NewsApril 19, 2022

SEMO mathematics professor talks research

SEMO mathematics professor Daniel Daly calls himself a pure mathematician. For Daly, research in mathematics means proving theorems, properties and statements in mathematics no one has solved before. SEMO mathematics professor Daniel Daly calls himself a pure mathematician. For Daly, research in mathematics means proving theorems, properties and statements in mathematics no one has solved before.

Daly went into college as a computer science major and a mathematics minor. While completing his major, Daly took a class in abstract algebra, which steered him toward mathematics.

In Daly’s abstract algebra class, he was introduced to algebraic structures called groups, rings and fields. Daly said what interested him in abstract algebra was to name similarities and differences between object sets at the same time.

“I might be able to name commonalities and differences at the same time of whole numbers and polynomials,” Daly said.

Daly received his master's degree in mathematics and his Ph.D. in mathematics and computer science from the University of Denver.

For Daly’s doctoral thesis, he researched the Fibonacci numbers, which are permutation patterns or reordered numbers. To start his pattern, Daly adds the number one twice for a sum of two. Then, for the next sum, Daly adds one and two for a total of three. For the next term, he adds numbers two and three, for a sum of five. Daly continues this pattern until he reaches a solution.

However, Daly said he wanted to use the Fibonacci numbers to count the number of permutations occurring in a certain pattern.

“Here’s a permutation of the numbers one through four. There are several permutations from one up to that number. The container avoids certain subsequences. So for instance, three, two, five, one and four here have a subsequence that the first element is the biggest, the second element in the middle and the third element is the smallest. I call that a three-to-one pattern,” Daly said. “I can ask, ‘How many permutations, one through 10, contain this pattern three to one, how many of them don't contain [this pattern] or how many permutations avoid this pattern.”

Daly said he believes the most common misconception about mathematics is thinking “we know everything about mathematics” and there is “nothing new.” Daly said there are many open questions in mathematics and answers to questions we don’t have.

“A good research question in mathematics is one you've answered, opening up three or four more questions. You answer one question, and it inspires a couple more questions. And that's how mathematics grows,” Daly said.

Daly believes mathematics is one of the most creative and vast disciplines in academia. Daly said beyond algebra, geometry and calculus, “There's so much more.” Many will never experience the interesting topics in mathematics, he said, which is why he continues to research, ask questions, prove new theorems and find new mathematical properties.

For Daly, mathematical research is fun and challenging. Daly said finding a proof of a theorem or finding a counting result is fun.

“There's an aesthetic in mathematics and a notion of beauty. In mathematics, the proof of a theorem to a mathematician could be just as nice as a fine poem would be to an English major,” Daly said.

Daly said he enjoys how mathematical logic fits together, how steps follow one another and proofs or arguments lock into one another.

For more information or interest in mathematical research, visit Dr. Daly - Research in :60ish Seconds.

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