Namely, to design a random walk on a lattice is equivalent to constructing a set of generators for the ideal of a variety given implicitly; that is, solving a problem in classical elimination theory. This is of importance in the statistics of medium-sized data sets—for example, contingency tables—where classical methods give wrong answers. The classical methods of elimination theory are hard; the modern technique of Groebner bases is now often used.
Certainly one of the most important and surprising recent developments in mathematics has been its interaction with theoretical high-energy physics. Large swathes of geometry, representation theory, and topology have been heavily influenced by the interaction of ideas from quantum field theory and string theory, and, in turn, these areas of physics have been informed by advances in the mathematical areas.
Examples include the relationship of the Jones polynomial for knots with quantum field theory, and Donaldson invariants for 4-manifolds and the related Seiberg-Witten invariants. Then there is mirror symmetry, discovered by physicists, which, in each original formulation, led to the solution of one of the classical enumerative problems in algebraic geometry, the number of rational curves of a given degree in a quintic hypersurface in projective 4-space. This has been expanded conjecturally to a vast theory relating complex manifolds and symplectic manifolds.
A more recent example of the cross-fertilization. A very recent example involves the computation of scattering amplitudes in gauge theories, motivated by the practical problem of computing backgrounds for the Large Hadron Collider. These computations use the tools of algebraic geometry and some methods from geometric number theory.
Ideas of topology are very important in many areas of physics. Most notably, topological quantum field theories of the Chern-Simons form are crucial to understanding some phases of condensed matter systems. These are being actively explored because they offer a promising avenue for constructing quantum computers.
Kinetic theory is a good example of the interaction between areas of the mathematical sciences that have traditionally been seen as core and those that had been seen as applied. Mathematically, the Boltzmann equation involves the spatial interaction collisions of probability densities of particles travelling at different velocities.
The analytical properties of solutions—their existence, regularity, and stability, and the phenomenon of shock formation—were little understood until approximately 30 years ago. Hilbert and Carleman worked on these problems for many years with little success, and attempts to understand the analytical aspects of the equation—existence, regularity, stability of solutions, as well as possible shock formation—had not advanced very far.
In the s, the equation arose as part of the need for the modeling of the reentry dynamics of space flight through the upper atmosphere and it was taken up again by the mathematical community, particularly in France. That gave rise to 20 years of remarkable development, from the celebrated work of Di Perna-Lions showing the existence of solutions, to the recent contributions of Villani and his collaborators.
In the meantime, the underlying idea of the modeling of particles interacting at a rarefied scale appeared in many other fields in a more complex way: The sorts of connections exemplified here are powerful, because they establish alternative modes by which mathematical concepts may be explored. They often inspire further work because surprising connections hint at deeper relationships. It is clear that the mathematical sciences have benefited in recent years from valuable, and perhaps surprising, connections within the discipline itself.
For example, the Langlands program in. Fields in the mathematical sciences are mature enough so that researchers know the capabilities and limitations of the tools provided by their field, and they are seeking other tools from other areas. This trend seems to be flourishing, with the result that there is an increase in inter-disciplinarity across the mathematical sciences.
For example, there is greater interest in combinatorial methods which, 50 years ago, might not have been pursued because those methods may not have elegant structures and because computation may be required. The tendency decades ago was to make simplifying assumptions to eliminate the need for combinatorial calculations.
But many problems have a real need for a combinatorial approach, and many researchers today are willing to do those computations. Because of these interdisciplinary opportunities, more researchers are reaching out from areas that might in the past have been self-contained. Also, as discussed below, it is easier to collaborate these days because of the Internet and other communications technologies. In other research areas, opportunities are created when statistics and mathematics are brought together, in part because the two fields have complementary ways of describing phenomena.
An example is found in environmental sciences, where the synergies between deterministic mathematical models and statistics can lead to important insights. Such an approach is helpful for, say, understanding the uncertainties in climate models, because of the value in combining insight about deterministic PDE-based models with statistical insights about the uncertainties. Because of these exciting opportunities that span multiple fields of the mathematical sciences, the amount of technical background needed by researchers is increasing.
Education is never complete today, and in some areas older mathematicians may make more breakthroughs than in the past because so much additional knowledge is needed to work at the frontier. For this reason, postdoctoral research training may in the future become necessary for a greater fraction of students, at least in mathematics. The increase in postdoctoral study has been dramatic over the past 20 years, such that in the fall of , 40 percent of recent Ph.
