厦门大学数学科学学院：《高等代数》课程教学资源（大学数学竞赛题选）丘成桐大学生数学竞赛2010年试题

S.-T. Yau College Student Mathematics Contests 2010 Analysis and Differential Equations Please select 5 problems to solve) a)Let Ck, k= 1,.., n be real numbers from the interval(0, T) and define x=i. Show that sin k/sinr k=1 From calculate the integral Jo sin( 2)da 2. Let f: R-R be any function. Prove that the set of points z in R where f is continuous is a countable intersection of open sets S3. Consider the equation i=-x+f(t, x),where If(t, =)<o(t)zl for all (t,r)ERXR,o(t)dt oo. Prove that every solution approaches zero as t→o 4. Find a harmonic function f on the right half-plane such that when approaching any point in the positive half of the y-axis, the function has limit 1, while when approaching any point in the negative half of the y-axis, the function has limit -1 Let K(a, y)EC(0, 1]x[0, 1)). For all f E C[0, 1], the space of continuous functions on [0, 1], define a function Tf(r)=/K(a,g)f(y)dy Prove that Tf∈C(0,1]). Moreover={Tfll∫lp≤1} Is precom pact in C(0, 1), i.e. every sequence in S has a converging subsequence, here|f|lsp=sup{f(x川|∈[0,1} 6. Prove the poisson summation formula ∑x+2m)=2∑f()e

S.-T. Yau College Student Mathematics Contests 2010 Geometry and Topology Individual Please select 5 problems to solve) 玉、LetD={(x,y)∈R210o be the closed orientable surface of genus g. Show that if g>l, then 2g is a covering space of > 6. Let M be a smooth 4-dimensional manifold. A symplectic form is a closed 2-form w on M such that w A w is a nowhere vanishing 4-form a) Construct a symplectic form on Ri (b). Show that there are no symplectic forms on S

S.-T. Yau College Student Mathematics Contests 2010 Algebra, Number Theory and Combinatorics Individual (Please select 5 problems to solve) A Let v be a finite dimensional complex vector space. Let A, b be two linear endomorphisms of V satisfying AB-BA= B. Prove that there is a common eigenvector for A and B 2. Let M2(R)be the ring of 2 x 2 matrices with real entries. Its group of multiplicative units is GL2(R), consisting of invertible matrices in M2(R) (a) Find an injective homomorphism from the field C of complex numbers into the ring M2(R) (b) Show that if p1 and 2 are two such homomorphisms, then there exists ag E GL2 (R) such that 2 (a)=go(a)g-l for all E C (c) Let h be an element in GL 2(R)whose characteristic polynomial f(a) is irreducible over R. Let F C M2(R)be the subring generated by h and a· I for all a∈武, where I is the idenity matrix. Show that F is isomorphic to C (d)Let h' be any element in GL2(R) with the same characteristic polynomial f(a) as h in(c). Show that h and h' are conjugate in GL2(R) (e)If f(r)in(c) and(d)is reducible over R, will the same conclu sion on h and h' hold? Give reasons 3. Let G be a non-abelian finite group. Let c(g) be the number of conjugacy classes in G Define C(G):=c(G)/Gl,(IG= Card(G) (a) Prove that(G)≤ b)Is there a finite group H with c(H)=R? (c)(open ended question) Suppose that there exists a prime num- ber p and an element E G such that the cardinality of the conjugacy class of z is divisible by p. Find a good /sharp upper bound for CG) 4. Let F be a splitting field over Q the polynomial z-5 E Q(z]. Recall that f is the subfield of c generated by all roots of this polynomial

2) Find th [F: Q of the number field F b) Determine the Galois group Gal(F/Q) 5. Let T C Nso be a finite set of positive integers. For each integer n>0, define an to be the number of all finite sequences(ti,.,t, with m n, ti E T for all i=1,., m and t1+..+tm= n. Prove that the infinite series ∑anz”∈C刁 is a rational function in z, and find this rational function 6. Describe all the irreducible complex representations of the group S4 (the symmetric group on four letters)