Mathematics

Mathematical analysis quantifies and describes the infinitesimal and asymptotic behavior of mathematical objects, such as functions. Familiar examples from calculus include limits, differentiation, and integration. Such topics are often studied in the context of real numbers, complex numbers, and their functions. They can be also studied in a more general setting, provided there is an appropriate notion of “nearness” (a topological space) or “distance” (a metric space). 
 
The applications of analysis are vast within mathematics and touch areas as seemingly diverse as differential equations and number theory. The utility of analysis reaches all the sciences, including physics, engineering, and economics. Faculty in our department study a wide array of analysis topics, such as geometric optics/electromagnetism, nonlinear waves, and nonlinear differential equations. Techniques used in their research range from Fourier analysis, bifurcation theory, functional analysis, all the way to topological degree theory.

Applied mathematics connects mathematical concepts and techniques to various fields of science and engineering. It is the application of mathematics to real-world problems with the dual goal of explaining observed phenomena and predicting new, as yet unobserved, phenomena. The problems come from various applications, such as physical and biological sciences, engineering, and social sciences.
 
Faculty in our department study a wide array of applied mathematics topics, such as infectious disease modeling, Ecological competition modeling, population genetics analysis, ecotoxicological risk assessment, social network analysis, computational neuroscience, machine learning, neuro-network analysis and viscoelasticity. Techniques used in their research include mathematical analysis, coalescence theory, fractional calculus, dynamical system, bifurcation analysis, differential equations, partial differential equations, stochastic differential equations, graph theory, simulation, and linear programming.

Discrete mathematics is a broad area of mathematics that deals with “individually separate and distinct” objects such as whole numbers (1, 2, 3, ...), partitions (breaking whole numbers into smaller pieces), permutations (shuffles of a list), and graphs (collections of edges and nodes). Some major themes of study are building objects from core objects (e.g. whole numbers are products of primes), understanding objects' underlying structures (e.g. cycles in graphs), studying various statistics on these objects (e.g. the distribution of prime numbers), and drawing connections between different objects (e.g. permutation and Young tableaux). Various tools and techniques are used in these studies: algebra such as polynomials and matrix representations, analysis such as generating functions and Fourier series, and probability such as expectation and variance.
 
Discrete mathematics has applications to computer science, cryptography, error-correcting codes, software engineering as well as to classical sciences such as chemistry, biology, and physics. Faculty in our department are especially interested in symmetric functions, posets and lattices, graph coloring, cycles in graphs, and analytic number theory.

Research in STEM (Science, Technology, Engineering, and Mathematics) Education explores a wide variety of issues related to teaching, learning, curriculum, assessment, and fostering change in STEM courses and programs. Faculty take an empirical approach (basic and applied) using qualitative, quantitative, or mixed-methods approaches that are informed by an expert understanding of STEM knowledge and practice.  Research in this area addresses complex problems that are investigated using a variety of interdisciplinary techniques to ultimately understand how students learn disciplinary ideas or how institutional structures and conditions facilitate broad implementation of research-based best practices.