Showing papers in "Notices of the American Mathematical Society in 2016"
TL;DR: The theory of aperiodic order is concerned with the development of ideas stimulated by the discovery of quasicrystals as discussed by the authors, and a more general introduction to some mathematical aspects of a periodic order can be found in this paper.
Abstract: The theory of aperiodic order is concerned with the development of ideas stimulated by the discovery of quasicrystals. We give a gentle introduction to some mathematical aspects of aperiodic order, aimed at a more general audience.
64 citations
TL;DR: In this paper, the authors describe how a powerful new constraint method yields many different extensions of the topological version of Tverberg's 1966 Theorem in the prime power case, and how the same method also was instrumental in the recent spectacular construction of counterexamples for the general case.
Abstract: We describe how a powerful new “constraint method” yields many different extensions of the topological version of Tverberg’s 1966 Theorem in the prime power case— and how the same method also was instrumental in the recent spectacular construction of counterexamples for the general case. © 2016. All rights reserved.
54 citations
TL;DR: In this article, a variety of physical systems and mathematical models, including randomly growing interfaces, certain stochastic PDEs, traffic models, paths in random environments, and random matrices all demonstrate the same universal statistical behaviors in their long-time/large-scale limit.
Abstract: Universality in Random Systems Universality in complex random systems is a striking concept which has played a central role in the direction of research within probability, mathematical physics and statistical mechanics. In this article we will describe how a variety of physical systems and mathematical models, including randomly growing interfaces, certain stochastic PDEs, traffic models, paths in random environments, and random matrices all demonstrate the same universal statistical behaviors in their long-time/large-scale limit. These systems are said to lie in the Kardar-Parisi-Zhang (KPZ) universality class. Proof of universality within these classes of systems (except for random matrices) has remained mostly elusive. Extensive computer simulations, nonrigorous physical arguments/heuristics, some laboratory experiments, and limited mathematically rigorous results provide important evidence for this belief.
37 citations
TL;DR: In this paper, the main problems and results concerning Diophantine m-tuples are discussed, as well as some open problems and unproved conjectures still remain in this area.
Abstract: The problem of the construction of Diophantine m-tuples, i.e. sets with the property that product of any two of its distinct elements is one less then a square, has a very long history. There are some new results in this area, but many open problems and unproved conjectures still remains. In this survey we explain the main problems and results concerning Diophantine m-tuples.
36 citations
TL;DR: The concept of Nash equilibria was introduced in game theory by as mentioned in this paper, where a Nash equilibrium is an array of strategies, one for each player, such that no player can obtain a higher payoff by switching to a different strategy while the strategies of all other players are held fixed.
Abstract: In game theory, aNash equilibrium is an array of strategies, one for each player, such that no player can obtain a higher payoff by switching to a different strategy while the strategies of all other players are held fixed. The concept is named after John Forbes Nash Jr. For example, if Chrysler, Ford, and GM choose production levels for pickup trucks, a commodity whose market price depends on aggregate production, an equilibrium is an array of production levels, one for each firm, such that none can raise its profits by making a different choice. Formally, an n-player game consists of a set I = {1,... ,n} of players, a set Si of strategies for each player i ∈ I, and a set of goal functions gi ∶ S1 × ⋯ × Sn → R that represent the preferences of each player i over the n-tuples, or profiles, of strategies chosen by all players. A strategy profile has a higher goal-function value, or payoff, than another if and only if the player prefers it to the other. Let S = S1×⋯×Sn denote the set of all strategy profiles, with generic element s, and let (ti, s−i) denote the strategy profile (s1,... , si−1, ti, si+1,... , sn) obtained from s by switching player i’s strategy to ti ∈ Si while leaving all other strategies unchanged. An equilibrium point of such a game is a strategy profile s∗ ∈ S with the property that, for each player i and each strategy ti ∈ Si, gi(s) ≥ gi(ti, s∗ −i).
32 citations
TL;DR: The authors conducted a survey of abstract algebra instructors to investigate typical teaching practices and, more specifically, faculty knowledge, goals, and orientation towards teaching and learning, finding that a majority of respondents appear quite content to lecture.
