For public audience
My research is part of a large collaborative project involving scientists around the world consisting mainly of algebraic geometers, mathematical physicists, and theoretical physicists. Just in the same manner that Newton developed the calculus of differentials and integrals as a language to express classical mechanics, this community of researchers are inventing the convenient mathematics in which modern theoretical physics can flourish. Specifically, the type of mathematics I am interested in is inspired by String Theory. For this theory to be consistent, there are lots of mathematical facts to be proved and for it to develop there are lots of mathematics yet to be created. [1, 2]
According to string theory the world we live in is not the 4-dimensional time-space
we see but has a minuscule 6-dimensional
curled shape attached to each point of it known
as a Calabi-Yau threefold (CY3).
What physicists would really like from a good theory,
is the capability of extracting quantitative
information, simply numbers, that can be checked through experiments. According to string theory,
such numbers are supposed to be independent of the minor changes that can happen to the CY3.
It is also possible to check this fact mathematically, hence these numbers are called invariants
assigned to a CY3 in mathematics.
Inside a CY3 physical objects called the BPS-branes live that encodes the states in which certain particles can exist. Physical interpretation suggests that there should be only a finite number of these BPS-states. These invariants translate mathematically to the analogue Donaldson-Thomas (DT) invariants [5] associated to the space of the BPS-states called moduli. Consistent definition of DT-invariants and studying their properties is a crucial ingredient of string theory, and a beautiful matheamtical challenge.
Often in mathematics, the reason a problem is hard to solve is that it is not stated in the correct language. One should restate the problem (in our case re-interpret the DT-invariants), probably in a richer mathematical theory. The richer theory is predictably more complicated but makes it easier to understand the solution of the problem. The challenges mathematicians face in this process, can be itemized as follows: (1) beyond some special cases, the moduli are complicated objects that are called stacks. (2) The invariants one hopes to associated to these spaces are supposed to behave very much like Euler characteristics. An important feature of Euler characteristic of a geometric shape is that it remains unchanged if one deforms the shape as if it was made of rubber. (3) A potential definition of such invariants, depends on extra factors (called stability conditions) and therefore this dependence also has to be understood thoroughly.[3, 4]
The status quo theories that suggest candidates for appropriate types of invariant as above, are not just conceptually complicated, but even hard to compute in concrete examples. Our mission is to make these invariant more understandable; in other words we are in the position of trying to state hard problems about the (generalized) DT-invariants in more appropriate language.
[1] Deligne P. and P. Etingof and D. S. Freed and L. C. Jeffrey and D. Kazhdan and J. W. Morgan and D. R. Morrison and E. Witten, Quantum Fields and Strings. A course for mathematicians, American Mathematical Society (1999)
[2] Hori K., S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, E. Zaslow, Mirror Symmetry
[3] Joyce, D., Song, Y. (2011) A theory of generalized Donaldson-Thomas invariants, 199 pages, Memoirs of the AMS
[4] Kontsevich and Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435
[5] Thomas R.P., A holomorphic Casson invariant for CalabiŠYau 3-folds, and bundles on K3 fibrations, J. Diff. Geom. 54 (2000), 367Š438. math.AG/9806111.