mathematics experts

mathematics experts

Advancements in Mathematical Research: A Comprehensive Overview

1. Introduction to Mathematical Research

The purpose of this introduction is to provide an overview of the state of mathematical research in 2020. Does this mean that in the past nobody has provided such an introduction? On the contrary, introductions are published, for instance, for the first book of a monograph series or for a special issue of a journal. But when we assemble a monograph series (and we have been doing this at regular intervals since 1993), we need to convince responsible mathematicians from several fields why their community deserves a new series of books and journal special issues and why mathematical research performed in a particular research area has reached a level of maturity that justifies this expense and effort. Thus there must be some comprehensive introduction to the field. If such an introduction exists, it is usually halfway towards an article of a special type about open problems and their behind-the-scene stories when a mathematician introduces an open problem at the beginning of a lecture to explain why these open problems are important and part of a particular scientific culture. There are also a number of books, roughly about ten pages of each book are devoted to the description of open problems in their research areas.

As part of our mission, we need to provide an overview of the state of the research at the beginning of each volume, an overview that has to be reliable and comprehensive. And with as many young researchers as there right now, this is a considerable challenge. With every new mathematical tool that is developed, we find several new enthusiastic researchers who try to apply this tool and thereby dig new tunnels and mines, thus opening new research areas. At the same time at every university worldwide, there is at least one class of first-year students that eagerly listens to the teacher, that notes down every word to be prepared for the next exam and to calculate, say, the volume of a solid of revolution around the x-axis. There are reasons to believe that the vast expansion of tentative new mines somewhere in the future is not matched by a corresponding discovery of new caverns, however. This is why when researchers in mathematics put their work in a larger context, they are always happy to do so. This work is going to be known and cited.

2. Cutting-Edge Techniques and Methods in Mathematics

Mathematical research has transcended numbers, with a vast array of mechanisms and tools guiding the way mathematicians, scientists, and engineers tackle new complex problems to advance their disciplines. Some of these methods and techniques aim at conveying information about properties of complex systems, solving difficult geometric and combinatorial problems, developing effective methods for solving difficult optimization problems, and establishing precise connections between parts of computer science and mathematics. Unveiling the inner workings of many innovative mathematical methods, from cellular automata in the 1940s to artificial intelligence in the 1950s, offers that the dynamic interactive networks these techniques rely on were originally rooted in the mathematically based theories of mathematical logic and discrete dynamical systems. Though diverse in their methodological techniques, they all coalesce on one unified and foundational vision: to pass from a class of techniques and methods that serve as “initial” methods for solving many problems in logic and computer science to a class of universal techniques whose expressive power fully captures the solutions to rich classes of problems in these disciplines.

These diverse techniques come in multiples. From linear programming – one of the oldest tools in optimization that has an intricate relation to polyhedra established by Dantzig in the 1940s – to dynamic programming applied to finding an appropriate global minimum as an effective tool in DNA and protein sequence alignment and transcription factor binding site prediction, to the divides at the crossroads of these techniques such as searching over cyclic orbits in discrete dynamical systems or exploiting precise connections between SAT and model checking, that move one way to model checking and formal verification, and yet another way to non-linear global optimization, these techniques together form a mature class of efficient, algorithmically-based techniques that support a broad-based set of problems representing at least three decades of core, underlying problems inherent to the discipline of computer science. More recently, research has led to the development of a new research base in these areas, primarily focused on decision problems, representation and indexing, for a broad class of counting and evaluating problems related to the conceptual structures of logic and computer science designed to perform effectively at solving NP-hard, computationally infeasible problems.

3. Applications of Mathematical Research in Various Fields

The applications of mathematics in various related fields have been addressed under some of the major branches of mathematics. Fields like medical science, engineering, astrophysics, computer graphics, etc. make use of mathematics in specific ways to accomplish their tasks. This begins right from the stage of designing the project, that is, one begins with the mathematics of and for the question. It is then followed by a phase of developing mathematical techniques and methods in order to solve the problem.

Mathematics occupies a prominent, sometimes central, role in a variety of fields. There has been an increasing tendency among applied mathematicians to work in, and for, interdisciplinary areas as well. The ways mathematics can be applied are numerous: in the development of new human diseases, the thought of mathematics comes to mind spontaneously. However, most of the areas of involvement of mathematics are carefully planned by man himself, the application often occurring through societal needs. In addition to undergraduate math majors, nearly every field of science and engineering will include people with strong mathematical foundations. Prominent among reasons that young scientists view mathematics as a central profession are the applications of mathematics that include intricate, unifying mathematics at frontiers in many sciences, and the appropriate use of high technology that allows numerical and graphical evaluations in nearly every applied study.

4. Challenges and Future Directions in Mathematical Research

Current advancements in mathematical research have been discussed. Implications and consequences were considered. In this section, the challenges faced in mathematical research, future directions, and emerging trends are discussed. In addition, the role of mathematics in the future, methods of increasing mathematical literacy, concise guidelines to readers interested in mathematical research and technology, the impact of information technology, and mathematical concepts and applications in molecular science were reviewed.

Challenges and future directions. Some challenging problems in mathematics are: How many primes are perfect numbers? As every even perfect number needs a prime, a key question is how infinitely many of these exist? Will every even perfect number be given by the formula from its Euler character? Or is this property only satisfied for 2, 3, 5, 7, and 13? Can the formula be checked, be used to prove for larger p that coordinates are abelian functions and their image points? Can there be an important (arbitrary) expansion vertex that most points can be described?