Through abstraction and logical reasoning, mathematics evolved from counting, calculating, measuring, and the systematic study of the shapes and movements of physical objects. Most mathematical activity involves discovering and demonstrating, through pure reasoning, the properties of abstract objects. These objects are abstractions of nature, such as natural numbers or lines, or, in modern mathematics, entities that are stipulated with certain properties, called axioms. 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. Mathematics is the science that deals with the logic of form, quantity and arrangement. Mathematics is all around us, in everything we do. It is the cornerstone of everything in our daily lives, including mobile devices, 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 look for patterns and develop new ideas and theories using pure logic and mathematical reasoning. Instead of experiments or observations, mathematicians use tests to support their ideas. 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.

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. Although complex mathematics involving pure and applied mathematics is beyond the comprehension of most people, 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. Lawrie adds that being able to work directly with mathematicians is also useful in their field, since understanding new mathematical research can be a challenge, even for researchers in theoretical physics.

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 and primitive cultures. The common approach in applied mathematics is to build a mathematical model of a phenomenon, solve the model, and develop recommendations for improving performance. 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.

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. Primitive tribes needed little more than the ability to count, but they also used mathematics to calculate the position of the sun and the physics of hunting. Rodrigo Barbosa also knows what it's like to work in different fields, in his case, 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.

Research in mathematics that included first and final year professors, as well as postdoctoral and graduate students in physics. 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. .