower bound on Ramsey. (Karp)
</li>
<li class="question">
State Lovas local theorem and Alan's Theorem. (Karp)
What is $R_1(n)$? (Karp)
</li>
<li class="question">
When is the Speiner bound tight? (Karp)
</li>
<li class="question">
What is the net work flow problem? (Sinclair)
</li>
<li class="question">
What is an algorithm to solve the problem? How do we know it terminates and what is a bound on the running time? (Sinclair)
</li>
<li class="question">
What is an algorithm with a better bound? (Sinclair)
</li>
<li class="question">
How can we use the algorithm to find a minimum cut? (Sinclair)
</li>
<li class="question">
What is a randomized algorithm for finding a minimal cut? (Sinclair)
</li>
<li class="question">
What is a bound on the error probability? (Sinclair)
</li>
<li class="question">
What does this tell us about how many minimal cuts there can be in a 1-graph? (Sinclair)
</li>
<li class="question">
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)
</li>
<li class="question">
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)
</li>
<li class="question">
Talk about the Incidence Algebra on a poset. (Klass)
</li>
<li class="question">
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)
</li>
<li class="question">
When is a function in the Incidence Algebra invertible? Prove it. (Bergman)
</li>
<li class="question">
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)
</li>
<li class="question">
Prove that a finite meet-semilattice with 1 is a lattice. (Bergman)
</li>
<li class="question">
Is an infinite meet-semilattice with 1 necessarily a lattice? If not, find a counterexample. (Bergman)
<li class="question">
Prove that in a finite poset with a unique maximal element, that element is 1; find a counterexample in the infinite case. (Bergman)
</li>
<li class="question">
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))
</li>
<li class="question">
State the finite field method for the characteristic polynomial of a hyperplane arrangement. Sketch a proof. (Ardila)
</li>
<li class="question">
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)
</li>
<li class="question">
What is the Lagrange inversion formula? (Serganova)
</li>
<li class="question">
How do you use this formula? (Serganova)
</li>
<li class="question">
What is the proof? (Sturmfels)
</li>
<li class="question">
What is a Schur polynomial? (Serganova)
</li>
<li class="question">
How do you show that it is symmetric? (Serganova)
</li>
<li class="question">
What is the definition of the Schur polynomial using determinants? (Serganova)
</li>
<li class="question">
Why is the Schur polynomial important? (Serganova)
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<div class="hint">
Inner products.
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</li>
<li class="question">
Can you use Schur polynomials in computational biology? (Sturmfels)
</li>
</ol>
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<h5 class="sectionTitle">Commutative Algebra</h5>
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<ol class="subject">
<li class="question">
What is Spec $\mathbb{C}[[x]]$?
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$\mathbb{C}[[x]]$ is a DVR.
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</li>
<li class="question">
State the Noether normalization theorem.
</li>
<li class="question"> 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)
</li>
<li class="question"> What is $\mathbb{Z}^*_{q-1}$?
</li>
<li class="question"> Give an example of a commutative ring with unity which has
prime ideals which are not maximal.
</li>
<li class="question"> Give two examples of UFD's which are not PID's.
</li>
<li class="question"> 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 $q(x)\in F[x]$.
</li>
<li class="question"> Show that an element $a$ is irreducible if and only if
$(a)$ is maximal, where $a\in R$ and $R$ is a PID.
</li>
<li class="question"> 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?
</li>
<li class="question"> Is there a general class of rings in which maximal ideals
and prime ideals are the same?
</li>
<li class="question"> 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.
</li>
<li class="question"> Show that in a commutative ring with unit every maximal
ideal must be prime.
</li>
<li class="question"> Describe the ring of endomorphisms of the integers.
</li>
<li class="question"> If $R$ is a UFD, prove that $R[X]$ is a UFD.
</li>
<li class="question"> 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.
</li>
<li class="question"> Let $R$ be the ring of real quaternions. Does $R[X]$
satisfy the division algorithm property?
</li>
<li class="question"> Distinguish algebraically between $\operatorname{GL}(3,\mathbb{R})$ and
$\operatorname{GL}(2,\mathbb{R})$, not using topology.
</li>
<li class="question"> What is a Dedekind ring?
