( also available)
( also available) |
Full-time: 3–4 years Part-time: 6–8 years |
February and October |
January to April |
The Pure Mathematics Group in the School of Mathematics and Statistics is made up of the Combinatorics and Algebra Group and the Dynamical Systems Groups.
Minimum 2:1 undergraduate degree (or equivalent). If you are not a UK citizen, you may need to prove your knowledge of English .
Applicants are strongly encouraged to apply for one of the research projects listed on the School of Mathematics and Statistics PhD recruitment page . The themes listed below indicate general topics where research projects may be available.
Dynamical systems
Combinatorics and algebra
UK fee | International fee |
---|---|
Full time: £4,786 per year | Full time: £15,698 per year |
Part time: £2,393 per year | Part time: £7,849 per year |
Some of our research students are funded via the EPSRC Mathematical Sciences Doctoral Training Partnership , some are funded by University studentships, others are self-funded.
For detailed information about fees and funding, visit Fees and studentships .
To see current funded studentship vacancies across all research areas, see Current studentships .
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Department members engage in cutting-edge research on a wide variety of topics in mathematics and its applications. Topics continually evolve to reflect emerging interests and developments, but can roughly grouped into the following areas.
Algebra, combinatorics, and geometry are areas of very active research at the University of Pittsburgh.
The research of the analysis group covers functional analysis, harmonic analysis, several complex variables, partial differential equations, and analysis on metric and Carnot-Caratheodory spaces.
The department is a leader in the analysis of systems of nonlinear differential equations and dynamical systems that arise in modeling a variety of physical phenomena. They include problems in biology, chemistry, phase transitions, fluid flow, flame propagation, diffusion processes, and pattern formation in nonlinear stochastic partial differential equations.
The biological world stands as the next great frontier for mathematical modeling and analysis. This group studies complex systems and dynamics arising in various biological phenomena.
A rapidly growing area of mathematical finance is Quantitative Behavioral Finance. The high-tech boom and bust of the late 1990s followed by the housing and financial upheavals of 2008 have made a convincing case for the necessity of adopting broader assumptions in finance.
The diversity of this group is reflected in its research interests: numerical analysis of partial differential equations , adaptive methods for scientific computing, computational methods of fluid dynamics and turbulence, numerical solution of nonlinear problems arising from porous media flow and transport, optimal control, and simulation of stochastic reaction diffusion systems.
Research in analytic topology continues in the broad area of generalized metric spaces. This group studies relativity theory and differential geometry, with emphasis on twistor methods, as well as geometric and topological aspects of quantum field theory, string theory, and M-theory.
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Applied mathematics is the application of mathematical techniques to describe real-world systems and solve technologically relevant problems. This can include the mechanics of a moving body, the statistics governing the atoms in a gas or developing more efficient algorithms for computational analysis. These ideas are closely linked with those of theoretical physics.
In recent years, there has been a surge of interest in the design of mechanical metamaterials for different science and engineering applications. In particular, various computational approaches have been developed to facilitate the systematic design of art-inspired metamaterials including origami and kirigami metamaterials. In this Comment, we highlight the recent advances and discuss the outlook for the computational design of art-inspired metamaterials.
50 years ago Roger Penrose described a set of aperiodic tilings, now named after him, that have fascinated artists, mathematicians and physicists ever since.
Spatial heterogeneity in disease transmission rates and in mixing patterns between regions makes predicting epidemic trajectories hard. Quantifying the mixing rates within and between spatial regions can improve predictions.
Morphing soft matter, which is capable of changing its shape and function in response to stimuli, has wide-ranging applications in robotics, medicine and biology. Recently, computational models have accelerated its development. Here, we highlight advances and challenges in developing computational techniques, and explore the potential applications enabled by such models.
Long ties that bridge socially separate regions of networks are critical for the spread of contagions, such as innovations or adoptions of new norms. Contrary to previous thinking, long ties have now been found to accelerate social contagions, even for behaviours that involve the social reinforcement of adoption by network neighbours.
The area under the receiver operating characteristic curve (AUROC) of the test set is used throughout machine learning (ML) for assessing a model’s performance. However, when concordance is not the only ambition, this gives only a partial insight into performance, masking distribution shifts of model outputs and model instability.
Mathematics is a vast and fascinating field that encompasses a wide range of topics and research areas. Whether you are an undergraduate student, graduate student, or a professional mathematician, engaging in math research opens doors to exploration, discovery, and the advancement of knowledge. The world of math research is filled with exciting challenges, unsolved problems, and groundbreaking ideas waiting to be explored.
In this guide, we will delve into the realm of math research topics, providing you with a glimpse into the diverse areas of mathematical inquiry. From pure mathematics to applied mathematics, this guide will present a variety of research areas that span different branches and interdisciplinary intersections. Whether you are interested in algebra, analysis, geometry, number theory, statistics, or computational mathematics, there is a wealth of captivating topics to consider.
