Category Archives: mathematics

Mathematics

Prospects for mathematics

The UK College of Arts & Sciences says that “Mathematics is challenging, rewarding and fun. It is both logical and creative. Students who major in mathematics have a variety of opportunities. The mathematics major prepares students for traditional pursuits such as graduate study, teaching and work as an actuary” (https://math.as.uky.edu/where-can-math-take-me). The reason it specifically mentions actuary is that this discipline is built around mathematics. You could say the same for what is these days called ‘Data Science’, although Statistics too is involved here. Asking ‘What Can You Do With a Mathematics Degree?’, https://www.topuniversities.com/student-info/careers-advice/what-can-you-do-mathematics-degree goes on to describe various areas directly and indirectly linked to mathematics. In an article dated February 20, 2015, they say: “Thanks to the growing importance placed on technology, big data and economic efficiency by all kinds of organizations, expert number crunchers are increasingly in demand.

In fact, according to the US Bureau of Labor Statistics, between 2012 and 2022, the job market for mathematicians is expected to grow by a whopping 23%, with a predicted median salary of US$110,000”. Students should find out what is the current scenario. An excellent resource is http://mathcareers.maa.org/, a site run by the Mathematics Association of America, which talks about the 101 things you can do with mathematics. The interesting aspect of this site is that it allows employers to post their requirements. UC Davis offers a useful compilation on the career options open to someone with a mathematics background (https://www.math.ucdavis.edu/~kouba/MathJobs.html), listing a whole range of careers. So does the University of Georgia (http://www.onedayonejob.com/majors/mathematics/) differentiating between entry-level jobs and others. You will come across mathematicians working in various fields talking about their experience in http://www.mathscareers.org.uk/environment/. A publication called De Gruyter has invited contributions to ‘Open Engineering: Recent Trends in Mathematical Analysis and their Applications’ and the list of topics should give students a good idea of mathematical research (https://www.degruyter.com/dg/page/Recent-Trends-Mathematical-Analysis-Applications/open-engineering-recent-trends-in-mathematical-analysis-and-their-applications). Lashi Bandara talks about what a pure mathematician does in http://theconversation.com/explainer-the-point-of-pure-mathematics-2385. Students will find an engaging and lively discussion in “Why Do We Pay Pure Mathematicians?” https://mathwithbaddrawings.com/2015/02/24/why-do-we-pay-mathematicians/.

The University of Cambridge offers, in its Masters in Pure Mathematics, about 80 courses, which gives you an idea of the range (https://www.graduate.study.cam.ac.uk/courses/directory/mapmaspmm).

Going into some details, Ireland’s Maynooth University Department of Mathematics and Statistics, explains the attraction of pure mathematics (https://www.maynoothuniversity.ie/mathematics-and-statistics/undergraduate-studies/why-do-research-pure-mathematics). Among the most resourceful sites is www.worldscientific.com which is a great place to explore the potential in science in general.

In a wonderful article titled “Computational Knowledge and the Future of Pure Mathematics”, Stephen Wolfram (August 12, 2014) discusses how pure mathematics was done more than a 100 years ago and how technology can assist (http://blog.stephenwolfram.com/2014/08/computational-knowledge-and-the-future-of-pure-mathematics/). It is an invaluable discussion for those with a deep interest in mathematics.

Asking ‘Why study Pure and Applied Mathematics?’ The University of Concordia (Montreal) explains: “Mathematics is a universal language that explains the currents of the ocean, string theory, the spiral of a snail’s shell or the growth of a fern. When you study pure and applied mathematics, you enter a field that has both a rich history and many future career possibilities. As a mathematician, you’ll design and analyze mathematical models and develop systems for testing and evaluation. In essence, you will use mathematics to find creative solutions for systems such as communications, software development, encryption technologies, banking and drug testing” (http://www.concordia.ca/academics/undergraduate/pure-and-applied-mathematics.html

