We define the toplogical generating rank dtop(G) of a topological group G as the minimal number of elements of G which generate a dense subgroup of G. I will present joint work with Gena Noskov, partly in progress, about our attempts to compute the topological generating rank for connected Lie groups. The cases that G is semisimple, where dtop(G) is slightly bigger than one, and that G is nilpo...

In the course of Deligne’s proof of the purity theorem, he makes certain monodromy constructions which do not a priori yield legitimate `-adic sheaves. In order to make the proof work, it therefore becomes necessary to slightly enlarge the category of sheaves considered from `-adic (constructible) sheaves to Weil sheaves. These satisfy very similar formal properties to ordinary `-adic sheaves, ...

We study spin-1/2 Heisenberg XXX antiferromagnet. The spectrum of the Hamiltonian was found by Hans Bethe in 1931, [1]. We study the probability of formation of ferromagnetic string in the antiferromagnetic ground state, which we call emptiness formation probability P (n). This is the most fundamental correlation function. We prove that for the short strings it can be expressed in terms of the ...

Living organisms are the most complex, interesting and significant objects regarding all substructures of the universe. Life science is regarded as a science of the 21st century and one can expect great new discoveries in the near futures. This article contains an introductory brief review of genetic information, its coding and translation of genes to proteins through the genetic code. Some the...

Following footsteps of Gauss, Euler, Riemann, Hurwitz, Smith, Hardy, Littlewood, Hedlund, Khinchin and Chebyshev, we visit some topics in elementary number theory. For matrices defined by Gaussian primes we observe a circular spectral law for the eigenvalues. We experiment then with various Goldbach conjectures for Gaussian primes, Eisenstein primes, Hurwitz primes or Octavian primes. These con...

While most algebra is done by writing text and formulas, diagrams have always been used to present structural information clearly and concisely. Text shells are the de facto interface for computational algebraic number theory, but they are incapable of presenting structural information graphically. We present GiANT, a newly developed graphical interface for working with number fields. GiANT off...

L-functions in Number Theory Yichao Zhang Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2010 As a generalization of the Riemann zeta function, L-function has become one of the central objects in Number Theory. The theory of L-functions, which produces a large family of consequences and conjectures in a unified way, concerns their zeros and poles, functional equat...