Application of the Probability Distribution of Distances in a Layered Sphere to Electrostatic Energy

This thesis works on a problem that involves calculating the total electrostatic interaction energy of two charged particles uniformly distributed in a sphere. It determines that traditional methods and the use of probability techniques yield equivalent results for the electrostatic energy in a charged layered sphere

Moduli Spaces of Flat GSp-Bundles

A classical problem in the theory of differential equations is the classification of first-order singular differential operators up to gauge equivalence. A related algebro-geometric problem involves the construction of moduli spaces of meromorphic connections. In 2001, P. Boalch constructed well-behaved moduli spaces in the case that each of the singularities are diagonalizable. In a recent series of papers, C. Bremer and D. Sage developed a new approach to the study of the local behavior of meromorphic connections using a geometric variant of fundamental strata, a tool originally introduced by C. Bushnell for the study of p-adic representation theory. Not only does this approach allow for the generalization of diagonalizable singularities, but it is adaptable to the study of flat G-bundles for G a reductive group. In this dissertation, the objects of study are irregular singular flat GSp-bundles. The main results of this dissertation are two-fold. First, the local theory of fundamental strata for GSp-bundles is made explicit; in particular, the fundamental strata necessary for the construction of well-behaved moduli spaces are shown to be associated to uniform symplectic lattice chain filtrations. Second, a construction of moduli spaces of flat GSp-bundles is given which has many of the geometric features that have been important in the work of P. Boalch and others

The Inner Power of a Graph

We define a new graph operation called the inner power of a graph. The construction is similar to the direct power of graphs, except that factors are intertwined in such a way that certain structural properties of graphs are more clearly reflected in their inner powers. We investigate various properties of inner powers, such as connectivity, bipartiteness, and their interaction with the direct product. We explore possible connections between inner powers and the problem of cancellation over the direct product of graphs

The Deligne-Simpson problem for connections on Gm\mathbb{G}_m with a maximally ramified singularity

The classical additive Deligne-Simpson problem is the existence problem for Fuchsian connections with residues at the singular points in specified adjoint orbits. Crawley-Boevey found the solution in 2003 by reinterpreting the problem in terms of quiver varieties. A more general version of this problem, solved by Hiroe, allows additional unramified irregular singularities. We apply the theory of fundamental and regular strata due to Bremer and Sage to formulate a version of the Deligne-Simpson problem in which certain ramified singularities are allowed. These allowed singular points are called toral singularities; they are singularities whose leading term with respect to a lattice chain filtration is regular semisimple. We solve this problem in the important special case of connections on $\mathbb{G}_m$ with a maximally ramified singularity at $0$ and possibly an additional regular singular point at infinity. We also give a complete characterization of all such connections which are rigid, under the additional hypothesis of unipotent monodromy at infinity.Comment: 27 pages. Minor correction

GME: GPU-based Microarchitectural Extensions to Accelerate Homomorphic Encryption

Fully Homomorphic Encryption (FHE) enables the processing of encrypted data without decrypting it. FHE has garnered significant attention over the past decade as it supports secure outsourcing of data processing to remote cloud services. Despite its promise of strong data privacy and security guarantees, FHE introduces a slowdown of up to five orders of magnitude as compared to the same computation using plaintext data. This overhead is presently a major barrier to the commercial adoption of FHE. In this work, we leverage GPUs to accelerate FHE, capitalizing on a well-established GPU ecosystem available in the cloud. We propose GME, which combines three key microarchitectural extensions along with a compile-time optimization to the current AMD CDNA GPU architecture. First, GME integrates a lightweight on-chip compute unit (CU)-side hierarchical interconnect to retain ciphertext in cache across FHE kernels, thus eliminating redundant memory transactions. Second, to tackle compute bottlenecks, GME introduces special MOD-units that provide native custom hardware support for modular reduction operations, one of the most commonly executed sets of operations in FHE. Third, by integrating the MOD-unit with our novel pipelined $64$-bit integer arithmetic cores (WMAC-units), GME further accelerates FHE workloads by $19\%$. Finally, we propose a Locality-Aware Block Scheduler (LABS) that exploits the temporal locality available in FHE primitive blocks. Incorporating these microarchitectural features and compiler optimizations, we create a synergistic approach achieving average speedups of $796\times$, $14.2\times$, and $2.3\times$ over Intel Xeon CPU, NVIDIA V100 GPU, and Xilinx FPGA implementations, respectively