Abstract: For permission to reprint this article, please contact: reprint-permission@ams.org. DOI: http://dx.doi.org/10.1090/noti1339 Teaching matters. It is arguably the most important factor affecting student learning. Efforts to improve teaching have led to reform initiatives being proposed and tested throughout the college mathematics curriculum. Abstract algebra specifically has been the subject of such reform, including new curricula and pedagogies, since at least the 1960s, yet there is little evidence that these change initiatives have widely influenced the way abstract algebra is taught. We conducted a survey of abstract algebra instructors to investigate typical teaching practices and, more specifically, faculty knowledge, goals, and orientation towards teaching and learning. Results revealed that a majority of respondents appear quite content to lecture. Even among those who indicated a willingness to consider a change of pedagogical strategy, there is very little usage of existing reform materials or interaction with pedagogical research results. There appears to be an impermeable barrier between the pedagogical researchers' findings and recommendations and practitioners who might implement them.
25 citations
TL;DR: New Century Maths as mentioned in this paper is an outcome-based syllabus for mathematics in New South Wales, which contains work from a number of stages to accommodate the mixed-ability classroom and to cater for studentsa individual differences.
Abstract: New Century Maths raises the benchmark for mathematics in New South Wales. Each text contains work from a number of stages to accommodate the mixed-ability classroom and to cater for studentsa individual differences. Texts structured in this way encourage flexible teaching and learning plans and truly reflect the intention of an outcomes-based syllabus. To fully cater for a wide range of abilities and needs, each text at years 9 and 10 is published in two versions, stages 5.1/5.2 and stages 5.2/5.3, both providing different pathways of learning. This structure enables students to follow the pathway into the stage 6 mathematics course that best suits their abilities and needs.
22 citations
TL;DR: Bressoud et al. as mentioned in this paper pointed out that for many of our students, their educational system delivers neither the skills required for post-secondary education nor the access to universities.
Abstract: ropean expectation is that university education is reserved for the elite. Their secondary schools distinguish between those they are preparing for university and those who receive skills training that prepares them for employment but blocks them from access to universities. Post-World War II, the American belief has been that university education should be available to all. A corollary of this belief is that everyone should enroll in the courses that will enable further study. Hacker wants it all. He wants to see secondary programs that focus on targeted workplace skills while preparing everyone for postsecondary education. I am not willing to claim that this is a circle that cannot be squared, but building such an educational system is going to be far harder than he implies. The sad fact is that for many of our students our educational system delivers neither. The blame cannot be placed entirely at the foot of mathematics. While over 60 percent of entering community college students need remediation in mathematics, about half of these also need remediation in reading, a far more serious impediment. Nevertheless, mathematics is a stumbling block for many students who could otherwise succeed. What high school mathematics do they really need? This is an important question that requires a thoughtful response. On page 8, Hacker admits that “basic algebra is definitely necessary for everyone.” Where he draws the line is at what he calls “advanced algebra,” which is, in fact, Algebra 2. According to the Common Core State Standards for Mathematics, this is the course where, among other skills, students are expected to learn to perform arithmetic operations on polynomials and understand the connections between their zeros and factors; construct and compare linear, quadratic, and exponential models; understand the general role of functions in modeling a David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College. His email address is bressoud@macalester.edu.
19 citations
TL;DR: In this article, the authors focus on the model of a universal quantum computer that allows the full computational potential for quantum systems, and on the restricted model, called "BosonSampling" based on noninteracting bosons.
Abstract: Q uantum computers are hypothetical devices, based on quantum physics, which would enable us to perform certain computations hundreds of orders ofmagnitude faster than digital computers. This feature is coined “quantum supremacy”, and one aspect or another of such quantum computational supremacy might be seen by experiments in the near future: by implementing quantum error-correction or by systems of noninteracting bosons or by exotic new phases of matter called anyons or by quantum annealing, or in various other ways. We concentrate in this paper on the model of a universal quantum computer that allows the full computational potential for quantum systems, and on the restricted model, called “BosonSampling”, based on noninteracting bosons. A main reason for concern regarding the feasibility of quantum computers is that quantum systems are inherently noisy. We will describe an optimistic hypothesis regarding quantum noise that will allow quantum computing and a pessimistic hypothesis that won’t. The
19 citations
TL;DR: The aim of the introduction is to generalize familiar notions, such as rank, eigenvectors and singular vectors, from matrices to tensors from linear algebra.