</li>
<li class="question"> Prove that $k[X,Y]$ is not Dedekind.
</li>
<li class="question"> 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?
</li>
<li class="question"> Can you give an example of a ring $R$ which is not
Cohen-Macauley? (Ogus)
</li>
<li class="question"> Can you give an example of a ring $R$ which is
Cohen-Macauley but not Gorenstein? (Ogus)
</li>
<li class="question"> 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)
</li>
<li class="question"> Given $R$ a domain, Noetherian, dimension 1, what can we
say about $\tilde{R}$, its integral closure? (Lenstra)
</li>
<li class="question"> Let's prove that in the above case, $\tilde{R}$ is
Noetherian. (Lenstra)
</li>
<li class="question"> Define "Hilbert function". (Sturmfels)
</li>
<li class="question"> 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)
</li>
<li class="question"> What sorts of invariants appear in the Hilbert polynomial?
(Sturmfels)
</li>
<li class="question"> Given an ideal $I$, how would one compute its Hilbert
function? (Sturmfels)
</li>
<li class="question"> Define Gr&ouml;bner basis, term order and initial
ideal. (Sturmfels)
</li>
<li class="question"> 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)
</li>
<li class="question"> Give an example of a tegrm order that refines the partial
order by degree. (Casson)
</li>
<li class="question"> 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)
</li>
<li class="question"> 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 Gr&ouml;bner basis for $I(X)$ but all three are.
</li>
<li class="question">
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 $X$.
</li>
<li class="question"> Tell me about integral extensions. (Hartshorne)
</li>
<li class="question"> 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)
</li>
<li class="question"> 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)
</li>
<li class="question"> Can you give an example of a surjective morphism of rings
which is not finite? (Hartshorne)
</li><li class="question"> State Nakayama's Lemma. Give an example of a ring and a nonzero ideal that
satisfy the hypothesis. (Bergman)
</li><li class="question"> 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.
</li>
<li class="question"> Show that a PID has dimension $0$ or $1$.
</li>
<li class="question"> Let $A$ be a noetherian valuation ring which is not a
field. Show that $A$ is a DVR.
</li>
<li class="question"> 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 $$\prod_{i=1}^r I_i = \bigcap_{i=1}^r I_i.$$
</li>
<li class="question"> 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?
</li>
<li class="question"> Let $M$ be an $A$-module, $m$ a maximal ideal
of $A$. Prove that $M/mM\cong M_m/mM_m$.
</li>
<li class="question"> Let $M$ be a finitely generated $A$-module. If $M=mM$
for every maximal ideal ${\bf m}$ of $A$ show that $M=0$.
</li>
<li class="question"> 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 $M$.
</li>
<li class="question">Let $B\supset A$ be commutative rings, with $B$ integral
over $A$. Let $x\in A$, $x\in B^*$. Show that $x\in A^*$.
</li>
<li class="question"> Prove that any ideal in a Dedekind ring is generated by at
most two elements.
</li>
<li class="question"> Show that $\mathbb{Z}_p$ is a DVR.
</li>
<li class="question"> Show that an Artinian ring has only finitely many maximal
ideals.
</li>
<li class="question"> 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 $A$.
</li>
<li class="question"> Show that if $f\colon A\to A$ is a surjective endomorphism
of a noetherian ring $A$, then $f$ is an automorphism.
