"As the result of solving new problems, new questions arise; some of these questions can be solved with the existing knowledge, but some of these questions also push the boundaries of knowledge and become conjectures that may take anywhere from one to 300 years, or even greater, to solve."
Roselyn Elaine Williams is an American mathematician who is an Associate Professor and former chair of the mathematics department at Florida Agricultural and Mechanical University. Her decades long involvement in the National Association of Mathematicians includes a 14-year term as secretary-treasurer.
"Well, Hopf algebras is an area of algebra named after a mathematician whose last name is Hopf [Heinz Hopf], and it is a set with dual algebraic structures and algebraic structure is a set with binary operations such as addition and multiplication, and so this is fundamental mathemat--algebraic structure."
"But Hopf algebras is one in which it plays a dual role, okay? It plays a role as a vector field, and then it plays the role as the operators on a vector field so in mathematics terminology, it plays the role of elements as well as functions on elements. Now the application of Hopf algebras--its an area of pure mathematics."
"Mathematics is usually thought of in two ways, pure mathematics and applied mathematics. Pure mathematics is the extension of known mathematics because as you are learning more about math or science, it always opens up new questions, and thats how the sciences and the mathematics grow."