Abstract: Engineers and scientists spice up their linear algebra toolbox with a pinch of algebraic geometry. Eigenvectors and singular vectors are familiar from linear algebra, where they are taught in concert with eigenvalues and singular values. Linear algebra is the foundation of applied mathematics and scientific computing. Specifically, the concept of eigenvectors and numerical algorithms for computing them became a key technology during the twentieth century. However, in our day and age of Big Data, the role of matrices is increasingly often played by tensors, that is, multidimensional arrays of numbers. Principal component analysis tells us that eigenvectors of matrices point to directions in which the data is most spread. One hopes to identify similar features in higher-dimensional data. This has encouraged engineers and scientists to spice up their linear algebra toolbox with a pinch of algebraic geometry. The spectral theory of tensors is the theme of the AMS Invited Address at the SIAM Annual Meeting, held in Boston on July 11–15, 2016. This theory was pioneered around 2005 by Lek-Heng Lim and Liqun Qi. The aim of our introduction is to generalize familiar notions, such as rank, eigenvectors and singular vectors, from matrices to tensors. Specifically, we address the following questions. The answers are provided in Examples 5 and 10 respectively.
16 citations
TL;DR: In a recent workshop at the American Statistical Association (ASA) headquarters in Alexandria, Virginia, participants came together to develop a Common Vision of the pressing need to modernize undergraduate programs in the mathematical sciences, with particular attention to the first two years.
Abstract: Higher education is under intense public scrutiny. Revenue streams are diminishing. Mathematics is often at the center of discussions, lambasted for low student success rates. Change is coming, and the mathematical sciences community must come together to ensure that the change happens coherently and in a mathematically sound way. More mathematicians must become involved, working with colleagues from partner disciplines to modernize curricula and adopt evidence-based active learning strategies in mathematics classrooms. In May 2015, we attended a remarkable workshop at the American Statistical Association headquarters in Alexandria, Virginia. What was truly extraordinary was the genuine collaboration of members of the five professional societies concerned with undergraduate teaching in the mathematical sciences: the American Mathematical Association of Two-Year Colleges (AMATYC), the American Mathematical Society (AMS), the American Statistical Association (ASA), the Mathematical Association of America (MAA), and the Society for Industrial and Applied Mathematics (SIAM). Workshop attendees represented not only the five mathematical sciences associations but also partner STEM disciplines, higher education advocacy organizations, and industry. Workshop participants came together to develop a Common Vision of the pressing need to modernize undergraduate programs in the mathematical sciences, with particular attention to the first two years. The main themes emerging from the workshop are that all of us in the mathematical sciences community face similar challenges, that the status quo is unacceptable, and that the most effective solutions will involve cooperation and collaboration. Research on “collective impact” [8] suggests that, in achieving significant and lasting change in any area, a coordinated effort supported by major players from all existing sectors is more effective than an array of new initiatives and organizations. Common Vision encourages such action by highlighting existing efforts and draws on the collective wisdom of a diverse group of stakeholders to articulate a shared vision for modernizing the undergraduate mathematics program.
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TL;DR: In this article, the authors define the construction of a blender and the role it plays in the study of smooth dynamical systems, and present an illustrated discussion of the role of the blender.
Abstract: What is a blender? In six illustrated pages we define the construction of a blender and the role it plays in the study of smooth dynamical systems.
TL;DR: In this article, it was shown that a priori solutions are always twice differentiable classical solutions, and that their second derivative is continuous only in very rigid situations that have a simple geometric interpretation.
Abstract: Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to one of the classical degenerate nonlinear second order differential equations on Euclidean space. One naturally wonders "what is the regularity of solutions?" A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry. Without deeply understanding the underlying geometry, it is impossible to prove fine analytical properties.
Journal Article•
TL;DR: This paper presented a statistical profile of recipients of doctoral degrees awarded by departments in the mathematical sciences at universities in the United States during the period July 1, 2012, through June 30, 2013.