</li>
<li class="question"> Give an example of an ideal in $k[x,y,z]$ whose associated
primes are $(x,y)$ and $(x,y,z)$. (Eisenbud)
</li>
<li class="question"> Give an example of a surface that has only one singular
point but is not normal. (Eisenbud)
</li>
<li class="question"> 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)
</li>
<li class="question"> How do you decide if a given polynomial belongs to an
ideal $I$? (Sturmfels)
</li>
<li class="question"> State the main theorem of Elimination Theory. Why is it
called this? (Sturmfels)
</li>
<li class="question"> 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)
</li>
<li class="question"> Give an example of an injective ring homomorphism which is not flat. (Eisenbud)
</li>
<li class="question">
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)
</li>
<li class="question">
Try computing a Gr&ouml;bner 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)
</li>
<li class="question">
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)
</li>
<li class="question"> 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)
</li>
<li class="question"> Compute a primary decomposition of $(x+y,x-y)$ over $\mathbb{Z}[x,y]$. (Sturmfels)
</li>
<li class="question"> State the Nullstellensatz. (Sturmfels)</li>
<li class="question"> Do you know a ring which isn't Jacobsen? (Sturmfels)</li>
<li class="question"> Prove that the more geometric versions of the Nullstellensatz follow from the general one. (Sturmfels)
</li>
<li class="question"> If $f\in R$, what familiar ring is $R[y]/(fy-1)$? (Eisenbud)
</li>
<li class="question"> Let $I=(x^2-y^2,x^3-y^3)$ . Find a primary decomposition of $I$, compute a Gr&ouml;bner basis, compute the Hilbert function. Is this ideal Cohen Macaulay? (Sturmfels)
</li>
</ol>
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<h5 class="sectionTitle">Algebraic Geometry</h5>
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<ol class="subject">
<li class="question"> State and prove the Riemann-Hurwitz formula.
<span class="hintButton" onclick="toggleSibling(this)">[Hint]</span>
<div class="hint">
Consider the conormal sequence.
</div>
</li>
<li class="question">
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)
</li>
<li class="question">
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)
</li>
<li class="question">
Calculate the Picard group of $k[t^2,t^3]\subset k[t]$. (Ogus)
</li>
<li class="question">
Give an example of a projective curve that is not rational.
</li>
<li class="question">
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 $\mathbf{P}^1\times\mathbf{P}^1\subset \mathbf{P}^3$.
</li>
<li class="question">
How do you use Hurwitz's formula to calculate the geneus of a give curve? (Coleman)
</li>
<li class="question">
What can you say about curves over perfect fields? (Coleman)
</li>
<li class="question">
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)
</li>
<li class="question">
What does the degree of a hypersurface have to do with the line bundles on $\mathbf{P}^1$? (Ogus)
</li>
<li class="question">
Let $X$ be the twisted cubic in $\mathbf{P}^3$, is $X$ a set-theoretical intersection of two surfaces in $\mathbf{P}^3$? (Ogus)
</li>
<li class="question">
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)
</li>
<li class="question">
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?
</li>
<li class="question">
Define differentials. Are differentials quasicoherent? (Ogus)
</li>
<li class="question">
What does the going up theorem mean in algebraic geometry? (Hartshorne)
</li>
<li class="question">
What can you say about the dimension of the image of a map from $\mathbf{P}^n$ to $\mathbf{P}^m$? (Hartshorne)
</li>
<li class="question">
What is the genus of a curve? Does the genus of a curve depend on the embedding? (Hartshorne)
</li>
<li class="question">
When is a canonical divisor very ample? (Wodzicki)
</li>
<li class="question">
State Riemann-Roch. (Wodzicki)
</li>
<li class="question">
Compute the dimension of the space of holomorphic differentials on a Riemann surface of genus $g$. (Wodzicki)
</li>
<li class="question">
State Abel's theorem. (Wodzicki)
</li>
<li class="question">
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)
</li>
<li class="question">
What is the connection between $H^1$ and line bundles? (Wodzicki)
</li>
<li class="question">
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)
</li>
<li class="question">
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)
</li>
<li class="question">
Calculate $H^0(\mathbf{P}^1,\Omega_{\mathbf{P}^1})$. (Poonen)
</li>
<li class="question">
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?
