Research in Theoretical Computer Science: Passing the Torch to Next-Generation Mathematicians
Imagine the problem: Stranded in the middle of the sea. It’s dark. Somewhere out there is a shoreline – no clue where. Navigational instruments are dead, but a few search robots on hand. You’ll send them out. One will find the shoreline. What’s the most efficient trajectory: straight line, spiral or other curve?
It’s a theoretical problem, but a typical one that combines operations research, search theory and optimization. Associate Professor Konstantinos Georgiou’s speciality is in finding mathematical solutions.
Much of Georgiou’s work straddles abstract mathematics and theoretical computer science. It’s a fast-paced field where competition is stiff and impressive results are no guarantee of visibility. Yet, amid this climate, Georgiou is producing a steady stream of published works – and he’s accomplishing it alongside research trainees at all levels of university studies.
Exploration at the Mathematics – Computer Science Interface
Georgiou is part of Graphs@Ryerson, a group that researches topics under the umbrella of discrete mathematics, the underlying language of computer science. His research encompasses combinatorics, convex optimization, distributed algorithms and game theory.
One of Georgiou’s main themes is optimization. The concept has ancient roots, but the technological explosion over the last century has given it new meaning and application. Optimization problems arise in environments where a function must be minimized or maximized in the face of limited resources. It could be maximizing output or minimizing time despite constraints on computational power, storage capacity, entities that don’t communicate well or other impediments to efficiency.
“Any of these can become overwhelming obstacles,” Georgiou explains. “Our job is to evaluate proposed solutions, quantify the outermost bounds of potential effectiveness, and then identify the best option to move forward. But the other important piece is knowing exactly why we cannot improve beyond that.”
This latter objective was the thrust of Georgiou’s most recently published work (external link) for the Conference on Principles of Distributed Systems (OPODIS). With student collaborators at the Bachelor’s, Master’s and PhD levels, Georgiou’s team closed the book on a decades-old mathematical problem – the shoreline scenario described at the outset.
Student Collaborators on the Conference Circuit
The original research question was how to minimize time required in finding a hidden object. The shoreline and robots were metaphors, but applications abound in computer science or even literal scenarios such as sending out mobile agents to locate concealed intruders.
A solution was proposed back in the early 90’s, but no one had since verified if it could be improved upon. Georgiou’s team re-examined the problem and established mathematically that the solution held and no further optimization was possible. The case could now be closed – always a satisfying feeling for mathematicians.
But the real success for Georgiou was in the team’s diverse composition and development. “I invest a lot in my students. The majority of them – including undergrads – publish with me. Each one brings different gifts to the whiteboard: skills, experience, fresh ideas. That’s how collaboration works and people evolve as researchers.”
Considering the rigour of the conference’s double-blind, peer review process, Georgiou reflects: “The paper doesn’t show the countless hours we all spent making progress together. The first few months, they see difficulties and may have no clue. Over time, an amazing transformation occurs. They develop better reasoning, and problems become interesting. They start as my students, but they turn into my collaborators. It’s a very nice process.”
Optimism for Research and Collaboration
Mathematics and computer science will only grow in importance through the 21st century, and Georgiou is poised to continue pushing results further. Looking ahead, he sees a bright future for research at Ryerson: “In the Mathematics Department, we have a very friendly environment,” he reflects appreciatively. “Although mathematicians sometimes spend many hours working in isolation, we enjoy very good relationships here. I couldn’t do what I do without the support of my colleagues, the University and research funding agencies.”
Meanwhile, Georgiou continues dedicating time and attention to nurturing next-generation mathematicians. “One thing we keep improving here is the quality of our students. When I was a trainee and finished a paper, I felt really proud. I want to offer the same to my students. There’s more pleasure discussing ideas with others. For me, that’s the reward at the end of the day.”
