Let R be a commutative Noetherian ring that is a smooth ℤ-algebra. For each ideal a of R and integer k, we prove that the local cohomology module has finitely many associated prime ideals. This settles a crucial outstanding case of a conjecture of Lyubeznik asserting this finiteness for local cohomology modules of all regular rings.
We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with n≥3 strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius l tends to 1 exponentially quickly as l tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the conjugacy search problem can be solved in quadratic time. The idea behind both results is that generic braids can be conjugated ''easily'' into a rigid braid.
The ternary Goldbach conjecture (1742) asserts that every odd number greater than 5 can be written as the sum of three prime numbers. Following the pioneering work of Hardy and Littlewood, Vinogradov proved (1937) that every odd number larger than a constant C satisfies the conjecture. In the years since then, there has been a succession of results reducing C, but only to levels much too high for a verification by computer up to C to be possible. My recent work proves the conjecture. We will go over the main ideas of the proof.
A d-regular graph is Ramanujan if its non-trivial eigenvalues in absolute value are bounded by 2√(d-1). Recently Adam Marcus, Daniel Spielman and Nikhil Srivastava gave a positive answer to this question by showing that any bipartite d-regular Ramanujan graph has a 2-fold cover that is also Ramanujan. In this talk we shall discuss their approach and mention similarities with function field towers.
To apply the technique of virtual fundamental cycle (chain) in the study of pseudo-holomorphic curve, we need to construct certain structure, which we call Kuranishi strucuture, on its moduli space. In this talk I want to review certain points of its construction.
To any convex integer polygon we associate a Poisson variety, which is essentially the moduli space of connections on line bundles on (certain) bipartite graphs on a torus. There is an underlying integrable Hamiltonian system whose Hamiltonians are weighted sums of dimer covers.
Fix a metric (Riemannian or Finsler) on a compact manifold M. The critical points of the length function on the free loop space 𝓛M of M are the closed geodesics on M. Filtration by the length function gives a link between the geometry of closed geodesics, and the algebraic structure given by the Chas-Sullivan product on the homology of 𝓛M and the 'dual' loop cohomology product.
If X is a homology class on 𝓛M, the 'minimax' critical level Cr(X) is a critical value of the length function. Gromov proved that if M is simply connected, there are positive constants k and K so that for every homology class X of degree > dim(M) on 𝓛M, k < deg(X)/Cr(X) < K. When M is a sphere, we prove there are positive constants a and b so that for every homology class X on 𝓛M, aCr(X)-b < deg(X) < aCr(X)+b. There are interesting consequences for the length spectrum.
In 1985 Misha Gromov proved his Nonsqueezing Theorem, and hence constructed the first symplectic 1-capacity. In 1989 Helmut Hofer asked whether symplectic d-capacities exist if 1 < d < n. I will discuss the answer to this question and its relevance in symplectic geometry.
We consider Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field. Motivated by work of Kato and others for n=3, we show that local-global principles hold for Hn(F,ℤ/mℤ(n−1)) for all n>1. In the case n=1, a local-global principle need not hold. Instead, we will see that the obstruction to a local-global principle for H1(F,G), a Tate-Shafarevich set, can be described explicitly for many (not necessarily abelian) linear algebraic groups G. Concrete applications of the results include central simple algebras and Albert algebras.
For a geometrically finite hyperbolic group with small critical exponent, the spectral
method for counting is not available, as there is no point eigenvalue of the Laplace operator on the L2-spectrum. We will explain counting results for orbits of a big class of thin groups acting on a symmetric variety of the real hyperbolic group, which are obtained via ergodic approach.
We will describe a recent effective counting result for Apollonian circle packings. The main ingredient of this result is an effective equidistribution of closed horospheres in an infinite volume hyperbolic 3-manifold whose fundamental group has critical exponent bigger than one. We will explain how the spectral theory of Lax and Phillips can be used for such equidistribution results.
In this talk, I will present a formulation of the Gross-Zagier formula over Shimura curves using automorphic representations with algebraic coefficients. It is a joint work with Shou-wu Zhang and Wei Zhang.
An elementary homomorphism from a free group to the pure braid group yields interesting connections between braid groups, homotopy theory, and low dimensional topology. This map induces a map on the Lie algebra obtained from the descending central series. Further, this map induces a morphism of simplicial groups. All of these maps are shown to be injective.
Brunnian braids are discussed. The analogous maps of Lie algebras induced on the filtration quotients of the mod-p descending central series is again an injection. Using these facts it turns out that the homotopy groups of this simplicial group, those of the 2-sphere, are isomorphic to natural subquotients of the pure braid group. In addition, the mod-p analogues give a connection between the classical unstable Adams spectral sequence, and the mod-p analogues of Vassiliev invariants of pure braids.
