UPDATED REMINDER - MATHS Seminars this week at ICTP

[ICTP Math Section]

MATH EVENTS THIS WEEK AT ICTP


    _Today, Tuesday 5 December_

*at 14:00
in the ***Budinich Lecture Hall** (ICTP Leonardo da Vinci Building)

Second lecture of the _MINI-COURSE on __Quantum Black Holes for 
Mathematicians_

By Atish Dabholkar (ICTP, HECAP)

Description of the course:  Study of black holes in string theory has 
revealed a beautiful and precise connection between the physics of 
quantum black holes and topics in number theory and geometry. The aim of 
these lectures is to outline these connections through examples starting 
with basic concepts and motivations from physics.

*


*

    **_Tomorrow, Wednesday 6 December _

*at 14:00 *
**in the Luigi Stasi Seminar Room **(ICTP Leonardo da Vinci Building, 
First floor)

_SEMINAR
_
by Pavel Putrov (IAS, USA)

Title:
Towards a categorification of WRT invariant of 3-manifolds

Abstract:
In my talk I will conjecture existence of certain new homological 
invariants of closed 3-manifolds. The invariants provide a 
categorification of the analytically continued Witten-Reshetikhin-Turaev 
(WRT) invariant, have some similarities with Heegaard/monopole Floer 
homology and generalize Khovanov homology of knots to compact 
3-manifolds. Physically such invariants can be defined as Hilbert spaces 
of M-theory in certain backgrounds. I will describe mathematical 
properties of such invariants, and how to calculate them for some 
specific types of 3-manifolds. The talk is mainly based on a joint work 
with S. Gukov, D. Pei and C. Vafa.




    _On Thursday 7 December_

*at 14:00*
*in the ***Budinich Lecture Hall** (ICTP Leonardo da Vinci Building)

First lecture of the _MINI-COURSE on Knot invariants, units, K-theory 
and modular form__s _

by Professor Don B. Zagier (MPI, Bonn and ICTP)

Description of the course: I will present joint work with Frank Calegari 
and Stavros Garoufalidis in which elements of the Bloch group of a 
number field (a more elementary version of algebraic K-theory that will 
be explained in detail in the lectures) produce units in cyclotomic 
extensions of this field. The result was motivated by experimental 
discoveries relating to quantum invariants (specifically, Jones 
polynomials and the Kashaev invariant) of knots, and had an unexpected 
application to a proof of a well-known conjecture of Werner Nahm 
relating the modularity of certain q-hypergeometric series to the 
vanishing of a certain invariant in the Bloch group.
_


_

    _On Friday 8 December
    _

_Two events_ will take place *_in the _**_Luigi Stasi Seminar Room_* 
(ICTP Leonardo da Vinci Building, first floor)


1)  From 15:00 to 16:00

_Course on Vertex algebras and modular forms_

By Don B. Zagier (MPI, Bonn/ICTP)

The course normally takes place at SISSA. It has been rescheduled at 
ICTP as SISSA will be closed on Friday




2) From 16:00 to 17:00_
____
__SEMINAR_*

*By Kiyokazu Nagatomo  (Osaka University)

Title:
Modular forms of rational weights and the minimal models*

*Abstract:
After the modular forms of rational weights on Γ(5) (and Γ(7)) were dis- 
covered, T. Ibukiyama formulated modular forms of weights (N − 3)/2N (N 
 > 3 and odd) on Γ(N) in the millennium, which have remained mysterious 
until now. In this talk I will gives a new point of view, which has 
advantages of understanding the factional weights and congruence groups 
that appear in the theory of Ibukiyama.
I (we) have been working on the minimal models and the associated 
differential equations which are a higher order generalization of 
Kaneko-Zagier equation. Recently, we found that the special case of the 
minimal models “essentially” gives these modular forms of fractional 
weights, where “essentially” means “after multiplying a power of eta 
function.” The characters (one-point functions) of (rational) conformal 
field theories may have negative powers of q when they are expanded as 
Fourier series. Of course, we can have only non-negative powers by 
multiplying a power of q. However, the results lose almost all good 
properties which characters have (including modular invariance 
property). Now, since the eta function commutes with the Serre 
derivation, we multiply a power of the eta function to the characters. 
Moreover, the power must be the so-called effective central charge in 
the Physics literature. Then the result we will prove is that modular 
forms of rational weights are obtained by multiplying ηceff to characters.
In a point of view of differential equations such as the Kaneko-Zagier 
equation, special functions would be defined as solutions of 
differential equations with regular singularities. Therefore, we may 
think that modular forms of rational weights would be “special functions.”
This talk requires elementary knowledge of (modular forms), vertex 
operator algebras, minimal models and modular linear differential 
equations, which have been (will be) given in series of lectures of 
Prof. Zagier.
Finally, this is a joint work with Y. Sakai (who is a number theorist) 
at Kyushu University.