</li>
<li class="question">
Give two criteria for a curve to be nonsingular (over an algebraically closed field.) (Ogus)
</li>
<li class="question">
What is a normal domain? How is this related to regular local rings? (Ogus)
</li>
<li class="question">
Find the singularities of the curve in $\mathbf{P}^2$ defined by the equation $X^3+y^3+z^3 = 3cxyz$. (Ogus)
</li>
<li class="question">
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)
</li>
<li class="question">
What is the degree of a divisor? (Ogus)
</li>
<li class="question">
Does there exist a variety $V$ with Pic$(V) =\mathbb{Z}/3$? (Poonen)
</li>
<li class="question">
Is the complement of a hypersurface in $\mathbf{P}^2$ affine? (Poonen)
</li>
<li class="question">
Define the geometric genus. (Poonen)
</li>
<li class="question">
What might be the geometric genus of a singular curve? (Poonen)
</li>
<li class="question">
Find the arithmetic genus of $y^3 = x^2z$. (Frenkel)
</li>
<li class="question">
Define sheaf cohomology. What's a right derived functor? (Olsson)
</li>
<li class="question">
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)
</li>
<li class="question">
Define projective morphisms and what are they good for? What's a morphism that is not projective? (Eisenbud)
</li>
<li class="question">
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)
</li>
<li class="question">
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)
</li>
<li class="question">
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)
</li>
</ol>
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<h5 class="sectionTitle">Representation Theory</h5>
</div>
<ol class="subject">
<li class="question">
Prove Engel's theorem. (Serganova)
</li>
<li class="question">
Prove Lie's theorem. What would happen if the hypothesis was not that $g$ is solvable but that $g \ne [g,g]$? (Serganova)
</li>
<li class="question">
Why is $g = [g,g]$ for $g$ semisimple? (Weinstein)
</li>
<li class="question">
What is the exponential map, and what is it good for? (Serganova)
</li>
<li class="question">
Classify the real connected abelian Lie groups. (Serganova)
</li>
<li class="question">
Prove that a Lie group homomorphism $\phi\colon H \to G$ for $H$ connected is determined by the derivative at the identity. (Serganova)
</li>
<li class="question">
Give an example of a Lie group G where the exponential map is not surjective. (Weinstein)
</li>
<li class="question">
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)
</li>
<li class="question">
Decompose $\operatorname{Sym}_n(V)\otimes \operatorname{Sym}_m(V)$ where $V$ is the 2-dimensional irreducible representation of $sl_2({\mathbb C})$. (Frenkel)
</li>
<li class="question">
Do the calculation above using a character formula. (Reshetikhin)
</li>
<li class="question">
State and explain the Harish-Chandra isomorphism. (Wodzicki)
</li>
<li class="question">
Explain how to write down the Weyl group of $SL_n$ using generators and relations. (Frenkel)
</li>
<li class="question">
What is a Verma module? (Reshetikhin)
</li>
<li class="question">
When is a Verma module finite-dimensional? (Wodzicki)
</li>
<li class="question">
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)
</li>
<li class="question">
What is Weyl's Integration formula? How do you use it to prove Weyl character formula? (Reshetikhin)
</li>
<li class="question">
What is the dimension of $E_8$? (Borcherds)
</li>
<li class="question">
Decompose $E_8$ as a representation of $E_7$. (Borcherds)
</li>
<li class="question">
Given a point in a semisimple Lie-algebra, how can we tell whether it lies in a Cartan subalgebra? (Knutsen)
</li>
<li class="question">
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)
</li>
<li class="question">
Under what condition on $G$ is every discrete normal subgroup of a Lie group $G$ contained in the center of $G$? Prove this. (Reshetikhin)
</li>
<li class="question">
List all the irreducible complex representations of the Lie group $SO(3,\mathbb{R})$. (Knutsen)
</li>
<li class="question">
Decompose the square of the adjoint representation of $sl(3)$ into irreducibles. (Hint: Weyl Character Formula.) (Haiman)
</li>
<li class="question">
State a theorem that explains the basic relationship between Lie algebras and Lie groups. Say some words about the proof. (Haiman)
</li>
<li class="question">
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)
</li>
<li class="question">
How many Lie groups are there with Lie algebra $sl_n$? (Serganova)
</li>
<li class="question">
What can you tell me about Bruhat decomposition and Bruhat cells? (Reshetikhin)
</li>
<li class="question">
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)
</li>
<li class="question">
Suppose a finite group $G$ has only 1-dimensional representations. Is $G$ necessarily abelian (over $\mathbb{C}$, over $\mathbb{R}$)? (Serganova)
</li>
<li class="question">
What can you say about the multiplicities of irreducibles in an induced representation? (Reshetikhin)
</li>
<li class="question">
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)
</li>
</ol>
</div>

<!-- Analysis -->
<div class="expander" onclick="toggleSibling(this);flipArrow(this.childNodes[1])">
<img src="img/arrow-down.png" class="arrow" />
<h4 class="sectionTitle">Mathematical Analysis</h4>
</div>
<div class="area">
<div class="expander" onclick="toggleSibling(this);flipArrow(this.childNodes[1])">
<img src="img/doublearrow-down.png" class="arrow doublearrow" />
<h5 class="sectionTitle">Banach spaces and Spectral Theory</h5>
</div>
<ol class="subject">
<li class="question"> What is a nuclear operator? (Coleman)
</li><li class="question"> Give an example of an integral operator which is
nuclear. (Coleman)
</li><li class="question"> What can you say about the specturm of a nuclear operator?