Math research topics are not only intellectually stimulating but also have significant real-world applications. Mathematical discoveries and advancements underpin various fields such as engineering, physics, computer science, finance, cryptography, and data analysis. By immersing yourself in math research, you have the opportunity to contribute to the development of these applications and make a meaningful impact on society.
Throughout this guide, we will explore different research areas, discuss their significance, and provide insights into potential research questions and directions. However, keep in mind that this is not an exhaustive list, and there are countless other exciting topics awaiting exploration.
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Table of Contents
Math research is the process of investigating new mathematical problems and developing new mathematical theories. It is a vital part of mathematics, as it helps to expand our understanding of the world and to develop new mathematical tools that can be used in other fields, such as science, engineering, and technology.
Math research is a challenging but rewarding endeavor. It requires a deep understanding of mathematics and a strong ability to think logically and creatively. Math researchers must be able to identify new problems, develop new ideas, and prove their ideas correct.
There are many different ways to get involved in math research. One way is to attend a math research conference. Another way is to join a math research group. You can also get involved in math research by working on a math research project with a mentor.
A few examples of math research topics:
Number theory is a branch of mathematics that studies the properties of integers and other related objects. It is a vast and active field of research, with many open problems that have yet to be solved. Some of the current research topics in number theory include:
This is one of the most important unsolved problems in mathematics. It states that the non-trivial zeros of the Riemann zeta function have real part 1/2.
This conjecture relates the zeta function of an elliptic curve to the behavior of its rational points.
This is a vast program in number theory that seeks to unify many different areas of the field.
This is a complete classification of all finite simple groups, which are the building blocks of all other finite groups.
Number theory is used in many cryptographic algorithms, such as RSA and Diffie-Hellman.
Prime numbers are fundamental to number theory, and there are many open problems related to them, such as the Goldbach conjecture and the twin prime conjecture.
This is a branch of number theory that studies the properties of algebraic numbers, which are roots of polynomials with integer coefficients.
This is a branch of number theory that uses tools from combinatorics to study problems in number theory.
This is a branch of number theory that uses computers to solve problems in number theory.
These are just a few of the many research topics in number theory. The field is constantly evolving, and new problems are being discovered all the time.
Topology is a branch of mathematics that studies the properties of spaces that are preserved under continuous deformations. Some of the most important research topics in topology include:
This branch of topology studies topological spaces using algebraic tools, such as homology and cohomology. Algebraic topology has been used to great effect in the study of knot theory, 3-manifolds, and other important topological spaces.
This branch of topology studies topological spaces using geometric tools, such as triangulations and manifolds. Geometric topology has been used to great effect in the study of surfaces, 3-manifolds, and other important topological spaces.
This branch of topology studies topological spaces using differential geometry. Differential topology has been used to great effect in the study of manifolds, including the study of their smooth structures and their underlying topological structures.
This branch of topology studies knots, which are closed curves in 3-space. Knot theory has applications in many other areas of mathematics, including physics, chemistry, and computer science.
This branch of topology studies topological spaces of low dimension, such as surfaces and 3-manifolds. Low-dimensional topology has been used to great effect in the study of knot theory, 3-manifolds, and other important topological spaces.
This branch of mathematics studies the relationship between topology and quantum field theory. Topological quantum field theory has applications in many areas of physics, including string theory and quantum gravity.
This branch of mathematics studies the use of topological methods to analyze data. Topological data analysis has applications in many areas, including machine learning, computer vision, and bioinformatics.
These are just a few of the many research topics in topology. Topology is a vast and growing field, and there are many exciting new directions for research.
Differential geometry is a branch of mathematics that studies the geometry of smooth manifolds. Some of the most important research topics in differential geometry include:
This branch of differential geometry studies Riemannian manifolds, which are smooth manifolds equipped with a Riemannian metric. Riemannian geometry has applications in many areas of mathematics, including physics, chemistry, and computer science.
This branch of differential geometry studies complex manifolds, which are smooth manifolds that are holomorphically equivalent to a complex vector space. Complex geometry has applications in many areas of mathematics, including physics, chemistry, and computer science.
This branch of differential geometry studies the interplay between differential geometry and analysis. Geometric analysis has applications in many areas of mathematics, including physics, chemistry, and computer science.
This branch of mathematics uses differential geometry to study physical systems. Mathematical physics has applications in many areas of physics, including general relativity, quantum field theory, and string theory.
This field of computer science uses differential geometry to create realistic images and animations. Computer graphics has applications in many areas, including video games, movies, and simulations.
This field of medicine uses differential geometry to create images of the human body. Medical imaging has applications in many areas, including diagnosis, treatment, and research.
These are just a few of the many research topics in differential geometry. Differential geometry is a vast and growing field, and there are many exciting new directions for research.