As the University of Luxemburg’s Mathematics Research Unit says, “During the last half of the twentieth century, striking applications of Mathematics appeared in all natural sciences, even in behavioural and social sciences. Mathematics is a universal tool to gain insight in highly complex systems. But mathematics is also a science of its own. It is highly alive, powered by its internal driving forces and by inspirations coming from new challenges in other fields. As the University of Waterloo adds “Research in Applied Mathematics has changed dramatically over the past 30 years, with revolutionary developments in traditional areas, together with the emergence of exciting new areas. These changes have been triggered by the development of more powerful computers allowing researchers to address previously intractable problems, and developments in other fields which have led to new mathematical problems”. The Mathematics Research Unit (Unité de Recherche en Mathématiques) RMATH carries out research in mathematics, both on its fundamental and its applied aspects”. (https://wwwen.uni.lu/research/fstc/mathematics_research_unit).

For a look at the sheer variety of areas of interest rooted in mathematics, take a look at http://mathematics-in-europe.eu/. This is a really interesting window to the immense possibilities in mathematics. You might also like to examine the new world of informatics and mathematics through https://www.ercim.eu/, the European Consortium for Research in Informatics and Mathematics. To some student’s question on ‘What are the most active areas of research in Mathematics?’ you will find interesting answers in https://www.quora.com/What-is-the-most-active-research-area-in-mathematics.

There are also industry-sponsored and industry-oriented research programmes. Worcester Polytechnic Institute has a Center for Industrial Mathematics and Statistics which offers research experiences for undergraduates in industrial mathematics and statistics as an eight week programme (https://labs.wpi.edu/cims/initiatives/reu/). University of East Anglia offers research in industrial mathematics (https://www.uea.ac.uk/mathematics/research/applied/industrial).  There are many more which you can search for using the search clue ‘industry-sponsored research in mathematics’. To examine mathematics in industry around the world, look at (https://sinews.siam.org/Details-Page/mathematics-in-industry-around-the-world).

You may like to read this document to have a sense of Mathematics in Europe (https://ec.europa.eu/futurium/en/system/files/ged/finalreport_maths.pdf).

Research in universities

As you would expect, areas and emphasis differ across universities. What follows is just a broad idea of what you can do.

The University of Nebraska has the following areas of research:

Commutative Algebra and Algebraic Geometry

The commutative algebra group has research interests which include algebraic geometry, algebraic and quantum coding theory, homological algebra, representation theory, and K-theory.

Discrete Mathematics and Coding Theory

Research interests in this group center around structural problems in combinatorics, and coding theory, the study of schemes for encoding data to, for example, efficiently detect errors in transmission.

Groups, Semigroups and Topology

The interplay between topology, group theory and semigroup theory has yielded a wealth of information in all three mathematical fields. These connections are central to the research of our faculty working in this area.

Applied Mathematics and Differential Equations

The Applied Mathematics and Differential Equations group within the Department of Mathematics have a great diversity of research interests, but a tying theme in each respective research program is its connection and relevance to problems or phenomena which occur in the engineering and physical sciences.

Functional Integration

Functional integration deals with the mathematical foundations of the Feynman Integral, originally introduced in the 1950s by Richard Feynman. Research with this group involves placing this work on a rigorous foundation.

Operator Theory/Operator Algebras

Operator Theory and Operator Algebras are concerned with the study of linear operators, usually on vector spaces whose elements are functions. The subject is analysis, but because the vector spaces are usually infinite dimensional, the subject has a nice blend of techniques from other areas of mathematics, ranging from algebra to topology to dynamical systems.

Mathematical Biology

Several faculty in the department have a strong interest in problems originating in the life sciences, especially from ecology and neuroscience. The group has significant collaborative relationships with colleagues in the life sciences across campus and at other institutions. Group members have mathematical backgrounds in several areas of pure and applied mathematics, including dynamical systems, partial differential equations, algebraic and differential geometry, topology, control theory, and game theory.

