1 Introduction — Research Vision

My research is centered on algorithmic and complexity-theoretic questions motivated by hardness amplification, number-theoretic constructions, and combinatorial optimization problems such as orienteering variants. I aim to bridge deep theoretical advances (hardness magnification, recovery reductions) with algorithmic design and applied optimization, training students to contribute both theory and implementable methods. The core ideas and recent work are summarized below; full details and references appear in my curriculum vitae.

2 Current Research

2.1 Special Integrals

My research monograph Special Integrals (Springer Nature, 2025) [26R] grew from my explorations of integral representations for series and special functions in 1997 [01M]. It collects novel integral identities and techniques accessible to readers with calculus and complex analysis background; several identities appear to be new. The monograph establishes analytic frameworks that I am now leveraging in algorithmic number-theoretic investigations (see below).

2.2 Mathematical Techniques in Computational Complexity

Recent collaborative work develops new families of reductions and constructions that strengthen our understanding of average-case and rare-case hardness.

• Recovery reductions and random-noise model: In joint work (Nareddy & Mishra, 2025) [27T] we defined a random-noise model and used group-theoretic constructions (symmetric groups, permutation group algorithms) to produce robust deterministic recovery reductions for canonical NP-complete problems (SAT, k-SAT, CLIQUE). These reductions unify several hardness phenomena via a single group-theoretic black-box approach.

• Rare-case hard functions via number theory: Building on distribution of primes and Chinese remainder techniques, we constructed number-theoretic polynomials that are rare-case hard under standard complexity assumptions (Nareddy & Mishra, 2025) [25P]. This line connects analytic number theory with complexity-theoretic constructions.

• Hardness amplification through group theory: Our 2024 work [23P] explores group-action based amplification, resolving portions of an open question posed by Goldreich (2020) for counting modulo problems and providing limits for our techniques.

2.3 Multiple Set Orienteering Problem (mSOP) and Optimization

I have supervised and co-authored a string of papers developing modeling and algorithmic strategies for set-based orienteering variants and their multi-depot / multi-salesperson generalizations [24P, 22P, 21C, 20C, 19C, 18C]. The work includes integer-linear programming formulations, compact encodings, and heuristic / metaheuristic comparisons (recent ICORES and FICTA proceedings). These applied optimization projects provide fertile ground for student projects.

3 Future Research Plan (Five-Year Roadmap)

I plan a coordinated program along four themes that leverage existing results and expand into new directions:

3.1 Hardness Magnification Beyond Locality Barriers

Hardness magnification offers a pathway from modest lower bounds to dramatic complexity separations. I will seek new magnification results that bypass known locality barriers using structured reductions and combinatorial constructions, focusing on sparse instances of natural problems. Potential milestones: (i) identify candidate sparse problems amenable to magnification; (ii) design reductions preserving sparsity and structure; (iii) target a breakthrough conditional result linking modest lower bounds to circuit lower bounds.

3.2 Conditional Lower Bounds via Network Coding Conjecture

I will explore conditional lower bounds derived from the Network Coding Conjecture (NCC) and related communication / throughput conjectures. Applications include lower bounds for sorting, multiplication circuits, and function inversion in memory and external-storage models. This connects to data-structure lower bounds and practical implications for parallel / distributed systems.

3.3 Quantum Number-Theoretic Algorithms

Leveraging my mathematical background, I plan to investigate quantum algorithms for number-theoretic tasks (e.g., Gauss sum estimation, exponential congruences) and to adapt classical analytic techniques to hybrid classical–quantum algorithm design.

3.4 Additive Combinatorics and Average-Case Hardness

I will apply additive combinatorics and probabilistic constructions to push average-case hardness results, aiming to produce explicit worst-case to average-case reductions for families of algebraic problems. This theme complements the number-theoretic constructions and the hardness magnification agenda.

4 Conclusion

My research aims to integrate mathematical depth with algorithmic applicability — advancing complexity theory while producing methods that can be implemented and taught. Full references and an extended list of publications appear in my curriculum vitae.