Varsha Gupta

Research Projects

1. Independent Research

The Superposition Principle and the Riemann Hypothesis: A New Perspective on a Classic Problem

This work explores a novel function, G(s), defined through prime numbers, which directly connects the distribution of primes and the non-trivial zeros of the Riemann zeta function. Through mathematical theorems and properties, the paper introduces a wave interference perspective to analyze the symmetry and cancellations in the Riemann zeta function. The research provides insights into fractal-like structures of critical zeros and their implications for proving the Riemann Hypothesis.

Publication: Under review in the Ramanujan Journal. Available on Zenodo.

Unified Framework for Total Variation Regularized Optimization in Fluid Dynamics and Related Physical Systems

This project presents an optimization framework for minimizing the energy functional governing physical systems such as fluid dynamics, particle transport, and phase transitions. The method introduces a Total Variation (TV) regularization term to analyze convergence in convection-dominated problems, and its application is demonstrated across systems like the Navier-Stokes, Boltzmann, and Maxwell equations.

Publication: Under review in the Journal of Non-Linear Mathematical Physics. Preprint available on ArXiv.

Presentation: Presented at AIChE 2024 in the session 213h - Computational Analyses of Mixing Processes III. Session Details.

Modular Forms in Combinatorial Optimization

This paper introduces a theoretical framework for applying modular forms to combinatorial optimization problems, with a focus on the Asymmetric Traveling Salesman Problem (ATSP). By translating the problem into the complex domain, the research uncovers a modular structure and leverages modular invariance to inform both unconstrained and constrained optimal solutions. The findings lay a foundation for future algorithmic development at the intersection of number theory and combinatorial optimization.

Publication: Preprint available on ArXiv.

Presentation: Presented at AIChE 2024 in Poster Session 371 - 10A: Interactive Session: Systems and Process Design. Session Details.

Stochastic Optimization Using Ricci Flow

This paper proposes a novel theoretical framework for optimizing bounded functions using Fourier series approximations and Ricci flow. The approach combines sampling methods along circles defined by Riemannian metrics with "inverse Ricci flow" to identify potential global optima through the accentuation of high curvature regions. The technique is applicable to a variety of optimization problems, offering insights into high-dimensional and non-convex landscapes.

Publication: Preprint available on ArXiv.

Presentation: Presented at the inaugural meeting of the Purdue Center for Operations and Optimization in Process Systems (COOPS), Poster Session: September 26, 2024. Session Details.

Ph.D. Research

1. Mathematical Modeling of Spatiotemporal Shadow Distribution (SSD)

Developed a precise mathematical framework to model shading patterns dynamically over an agrivoltaic domain D ⊂ ℝ² across time t ∈ [0, T]. Introduced the concept of Spatiotemporal Shadow Distribution (SSD), mapped onto a toroidal surface T² = S¹ × S¹, capturing spatial and temporal periodicity due to panel configurations and diurnal cycles.

Key SSD Formulation:

Φ(x, t) = {
    1, if x is shaded at time t,
    0, otherwise.
}

2. Stochastic Optimization on the Torus

Incorporated stochastic models to address environmental uncertainties (e.g., cloud cover) by defining probability density functions 𝒫(θ(t), t) over T². Applied calculus of variations on T² to determine geodesic paths that simultaneously optimize agricultural yield Yₜₒₜₐₗ and energy output Eₑₙₑᵣgᵧ.

Combined Optimization Objective:

max{θ(t) ∈ S¹} E = αEₑₙₑᵣgᵧ + βYₜₒₜₐₗ

3. Non-Monotonicity in Yield Functionality

Proved that crop yield exhibits a non-monotonic relationship with total radiation Rₜₒₜₐₗ. Formulated the yield functional Y[x; Φ] and validated non-monotonicity using Jensen's inequality:

Y[x; Φ] = ∫₀ᵀ f((1 - Φ(x, t)) ⋅ Rₛₒₗₐᵣ(t)) dt,
f(R) = 1 - e⁻⁽ᵅ⁾R

4. Differential Geometry and Shadow Continuity

Ensured smooth transitions in shadow profiles P(t) through differential geometry techniques:

P(t) = ⋃q=1ⁿ Uq(t),
Uq(t) = {x ∈ D | xqi ≤ x ≤ xqf}

Represented shadow dynamics as geodesic paths on the torus, reducing the complexity of combinatorial optimization in system design.

5. Integration of Thermal Dynamics

Developed coupled energy balance models to integrate the thermal effects of shading, highlighting its dual role in mitigating plant heat stress and enhancing PV panel efficiency. These models provide a deeper understanding of agrivoltaic system interactions.

Research Significance

This research introduces advanced mathematical tools to address the challenges of agrivoltaic system design, enabling:

Precise optimization of spatiotemporal solar resource distribution across higher-dimensional spaces.
Quantitative analysis of trade-offs between crop yield and renewable energy output under varying environmental conditions.
Integration of plant dynamics, radiation physics, and energy balance models into scalable agrivoltaic frameworks.