Rose, , Report on the new doctoral recipients. Notices of the AMS 58 7: While these trends clearly create researchers with stronger backgrounds, the length of time for becoming established as a researcher could lessen the attractiveness of this career path. At the same time, more mathematical scientists are now addressing applications, such as those in computer science.
This work builds successfully on the deep foundations that have been constructed within mathematics. For example, results of importance to computer science have been achieved by individuals who are grounded in discrete mathematics and combinatorics and who may not have had previous exposure to the particular application.
These increasing opportunities for interdisciplinary research pose some challenges for individuals and the community. Interdisciplinary work is facilitated by proximity, and even a walk from one corridor to another can be a hindrance. So attention must be paid to fostering collaboration, even within a single department.
When the connections are to be established across disciplines, this need is even more obvious. Ideally, mathematical scientists working in biology, for example, will spend some of their time visiting experimental labs, as will mathematical scientists working with other disciplines. But to make this happen, improved mechanisms for connecting mathematical scientists with potential collaborators are needed, such as research programs that bring mathematical scientists and collaborators together in joint groups.
Such collaborations work best when the entire team shares one primary goal—such as addressing a question from biology—even for the team members who are not biologists per se. But to make this work we need adjustments to reward systems, especially for the mathematical scientists on such teams. One leading researcher who spoke with the committee observed how mathematicians at Microsoft Research are often approached by people from applied groups, which is a fortunate result of the internal culture.
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One value that mathematicians can provide in such situations that is often underestimated is that they can prove negative results: This ultimately increases productivity because it helps the organization to focus resources better. This is a contribution of mathematics beyond product-focused work or algorithm development.
Another set of challenges to interdisciplinary students and researchers stems from their lack of an obvious academic home. Who is in their community of peers? Who judges their contributions? How are research proposals and journal submissions evaluated? At the National Institutes of Health, for example, the inclusion of mathematical scientists in the study sections that review proposals with mathematical or statistical content is an important step, though it is not a perfect process.
Tenure review in large. While it takes longer to build a base of interdisciplinary knowledge, once that base is built it can open doors to very productive research directions that are not feasible for someone with a more conventional background. Universities are changing slowly to recognize that interdisciplinary faculty members can produce both better research and better education. There is a career niche for such people, but it could be improved.
For example, it might be necessary to relax the tenure clock for researchers pursuing interdisciplinary topics, and a proper structure must be in place in order to conduct appropriate tenure reviews for them.
Workshop on Future Directions in Mathematics
This is one way to break down academic silos. A major change in the mathematical sciences over the past decade and more has been the increasing number of mathematical science institutes and their increased influence on the discipline and community. In there was only one such institute in the United States—the Institute for Advanced Study in Princeton, which has a very different character from the institutes created after it. In the last 20 years we have seen new institutes appear in Japan, England, Ireland, Canada, and Mexico, to name just a few countries, joining older institutes in France, Germany, and Brazil.
Overall, there are now some 50 mathematical science institutes in 24 different countries. The institutes promote research and collaboration in emerging areas, encourage continued work on important problems, tackle large research agendas that are outside the scope of individual researchers, and help to maintain the pipeline of qualified researchers for the future. Many of the institute programs help researchers broaden their expertise, addressing the need for linking multiple fields that was emphasized above and in Chapter 3.
For example, every year the Institute for Mathematics and its Applications IMA offers intensive two-week courses aimed at helping to introduce established researchers to new areas; recent courses have focused on mathematical neuroscience, economics and finance, applied algebraic topology, and so on. All of these institutes have visiting programs, often around a specific theme that varies from year to year, and they invite mathematical scientists from around the world to visit and participate in the programs. This has led to an enormous increase in the cross-fertilization of ideas as people from different places and different disciplines meet and exchange ideas.
In addition, it is quite common for these institutes to record the lectures and make them freely available for downloading. Real-time streaming of lectures is just starting to emerge. All of these steps help to strengthen the cohesiveness of the community. Beyond this, the institutes frequently allow researchers to meet people they would not otherwise meet. This is especially crucial in connecting researchers from other disciplines with the right mathematical scientists.
Often, scientists, engineers and medical researchers do not know what mathematics and statistics are available that might be relevant to their problem, and they do not know whom to turn to. Likewise, mathematical scientists are often sitting on expertise that would be just what is needed to solve an outside problem, but they are unaware of the existence of these problems or of who might possess the relevant data. Arguably, the institutes have collectively been one of the most important vehicles for culture change in the mathematical sciences.