Algebraic geometry is a branch of mathematics that studies geometric objects using the tools of abstract algebra. Some of the most important research topics in algebraic geometry include:
This branch of algebraic geometry studies curves, which are one-dimensional algebraic varieties. Algebraic curves have applications in many areas of mathematics, including number theory, representation theory, and mathematical physics.
This branch of algebraic geometry studies surfaces, which are two-dimensional algebraic varieties. Algebraic surfaces have applications in many areas of mathematics, including topology, differential geometry, and number theory.
This branch of algebraic geometry studies threefolds, which are three-dimensional algebraic varieties. Algebraic threefolds have applications in many areas of mathematics, including topology, differential geometry, and number theory.
This branch of algebraic geometry studies varieties, which are arbitrary-dimensional algebraic sets. Algebraic varieties have applications in many areas of mathematics, including topology, differential geometry, and number theory.
This branch of algebraic geometry studies groups that are also algebraic varieties. Algebraic groups have applications in many areas of mathematics, including number theory, representation theory, and mathematical physics.
This branch of algebraic geometry studies moduli spaces, which are spaces that parameterize objects of a certain type. Moduli spaces have applications in many areas of mathematics, including number theory, representation theory, and mathematical physics.
This branch of algebraic geometry studies the intersection of algebraic geometry and number theory. Arithmetic geometry has applications in many areas of mathematics, including number theory, representation theory, and mathematical physics.
This branch of algebraic geometry studies algebraic varieties over the complex numbers. Complex algebraic geometry has applications in many areas of mathematics, including topology, differential geometry, and mathematical physics.
This branch of algebraic geometry studies the intersection of algebraic geometry and combinatorics. Algebraic combinatorics has applications in many areas of mathematics, including combinatorics, computer science, and mathematical physics.
These are just a few of the many research topics in algebraic geometry. Algebraic geometry is a vast and growing field, and there are many exciting new directions for research.
Mathematical physics is a field of study that uses the tools of mathematics to study physical systems. Some of the most important research topics in mathematical physics include:
This branch of physics studies the behavior of matter and energy at the atomic and subatomic level. Quantum mechanics has applications in many areas of physics, including chemistry, biology, and engineering.
This branch of physics studies the relationship between space and time. Relativity has applications in many areas of physics, including cosmology, astrophysics, and nuclear physics.
This branch of physics studies the behavior of systems of many particles. Statistical mechanics has applications in many areas of physics, including thermodynamics, chemistry, and biology.
This branch of physics studies the behavior of systems that are sensitive to initial conditions. Chaos theory has applications in many areas of physics, including meteorology, economics, and biology.
This field of mathematics uses the tools of mathematics to study financial markets. Mathematical finance has applications in many areas of finance, including investment banking, insurance, and risk management.
This field of mathematics uses the tools of mathematics to solve physical problems. Computational physics has applications in many areas of physics, including materials science, engineering, and medicine.
This field of mathematics uses the tools of mathematics to study biological systems. Mathematical biology has applications in many areas of biology, including genetics, ecology, and evolution.
This field of mathematics uses the tools of mathematics to study chemical systems. Mathematical chemistry has applications in many areas of chemistry, including materials science, biochemistry, and pharmacology.
This field of mathematics uses the tools of mathematics to study engineering systems. Mathematical engineering has applications in many areas of engineering, including civil engineering, mechanical engineering, and electrical engineering.
These are just a few of the many research topics in mathematical physics. Mathematical physics is a vast and growing field, and there are many exciting new directions for research.
Mathematical biology is a field of study that uses the tools of mathematics to study biological systems. Some of the most important research topics in mathematical biology include:
This branch of mathematical biology uses mathematical models to study the behavior of biological systems. Mathematical models can be used to understand the dynamics of biological systems, to predict how they will respond to changes in their environment, and to design new interventions to improve their health.
This field of mathematical biology uses computational methods to study biological systems. Computational methods can be used to analyze large amounts of biological data, to simulate biological systems, and to design new experiments.
This field of mathematical biology uses statistical methods to study biological data. Biostatistical methods can be used to identify patterns in biological data, to test hypotheses about biological systems, and to design clinical trials.
This field of mathematical biology uses mathematical models to study the spread of diseases. Mathematical models can be used to predict the course of an epidemic, to design public health interventions, and to assess the effectiveness of those interventions.
This field of mathematical biology uses mathematical models to study the interactions between species in an ecosystem. Mathematical models can be used to predict how ecosystems will respond to changes in their environment, to design conservation strategies, and to assess the effectiveness of those strategies.
This field of mathematical biology uses mathematical models to study the nervous system. Mathematical models can be used to understand how the nervous system works, to design new treatments for neurological disorders, and to assess the effectiveness of those treatments.
