By Mo-Lin Ge, Weiping Zhang

This volumes offers a entire assessment of interactions among differential geometry and theoretical physics, contributed through many prime students in those fields. The contributions promise to play an incredible position in selling the advancements in those interesting parts. along with the plenary talks, the insurance contains: types and comparable subject matters in statistical physics; quantum fields, strings and M-theory; Yang-Mills fields, knot idea and similar issues; K-theory, together with index thought and non-commutative geometry; replicate symmetry, conformal and topological quantum box conception; improvement of integrable platforms; and random matrix thought.

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L. Ge, K. -B. Zhang {V>+(z), Mv))+ = S(x - y)5a0. 13) satisfies the commuting relations for Y(SO(5)). C). 18) + 2£ 2 /]. 19) and the -R(u)-matrix reads R{u) = f(u)[u2P + v£(A -2P-I) The RTT relation gives 4+4+441 + 315+225 more relations. 22) 3. Applications of Yangian The first example was given by Belavin ([38]) in deriving the spectrum of nonlinear a model. Here we only show briefly some interpretations of Yangian through the particular realizations of Yangian. 1. 1) that contains the (antisymmetric) tensor interaction between Si and S2.

Pressley A guide to quantum groups (Cambridge Univ. Press, Cambridge, 1994). 25. E. Ragoucy, P. Sorba, Comm. Math. Phys. 203 (1999) 551. 26. M. Mintchev, E. Ragoucy, P. Sorba, Ph. Zaugg, J. Phys. A. 32 (1999) 5885. 27. S. Murakami, M. Wadati, J. Phys. Soc. Japan 62 (1993) 4203. 28. B. E. Korepin, Phys. Lett. A 190 (1994) 238. 29. F. E. Korepin, K. Schoutens, Nucl. Phys. B 372 (1992) 559. 30. F. E. Korepin, K. Schoutens, Nucl. Phys. B 384 (1992) 431. 31. F. E. Korepin, Phys. Rev. Lett. 72 (1994)908.

18) It turns out that there is a vibration between s = 0 and s = 1. 19) LLC < s2 >= 2 at t = - (total spin = 1). 20) u> Under adiabatic approximation it can be proved that it appears Berry's phase. Obviously, only spin vector can make the stereo angle. The role of spin singlet here is a witness that shares energy of spin=l state. 24) and A (±) = [2(a 2 + 2 5 S 0 V+M-)]- 1 / 2 [(« 2 + < ? 2 B 0 W - ) 1 / 2 ± ^ ' ^ f™ = [2(a2+g2B2^^)}-'/2[^±(a2+g2B^+^)l/2Ta]1/2. -M. -L. Ge, K. -B. Zhang where |Xu(t)> = cos2 ^ | x i i ) + - ^ s i n f l e - ^ l x i o ) + s i n 2 ie-^lxi-i), |Xi-iW> = sin2 ^ - o ' l x i i ) - - ^ s i n ^ e - ^ l x i o ) +cos 2 °-\Xi-i), \x±(t)) = ^ / P H - s i n ^ e ^ ^ l x i i ) + V^cos^Xio) + s i n 0 e - i - o t | x 1 _ 1 ) } +#)|Xoo>.