(http://www.math.unl.edu/research/areas)

The University of California, Berkeley, has six loosely defined research area groups, with overlapping memberships:

Algebra Research, Mathematical Analysis Research, Applied Mathematics Research, Geometry/Topology Research, Mathematical Logic Research and Probability Research (https://math.berkeley.edu/research/areas).

At Stanford University, you can engage in research in these areas: Algebraic Geometry, Analysis/PDE, Applied Math, Combinatorics

Financial Math, Geometry, Number Theory, Probability, Representation Theory, Symplectic Geometry and Topology (https://mathematics.stanford.edu/research-areas/).

Mathematics at MIT is administratively divided into two categories: Pure Mathematics and Applied Mathematics.

Pure Mathematics involves research in Algebra & Algebraic Geometry, Algebraic Topology, Analysis & PDEs, Geometry, Mathematical Logic & Foundations, Number Theory, Probability & Statistics and Representation Theory.

In Applied Mathematics, MIT looks 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”. Given this, the corresponding areas are Combinatorics, Computational Biology, Physical Applied Mathematics, Computational Science & Numerical Analysis, Theoretical Computer Science and Theoretical Physics (http://math.mit.edu/research/index.php).

The Mathematics department at the University of Waterloo has strong research programs in Control and Dynamical Systems (including differential equations), Fluid Mechanics, Mathematical Medicine and Biology, Mathematical Physics and Scientific Computing. Some examples: Math and Water, Carbon Nanotubes, Mathematics and medicine are powerful partners, Saving the whales with mathematics, Quantum sounds could reveal the shape of the universe and Using social media to help prevent the spread of disease (https://uwaterloo.ca/applied-mathematics/research-areas).

Research in the Faculty of Mathematics at the University of Cambridge is divided in to two: Research in Applied Maths and Theoretical Physics (Including: fluid and solid mechanics, waves, nonlinear dynamics, numerical analysis, mathematical and computational biology, high energy physics, quantum information, relativity, and cosmology) and Research in Pure Maths and Mathematical Statistics (Including: algebra, analysis, category theory and logic, combinatorics, geometry and topology, number theory, operation research and financial mathematics, probability, and statistics) (https://www.maths.cam.ac.uk/research).

Newcastle University has a variety of research programmes (http://www.ncl.ac.uk/maths-physics/postgraduate/research-projects/). At the University of York, you can get to do Algebra, Number theory, Mathematical finance and stochastic analysis, geometry and analysis, mathematical physics, statistics and probability and mathematical biology and chemistry

(https://www.york.ac.uk/maths/research/).

If you are interested in Nonlinear differential equations, Painlevé equations, Mathematical biology, Quantum integrable systems, Topological solitons, Algebra and representation theory, Algebraic topology, Invariant theory, Financial mathematics, you may look at the University of Kent (https://www.kent.ac.uk/courses/postgraduate/149/mathematics#research-areas). The University of Leeds has a different approach (https://physicalsciences.leeds.ac.uk/info/35/research_and_innovation).

At RMIT University, Australia, research is in algebraic coding theory, complex networks and epidemiology, computational fluid dynamics, environmental and resource modelling, general mathematical modelling, information security, integer programming and discrete mathematics, mathematical biology, networks, complexity and graph theory, numerical analysis, optimisation theory – algorithms and applications (https://www.rmit.edu.au/study-with-us/levels-of-study/research-programs/phd/dr222).