This field of mathematical biology uses mathematical models to study genetics. Mathematical models can be used to understand how genes work, to design new treatments for genetic disorders, and to assess the effectiveness of those treatments.
This field of mathematical biology uses mathematical models to study evolution. Mathematical models can be used to understand how evolution works, to design new conservation strategies, and to assess the effectiveness of those strategies.
These are just a few of the many research topics in mathematical biology. Mathematical biology is a vast and growing field, and there are many exciting new directions for research.
Mathematical finance is a field of study that uses the tools of mathematics to study financial markets. Some of the most important research topics in mathematical finance include:
This branch of mathematical finance studies the prices of assets, such as stocks, bonds, and options. Asset pricing models are used to price new financial products, to manage risk, and to make investment decisions.
This branch of mathematical finance studies how to allocate money between different assets in a portfolio. Portfolio optimization models are used to maximize returns, to minimize risk, and to achieve other investment goals.
This branch of mathematical finance studies the prices of derivatives, such as options and futures. Derivatives are used to hedge risk, to speculate on future prices, and to generate income.
This branch of mathematical finance studies how to measure and manage risk. Risk management models are used to identify and quantify risks, to develop strategies to mitigate risks, and to comply with regulations.
This branch of mathematical finance studies the structure and dynamics of financial markets. Market microstructure models are used to understand how markets work, to design new trading systems, and to improve market efficiency.
This branch of mathematical finance uses statistical methods to study financial data. Financial econometrics models are used to identify patterns in financial data, to test hypotheses about financial markets, and to forecast future prices.
This field of mathematical finance uses computational methods to solve financial problems. Computational finance methods are used to price financial products, to manage risk, and to simulate financial markets.
This field of mathematical finance uses machine learning methods to study financial markets and to make financial predictions. Machine learning methods are used to identify patterns in financial data, to predict future prices, and to develop new trading strategies.
These are just a few of the many research topics in mathematical finance. Mathematical finance is a vast and growing field, and there are many exciting new directions for research.
Numerical analysis is a branch of mathematics that deals with the approximation of functions and solutions to differential equations using numerical methods. Some of the most important research topics in numerical analysis include:
This branch of numerical analysis studies the errors that are introduced when approximate solutions are used to represent exact solutions. Error analysis is used to design numerical methods that are accurate and efficient.
This branch of numerical analysis studies the stability of numerical methods. Stability analysis is used to design numerical methods that are guaranteed to converge to the correct solution.
This branch of numerical analysis studies the convergence of numerical methods. Convergence analysis is used to design numerical methods that will converge to the correct solution in a finite number of steps.
This branch of numerical analysis studies adaptive methods. Adaptive methods are numerical methods that can automatically adjust their step size or mesh size to improve accuracy.
This branch of numerical analysis studies parallel methods. Parallel methods are numerical methods that can be used to solve problems on multiple processors.
This branch of numerical analysis studies heterogeneous computing. Heterogeneous computing is the use of multiple processors with different architectures to solve problems.
This branch of numerical analysis studies nonlinear problems. Nonlinear problems are problems that cannot be solved using linear methods.
This branch of numerical analysis studies methods for finding the best solution to a problem. Optimization methods are used to find the best parameters for a numerical method, to find the best solution to a problem, and to find the best way to solve a problem.
This branch of numerical analysis studies the use of numerical methods to solve problems in science and engineering. Scientific computing is used to solve problems in areas such as physics, chemistry, biology, and engineering.
This branch of numerical analysis studies the use of numerical methods to solve problems in physics. Computational physics is used to solve problems in areas such as fluid dynamics, solid mechanics, and quantum mechanics.
This branch of numerical analysis studies the use of numerical methods to solve problems in chemistry. Computational chemistry is used to solve problems in areas such as molecular dynamics, quantum chemistry, and materials science.
This branch of numerical analysis studies the use of numerical methods to solve problems in biology. Computational biology is used to solve problems in areas such as genetics, molecular biology, and neuroscience.
These are just a few of the many research topics in numerical analysis. Numerical analysis is a vast and growing field, and there are many exciting new directions for research.
Probability is a branch of mathematics that deals with the analysis of random phenomena. Some of the most important research topics in probability include:
This branch of probability studies the axioms and foundations of probability theory. Foundations of probability is important for understanding the basic concepts of probability and for developing new probability theories.
This branch of probability studies the evolution of random phenomena over time. Stochastic processes are used to model a wide variety of phenomena, such as stock prices, traffic patterns, and disease outbreaks.
This branch of probability studies graphs whose vertices and edges are chosen randomly. Random graphs are used to model a wide variety of networks, such as social networks, computer networks, and biological networks.
This branch of probability studies stochastic processes whose future state depends only on its current state. Markov chains are used to model a wide variety of phenomena, such as queuing systems, genetics, and epidemiology.
