More Like this

1. The companion section for classical groups

   https://doi.org/10.1142/S0129167X24410106  

   Hameister, Thomas; Ngô, Bao Châu (August 2024, International Journal of Mathematics)

   We use the companion matrix construction for [Formula: see text] to build canonical sections of the Chevalley map [Formula: see text] for classical groups [Formula: see text] as well as the group [Formula: see text]. To do so, we construct canonical tensors on the associated spectral covers. As an application, we make explicit lattice descriptions of affine Springer fibers and Hitchin fibers for classical groups and [Formula: see text]. 

   more » « less
2. Solving PDEs in space-time: 4D tree-based adaptivity, mesh-free and matrix-free approaches

   https://doi.org/10.1145/3295500.3356198  

   Ishii, Masado; Fernando, Milinda; Saurabh, Kumar; Khara, Biswajit; Ganapathysubramanian, Baskar; Sundar, Hari (November 2019, SC '19: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis)

   Numerically solving partial differential equations (PDEs) remains a compelling application of supercomputing resources. The next generation of computing resources - exhibiting increased parallelism and deep memory hierarchies - provide an opportunity to rethink how to solve PDEs, especially time dependent PDEs. Here, we consider time as an additional dimension and simultaneously solve for the unknown in large blocks of time (i.e. in 4D space-time), instead of the standard approach of sequential time-stepping. We discretize the 4D space-time domain using a mesh-free kD tree construction that enables good parallel performance as well as on-the-fly construction of adaptive 4D meshes. To best use the 4D space-time mesh adaptivity, we invoke concepts from PDE analysis to establish rigorous a posteriori error estimates for a general class of PDEs. We solve canonical linear as well as non-linear PDEs (heat diffusion, advection-diffusion, and Allen-Cahn) in space-time, and illustrate the following advantages: (a) sustained scaling behavior across a larger processor count compared to sequential time-stepping approaches, (b) the ability to capture "localized" behavior in space and time using the adaptive space-time mesh, and (c) removal of any time-stepping constraints like the Courant-Friedrichs-Lewy (CFL) condition, as well as the ability to utilize spatially varying time-steps. We believe that the algorithmic and mathematical developments along with efficient deployment on modern architectures shown in this work constitute an important step towards improving the scalability of PDE solvers on the next generation of supercomputers. 

   more » « less
3. Supermartingale shadow couplings: The decreasing case

   https://doi.org/10.3150/23-BEJ1592  

   Bayraktar, Erhan; Deng, Shuoqing; Norgilas, Dominykas (February 2024, Bernoulli)

   For two measures that are in convex-decreasing order, Nutz and Stebegg (Canonical supermartingale couplings, Ann. Probab. 46(6) 3351–3398, 2018) studied the optimal transport problem with supermartingale constraints and introduced two canonical couplings, namely the increasing and decreasing transport plans, that are optimal for a large class of cost functions. In the present paper we provide an explicit construction of the decreasing coupling by establishing a Brenier-type result: (a generalised version of) this coupling concentrates on the graphs of two functions. Our construction is based on the concept of the supermartingale shadow measure and requires a suitable extension of the results by Juillet (Stability of the shadow projection and the left-curtain coupling, Ann. Inst. H. Poincaré Probab. Statist. 52(4) 1823–1843, November 2016) and Beiglböck and Juillet (Shadow couplings, Trans. Amer. Math. Soc. 374 4973–5002, 2021) established in the martingale setting. In particular, we prove the stability of the supermartingale shadow measure with respect to initial and target measures introduce an infinite family of lifted supermartingale couplings that arise via shadow measure, and show how to explicitly determine the martingale points of each such coupling. 

   more » « less
4. Congruence modules in higher codimension and zeta lines in Galois cohomology

   https://doi.org/10.1073/pnas.2320608121  

   Iyengar, Srikanth B.; Khare, Chandrashekhar B.; Manning, Jeffrey; Urban, Eric (April 2024, Proceedings of the National Academy of Sciences)

   This article builds on recent work of the first three authors where a notion of congruence modules in higher codimension is introduced. The main results are a criterion for detecting regularity of local rings in terms of congruence modules, and a more refined version of a result tracking the change of congruence modules under deformation. Number theoretic applications include the construction of canonical lines in certain Galois cohomology groups arising from adjoint motives of Hilbert modular forms. 

   more » « less
5. Persuading Risk-Conscious Agents: A Geometric Approach

   https://doi.org/10.1007/978-3-030-35389-6  

   Anunrojwong, Jerry; Iyer, Krishnamurthy; Lingenbrink, David. (December 2019, Lecture notes in computer science)

   Motivated by practical concerns in applying information design to markets and service systems, we consider a persuasion problem between a sender and a receiver where the receiver may not be an expected utility maximizer. In particular, the receiver’s utility may be non-linear in her belief; we deem such receivers as risk-conscious. Such utility models arise, for example, when the receiver exhibits sensitivity to the variability and the risk in the payoff on choosing an action (e.g., waiting time for a service). In the presence of such non-linearity, the standard approach of using revelation-principle style arguments fails to characterize the set of signals needed in the optimal signaling scheme. Our main contribution is to provide a theoretical framework, using results from convex analysis, to overcome this technical challenge. In particular, in general persuasion settings with risk-conscious agents, we prove that the sender’s problem can be reduced to a convex optimization program. Furthermore, using this characterization, we obtain a bound on the number of signals needed in the optimal signaling scheme. We apply our methods to study a specific setting, namely binary per-suasion, where the receiver has two possible actions (0 and 1), and the sender always prefers the receiver taking action 1. Under a mild convexity assumption on the receiver’s utility and using a geometric approach,we show that the convex program can be further reduced to a linear program. Furthermore, this linear program yields a canonical construction of the set of signals needed in an optimal signaling mechanism. In particular, this canonical set of signals only involves signals that fully reveal the state and signals that induce uncertainty between two states.We illustrate our results in the setting of signaling wait time information in an unobservable queue with customers whose utilities depend on the variance of their waiting times. 

   more » « less