Most undergraduate students expect to wait until graduate school before tackling unsolved problems in mathematics. Manish Acharya didn’t want to wait.
Acharya, a rising senior studying computer science and mathematics, is spending his summer as a research intern at Aalto University in Finland, where he is working on fundamental open questions in theoretical computer science and computational geometry.
“Finding a project in theoretical computer science felt almost too good to be true,” Acharya said.“I get to spend my summer working on open research questions alongside experts in the field before even starting my Ph.D. journey.”
His work centers on a set of open problems from a recent algorithms paper on smallest convex intersecting polygons. Rather than building applications, he and his collaborators are investigating whether several geometric optimization problems admit exact polynomial-time solutions, whether some are NP-hard, and how these ideas might extend into three-dimensional space.The questions are, in a word, unsolved—and that’s exactly the point.
From empirical evidence to mathematical proof
Acharya’s turn toward theory grew out of a puzzle he couldn’t shake. During his junior year at Vanderbilt, he worked with Professor David Hyde on Bayesian Optimization for the Sliced Wasserstein Distance, research that was accepted to the International Conference on Learning Representations (ICLR) in 2026.
“We had strong empirical evidence that our method worked, but we couldn’t fully prove why,” Acharya said. “That experience stayed with me.”
That gap—between knowing something works and being able to prove why it works—is what pushed him toward theoretical computer science. He had always enjoyed the subject, but that project made the pursuit feel urgent.
Without a dedicated theoretical computer science research group at Vanderbilt, he had assumed open problems like these were out of reach until graduate school.
“As an undergraduate at Vanderbilt, I assumed I would have to wait until graduate school before working on these kinds of problems,” he said. This experience changed that.
No roadmap, just mathematics
At Aalto, a typical day looks less like writing code and more like filling whiteboards. Acharya begins each morning with a walk through the forest near his apartment before heading to the lab, where most of the day revolves around reading papers, discussing ideas with mentors, and testing approaches that may or may not hold up.
“Sometimes we spend hours pursuing an approach only to discover why it cannot work,” Acharya said.
“Other days, a small observation opens up an entirely new direction. It’s a constant process of learning, questioning, and refining ideas.”
He works closely with his supervisor, Professor Sándor Kisfaludi-Bak, and a postdoctoral researcher—a dynamic he describes as more collaborative than he expected.
“One of the most valuable lessons I’ve learned is that even when someone else’s idea doesn’t work, understanding why it fails often leads to a better idea,” he said. “Research feels much less like solving problems alone and much more like solving puzzles as a team.”
Mentors who made the path possible
The road to Finland ran through Vanderbilt’s research community.
Acharya began doing research during his freshman year under Dr. Yifan Zhang in Professor Yu Huang’s MIND Lab, where he published his first co-authored paper. Zhang, he said, taught him how to think like a researcher rather than simply solve problems. Driven by a growing interest in optimization research, he joined Professor Hyde’s SOL Lab. The opportunity to conduct independent research and participate in weekly graduate lab meetings alongside Ph.D. students gave him a unique perspective on academic life and elevated the depth of his research inquiries.
The SyBBURE Searle Undergraduate Research Program has also been central to his journey. Its support allowed him to attend conferences and pursue projects that would otherwise have been out of reach, a fact he says is especially meaningful as an international student.
The interdisciplinary path that led to theory
The College of Connected Computing’s emphasis on interdisciplinary exploration mirrors Acharya’s own trajectory. He began with AI research touching multiple scientific domains, and those early experiences helped him identify what he was actually drawn to: the underlying mathematics, not the applications.
“Connected Computing encourages students to explore different disciplines before finding where they can make the biggest impact,” Acharya said. “That’s exactly what happened to me, and it ultimately led me toward theoretical computer science.”
He plans to pursue a Ph.D. focused on algorithms and computational complexity and is hopeful the summer’s work will lead to a publication at a top theoretical computer science conference. When he returns to Vanderbilt, he will begin his Honors Thesis with Professor Mark Ellingham, continuing within discrete mathematics and theoretical computer science.
As for students who find the idea of research intimidating, Acharya’s advice is direct.
“Don’t wait until you feel ready,” Acharya said. “Research isn’t about already knowing the answers. It’s about being curious enough to ask questions and persistent enough to keep searching when the answers aren’t obvious.”