(Coleman)
Could it be the empty set? (Arveson)
</li><li class="question"> Give an example of an operator on a real Banach space with
no specturm. (Arveson)
</li><li class="question"> Does the sum of the elements of the spectrum of a nuclear
operator converge? (Coleman)
</li><li class="question"> What is a trace class operator? (Coleman)
</li><li class="question"> What is a Hilbert-Schmidt operator? Can you give an
example over $\mathcal{ L}^2$ of the unit interval? (Coleman)
</li><li class="question"> Can $[0,1]$ be the spectrum of a compact operator? (
Arveson)
</li><li class="question"> What is the spectrum of $M_{e^{2\pi it}}$? How could you
know that it is invertible? What is the inverse? (Arveson)
</li><li class="question"> If $T$ is an operator on a Banach space, what is $\cos^2T + \sin^2T$? (Arveson)
</li><li class="question"> What is $\cos T$ (Arveson)
</li><li class="question"> If $f$ is an entire function, what is $fT$? (
Arveson)
</li><li class="question"> List the properties of the functional calculus. (
Arveson)
</li><li class="question"> Consider $\mathcal{ C}\bigl([0,1], \mathbb{R}\bigr)$. Is there a
natural topology on this space? (Arveson)
</li><li class="question"> 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)
</li><li class="question"> Let $S_0=\{f\in S\,|\, f(0)=0\}$. What properties does it
have (e.g. closed, complete, bounded, compact) ? (Arveson)
</li><li class="question"> What is the Riesz theory of compact operators?
</li><li class="question"> 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?
</li><li class="question"> Suppose you have an operator $x$ on a Hilbert space such that
$x - x^2$ is compact. What can you tell me about it?
</li><li class="question"> 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?
</li><li class="question"> What is the polar decomposition? What can you say about it?
</li>
</ol>
<div class="expander" onclick="toggleSibling(this);flipArrow(this.childNodes[1])">
<img src="img/doublearrow-down.png" class="arrow doublearrow" />
<h5 class="sectionTitle">$C^*$ and Von Neumann Algebras</h5>
</div>
<ol class="subject">
<li class="question"> What are Fredholm operators?
</li><li class="question"> What do they have to do with $K$-theory for operator
algebras?
</li><li class="question"> Could you give some examples of interesting $C^*$-algebras
with nontrivial $K$-theory?
</li><li class="question"> How does one recognize a compact operator? Give examples.
</li><li class="question"> Prove that the Hilbert-Schmidt integral operators are
compact.
</li><li class="question"> 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?
</li><li class="question"> State Kaplansky's Density Theorem. ( Jones)
</li><li class="question"> What is it good for? (e.g. in $L^\infty(S^1)$) ( Jones)
</li><li class="question"> 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)
</li><li class="question"> Define the index of a subfactor. ( Jones)
</li><li class="question"> What are all the hyperfinite subfactors of index $< 4$?
( Jones)
</li><li class="question"> Let $S$ be the unilateral shift. What is the commutant of
$C^*(S^2)$?
</li><li class="question"> Do the Hilbert-Schmidt and trace class operators constitute
$C^*$ algebras under the Hilbert-Schmidt and trace norms, respectively?
</li>
</ol>
<div class="expander" onclick="toggleSibling(this);flipArrow(this.childNodes[1])">
<img src="img/doublearrow-down.png" class="arrow doublearrow" />
<h5 class="sectionTitle">Complex Analysis</h5>
</div>
<ol class="subject">
<li class="question">
Given a function continuous in a disk and analytic everwhere but at the center, prove that the function is analytic in the entire disk.
</li>
<li class="question">
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.
