Geometric structure in smooth dual and local Langlands conjecture

This expository paper first reviews some basic facts about p-adic fields, reductive p-adic groups, and the local Langlands conjecture. If G is a reductive p-adic group, then the smooth dual of G is the set of equivalence classes of smooth irreducible representations of G. The representations are on vector spaces over the complex numbers. In a canonical way, the smooth dual is the disjoint union of subsets known as the Bernstein components. According to a conjecture due to ABPS (Aubert–Baum–Plymen–Solleveld), each Bernstein component has a geometric structure given by an appropriate extended quotient. The paper states this ABPS conjecture and then indicates evidence for the conjecture, and its connection to the local Langlands conjecture.     

关于自守型式的Langlands函子性

本文系该著者有关自守型式的Langlands函子性的演讲稿的细述及扩充文稿.特别强调了与该著者的近期工作相关的课题.     

The coarse geometric Novikov conjecture and uniform convexity

The coarse geometric Novikov conjecture provides an algorithm to determine when the higher index of an elliptic operator on a noncompact space is nonzero. The purpose of this paper is to prove the coarse geometric Novikov conjecture for spaces which admit a (coarse) uniform embedding into a uniformly convex Banach space.     

An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg

We present a direct analytic approach to the Guillemin-Sternberg conjecture [GS] that `geometric quantization commutes with symplectic reduction', which was proved recently by Meinrenken [M1], [M2] and Vergne [V1], [V2] et al. Besides providing a new proof of this conjecture, our methods also lead immediately to further extensions in various contexts. Oblatum 3-IX-1996 & 4-VIII-1997     

On a conjecture of D. Styer regarding univalent geometric and annular starlike functions

The aim of this paper is two-fold. First, to give a direct proof for the already established result of Styer which states that a univalent geometrically starlike function is a univalent annular starlike function if is bounded. Second, to show that the boundedness condition of is necessary, thus disproving a conjecture of Styer.

     

On the Cameron-Praeger conjecture

This paper takes a significant step towards confirming a long-standing and far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They conjectured in 1993 that there are no non-trivial block-transitive 6-designs. We prove that the Cameron-Praeger conjecture is true for the important case of non-trivial Steiner 6-designs, i.e. for 6-(v,k,λ) designs with λ=1, except possibly when the group is PΓL(2,pe) with p=2 or 3, and e is an odd prime power.     

On the multiplier conjecture

The Multiplier Theorem is a celebrated theorem in the Design theory. The conditionp>λ is crucial to all known proofs of the multiplier theorem. However in all known examples of difference sets μ _p . is a multiplier for every primep with (p, v)=1 andp‖n. Thus there is the multiplier conjecture: “The multiplier theorem holds without the assumption thatp>λ”. The general form of the multiplier theorem may be viewed as an attempt to partially resolve the multiplier conjecture, where the assumption “p>λ” is replaced by “n ₁>λ”. Since then Newman (1963), Turyn (1964), and McFarland (1970) attempted to partially resolve the multiplier conjecture (see [7], [8], [9]). This paper will prove the following result using the representation theory of finite groups and the algebraic number theory: LetG be an abelian group of orderv,v ₀ be the exponent ofG, andD be a (v, k, λ)-difference set inG. Ifn=2_{n 1}, then the general form of the multiplier theorem holds without the assumption thatn ₁>λ in any of the following cases:

On the points-lines-planes conjecture

It has been conjectured that in any matroid, if W₁, W₂, W₃ denote the number of points, lines, and planes respectively, then W₂² ≥ W₁W₃. We prove this conjecture (and some strengthenings) for matroids in which no line has five or more points, thus generalizing a result of Stonesifer, who proved it for graphic matroids.     

On the Vergne conjecture

Consider a Hamiltonian action by a compact Lie group on a possibly non-compact symplectic manifold. We give a short proof of a geometric formula for the decomposition into irreducible representations of the equivariant index of a \({{\mathrm{{{\mathrm{Spin}}}^c}}}\)-Dirac operator in this context. This formula was conjectured by Vergne in (Eur Math Soc Zürich I:635–664, 2007) and proved by Ma and Zhang in (Acta Math 212:11–57, 2014).     

On the direct sum conjecture

We prove the direct sum conjecture for various sets of systems of bilinear forms. Our results depend on a priori knowledge of the complexity of at least one of the direct summands and its underlying algebraic structure. We also briefly survey some previous results concerning the complexity and structure of minimal algorithms for various direct sum systems.     

On checking the Goldbach conjecture

This paper describes an attempt to check Goldbach's conjecture on a digital computer. The validity of the conjecture has been numerically verified up toN=33,000,000.