Let F̄p be an algebraic closure of a finite field of characteristic p. Let ρ be a continuous homomorphism from the absolute Galois group of Q to GL(3, F̄p) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the Serre conductor of ρ is squarefree, we prove that ρ is attached to a Hecke eigenclass in the homology of an a...

Problems. Problem 1. Let Y be a finite type, separated k-scheme. Let E be a locally free OY -module of rank r + 1. Let πE : PY (E)→ Y, φ : π∗E∨ → O(1) be a universal pair of a morphism to Y together with an invertible quotient of the pullback of E∨ (to help calibrate conventions, this is covariant in E with respect to locally split monomorphisms of locally free sheaves). Recall in the proof of ...

We show that certain canonical realizations of the complexes Hom(G,H) and Hom+(G,H) of (partial) graph homomorphisms studied by Babson and Kozlov are in fact instances of the polyhedral Cayley trick. For G a complete graph, we then characterize when a canonical projection of these complexes is itself again a complex, and exhibit several well-known objects that arise as cells or subcomplexes of ...

(ii) (Lazard’s Theorem) L is the polynomial algebra Z[x1, x2, ...]. We proved Lazard’s theorem last semester. We also proved that MU∗ is isomorphic to Z[x1, x2, ...], so MU∗ and L are both polynomial algebras with generators in even degrees (positive even degrees in MU∗, and negative even degrees in L). The usual complex orientation on MU gives rise to a formal group law, and thus we have a hom...

We defined the Grothendieck groups KX and K0X. They are vector bundles, respectively coherent sheaves, modulo the relation [E] = [E ] + [E ]. We have a pullback on K: f : KX → KY. KX is a ring: [E] · [F] = [E ⊗ F]. We have a pushforward on K0: f∗[F ] = ∑ i≥0(−1) [Rf∗F ]. We obviously have a homomorphism KX → K0X. K0X is aK X-module: KX⊗K0X → X is given by [E] · [F ] = [E⊗ F ]. Unproved fact: If...

Introduction 145 1. Quantum affine algebra 150 2. Quiver variety 155 3. Stratification of M0 163 4. Fixed point subvariety 167 5. Hecke correspondence and induction of quiver varieties 169 6. Equivariant K-theory 174 7. Freeness 178 8. Convolution 185 9. A homomorphism Uq(Lg)→ KGw×C ∗ (Z(w))⊗Z[q,q−1] Q(q) 192 10. Relations (I) 194 11. Relations (II) 202 12. Integral structure 214 13. Standard m...

We prove the following theorem: Let F be an algebraic closure of a finite field of characteristic p. If ρ is a continuous homomorphism from the absolute Galois group of Q to GLn(F) which is isomorphic to a direct sum of one-dimensional representations, and if p > n+1 and the conductors of the one-dimensional representations are pairwise relatively prime, and if a certain parity condition holds,...

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Let Gn,k denote the Kneser graph whose vertices are the n-element subsets of a (2n + k)-element set and whose edges are the disjoint pairs. In this paper we prove that for any non-negative integer s there is no graph homomorphism from G4,2 to G4s+1,2s+1. This confirms a conjecture of Stahl in a special case.

The Hilbert polynomial can be defined axiomatically as a group homomorphism on the Grothendieck group K(X) of a projective variety X, satisfying certain properties. The Chern polynomial can be similarly defined. We therefore define these rather abstract notions to try and find a nice description of this relationship. Introduction Let X = P be a complex projective variety over an algebraically c...