Mathematics Seminars at ICTP - next week

[Math]

M A T H E M A T I C S   S E M I N A R S   2007


Tuesday, 14 August at 14:00 hrs.


Professor Augustin Banyaga
(The Pennsylvania State University, University Park, USA)

"A new proof of the Morse-Bott inequalities"

Abstract: Let $f$ be a Morse-Bott function on a compact oriented  
finite dimensional
manifold $M$. The polynomial Morse inequalities and a perturbation  
technique approximating any Morse-Bott function by a Morse function  
show immediately that if
$MB(f)(t)$ is the Morse-Bott polynomial of $f$, and $P(M)(t)$ is the  
Poincare polynomial of $M$, there exists a polynomial $R(t)$ such that
$MB(f)(t) = P(M)(t)  + (1+t)R(t)$. We prove that $R(t)$ is a  
polynomial with nonnegative integer coefficients, using the Morse  
Homology Theorem, and by studying the ranks of the kernels of the  
differentials of the Morse-Smale-Witten complexes of the Morse  
functions involved in the approximation scheme.
Our method works when all the critical submanifolds are oriented or  
when $Z_2$
coefficients are used. Our proof is much more  direct than all  
previous proofs.
This is a joint work with David  E. Hurtubise.


VENUE:  SEMINAR ROOM (ICTP Main Building, first floor)

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Thursday, 16 August at 15:00 hrs.


Professor Tatiana Gateva-Ivanova
(Bulgarian Academy of Sciences, Sofia, and
American University in Bulgaria, Blagoevgrad, Bulgaria)


"Set theoretic solutions of the Yang-Baxter equation, graphs and
computations"

Abstract:  It is well-known that certain matrix solutions of the  
braid or Yang-Baxter equations lead to braided categories, knot  
invariants, quantum groups and other important constructions.  
However, these equations are also very interesting at the level of  
set maps r : X x X --> X x X, where X is a set and r is a bijection,  
a line of study proposed by Drinfeld. Solutions extend linearly to  
very special linear solutions but also lead to a great deal of  
combinatorics, and to algebras with very nice algebraic and  
homological properties including those relating to Artin-Schelter  
regularity, Koszulity, being Noetherian domain, the existence of  
noncommutative Groebner bases.
We extend our recent work on set-theoretic solutions of the Yang- 
Baxter or braid relations with new results about their automorphism  
groups, strong twisted unions of solutions and multipermutation  
solutions. We introduce and study graphs of solutions and use our  
graphical methods for the computation of solutions of finite order  
and their automorphisms. Results include a detailed study of  
solutions of multipermutation level 2.



VENUE:  SEMINAR ROOM (ICTP Main Building, first floor)