This branch of probability studies the behavior of queues. Queues are used to model a wide variety of systems, such as call centers, hospitals, and traffic systems.
This branch of probability studies the problem of choosing when to stop a stochastic process. Optimal stopping theory is used to make decisions in a wide variety of situations, such as gambling, investing, and medical diagnosis.
This branch of probability studies the quantification and manipulation of information. Information theory is used in a wide variety of fields, such as communication, cryptography, and machine learning.
This branch of probability studies the use of computers to solve probability problems. Computational probability is used to solve a wide variety of problems, such as simulating random phenomena, computing probabilities, and designing algorithms .
This branch of probability studies the use of probability in other fields, such as physics, chemistry, biology, and economics. Applied probability is used to solve a wide variety of problems in these fields.
These are just a few of the many research topics in probability. Probability is a vast and growing field, and there are many exciting new directions for research.
Statistics is a field of study that deals with the collection, analysis, interpretation, presentation, and organization of data. Some of the most important research topics in statistics include:
This branch of statistics studies the analysis of large and complex datasets. Big data is used in a wide variety of fields, such as business, finance, healthcare, and government.
This branch of statistics studies the development of algorithms that can learn from data without being explicitly programmed. Machine learning is used in a wide variety of fields, such as natural language processing, computer vision, and fraud detection.
This branch of statistics studies the extraction of knowledge from data. Data mining is used in a wide variety of fields, such as marketing, customer relationship management, and fraud detection.
This branch of statistics uses Bayes’ theorem to update beliefs in the face of new evidence. Bayesian statistics is used in a wide variety of fields, such as medical diagnosis, finance, and weather forecasting.
This branch of statistics uses methods that do not make assumptions about the distribution of the data. Nonparametric statistics is used in a wide variety of fields, such as social science, medical research, and environmental science.
This branch of statistics studies the analysis of data that has multiple variables. Multivariate statistics is used in a wide variety of fields, such as marketing, finance, and environmental science.
This branch of statistics studies the analysis of data that changes over time. Time series analysis is used in a wide variety of fields, such as economics, finance, and meteorology.
This branch of statistics studies the analysis of data that records the time until an event occurs. Survival analysis is used in a wide variety of fields, such as medical research, epidemiology, and finance.
This branch of statistics studies the methods used to ensure that products or services meet a certain level of quality. Quality control is used in a wide variety of fields, such as manufacturing, healthcare, and government.
These are just a few of the many research topics in statistics. Statistics is a vast and growing field, and there are many exciting new directions for research.
Here are some tips on how to find math research topics:
They will be able to give you insights into current research in your area of interest and help you identify potential topics.
This will help you stay up-to-date on the latest research and identify areas where you could make a contribution.
This is a great way to meet other mathematicians and learn about their research.
What are you curious about? What do you want to learn more about? These can be great starting points for research topics.
Don’t be afraid to ask for help. If you’re struggling to find a research topic, talk to your professors, advisors, or other mathematicians. They will be happy to help you get started.
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Getting started with math research can be daunting, but it doesn’t have to be. Here are some tips to help you get started:
A mentor can help you find a research topic, develop your research skills, and navigate the research process. Talk to your professors, advisors, or other mathematicians to find someone who is interested in your research interests.
Read articles, books, and papers on your topic. Talk to experts in the field. The more you know about your topic, the better equipped you will be to conduct research.
A research plan will help you stay organized and on track. It should include your research goals, methods, and timeline.
Research can be a slow and challenging process. Don’t get discouraged if you don’t make progress immediately. Just keep working hard and you will eventually reach your goals.
Don’t try to tackle too much at once. Start with a small research project that you can complete in a reasonable amount of time.
Share your work with others and get their feedback. This will help you identify areas where you can improve.
If you’re struggling with something, don’t be afraid to ask for help from your mentor, advisor, or other mathematicians.
Research can be a rewarding experience. By following these tips, you can increase your chances of success.
In conclusion, exploring math research topics provides an opportunity to delve into the fascinating world of mathematics and contribute to its advancement.
The wide range of potential research areas ensures that there is something for everyone, whether you are interested in pure mathematics, applied mathematics, or interdisciplinary studies. By engaging in math research, you can deepen your understanding of mathematical principles, develop problem-solving skills, and contribute to the collective knowledge of the field.
Remember to choose a research topic that aligns with your interests and goals, and seek guidance from mentors and experts in the field to maximize your research potential. Embrace the challenge, curiosity, and creativity that math research offers, and embark on a journey that can lead to exciting discoveries and breakthroughs in the realm of mathematics.
How do i choose a math research topic.
When choosing a math research topic, consider your interests, background knowledge, and future goals. Explore various branches of mathematics and identify areas that intrigue you. Additionally, consult with professors, mentors, and professionals in the field for guidance and suggestions.