</li>
<li class="question">
Show that the mapping group of the torus is $SL(2,\mathbb{Z})$.
</li>
<li class="question">
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)
</li>
<li class="question">
Is there one with $f(0)=0$? (McMullen)
</li>
<li class="question">
How can we arrange for a unique $f$ with $f(0)=0$? (McMullen)
</li>
<li class="question">
What can you say about the coefficients $a_i$ of the power series expansion $f=\sum a_jz^j$? (McMullen)
</li>
<li class="question">
For $f$ with $f'(0)\in \mathbb{R}$, $f'(0)\geq 0$, what ring do the $a_i$ lie in? (McMullen)
</li>
<li class="question">
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)
</li>
<li class="question">
Calculate $a_i$. Now what ring do the $a_i$ lie in? (McMullen)
</li>
<li class="question">
How would you write down the power series for $\tan z$? (McMullen)
</li>
<li class="question">
What is its radius of convergence? (McMullen)
</li>
<li class="question">
Can you prove what the zeroes of $\cos z$ are? (McMullen)
</li>
<li class="question">
Why does the radius of convergence correspond this way? (McMullen)
</li>
<li class="question">
What is the area of a spherical triangle? Can you prove it? (McMullen)
</li>
<li class="question">
Same for hyperbolic triangle. (McMullen)
</li>
<li class="question">
Define a complex torus. (McMullen)
</li>
<li class="question">
What is the automorphism group of a complex torus? (McMullen)
</li>
<li class="question">
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.
</li>
<li class="question">
What is the area of a spherical triangle?
</li>
<li class="question">
What is the automorphism group of a complex torus?
</li>
<li class="question">
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 $f_n'$?
</li>
<li class="question">
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?
</li>
<li class="question">
Why is there no conformal automorphism from the punctured disk to an annulus?
</li>
<li class="question">
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.
</li>
<li class="question">
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 $D$.
</li>
<li class="question">
Find the poles and residues of $1/\sin(z)$.
</li>
<li class="question">
Give the formula for a conformal map from the unit disk to the inside of a polygon with angles $2\pi-\beta_i\pi$.
</li>
<li class="question">
Show that a continuous real-valued function $u$ on some region $\Omega$ which has the mean-value property is harmonic.
</li>
<li class="question">
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)
</li>
<li class="question">
What can you say about the growth of the $a_n$'s?
</li>
<li class="question">
Relate this to radii of convergence.
</li>
<li class="question">
If $f$ is bounded, what additional things can you say?
</li>
<li class="question">
How would you compute the integral $\int_0^{\infty}{x^{1\over 2}\over {1+x^2}}\,dx$?(McMullen)
</li>
<li class="question">
Is the top half of a disk conformally isomorphic to the whole disk? What is the isomoprhism? (McMullen)
</li>
<li class="question">
What is the argument principle? (Sarason)
</li>
<li class="question">
Why is it called the argument principle? (Sarason)
</li>
<li class="question">
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)
</li>
<li class="question">
You don't want to calculate any integrals do you? (Sarason)
</li>
<li class="question">
How do you prove the uniqueness part of the Riemann Mapping Theorem? (Sarason)
</li>
<li class="question">
What are the conformal automorphisms of the disk? (Sarason)
</li>
<li class="question">
What are the conformal automorphisms of the upper half plane? (Sarason)
</li>
<li class="question">
What is the modular group? (Sarason)
</li>
<li class="question">
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)
</li>
<li class="question">
What is the Riemann zeta function? (Poonen)
</li>
<li class="question">
State the Riemann hypothesis. (Poonen)
</li>
<li class="question">
What is analytic continuation? Why is it unique?
</li>
<li class="question">
Does every complex analytic function have a power series?
</li>
<li class="question">
What is a a holomorphic function? What have properties do they have? (Sarason)
</li>
<li class="question">
How can you prove they have a Taylor series expansion? (Sarason)
</li>
<li class="question">
What connection does complex analysis have to algebraic geometry? (Sarason)
</li>
</ol>
</div>

<!-- Geometry and Topology -->
<div class="expander" onclick="toggleSibling(this);flipArrow(this.childNodes[1])">
<img src="img/arrow-down.png" class="arrow" />
<h4 class="sectionTitle">Geometry and Topology</h4>
</div>
<div class="area">
<div class="expander" onclick="toggleSibling(this);flipArrow(this.childNodes[1])">
<img src="img/doublearrow-down.png" class="arrow doublearrow" />
<h5 class="sectionTitle">Algebraic Topology</h5>
</div>
<ol class="subject">
<li class="question"> Compute the homotopy group $\pi_3(S^2)$.
