Talk

The notion of compatible R-matrices was introduced by Isaev-Ogievetsky-Pyatov.
It enables one to introduced a large family of new quantum algebras and the
corresponding Bethe subalgebras. Also, this notion  is useful for defining
the so-called half-quantum algebras (Isaev-Ogievetsky) which generalize the
notions of the Manin and q-Manin matrices. In my talk I plan to introduce  all
these  objects and to exhibit the role of  the mentioned algebras in
studying the  Yangians and their different generalizations.

State integrals (and their building block, the quantum dilogarithm) express the partition function of complex Chern-Simons theory of triangulated 3-manifolds with boundary. I will give an introduction to the subject, focusing on examples, as well as recent results on expressing state integrals in terms of Neumann-Zagier data and in terms of q-series of Nahm type. This is work joint in parts with R. Kashaev and D. Zagier.

In this talk, we consider some asymptotic behaviour of binary additive
problems with prime variables in short intervals. We start with a
brief review on exceptional set estimates in short intervals and
continue to the recent work of Languasco and Zaccagnini on the short
interval asymptotic formulas. Although the preceding results were
mainly obtained via the circle method, we see that a more direct
method gives better results for some cases.

We show that the number of partitions into squares with an even number of parts is asymptotically equal to that of partitions into squares with an odd number of parts. We further prove that, for $ n $ large enough, the two quantities are different and which of the two is bigger depends on the parity of $ n. $ This solves a recent conjecture formulated by Bringmann and Mahlburg (2012)