Abstract: This report presents a statistical profile of recipients of doctoral degrees awarded by departments in the mathematical sciences at universities in the United States during the period July 1, 2012, through June 30, 2013. Information in the report was provided by the departments that awarded the degrees with additional information provided by the individual new doctoral recipients. The report includes an analysis of the fall 2013 employment plans of 2012–2013 doctoral recipients and a demographic profile summarizing characteristics of citizenship status, gender, and racial/ethnic group. This report is based on a complete census of the 2012–2013 new doctorates and includes information about 2012-2013 doctoral recipients that were not included in the preliminary report in the June/July 2014 issue of Notices.
Journal Article•
TL;DR: The recent detection of gravitational waves by the LIGO/VIRGO team is an incredibly impressive achievement of experimental physics and is also a tremendous success of the theory of General Relativity as mentioned in this paper.
Abstract: The recent detection of gravitational waves by the LIGO/VIRGO team is an incredibly impressive achievement of experimental physics. It is also a tremendous success of the theory of General Relativity. It confirms the existence of black holes; shows that binary black holes exist; that they may collide and that during the merging process gravitational waves are produced. These are all predictions of General Relativity theory in its fully nonlinear regime. The existence of gravitational waves was predicted by Albert Einstein in 1916 within the framework of linearized Einstein theory. Contrary to common belief, even the very \emph{definition} of a gravitational wave in the fully nonlinear Einstein theory was provided only after Einstein's death. Actually, Einstein had arguments against the existence of nonlinear gravitational waves (they were erroneous but he did not accept this), which virtually stopped development of the subject until the mid 1950s. This is what we refer to as the \emph{Red Light} for gravitational waves research. In the following years, the theme was picked up again and studied vigorously by various experts, mainly Herman Bondi, Felix Pirani, Ivor Robinson and Andrzej Trautman, where the theoretical obstacles concerning gravitational wave existence were successfully overcome, thus giving the `Green Light' for experimentalists to start designing detectors, culminating in the recent LIGO/VIRGO discovery. In this note we tell the story of this theoretical breakthrough. Particular attention is given to the fundamental 1958 papers of Trautman, which seem to be lesser known outside the circle of General Relativity experts. A more detailed technical description of these 2 papers is given in the Appendix.
TL;DR: Grothendieck (1922-2014) as discussed by the authors was one of the maiores matemáticos do século 20 and um dos mais atípicos.
Abstract: Este artigo está licenciado sob forma de uma licença Creative Commons Atribuição 4.0 Internacional, que permite uso irrestrito, distribuição e reprodução em qualquer meio, desde que a publicação original seja corretamente citada. http://creativecommons.org/licenses/by/4.0/deed.pt_BR Resumo: Alexandre Grothendieck (1922-2014) foi um dos maiores matemáticos do século 20 e um dos mais atípicos. Nascido na Alemanha a um pai anarquista de origem russa, sua infância foi marcada pela militância política dos seus pais, assim passando por revoluções, guerras e sobrevivência. Descoberto por sua precocidade matemática por Henri Cartan, Grothendieck fez seu doutorado sob orientação de Laurent Schwartz e Jean Dieudonné. As principais contribuições dele são na área da topologia e na geometria algébrica, assim como na teoria das categorias. No final dos anos de 1960, ele se dedicou à militância política e ecológica, organizando a revista Survivre durante três anos. Em 1986, publicou um manuscrito autobiográfico de 1000 páginas, Récoltes et semailles, em que ele descreve sua experiência e sua prática da matemática, assim suas contribuições à comunidade matemática francesa. Pouco comentado na filosofia, as implicações dos seus descobrimentos fora mais recentemente discutidas por Alain Badiou na sua \"fenômeno-lógica\", em Logiques des mondes (2016) e Arkady Plonitsky, Mathgematics, Science and postclassical Theory (1997), pesquisa trata da semelhança entre os aspectos formais da filosofia de Gilles Deleuze e da topologia de Grothendieck.
TL;DR: Castillo-Chavez et al. as mentioned in this paper presented the possibility that quarantine can cause increased levels of Ebola transmission in STEM students at Arizona State University, where they showed that the possibility of quarantine can increase the risk of transmission of the virus.