</li>
<li class="question"> 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?
</li>
<li class="question"> What can you say about the homotopy type of the "dunce
cap"?
</li>
<li class="question"> Tell us about the Van Kampen theorem. (Stallings)
</li>
<li class="question"> Can you use the Van Kampen theorem to compute the
fundamental group of the Hawaiian earring? (Kirby)
</li>
<li class="question"> Is the fundamental group of the Hawaiian earring finite?
Free? Countable or Uncountable? (Kirby)
</li>
<li class="question"> Why do you have the Fundamental Theorem of Algebra under
algebraic topology in your syllabus? (Stallings)
</li>
<li class="question"> How do you know the fundamental group of $S^1$ is $\mathbb{Z}$?
(Stallings)
</li>
<li class="question"> Find all $2$-fold coverings of the figure $8$.
</li>
<li class="question"> 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
$H_*(X)\not=H_*(Y)$.
</li>
<li class="question"> The example to (2) above seems to contradict Whitehead's
Theorem. Do you know why it doesn't contradict it?
</li>
<li class="question"> Compute $H_*(\mathbf{P}_\mathbb{C}^n)$ and $H^*(\mathbf{P}_\mathbb{C}^n)$.
</li>
<li class="question"> Define Chern classes and compute them for some examples
like $\mathbf{P}_\mathbb{C}^n$.
</li>
<li class="question"> Does "Euler Class" classify all disk bundles over $S^2$?
</li>
<li class="question"> Does $C_1$, the first Chern class, classify all complex
line bundles over $T^2$?
</li>
<li class="question"> Compute the intersection form from the framed link which
represents the $4$-manifold.
</li>
<li class="question"> What is $\pi_2(S^2\vee S^2)$?
</li>
<li class="question"> Give an example of two spaces which are not homotopy
equivalent, but have the same homology.
</li>
<li class="question"> What is $\pi_2$ of $S^2\vee S^1$?
</li>
<li class="question"> 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 $(1,0)\mapsto (2,1)$, $(0,1)\mapsto (1,1)$.
</li>
<li class="question"> Show that the free group on two generators contains the
free group on $n$ generators with finite index.
</li>
<li class="question"> Show that every subgroup of a free group is free.
</li>
<li class="question"> Given an example of a pair $(X,A)$ such that
$\pi_i(X,A)\ne \pi_i(X/A)$ for some $i$.
</li>
<li class="question"> Give an example of two spaces with the same cohomology
groups but with a different ring structure.
</li>
<li class="question"> Show that a compact surface with sectional curvature
positive everywhere is homeomorphic to $S^2$.
</li>
<li class="question"> Calculate the homology with coefficients in $\mathbb{Z}$ of the
Lens space $L(a,b)$.
</li>
<li class="question"> 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 $M$.
</li>
<li class="question"> Show that an element of $H_{n-1}$ for an orientable
$n$-manifold is represented by a smoothly embedded $n-1$-manifold.
</li>
<li class="question"> 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 $S^2\vee S^2$.
</li>
<li class="question"> Show that if $G$ is a finitely generated finitely
presented group, then $G$ is the fundamental group of some compact
$4$-manifold.
</li>
<li class="question"> Show that a simply connected differentiable manifold is
orientable.
</li>
<li class="question"> Classify $S^3$ bundles over $S^5$.
</li>
<li class="question"> Show that any two embeddings of a connected closed set
$X$ in $S^2$ has homeomorphic complements $C_1$, $C_2$.
</li>
<li class="question"> Show that $\mathbf{P}_\mathbb{C}^2$ does not cover any manifold other than
itself.
