2 4 N ov 2 00 4 An algebraic proof of Deligne ’ s regularity criterion

N ov 2 00 6 SPHERICAL DESIGNS AND ZETA FUNCTIONS OF LATTICES

We set up a connection between the theory of spherical designs and the question of minima of Epstein’s zeta function. More precisely, we prove that a Euclidean lattice, all layers of which hold a 4-design, achieves a local minimum of the Epstein’s zeta function, at least at any real s > n 2 . We deduce from this a new proof of Sarnak and Strömbergsson’s theorem asserting that the root lattices ...

N ov 2 00 1 Random low rank mixed states are highly entangled Hao

We prove that for many low ranks r ≤ 2m − 3, random rank r mixed states in H A ⊗ H B have realtively high Schmidt numbers based on algebraic-geometric separability criterion proved in [1]. This also means that algebraic-geometric separability criterion can be used to detect all low rank entagled mixed states outside a measure zero set. Quantum entanglement was first noted as a feature of quantu...

1 4 N ov 2 00 5 Ramanujan ’ s Harmonic Number Expansion

An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the nth harmonic number.

ar X iv : m at h - ph / 0 31 10 01 v 4 3 N ov 2 00 4 Clifford Valued Differential Forms , Algebraic Spinor Fields

m at h . D S ] 1 3 N ov 2 00 3 Ratner ’ s Theorem on Unipotent Flows

Unipotent flows are well-behaved dynamical systems. In particular, Marina Ratner has shown that the closure of every orbit for such a flow is of a nice algebraic (or geometric) form. After presenting some consequences of this important theorem, these lectures explain the main ideas of the proof. Some algebraic technicalities will be pushed to the background. Chapter 1 is the main part of the bo...