Yes, many universities and research institutions offer opportunities for undergraduate students to engage in math research. Reach out to your professors or department advisors to inquire about available research programs or projects suitable for undergraduates.
Math research is a constantly evolving field. Some emerging areas include computational mathematics, data science, cryptography, mathematical biology, quantum computing, and mathematical physics. Staying updated with current research trends and attending conferences or seminars can help you identify new and exciting research avenues.
Effective math research involves a systematic approach. Start by thoroughly understanding the existing literature on your chosen topic. Develop clear research questions and hypotheses, and apply appropriate mathematical techniques and methodologies.
Absolutely! Math research has numerous real-world applications in fields such as engineering, finance, computer science, cryptography, data analysis, and physics. Mathematical models and algorithms play a crucial role in solving complex problems and optimizing various processes in diverse industries.
Utilize academic journals, online databases, research papers, books, and mathematical software to access relevant information and tools. Libraries, online platforms, and research institutions also provide access to valuable resources and databases specific to mathematical research.
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Pure mathematics, also known as abstract mathematics, is the study of mathematical concepts without regard to real-world application.
This is in contrast to uses of mathematics specifically for problem solving, as is the case for example in applied mathematics and concrete mathematics .
More things to try:
Weisstein, Eric W. "Pure Mathematics." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/PureMathematics.html
Faculty and students interested in the applications of mathematics are an integral part of the department of mathematics; there is no formal separation between pure and applied mathematics, and the department takes pride in the many ways in which they enrich each other. we also benefit tremendously from close collaborations with faculty and students in other departments at uc berkeley as well as scientists at lawrence berkeley national laboratory and visitors to the mathematical sciences research institute.
The Department regularly offers courses in ordinary and partial differential equations and their numerical solution, discrete applied mathematics, the methods of mathematical physics, mathematical biology, the mathematical aspects of fluid and solid mechanics, approximation theory, scientific computing, numerical linear algebra, and mathematical aspects of computer science. Courses in probability theory, stochastic processes, data analysis and bioinformatics are offered by the Department of Statistics, while courses in combinatorial and convex optimization are offered by the Department of Industrial Engineering and Operations Research. Our students are encouraged to take courses of mathematical interest in these and many other departments. Topics explored intensively by our faculty and students in recent years include scientific computation and the mathematical aspects of quantum theory, computational genomics, image processing and medical imaging, inverse problems, combinatorial optimization, control, robotics, shape optimization, turbulence, hurricanes, microchip failure, MEMS, biodemography, population genetics, phylogenetics, and computational approaches to historical linguistics. Within the department we also have a Laboratory for Mathematical and Computational Biology .
Chair: Per-Olof Persson
Senate faculty, graduate students, visiting faculty, meet our faculty, mina aganagic, david aldous, robert m. anderson, sunčica čanić, jennifer chayes, alexandre j. chorin, paul concus, james w. demmel, l. craig evans, steven n. evans, f. alberto grünbaum, venkatesan guruswami, ole h. hald, william m. kahan, richard karp, michael j. klass, hendrik w. lenstra, jr., lin lin (林霖), michael j. lindsey, c. keith miller, john c. neu, beresford n. parlett.
Mathematics is a broad field that branches into different areas, with Pure Mathematics and Applied Mathematics being two of the most distinct and significant divisions. Pure Mathematics focuses on abstract concepts, theories, and problems without immediate concern for real-world applications. On the other hand, Applied Mathematics takes these mathematical theories and uses them to solve practical problems in various fields such as engineering, physics, finance, and computer science.
In this article, we will discuss key differences between Pure and Applied Mathematics including career opportunities.
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Key concepts in pure mathematics, what is applied mathematics, key concepts in engineering mathematics, examples of applied mathematics, difference between pure and applied mathematics, career opportunities for pure and applied mathematics.
Pure Mathematics is the branch of mathematics that is concerned with abstract concepts and theoretical frameworks rather than practical applications.
Unlike applied mathematics, which focuses on solving real-world problems, pure mathematics is driven by the exploration of mathematical structures, relationships, and theories for their own sake.
Key areas in pure mathematics include algebra, geometry, number theory, topology, and analysis. These fields study fundamental mathematical concepts such as numbers, shapes, spaces, and functions, often without immediate concern for how they might be used in the physical world.
This section outlines the core areas of Pure Mathematics, which includes:
Applied Mathematics is a discipline within mathematics that focuses on using mathematical methods and techniques to solve practical problems in a wide range of fields, including science, engineering, economics, business, and technology.
Unlike pure mathematics, which is concerned with abstract concepts and theoretical frameworks, applied mathematics is grounded in real-world applications.
