Mathematics is a science that deals with the logic of form, quantity, and arrangement. It has evolved from counting, calculating, measuring, and the systematic study of the shapes and movements of physical objects. Through abstraction and logical reasoning, mathematicians look for patterns and develop new ideas and theories using pure logic and mathematical reasoning. Mathematics is all around us, in everything we do.
It is the cornerstone of our daily lives, from mobile devices to computers, software, architecture (ancient and modern), art, money, engineering, and even sports. On the contrary, mathematics focuses on abstract topics such as quantity (number theory), structure (algebra) and space (geometry). Mathematicians use tests to support their ideas instead of experiments or observations. A demonstration consists of a succession of applications of certain deductive rules to already known results, including theorems, axioms, and (in the case of abstraction from nature) some basic properties that are considered true starting points of the theory in question.
The result of a proof is called a theorem. Combinatorics has been used to study the enumeration problems that arise in pure mathematics within algebra, number theory, probability theory, topology and geometry, as well as many areas of applied mathematics. Solutions developed from processes have affected and improved the lives of many people. Modern areas of applied mathematics include mathematical physics, mathematical biology, control theory, aerospace engineering, and mathematical finance.
At Penn, long-standing collaborations between the departments of physics and astronomy and mathematics show the importance of interdisciplinary research that pushes traditional boundaries. Since the beginning of recorded history, mathematical discovery has been at the forefront of all civilized societies, and mathematics has been used by even the most primitive cultures. Primitive tribes needed little more than the ability to count but also used mathematics to calculate the position of the sun and the physics of hunting. The common approach in applied mathematics is to build a mathematical model of a phenomenon, solve the model, and develop recommendations for improving performance. Craig Lawrie and Ling Lin, a current and former postdoctoral student working with Cvetič and Heckman know firsthand the challenges and opportunities of working on a problem that combines cutting-edge mathematics and physics. In the mid-2000s, Donagi and Ovrut jointly directed a mathematics and physics program with Pantev and Grassi which was supported by the United States.
Faculty in both departments view the next generation of students and postdocs as “ambidextrous” with fundamental skills knowledge and intuition in both mathematics and physics. Rodrigo Barbosa also knows what it's like to work in different fields from mathematics to physics. Physics works by finding examples and describing solutions while in mathematics you try to see how general these equations are and how things fit together Barbosa says. For Donagi it was a chance encounter with Witten in the mid-1990s that led the mathematician to collaborate for the first time with a researcher outside of pure mathematics. Research in mathematics included first-year professors as well as postdoctoral and graduate students in physics.