Mathematics and the environment

The American Mathematical Society has studied the various dimensions of the application of mathematics to the study of environment (http://www.ams.org/books/psapm/032/psapm032-endmatter.pdf). The University of East Anglia has a special school for Environmental Mathematics within which it examines these topics: ocean, atmosphere and climate modelling; wave modelling and volcano modelling. In addition to research, it offers PhD projects in the subject (https://www.uea.ac.uk/mathematics/research/applied/environmental). University of Exeter also has a special focus on this subject (http://www.exeter.ac.uk/undergraduate/degrees/mathematics/mathsmsci-environment/). Grantham Institute, part of the Imperial College London discusses ‘Seven ways maths can save the world’ (https://granthaminstitute.com/2016/05/31/seven-ways-maths-can-save-the-world/). The Faculty of Environmental Science and Technology, Okayama University says “To improve the quality of life and create an environment in which human beings live in harmony with nature, it is necessary to evaluate the effects of human activities and predict the results under various conditions”, which calls for that mathematics and statistical tools along with computing techniques (http://www.est.okayama-u.ac.jp/contents/emsen.html). Scientific Computing World has an interesting discussion on Mathematics and the Environment through the work of Louis Gross, a mathematical ecologist (https://www.scientific-computing.com/feature/mathematics-and-environment). The Centre for Applications in Natural Resources Mathematics of The University of Queensland aims “to develop and apply mathematical and statistical theory to produce tools that directly impact the management of fisheries, forestry, water security, conservation, pest and disease management, and adaptation to global changes” (https://smp.uq.edu.au/research/centres/carm). The Department of Forest Resources Management of The Swedish University of Agricultural Sciences engages in research in mathematical statistics applied to forest sciences (https://www.slu.se/en/departments/forest-resource-management/sections/mathematical-statistics-applied-to-forest-sciences/). The National Center for Replacement, Refinement and Reduction of Animals in Research is engaged in research in applying mathematics to 3Rs problem (https://www.nc3rs.org.uk/applying-mathematics-3rs-problems).

One of the applications of mathematics in environment is in modelling. One of the earliest books (1999) in this area is titled ‘Modeling the environment: an introduction to system dynamics models of environmental systems’ by Andrew Ford (https://searchworks.stanford.edu/view/4104319). Envision, an institution committed to developing next generation leaders in environmental sciences has set up a research project (as a PhD) in ‘Geospatial modelling the spread of antimicrobial resistance in the environment’, to focus attention on the ‘spread of AMRs in the environment’. It says further: “Antimicrobials and antimicrobial resistant genes (ARGs) and organisms have sources in agriculture and wastewater treatment plants (WWTP), which are spread on land through slurry, manures or sewage sludge, or released directly into rivers. Soil and water polluted by antimicrobials and resistant bacteria can impact crops, animals and humans”(http://www.envision-dtp.org/2016/geospatial-modelling-the-spread-of-antimicrobial-resistance-in-the-environment/).  (Please check if the project is open when you read this article)

The Oxford Martin Programme on the Future of Food has set up a project on ‘Modelling the Relationship between the Food System and Health, Development, and the Environment’ (http://www.futureoffood.ox.ac.uk/project/modelling-relationship-between-food-system-and-health-development-and-environment). There are private institutions such as Indeco which focus on modelling environment-economy relationships (http://www.indeco.com/ideas/eemodel/).

We could go on but you can find out more based on your areas of interest. People study the impact of various water recycling technologies, patterns in agriculture and their impact on the environment, impact of upstream mining on downstream water and the health of people settled in the area surrounding the downstream. Fortunately, most field work produces large amounts of data. Assuming the integrity of the data, analysis can reveal a lot of hidden aspects. Literally, there is no limit to what you can study about the environment using mathematics.

Subject-specific research

Students who are interested in specific topics could explore https://www.bcps.org/offices/lis/researchcourse/subject_math.html to evaluate various possible areas of research. The Fields Institute for Research in Mathematical Sciences has a lecture series between July 1, 2017 and July 31, 2018 (http://www.fields.utoronto.ca/activities/lectures). The Mathematics Research Group of The International Centre for Theoretical Sciences has identified these as its focus areas: Analysis of partial differential equations and applications, Data assimilation, Differential geometry, Dynamical systems, Mathematical physics, Monsoon dynamics (https://www.icts.res.in/research/mathematics). The Department of Mathematics of Creighton University has a Center for Mathematics of Uncertainty (Fuzzy Math) says its mission is “to support the paradigm shift in the sciences involving uncertainty”, and supports visiting faculty from China, India, Saudi Arabia, Korea and Japan (https://www.creighton.edu/ccas/fuzzymath/).