Some illustrations of the impact of mathematical science institutes follow. To help the mathematical sciences build connections, the IMA reaches out so that some 40 percent of the participants in its programs come from. In this way, it has helped to nucleate new communities and networks in topics such as mathematical materials science, applied algebraic geometry, algebraic statistics, and topological methods in proteomics.
The institutes have had success in initiating new areas of research.
For example, IPAM worked for 9 years to nucleate and then nurture a new focused area of privacy research, starting with a workshop on contemporary methods in cryptography in That led to a workshop on statistical and learning-theoretic challenges in data privacy, which brought together data privacy and cryptography researchers to develop an approach to data privacy that is motivated and informed by developments in cryptography, one of them being mathematically rigorous concepts of data security. A second follow-on activity was a workshop on mathematics of information-theoretic cryptography, which saw algebraic geometers and computer scientists working on new approaches to cryptography based on the difficulty of compromising a large number of nodes on a network.
Another IPAM example illustrates that the same process can be important and effective in building connections within the discipline. Traditional methods for dealing with such problems involve techniques such as variable selection, ridge regression, and principal components regression. Beginning in the s, more modern methods such as lasso regression and wavelet thresholding were developed.
These ideas have now been extended in numerous directions and have attracted the attention of researchers in computer science, applied mathematics, and statistics, in areas such as manifold learning, sparse modeling, and the detection of geometric structure. This is an area with great potential for interaction among statisticians, applied mathematicians, and computer scientists. The Mathematical Sciences Research Institute MSRI is focused primarily on the development of fundamental mathematics, specifically in areas in which mathematical thinking can be applied in new ways.
In spite of its primary focus on the mathematical sciences per se, MSRI has long included a robust set of outreach activities. For example, its program Computational Aspects of Algebraic Topology explored ways in which the techniques of algebraic topology are being applied in various contexts related to data analysis, object recognition, discrete and computational geometry, combinatorics, algorithms, and distributed computing.
That program included a workshop focused on application of topology in science and engineering, which brought together people working in problems ranging from protein docking, robotics, high-dimensional data sets, and sensor networks. In , MSRI organized and sponsored the World Congress on Computational Finance, in London, which brought together both theoreticians and practitioners in the field to discuss its current problems. MSRI has also sponsored a series of colloquia to acquaint mathematicians with fundamental problems in biology. An example is the workshop jointly sponsored by MSRI and the Jackson Laboratories on the topic of mathematical genomics.
Both MSRI and SAMSI have helped build bridges between statisticians and climate scientists through at least six programs focused on topics such as new methods of spatial statistics for climate change applications, data assimilation, analysis of climate models as computer experiments, chaotic dynamics, and statistical methods for combining ensembles of climate models.
After a professor of Scandinavian languages at UCLA participated in a IPAM program on knowledge and search engines, which introduced him to researchers and methods from modern information theory, he went on to organize two workshops in and , on networks and network analysis for the humanities; the workshops were funded by the National Endowment for the Humanities and cosponsored by IPAM. They led to the exploration of new data analysis tools by many of the humanists who participated.
The IMA has a long history of outreach to industry, for instance through its Industrial Postdoctoral Fellowship program and other activities. Examples include uncertainty quantification in the automotive industry and numerical simulation of ablation surgery. Overall, IMA has trained over postdoctoral fellows since , and about 80 percent of them are now in academic positions. The IMA also offers programs for graduate students, most notably its regular workshops on mathematical modeling in industry, in which students work in teams under the guidance of industry mentors on real-world problems from their workplace.
Through this program, many mathematical scientists have been exposed early in their careers to industrial problems and settings. This led to hiring by NGA of several new mathematics Ph. The first thing that comes to mind when one thinks of interconnectivity these days is the Internet and the World Wide Web. These affect practically all human activity, including the way that mathematical scientists work.
The maturation of the Internet has led in the past years to the availability of convenient software tools that painlessly lead to the quick dissemination of research results consider for instance the widely used arXiv preprint server, http: These new tools have profoundly changed both the modes of collaboration and the ease with which mathematical scientists can work across fields.
The existence of arXiv has had a major influence on scholarly communication in the mathematical sciences, and it will probably become. The growth of such sites has already had a great impact on the traditional business model for scientific publishing, in all fields. It is difficult to say what mode s of dissemination will predominate in , but the situation will certainly be different from that of today. The widespread availability of preprints and reprints online has had a tremendous democratizing influence on the mathematical sciences. Face-to-face meetings between mathematical scientists remain an essential mode of communication, but the tyranny of geography has substantially lessened its sway.