This section provides an overview of the fundamental concepts in Engineering Mathematics, which are essential for solving complex engineering problems:
The difference between Pure Mathematics and Applied Mathematics lies in their focus, objectives, and applications. Some of the common differences are listed in the following table:
Concerned with abstract concepts, theories, and mathematical structures. | Focuses on practical application of mathematical techniques to solve real-world problems. | |
Develops and explores mathematical theories without necessarily seeking immediate practical use. | Utilizes mathematical methods to address and solve problems in various fields like science, engineering, economics, and technology. | |
Emphasizes rigor, proof, and the development of new mathematical theories. | Emphasizes the creation of mathematical models and methods for practical use and problem-solving. | |
Includes areas like algebra, number theory, topology, and analysis. | Includes areas like numerical analysis, statistics, optimization, and mathematical modeling. | |
Primarily academic and research-oriented roles such as university professor, research mathematician, or cryptographer. | Involves careers in industry, engineering, technology, and data science, such as data scientist, engineer, or operations researcher. | |
Rarely involves direct application to real-world problems; more focused on theoretical advancements. | Directly applied to solve practical problems in diverse industries, including engineering, finance, medicine, and technology. | |
Relies heavily on theoretical proofs and logical deductions. | Utilizes computational methods, simulations, and applied techniques like differential equations and statistical analysis. | |
Studying the properties of prime numbers or proving theorems in geometry. | Modeling the spread of diseases, optimizing supply chains, or designing algorithms for machine learning. |
There are various career opportunities in both Pure and Applied Mathematics:
Field | Pure Mathematics Career Opportunities | Applied Mathematics Career Opportunities |
Academic and Research | University Professor | University Professor (Applied) |
Research Mathematician | Research Scientist | |
Postdoctoral Researcher | Postdoctoral Researcher (Applied) | |
Data Science and Analytics | Data Scientist | Data Scientist |
Quantitative Analyst | Machine Learning Engineer | |
Cryptography | Cryptographer | Cryptographer |
Security Analyst | Security Analyst | |
Software Development | Algorithm Developer | Algorithm Developer |
Software Engineer | Software Engineer | |
Actuarial Science | Actuary | Actuary |
Risk Analyst | Risk Analyst | |
Government and Public Policy | Statistician | Statistician |
Policy Analyst | Operations Research Analyst | |
Engineering | N/A | Systems Engineer |
Industrial Engineer | ||
Finance and Economics | Financial Analyst | Financial Analyst |
Quantitative Finance Specialist | Quantitative Finance Specialist | |
Technology and Computer Science | N/A | Machine Learning Engineer |
Data Scientist | ||
Healthcare and Medicine | N/A | Biostatistician |
Medical Researcher | ||
Environmental Science | N/A | Climate Modeler |
Environmental Consultant | ||
Manufacturing and Industry | N/A | Quality Control Analyst |
Operations Research Analyst |
In comparing Pure Mathematics and Applied Mathematics , it’s clear that both are important, but they have different focuses. Pure Mathematics is all about exploring abstract ideas and theories. Applied Mathematics, on the other hand, takes those mathematical ideas and puts them to work in the real world. Both fields are essential, each contributing in their own way to the advancement of knowledge and technology.
Is pure math useful in engineering.
Yes, pure math is useful in engineering for developing theoretical models, solving complex problems, and enhancing analytical thinking.
Calculus, linear algebra, and differential equations are the most important types of math in engineering, as they help in designing and analyzing systems.
The role of ‘‘pure’’ mathematics is to study mathematical ideas and abstractions without regard to specific applications, which sometimes results in advances in many spheres.
Electrical engineering makes the most use of math and is evident in the areas of signal processing, control engineering, and circuit engineering among others that require a lot of math.
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Mathematical logic investigates the power of mathematical reasoning itself. The various subfields of this area are connected through their study of foundational notions: sets, proof, computation, and models. The period from the 1930s thru the 1970s saw great progress in logic. MIT was a major center in the field from the 1950s through the 1980s.
The exciting and active areas of logic today are set theory, model theory and connections with computer science. Set theory addresses various ways to axiomatize mathematics, with implications for understanding the properties of sets having large infinite cardinalities and connections with the axiomatization of mathematics. Model theory investigates particular mathematical theories such as complex algebraic geometry, and has been used to settle open questions in these areas. Theoretical computer science developed partially out of logic, and questions such as P =? NP are being pursued with techniques from logic.
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AI-driven research in pure mathematics and theoretical physics. Advances in artificial-intelligence-assisted mathematical investigations suggest that human-machine collaboration will be an ...
251+ Math Research Topics: Beginners To Advanced. Prime Number Distribution in Arithmetic Progressions. Diophantine Equations and their Solutions. Applications of Modular Arithmetic in Cryptography. The Riemann Hypothesis and its Implications. Graph Theory: Exploring Connectivity and Coloring Problems.
Explore the latest full-text research PDFs, articles, conference papers, preprints and more on PURE MATHEMATICS. Find methods information, sources, references or conduct a literature review on ...