</li>
<li class="question"> Compute the homology of $\mathbf{P}^n$. (Stallings)
</li>
<li class="question"> Compute the homology of $\mathbf{P}^n$ with $\mathbb{Z}/2$. (Stallings)
</li>
<li class="question"> Compute the cohomology of $\mathbf{P}^n$. (Stallings)
</li>
<li class="question"> 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)
</li>
<li class="question"> What are all of the $n$-fold covers of the genus $2$
surface.
</li>
<li class="question"> What is an $H$-space? What special property does $\pi_1$
of an $H$-space have? Prove it. (Casson)
</li>
<li class="question"> Why can't $S^2$ be an $H$-space? (Stallings)
</li>
<li class="question"> 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)
</li>
<li class="question"> 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)
</li>
<li class="question"> 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)
</li>
<li class="question"> 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)
</li>
<li class="question">
What is the homology of $\mathbb{R}^4 - S^1$? (Hutchings)
</li>
<li class="question">
How does Poincare duality show up in Morse theory? (Hutchings)
</li>
</ol>
<div class="expander" onclick="toggleSibling(this);flipArrow(this.childNodes[1])">
<img src="img/doublearrow-down.png" class="arrow doublearrow" />
<h5 class="sectionTitle">Differential Topology</h5>
</div>
<ol class="subject">
<li class="question"> What is Sard's Theorem?
</li><li class="question"> Give an application of Sard's Theorem.
</li><li class="question"> Give a smooth map from $S^3$ to $S^3$. Can "most" points
have an infinite number of preimages?
</li><li class="question"> Define the Lie bracket of two vector fields on a
$\mathcal{C}^{\infty}$ manifold. (Casson)
</li><li class="question"> What does it mean to compose two vector fields? i.e.
what does $XY-YX$ mean? (Casson)
</li><li class="question"> 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?
</li><li class="question"> When does a vector field determine a flow? (Casson)
</li><li class="question"> What does it mean for a vector field to have compact
support? (Casson)
</li><li class="question"> Define flow. (Casson)
</li><li class="question"> In what sense do the diffeomorphisms in a flow vay "in a
$\mathcal{C}^{\infty}$ fashion"? (Casson)
</li><li class="question"> Does a flow determine a vector field? (Casson)
</li><li class="question"> Give conditions on $M$ so that every vector field on $M$
determines a flow. (Casson)
</li><li class="question"> Relate "tangent vector to a curve at a point" to "point
derivation". (Casson)
</li><li class="question"> Give an example of a vector field on a manifold that does
not determine an everywhere-defined flow. (Casson)
</li><li class="question"> 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 $\mathbb{R}^3\to\mathbb{R}^3$?
<span class="hintButton" onclick="toggleSibling(this)">[Hint]</span>
<div class="hint">
The trefoil knot cannot be deformed into its mirror image.
</div>
</li><li class="question"> How many components does $\operatorname{Diff}(\mathbb{R}^3,\mathbb{R}^3)$ have, and
what is meant by this? (Casson)
</li><li class="question"> 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)
</li><li class="question"> 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)
</li><li class="question"> Define a Morse function. Define index. (Casson)
</li><li class="question"> What is
$h^{-1}(a,b)$ if $(a,b)$ does not contain any critical values? What
if it contains exactly one critical value? (Casson)
</li><li class="question"> If a morse function on a manifold $M$ has exactly two
critical points, what can you say about $M$? (Casson)
</li><li class="question"> What is the Frobenius Integrability Theorem? (Serganova)
</li><li class="question"> What is an integral submanifold? (Serganova)
</li><li class="question"> What is $[x,y]$? (Serganova)
</li><li class="question"> Can you give an example of a distribution which is not integrable? (Serganova)
</li><li class="question"> Explain how the characteristic classes of a vector bundle arise. What are all the characteristic classes of a vector bundle?
(Wodzicki)
</li><li class="question"> What is the Thom isomorphism? (Wodzicki)
</li><li class="question"> What is the Thom class? What is it an obstruction to?
(Wodzicki)
</li><li class="question"> Construct the Thom class explicitly for a trivial bundle.
(Frenkel)
</li><li class="question"> Prove that the cohomology of a compact Lie group is that of its Lie algebra. (Wodzicki)
</li><li class="question"> How does one use Sard's theorem to prove the Whitney embedding theorem? (Harrision)