Nonlinear inverse problems in holography and particle kinematics

全息术和粒子运动学中的非线性反问题

• 批准号: RGPIN-2022-03290
• 负责人: Balehowsky, Tracey
• 金额: $ 1.38万
• 依托单位: University of Calgary
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758903
• 关键词: Nonlinear; inverse; problems; holography; particle

By placing electrodes on a patient's chest, a medical technician may use an electrocardiogram to monitor a patient's heart. Could we use the electrical information on the skin of the patient to build a 3-dimensional image of the patient's heart? This imaging question can be mathematically formulated as an inverse problem. The goal of this project is to study inverse problems which arise in the physical fields of holography, condensed matter physics, and atmospheric physics. In these fields, we will seek to build mathematical objects describing the physical situation in a bulk region (analogous to the patient's chest) from measurement data (analogous to the electrical measurements on the patient's skin). The main postulate of holography is that our 4-dimensional universe is described by a 3-dimensional space, kind of how a 3-dimensional object can be represented in a 2-dimensional hologram image. Broadly, some of the questions this project will consider are (1) can the bulk spacetime modelling gravity be completely described by conformal field theory objects defined on a lower dimensional subspacetime? In the areas of condensed matter physics and atmospheric physics, we will address questions such as (2) Is it possible to determine the kinetic properties of a system of particles in a region from measurement data such as particle concentrations or estimated counts? The answers found to these and related questions over the course of this project aim to provide researchers in the fields of holography theory, condensed matter physics, and atmospheric physics analytical tools for the physical models in their fields. It also aims to equip researchers with mathematical techniques for the development of algorithms which compute key properties, such as the spacetime geometry describing quantum gravity or the operator describing how particles collide. Both geometric and analytic mathematical viewpoints will be employed towards the solutions of such problems. From this dual approach, novel combinations of theoretical techniques from across the disciplines of minimal surface theory, partial differential equations theory, microlocal analysis, and inverse problems theory will be created to develop new approaches for solving key problems in holography theory and condensed matter physics, as well as theoretical problems related to the Boundary Rigidity Problem in geometry.

通过将电极放置在患者的胸部，医疗技术人员可以使用心电图来监测患者的心脏。我们能否利用患者皮肤上的电信息来构建患者心脏的 3 维图像？这个成像问题可以用数学公式表示为反问题。该项目的目标是研究全息、凝聚态物理和大气物理领域中出现的反演问题。在这些领域中，我们将寻求根据测量数据（类似于患者皮肤上的电测量）建立描述大块区域（类似于患者胸部）的物理情况的数学对象。全息术的主要假设是，我们的 4 维宇宙是由 3 维空间描述的，类似于 3 维物体可以在 2 维全息图像中表示。从广义上讲，该项目将考虑的一些问题是（1）体时空建模重力是否可以通过在低维子时空上定义的共形场论对象来完全描述？在凝聚态物理和大气物理领域，我们将解决以下问题：（2）是否可以根据粒子浓度或估计计数等测量数据来确定某个区域中粒子系统的动力学特性？ 在本项目过程中找到的这些及相关问题的答案旨在为全息理论、凝聚态物理和大气物理领域的研究人员提供其领域物理模型的分析工具。它还旨在为研究人员提供数学技术，用于开发计算关键属性的算法，例如描述量子引力的时空几何或描述粒子如何碰撞的算子。 几何和解析数学观点都将用于解决此类问题。通过这种双重方法，将创建最小表面理论、偏微分方程理论、微局域分析和反问题理论等跨学科的理论技术的新颖组合，以开发解决全息理论和凝聚态物理中关键问题的新​​方法，以及与几何中的边界刚度问题相关的理论问题。