Advancing mathematics by guiding human intuition with AI. A framework through which machine learning can guide mathematicians in discovering new conjectures and theorems is presented and shown to ...
Research in Mathematics is a broad open access journal publishing all aspects of mathematics including pure, applied, and interdisciplinary mathematics, and mathematical education and other fields. The journal primarily publishes research articles, but also welcomes review and survey articles, and case studies. Topics include, but are not limited to:
The Riemann hypothesis, a Clay Millennium Problem, is a part of analytic number theory, which employs analytic methods (calculus and complex analysis) to understand the integers. Recent advances in this area include the Green-Tao proof that prime numbers occur in arbitrarily long arithmetic progressions. The Langlands Program is a broad series ...
Research » Research Areas » Research in Pure Mathematics. Algebra & Number Theory. Topics of interest include additive and analytic number theory, arithmetic algebraic geometry, automorphic forms, L-functions, quantum groups, cohomology of groups, representation theory of symmetric groups and related algebras. Combinatorics & Graph Theory.
Known as experimental mathematics, this kind of research has resulted in well-known conjectures, such as the Birch and Swinnerton-Dyer conjecture — one of six Millennium Prize Problems, the most well-known open problems in mathematics (with a US$1 million prize attached to each). While this approach has been successful and is fairly common ...
Mathematics, Pure and Applied Math | Explore the latest full-text research PDFs, articles, conference papers, preprints and more on MATHEMATICS. Find methods information, sources, references or ...
The department has strong research groups across the spectrum of Pure Mathematics. Major strengths include: Algebra Skip to main content. Jump to ... Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is co-ordinated within the Office of ...
Pure mathematics studies the properties and structure of abstract objects, [ 1] such as the E8 group, in group theory. This may be done without focusing on concrete applications of the concepts in the physical world. Pure mathematics is the study of mathematical concepts independently of any application outside mathematics.
The Pure Mathematics Group in the School of Mathematics and Statistics is made up of the Combinatorics and Algebra Group and the Dynamical Systems Groups. ... If you have an enquiry specific to this research topic, please contact: Postgraduate Research Tutor, School of Mathematics and Statistics Email: STEM-MS-PhD Phone: +44 (0)1908 655552.
Mathematics . Lists of mathematics topics cover a variety of topics related to mathematics. Some of these lists link to hundreds of articles; some link only to a few. ... Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss said ...
Department members engage in cutting-edge research on a wide variety of topics in mathematics and its applications. Topics continually evolve to reflect emerging interests and developments, but can roughly grouped into the following areas. Algebra, Combinatorics, and Geometry Algebra, combinatorics, and geometry are areas of very active research at the University of Pittsburgh.
If you are an undergraduate looking for some research topics for your next math paper, you will surely appreciate our list of interesting undergraduate math research topics: Methods to count discrete objects. The origins of Greek symbols in mathematics. Methods to solve simultaneous equations. Real-world applications of the theorem of Pythagoras.
See our Pure Mathematics Projects. Pure mathematics. Algebra and number theory. Combinatorics. Computational geometry and topology. Geometric and nonlinear analysis. Honours. Masters. PhD.
Applied mathematics is the application of mathematical techniques to describe real-world systems and solve technologically relevant problems. This can include the mechanics of a moving body, the ...
In applied mathematics, we look for important connections with other disciplines that may inspire interesting and useful mathematics, and where innovative mathematical reasoning may lead to new insights and applications. Combinatorics. Computational Biology. Physical Applied Mathematics. Computational Science & Numerical Analysis.
From pure mathematics to applied mathematics, this guide will present a variety of research areas that span different branches and interdisciplinary intersections. Whether you are interested in algebra, analysis, geometry, number theory, statistics, or computational mathematics, there is a wealth of captivating topics to consider.
Created, developed and nurtured by Eric Weisstein at Wolfram Research. Pure mathematics, also known as abstract mathematics, is the study of mathematical concepts without regard to real-world application. This is in contrast to uses of mathematics specifically for problem solving, as is the case for example in applied mathematics and concrete ...
Applied Mathematics. Faculty and students interested in the applications of mathematics are an integral part of the Department of Mathematics; there is no formal separation between pure and applied mathematics, and the Department takes pride in the many ways in which they enrich each other. We also benefit tremendously from close collaborations ...
Pure Mathematics is the branch of mathematics that is concerned with abstract concepts and theoretical frameworks rather than practical applications. Unlike applied mathematics, which focuses on solving real-world problems, pure mathematics is driven by the exploration of mathematical structures, relationships, and theories for their own sake.
Mathematical logic investigates the power of mathematical reasoning itself. The various subfields of this area are connected through their study of foundational notions: sets, proof, computation, and models. The period from the 1930s thru the 1970s saw great progress in logic. MIT was a major center in the field from the 1950s through the 1980s.