However, the committee is concerned about preserving the long-term accessibility of the results of mathematical research.
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Rapid changes in the publishing industry and the fluidity of the Internet are also of concern. This is a very uncertain time for traditional scholarly publishing, 6 which in turn raises fundamental concerns about how to share and preserve research results and maintain assured quality. Public archives such as arXiv play a valuable role, but their long-term financial viability is far from assured, and they are not used as universally as they might be.
The mathematical sciences community as a whole, through its professional organizations, needs to formulate a strategy for maximizing public availability and the long-term stewardship of research results. The NSF could take the lead in catalyzing and supporting this effort. Thanks to mature Internet technologies, it now is easy for mathematical scientists to collaborate with researchers across the world.
Not only do such projects contribute to advancing research, but they also serve to locate other researchers with the same interest and with the right kind of expertise; they represent an ideal vehicle for expanding personal collaborative networks. The New York Times.
Accessed March 19, Widespread dissemination of research results has made it easier for anyone to borrow ideas from other fields, thereby creating new bridges between subdisciplines of the mathematical sciences or between the mathematical sciences and other fields of science, engineering, and medicine.
For example, new research directions can be seen where ideas from abstract probability theory prove to have very deep consequences in signal processing and tantalizing applications in signal acquisition, and where tools from high-dimensional geometry can change the way we perform fundamental calculations, such as solving systems of linear equations. Effortless access to information has spurred the development of communities with astonishingly broad collective expertise, and this access has lowered barriers between fields.
In this way, theoretical tools find new applications, while applications renew theoretical research by offering new problems and suggesting new directions. This cycle is extremely healthy. A recent paper 9 evaluated an apparent shift in collaborative behavior within the mathematical sciences in the mids. At that time, the networks of researchers in core and applied mathematics moved from being centered primarily around a small number of highly prolific authors toward networks displaying more localized connectivity.
More and stronger collaboration was in evidence. Brunson and his collaborators speculated that a cause of this trend was the rise of e-communications and the Web—for example, arXiv went online in and MathSciNet in —because applied subdisciplines, which historically had made greater use of computing resources, showed the trend most strongly. The Internet provides a ready mechanism for innovation in communication and partnering, and novel mechanisms are likely to continue to appear. As just one more example, consider the crowdsourcing, problem-solving venture called InnoCentive. It is an example of another new Web-enabled technology that may have real impacts on the mathematical sciences by providing opportunities to learn directly of applied challenges from other disciplines and to work on them.
Of these, 13 were flagged as having mathematical or statistical content. Examples of the latter included challenges such as the development of an algorithm to identify underlying geometric features. Manuscript submitted for publication. There are debates about whether crowdsourcing is a healthy trend for a research community. However, crowdsourcing is one more Web-based innovation that may affect mathematical scientists, and the community should be aware of it.
In the Mathematics Subject Classification section 01Axx History of mathematics and mathematicians, the subsection 01A67 is titled Future prospectives. Given the support of research by governments and other funding bodies, concerns about the future form part of the rationale of the distribution of funding. How they could Change Education?
Krantz writes in "The Proof is in the Pudding. It seems plausible that in years we will no longer speak of mathematicians as such but rather of mathematical scientists. Experimental mathematics is the use of computers to generate large data sets within which to automate the discovery of patterns which can then form the basis of conjectures and eventually new theory.
The paper "Experimental Mathematics: Recent Developments and Future Outlook"  describes expected increases in computer capabilities: Doron Zeilberger considers a time when computers become so powerful that the predominant questions in mathematics change from proving things to determining how much it would cost: I can envision an abstract of a paper, c. In "Rough structure and classification",  Timothy Gowers writes about three stages: In , Peter Cameron in "Combinatorics entering the third millennium"  attempts to "throw some light on present trends and future directions.
I have divided the causes into four groups: It is exactly this that makes combinatorics very much alive. I have no doubt that combinatorics will be around in a hundred years from now. It will be a completely different subject but it will still flourish simply because it still has many, many problems".
On numerical analysis and scientific computing: In , Lloyd N. In , Mikhail Gromov in "Possible Trends in Mathematics in the Coming Decades",  says that traditional probability theory applies where global structure such as the Gauss Law emerges when there is a lack of structure between individual data points, but that one of today's problems is to develop methods for analyzing structured data where classical probability does not apply. Such methods might include advances in wavelet analysis , higher-dimensional methods and inverse scattering.