Research [PLACEHOLDER TEXT FOR NOW]

Geometric Decomposition for Spaces of Double Forms

My research focuses on geometric decomposition techniques for spaces of differential forms, with a particular emphasis on spaces of double forms and their tensor products. Differential forms are fundamental objects in mathematics that generalize scalars, vectors, and higher-dimensional analogues such as areas and volumes.

The key insight behind my work is the use of symmetric tensor products of differential forms to model physical and geometric quantities that exhibit tangential-tangential or normal-normal continuity across simplices in a triangulated domain. For instance, spaces like \( \Lambda_{\text{sym}}^{1,1} \) and \( \Lambda_{\text{sym}}^{2,2} \) represent such constructs, where the symmetric tensor combines lower-dimensional differential forms into higher-dimensional spaces.

Motivation

Applications of these methods range from elasticity, where stress tensors must respect normal-normal continuity, to general relativity, where discrete approximations like Regge calculus require accurate flux continuity across simplex boundaries. The overarching goal is to improve numerical simulations by developing finite element spaces that preserve key physical and geometric properties of the underlying equations.

Key Results

  • Developed a decomposition for \( \Lambda_{\text{sym}}^{2,2} \), showing it admits a geometric splitting into simpler subspaces: \[ \Lambda_{\text{sym}}^{2,2} \cong \Lambda_{0,\beta}^{2,2} \oplus \Lambda_{0,\gamma}^{2,2} \oplus \Lambda^4. \]
  • Established the relationship between trace-free spaces and reduced polynomial spaces, such as: \[ \mathring{P}_r\gamma_{ijkl} \cong P_{r-1}\gamma_{ijkl}, \quad \mathring{P}_r\beta_{ijk} \cong P_{r-2}\beta_{ijk}. \]
  • Applied these decompositions to enhance the numerical approximation of differential operators in finite element spaces.

Future Directions

Future work aims to generalize these decompositions to higher-dimensional simplices and more complex tensor products. This will provide a unified framework for solving partial differential equations arising in physics and geometry, with improved accuracy and computational efficiency.

Gamma-Ray Burst Afterglows

This project focuses on modeling the afterglow emission of gamma-ray bursts (GRBs) to improve our understanding of these powerful cosmic events. GRBs are brief but intense flashes of gamma rays, followed by an afterglow that evolves over time. Using theoretical models and computational techniques, we aim to predict the spectrum and flux density of GRB afterglows under various physical conditions.

So far, we have successfully recreated and verified the old code from Warren et al. (2022), ensuring consistency with established models like Granot & Sari (2002). This initial success allows us to confidently move forward with the next phase of our project: generating a large dataset for training a machine learning model. The dataset will be derived from simulated GRB afterglow scenarios, spanning a wide range of physical parameters.

Goals

  • Verify that the updated model matches predictions for key physical scenarios, such as energy distributions with \(k = 0\) and \(k = 2\).
  • Generate a comprehensive training dataset for a neural network to predict GRB afterglows.

Next Steps

With the foundational code successfully tested, the next steps include running simulations for the \(k = 2\) scenario and generating the training data. Once this phase is complete, we will integrate machine learning techniques to enhance the predictive capabilities of our model.