GME: GPU-based Microarchitectural Extensions to Accelerate Homomorphic Encryption

Fully Homomorphic Encryption (FHE) enables the processing of encrypted data without decrypting it. FHE has garnered significant attention over the past decade as it supports secure outsourcing of data processing to remote cloud services. Despite its promise of strong data privacy and security guarantees, FHE introduces a slowdown of up to five orders of magnitude as compared to the same computation using plaintext data. This overhead is presently a major barrier to the commercial adoption of FHE. While prior efforts recommend moving to custom accelerators to accelerate FHE computing, these solutions lack cost-effectiveness and scalability. In this work, we leverage GPUs to accelerate FHE, capitalizing on a well-established GPU ecosystem that is available in the cloud. We propose GME, which combines three key microarchitectural extensions along with a compile-time optimization to the current AMD CDNA GPU architecture. First, GME integrates a lightweight on-chip compute unit (CU)-side hierarchical interconnect to retain ciphertext in cache across FHE kernels, thus eliminating redundant memory transactions and improving performance. Second, to tackle compute bottlenecks, GME introduces special MOD-units that provide native custom hardware support for modular reduction operations, one of the most commonly executed sets of operations in FHE. Third, by integrating the MOD-unit with our novel pipelined 64-bit integer arithmetic cores (WMAC-units), GME further accelerates FHE workloads by 19%. Finally, we propose a Locality-Aware Block Scheduler (LABS) that improves FHE workload performance, exploiting the temporal locality available in FHE primitive blocks. Incorporating these microarchitectural features and compiler optimizations, we create a synergistic approach achieving average speedups of 796×, 14.2×, and 2.3× over Intel Xeon CPU, NVIDIA V100 GPU, and Xilinx FPGA implementations, respectively

Topological Deep Learning: Going Beyond Graph Data

Topological deep learning is a rapidly growing field that pertains to the development of deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize many domains encountered in scientific computations. In this paper, we present a unifying deep learning framework built upon a richer data structure that includes widely adopted topological domains. Specifically, we first introduce combinatorial complexes, a novel type of topological domain. Combinatorial complexes can be seen as generalizations of graphs that maintain certain desirable properties. Similar to hypergraphs, combinatorial complexes impose no constraints on the set of relations. In addition, combinatorial complexes permit the construction of hierarchical higher-order relations, analogous to those found in simplicial and cell complexes. Thus, combinatorial complexes generalize and combine useful traits of both hypergraphs and cell complexes, which have emerged as two promising abstractions that facilitate the generalization of graph neural networks to topological spaces. Second, building upon combinatorial complexes and their rich combinatorial and algebraic structure, we develop a general class of message-passing combinatorial complex neural networks (CCNNs), focusing primarily on attention-based CCNNs. We characterize permutation and orientation equivariances of CCNNs, and discuss pooling and unpooling operations within CCNNs in detail. Third, we evaluate the performance of CCNNs on tasks related to mesh shape analysis and graph learning. Our experiments demonstrate that CCNNs have competitive performance as compared to state-of-the-art deep learning models specifically tailored to the same tasks. Our findings demonstrate the advantages of incorporating higher-order relations into deep learning models in different applications

Accelerating Finite Field Arithmetic for Homomorphic Encryption on GPUs

© 2023. This manuscript version is made available under the CC-BY4.0 license http://creativecommons.org/licenses/by /4.0/ This document is the accepted version of a Published Work that appeared in final form in IEEE Micro. To access the final edited and published work see https://doi.org/10.1109/MM.2023.3253052Fully Homomorphic Encryption (FHE) is a rapidly developing technology that enables computation directly on encrypted data, making it a compelling solution for security in cloud-based systems. In addition, modern FHE schemes are believed to be resistant to quantum attacks. Although FHE offers unprecedented potential for security, current implementations suffer from prohibitively high latency. Finite field arithmetic operations, particularly the multiplication of high-degree polynomials, are key computational bottlenecks. The parallel processing capabilities provided by modern Graphical Processing Units (GPUs) make them compelling candidates to target these highly parallelizable workloads. In this article, we discuss methods to accelerate polynomial multiplication with GPUs, with the goal of making FHE practical