GENERALS 15 OCTOBER 2001 Committee: Prof. Yakov G. SINAI, Chair, Prof. John N. MATHER, Prof. Wee Teck GAN. Topics: Representation Theory, Ergodic Theory. Duration: 1 h 35 min. ALGEBRA (Gan) State the main theorem of Galois theory. What is a normal extension? (Sinai) Why can you solve the 3rd and the 4th but not the 5th? (Gan) Describe extensions of a finite field. Galois group? We get to the Frobenius automorphism. (Gan) What are the characteristic and minimal polynomial of the Frobenius automorphism? (many thanks to Akshay who had suggested the problem previously to me.) Answer: x^n-1, n is the degree of the extension, on assumption that n is not divisible by the characteristic of the field; Gan says it seems to him that the result is true even when n is divisible by the field characteristic but we do not go into that). (Gan) Take X^3+6X+3. Why is it irreducible over Q? Prove the Eisenstein Criterion. What is the Galois group of this polynomial? (Answer: since it has one real root and two complex conjugate roots, the group is symmetric). REAL ANALYSIS. Sinai: Suppose two random variables have a joint distribution density p(x,y). What is the density of the conditional distribution of one wrt the other? (This is a very easy question, but I was so entirely unprepared for it that it took me a while to write the correct answer). Mather: What are differentiability properties of a monotonic function? (a.e. differentiable). Example of a function with derivative a.e. 0 but not constant (the Cantor function). We talked a little about the fact that a continuous mapping exists from the Cantor set to any compact set, which paved the way to (Mather) Can you construct a Peano curve? (Gan) State the Riesz representation theorem. By the way, what about noncompact sets? (Sinai) Yes, could you give an example of a positive linear functional on continuous functions on the real line which is not given by a measure? I could not, and Sinai said that such functionals could be constructed using the Banach limit (as I later found, the construction is given in many FuncAn textbooks). (Sinai) What is weak convergence of measures? (Sinai) Suppose I have a sequence of measures \mu_n, say, on R^N, that converges weakly to some limiting measure \mu. For what closed C in R^N do I have \mu_n(C)\to \mu(C)? This again took me by surprise... I said that it is generally not true neither for open nor for closed sets, but that was, of course, not the answer. Sinai: hint: what about the one-dimensional case? Here I remembered that convergence of distribution functions of \mu_n must take place at all points of continuity of the distribution function of \mu, and we finally arrive to the general answer: The \mu-measure of the boundary of C is 0. (Sinai) Give an example of the use of weak convergence of measures in probability theory. I pause a little, and then come up with the Central Limit Theorem, which then Mather asks to formulate precisely and to explain in detail where the weak convergence of measures comes in. I write the formula (*) P{\frac{\xi_1+\dots+\xi_n}{\sqrt n}}\in C} \to \mu(C), where \mu is Gaussian measure, and C, for example, an open (or closed) set with measure of the boundary 0. (Sinai) Give an example of C and of \xi_n such that this fails. This I failed to do, and Sinai gave such an example: take \xi_n to take values -1,1, and let C be the set of all points of the form k/(sqrt n), k,n integers, n>0. Then the left part of (*) is identically 1 while the right part is identically 0 (and this contradicts nothing since C is dense in R). (Mather) What is the dual of L^p? (Sinai) Given a sequence a_n of real numbers, does there always exists a function whose nth derivative at 0 assumes the value a_n? Why? (here I said I did not know the precise proof and waved hands) COMPLEX ANALYSIS (Mather) What is a holomorphic function? Why can a holomorphic function be represented as a power series? Prove Cauchy formula. (Mather) Suppose you have an operator on the Hilbert space. How can you define a holomorphic function of this operator? (by using the Cauchy formula). (Gan) State the Liouville theorem. Deduce from it the fundamental theorem of algebra. Now a question on the Liouville theorem: (here comes a beautiful question) Let F, G be meromorphic functions such that F^n+G^n=1. Prove that F,G are constants. I am entirely at a loss. Gan: hint: take a mapping to CP^2 given by (1:F:G). Suppose we know that this is a mapping to a compact Riemann surface of genus at least 2. What then? (raise the mapping to the disk, which is the universal cover, and, by the Liouville theorem, we are done) Sinai: Compute the integral \int_R \frac{exp(ixt)}{1+x^2}dx (i.e., the Fourier transform of 1/1+x^2). SPECIAL TOPICS REPRESENTATION THEORY (Sinai) What is a Lie group and a Lie algebra? How much do you know about the group if you know the algebra? Give examples of two different groups with the same algebra. (Gan) What are induced representations? (Gan) Character orthogonality relations for finite groups: state and prove. What is the character of a permutation representation? (Gan) What are irreducible representations of a compact Lie group parametrized by? (The theorem on maximal weights; I formulated it in detail, introducing all definitions; this probably took at least 10 minutes). (Gan) How to integrate a class function on a Lie group? (The Weyl Integration Formula). (Gan) What form does the formula take for SU(2)? ERGODIC THEORY This was the shortest part of the exam, with two questions only: (Sinai) Let A_n be a sequence of subsets of $Z$; take the ergodic averages over A_n, that is, take \frac 1{#A_n} \sum_{k\in A_n}f\circ T^k Under what conditions does the Ergodic Theorem hold for such averages? (answer: when A_n is a Foelner sequence, convergence in the mean holds and a.e. convergence holds on a subsequence). (Sinai) When can a diffeomorphism of a circle be reduced to a rotation? (when the rotation number is irrational and the diffeomorphism is sufficiently smooth). After this I was asked to step out and shortly after was told that I passed. Sinai: "Congratulations. Now, there is a problem, I wonder if you'd be interested..." COMMENTS, SUGGESTIONS, AND ACKNOWLEDGEMENTS The Committee was unbelievably friendly and emotionally the exam was much more relaxed than I had feared it would be. Consider bringing a bottle of water at the exam. I have read this suggestion somewhere on the webpage, and am very glad to have followed it. I am deeply grateful to Aravind ASOK, Mirela CIPERJANI, Christopher HALL, Joachim KRIEGER, and Akshay VENKATESH, without whose generous help I would surely have failed the exam.