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Mathematics and the Environment

The American Mathematical Society has studied the various dimensions of the application of mathematics to the study of environment (http://www.ams.org/books/psapm/032/psapm032-endmatter.pdf). The University of East Anglia has a special school for Environmental Mathematics within which it examines these topics: ocean, atmosphere and climate modelling; wave modelling and volcano modelling. In addition to research, it offers PhD projects in the subject (https://www.uea.ac.uk/mathematics/research/applied/environmental). University of Exeter also has a special focus on this subject (http://www.exeter.ac.uk/undergraduate/degrees/mathematics/mathsmsci-environment/). Grantham Institute, part of the Imperial College London discusses ‘Seven ways maths can save the world’ (https://granthaminstitute.com/2016/05/31/seven-ways-maths-can-save-the-world/). The Faculty of Environmental Science and Technology, Okayama University says “To improve the quality of life and create an environment in which human beings live in harmony with nature, it is necessary to evaluate the effects of human activities and predict the results under various conditions”, which calls for that mathematics and statistical tools along with computing techniques (http://www.est.okayama-u.ac.jp/contents/emsen.html). Scientific Computing World has an interesting discussion on Mathematics and the Environment through the work of Louis Gross, a mathematical ecologist (https://www.scientific-computing.com/feature/mathematics-and-environment). The Centre for Applications in Natural Resources Mathematics of The University of Queensland aims “to develop and apply mathematical and statistical theory to produce tools that directly impact the management of fisheries, forestry, water security, conservation, pest and disease management, and adaptation to global changes” (https://smp.uq.edu.au/research/centres/carm). The Department of Forest Resources Management of The Swedish University of Agricultural Sciences engages in research in mathematical statistics applied to forest sciences (https://www.slu.se/en/departments/forest-resource-management/sections/mathematical-statistics-applied-to-forest-sciences/). The National Center for Replacement, Refinement and Reduction of Animals in Research is engaged in research in applying mathematics to 3Rs problem (https://www.nc3rs.org.uk/applying-mathematics-3rs-problems).

One of the applications of mathematics in environment is in modelling. One of the earliest books (1999) in this area is titled ‘Modeling the environment: an introduction to system dynamics models of environmental systems’ by Andrew Ford (https://searchworks.stanford.edu/view/4104319). Envision, an institution committed to developing next generation leaders in environmental sciences has set up a research project (as a PhD) in ‘Geospatial modelling the spread of antimicrobial resistance in the environment’, to focus attention on the ‘spread of AMRs in the environment’. It says further: “Antimicrobials and antimicrobial resistant genes (ARGs) and organisms have sources in agriculture and wastewater treatment plants (WWTP), which are spread on land through slurry, manures or sewage sludge, or released directly into rivers. Soil and water polluted by antimicrobials and resistant bacteria can impact crops, animals and humans”(http://www.envision-dtp.org/2016/geospatial-modelling-the-spread-of-antimicrobial-resistance-in-the-environment/).  (Please check if the project is open when you read this article)

The Oxford Martin Programme on the Future of Food has set up a project on ‘Modelling the Relationship between the Food System and Health, Development, and the Environment’ (http://www.futureoffood.ox.ac.uk/project/modelling-relationship-between-food-system-and-health-development-and-environment). There are private institutions such as Indeco which focus on modelling environment-economy relationships (http://www.indeco.com/ideas/eemodel/).

We could go on but you can find out more based on your areas of interest. People study the impact of various water recycling technologies, patterns in agriculture and their impact on the environment, impact of upstream mining on downstream water and the health of people settled in the area surrounding the downstream. Fortunately, most field work produces large amounts of data. Assuming the integrity of the data, analysis can reveal a lot of hidden aspects. Literally, there is no limit to what you can study about the environment using mathematics.