I will describe recent progress in describing 4-dimensional N = 8 supergravity using the framework of ambitwistor string theory.
Parallelizing the time domain in numerical simulations is non-intuitive, but has been proven to be possible using various algorithms like parareal, PFASST and RIDC. Temporal parallelizations adds an entire new dimension to parallelize and significantly enhances use of super computing resources. Exploiting this technique serves as a big step towards exascale computation.
Starting with relatively simple problems, the parareal algorithm (Lions et al, A ''parareal'' in time discretization of PDE's, 2001) has been successfully applied to various complex simulations in the last few years (Samaddar et al, Parallelization in time of numerical simulations of fully-developed plasma turbulence using the parareal algorithm, 2010). The algorithm involves a predictor-corrector technique.
Numerical studies of the edge of magnetically confined, fusion plasma are an extremely challenging task. The complexity of the physics in this regime is particularly increased due to the presence of neutrals as well as the interaction of the plasma with the wall. These simulations are extremely computationally intensive but are key to rapidly achieving thermonuclear breakeven on ITER-like machines.
The SOLPS code package (Schneider et al, Plasma Edge Physics with B2‐Eirene, 2006) is widely used in the fusion community and has been used to design the ITER divertor. A reduction of the wallclock time for this code has been a long standing goal and recent studies have shown that a computational speed-up greater than 10 is possible for SOLPS (Samaddar et al, Greater than 10x Acceleration of fusion plasma edge simulations using the Parareal algorithm, 2014), which is highly significant for a code with this level of complexity.
In this project, the aim is to explore a variety of cases of relevance to ITER and thus involving more complex physics to study the feasibility of the algorithm. Since the success of the parareal algorithm heavily relies on choosing the optimum coarse solver as a predictor, the project will involve studying various options for this purpose. The tasks will also include performing scaling studies to optimize the use of computing resources yielding maximum possible computational gain.
Firstly, we will discuss how the category of strict polynomial functors can be endowed with a monoidal structure, including concrete calculations. It is well-known that the above category is equivalent to the category of modules over the Schur algebra. The so-called Schur functor in turn relates the category of modules over the Schur algebra to the category of representations of the symmetric group which posseses a monoidal structure given by the Kronecker product. We show that the Schur functor is monoidal with respect to these structures.
Finally, we consider the right and left adjoints of the Schur functor. We explain how these can be expressed in terms of one another using Kuhn duality and the central role the monoidal structure on strict polynomial functors plays in this context.
The moduli space of tropical curves (and its variants) is one of the most-studied objects in tropical geometry. So far this moduli space has only been considered as an essentially set-theoretic coarse moduli space (sometimes with additional structure). As a consequence of this restriction, the tropical forgetful map does not define a universal curve
(at least in the positive genus case). The classical work of Knudsen has resolved a similar issue for the algebraic moduli space of curves by considering the fine moduli stacks instead of the coarse moduli spaces. In this talk I am going to give an introduction to these fascinating tropical moduli spaces and report on ongoing work with R. Cavalieri, M. Chan, and J. Wise, where we propose the notion of a moduli stack of tropical curves as a geometric stack over the category of rational polyhedral cones. Using this framework one can give a natural interpretation of the forgetful morphism as a universal curve. The coarse moduli space arises as the set of $\mathbb{R}_{\geq 0}$-valued points of the moduli stack. Given time, I will also explain how the process of tropicalization for these moduli stacks can be phrased in a more fundamental way using the language of logarithmic algebraic stacks.
It has been observed long time ago (by many people) that singular affine symplectic varieties come in pairs; that is often to an affine singular symplectic variety $X$ one can associate a dual variety $X^!$; the geometries of $X$ and $X^!$ (and their quantizations) are related in a non-trivial way. The purpose of the talk will be 3-fold:
1) Explain a set of conjectures of Braden, Licata, Proudfoot and Webster which provide an exact formulation of the relationship between $X$ and $X^!$
2) Present a list of examples of symplectically dual pairs (some of them are very recent); in particular, we shall explain how the symplectic duals to Nakajima quiver varieties look like.
3) Give a new approach to the construction of $X^!$ and a proof of the conjectures from part 1).
The talk is based on a work in progress with Finkelberg and Nakajima.
Let F be a global field and let A denote its adele ring. The usual Tamagawa number formula computes the (suitably normalized) volume of the quotient G(A)/G(F) in terms of values of the zeta-function of F at the exponents of G; here G is simply connected semi-simple group. When F is functional field, this computation is closely related to the Atiyah-Bott computation of the cohomology of the moduli space of G-bundles on a smooth projective curve.
I am going to present a (somewhat indirect) generalization of the Tamagawa formula to the case when G is an affine Kac-Moody group and F is a functional fiend. Surprisingly, the proof heavily uses the so called Macdonald constant term identity. We are going to discuss possible (conjectural) geometric interpretations of this formula (related to moduli spaces of bundles on surfaces).
This is joint work with D.Kazhdan.
The question that the Crepant Resolution Conjecture (CRC) wants to address is: given an orbifold X that admits a repant resolution Y, can we systematically compare the Gromov-Witten theories of the two spaces? That this should happen was first observed by physicists and the question was imported into mathematics by Y.Ruan, who posited it as the search for an isomorphism in the quantum cohomologies of the two spaces. In the last fifteen years this question has evolved and found different formulations which various degree of generality and validity. Perhaps the most powerful approach to the CRC is through Givental's formalism. In this case, Coates, Corti, Iritani and Tseng propose that the CRC should consist of the natural comparison of geometric objects constructed from the GW potential fo the space. We explore this approach in the setting of open GW invariants. We formulate an open version of the CRC using this formalism, and make some verifications. Our approach is well tuned with Iritani's approach to the CRC via integral structures, and it seems to suggest that open invariants should play a prominent role in mirror symmetry.
Due to its use in cryptographic protocols such as the Diffie--Hellman key exchange, the discrete logarithm problem attracted a considerable amount of attention in the past 40 years. In this talk, we summarize the key technical ideas and their evolution for the case of discrete logarithms in small characteristic finite fields. This road leads from the original belief that this problem was hard enough for cryptographic purpose to the current state of the art where the algorithms are so efficient and practical that the problem can no longer be considered for cryptographic use.