(4mq) Topological Analysis, Modeling, and Combinatorial Optimization Under Uncertainty

\section{Research Goals}

\subsection{Advanced Mathematical Tools}
My interests lie in the intersection of mathematics and physics, with a focus on creating sophisticated analytical frameworks for studying complex problems. The main idea is to convert the complex functions and series into a wave-superimposed form. We can then further study these superimposed structures to study the function behavior. This approach can be extensively used in the domain of optimization by assigning appropriate tasks to classical and quantum computers. This interdisciplinary approach will enable innovative solutions across various domains, including energy systems, supply chain analysis, mathematical concepts like the Riemann hypothesis, quantum physics, optimization techniques, and quantum chaos.\\

\subsection{Graphical Topological Analysis}
I am highly motivated to advance the development of rigorous and versatile techniques for solving non-convex optimization problems such as finding the optimal strategy to win a chess game. In the future, I plan to utilize the topological aspects of a more comprehensive graphical representation for a specific subset of non-convex optimization problems. \\
Additionally, I plan to deploy tools used in quantum mechanics such as the Fourier Transform, the superposition principle, and perturbation analysis to refine and expand our methods for exploring and optimizing the search space.\\

\subsection{Machine Learning in Optimization}
I also plan to develop advanced algorithms that exploit machine learning for optimization tasks. My approach involves utilizing sophisticated activation functions to more accurately approximate non-linearities. For handling discrete variables, I intend to employ logic gate circuits (LGC). Subsequently, I aim to integrate these activation functions and LGC to enhance the decision-making process. \\

\subsection{Long-Term Objectives: Study of Waveforms and Chaos}
In the long term, my ultimate objective is to establish a new field that bridges the gap between abstract formulations in both pure and applied mathematics and the structural analysis of waveforms. For instance, in the context of non-linear optimization problems, this entails a detailed investigation of superimposed structures, which govern the distribution of minima and maxima through destructive and constructive interference. Additionally, I aspire to develop advanced techniques for the study and prediction of chaos, with a particular emphasis on employing tools like the Fourier Transform and perturbation analysis.\\

Given the interdisciplinary nature of my Ph.D. research, I am open to collaborating with experts and students from diverse fields. As demonstrated by my involvement in writing NSF reports and grant proposals, I am confident in my ability to secure funding for my research through grants and proposals. Moreover, my prior industrial experience will also aid in improving my ability to collaborate with various industrial partners.\\

\section{Research Experience}
My research integrates rigorous mathematics and fundamental physics to dissect and solve complex problems. I aim to gain insights into such problems by recognizing patterns and drawing analogies across disciplines. \\

\subsection{Ph.D. Work}
{\hskip 2em} The novelty and interdisciplinary nature of my research work excite me to think of innovative and relatively original approaches to solving complex problems.\\
{\hskip 2em} As part of my Ph.D. research, I am working on topographical analysis, Artificial Intelligence (AI) based predictive modeling, and mathematical modeling and combinatorial optimization (Mixed Integer Non-Linear Programming framework) of Agriphotovoltaic Systems under uncertainty. These are systems designed for simultaneous food and energy production using elevated solar panels on agricultural land. Given the novelty of the field, I began by developing a 3-dimensional analytical shadow model that predicts the spatiotemporal radiation pattern for a given geometry of panels and a tracking algorithm on any given location on Earth. I have developed a topographical framework to study the problem by transforming it into a surface of genus one. In addition, I have deployed series analysis and perturbation analysis to gain insights into the nature of the solution. \\

{\hskip 2em} Working as a graduate student associate in the NSF-funded SFEWS project, I collaborate with faculty across various disciplines. I have also interacted with the faculty and students of Florida A\&M University (FAMU). Working in collaboration helped me understand and implement various aspects of this multidimensional problem. In addition, my involvement in grant writing, yearly NSF report writing, and collaboration with state government and solar companies for developing advanced versions of the existing research facility helped me gain a holistic understanding of the process. Recently, I have also been invited by the Purdue Lecture Hall Series as a distinguished speaker to give a talk on my academic journey~\cite{1}. I have also been involved in giving research overviews to media outlets~\cite{2,3,4,5}.

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%\section*{References}
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\begin{enumerate}
\item \label{1}\href{https://www.youtube.com/watch?v=VlkG7zrdyYM}{Purdue Lecture Hall Series Varsha Gupta, Purdue University.}
\item \label{2}\href{https://wisconsinwatch.org/2022/09/research-seeks-ways-to-grow-solar-and...}{Wisconsin Watch: A skeptical Corn Belt.}
\item\label{3}\href{https://www.indystar.com/story/news/environment/2022/09/13/purdue-leadin...}{Indianapolis Star: Can solar panels and row crops coexist?}
\item \label{4}\href{https://thelensnola.org/2022/09/09/research-seeks-ways-to-grow-solar-and...}{The Lens: Research on solar and crops.}
\item \label{5}\href{https://www.thegazette.com/news/research-seeks-ways-to-grow-solar-and-cr...}{The Gazette: Research on solar and crops.}

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