Abstract: Work with STEM students at MTBI advanced the possibility that quarantine can cause increased levels of Ebola transmission.1 C. Castillo-Chavez is regents professor and the Joaquín Bustoz, Jr. Professor of Mathematical Biology at Arizona State University, as well as the executive director of the Mathematical and Theoretical Biology Institute. His email address is ccchavez@asu.edu. K. Barley is a graduate student at the Simon A. Levin Mathematical, Computational and Modeling Sciences Center. His email address is Kamal.Barley@asu.edu. D. Bichara is a postdoctoral fellow at the Simon A. Levin Mathematical, Computational and Modeling Sciences Center. His email address is derdei.bichara@asu.edu. D. Chowell is a graduate student at the Simon A. Levin Mathematical, Computational and Modeling Sciences Center. His email address is Diego.Chowell-Puente@
TL;DR: In this paper, a metric on the space of parameterized surfaces that is degenerate in the direction of reparameterization is defined, which is called spherical surfaces, which are diffeomorphic to the unit sphere.
Abstract: Introduction Having applications to Form recognition in mind, we want to be able to compare shapes of surfaces in R3 in a way that does not depend on parameterizations. To accomplish such so-called gauge invariance, we defined a metric on the space of parameterized surfaces that is degenerate in the direction of reparameterization. 1 What are the surfaces under consideration? The surfaces we will consider in this note are surfaces which are diffeomorphic to the unit sphere. In other words, the unit sphere will be our model surface, and the surfaces we will consider will be those that can be modeled out of it. To be mathematically precise, these are orientable genus-0 smooth compact surfaces or, equivalently, orientable 2dimensional compact simply connected submanifolds of R3 and will be called spherical surfaces in this note. How is the unit sphere represented? The good thing about the unit sphere is that only one chart suffices to cover it almost completely. We will use spherical coordinates, with polar angle θ being greater than 0 (North Pole) and less than π (South Pole) and azimuthal angleφ being greater than or equal to 0 (Greenwich prime meridian) and less than 2π (Greenwich prime meridian again); see Figure 1.
TL;DR: The efforts of the SFSU-Colombia Combinatorics Initiative to build a research and learning community between California and Colombia seeks to broaden and deepen representation in mathematics.
Abstract: This article describes the efforts of the SFSU-Colombia Combinatorics Initiative to build a research and learning community between California and Colombia. It seeks to broaden and deepen representation in mathematics, based on four underlying principles:
1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
4. Every student deserves to be treated with dignity and respect.
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TL;DR: In this article, the authors highlight the production of doctorates in mathematics education for the past fifty years and highlight the different paths of doctoral programs in education, focusing on issues related to mathematics learning, teaching, or curriculum.
Abstract: Institutions in the United States have been producing PhDs in mathematics education for more than a century. Teachers College at Columbia University and the University of Chicago produced the first graduates in mathematics education in 1906 and 1912, respectively [1]. In those institutions, doctoral students in mathematics education typically took courses along with doctoral students in mathematics. However, the dissertation research took a different direction as doctorates in mathematics education focused on issues related to mathematics learning, teaching, or curriculum. Over the past fifty years the nature of doctoral programs in mathematics education has taken many different paths [2], [3], [4], [5]. While some doctoral programs remain in mathematics departments, today the majority of doctoral programs in mathematics education are in colleges/schools of education. Some doctoral programs in mathematics education continue to require a substantial amount of mathematics, while others require very little graduate-level mathematics. This article highlights the production of doctorates in mathematics education for the past fifty years.
TL;DR: Barry used to visit Israel regularly and set up base at the Hebrew University in Jerusalem and came for a day or two to give a seminar at the Technion as mentioned in this paper, and the visits were like my annual driving tests: if Barry simply shrugged and lost interest,
Abstract: Barry used to visit Israel regularly. He always set up base at the Hebrew University in Jerusalem and came for a day or two to give a seminar at the Technion. Barry made his itinerary early, which meant that I had plenty of time to get ready for his visit, which really meant that I had plenty of time to worry what worthwhile observation I had to impress Barry with. Barry’s visits were like my annual driving tests: if Barry simply shrugged and lost interest,
Journal Article•
TL;DR: In this paper, a short exposition of my ideas about producing mathematical documents is given, along with a short introduction to mathematical typesetting in the AMS Notices, and a short summary of the typesetting process.
Abstract: Richard Palais ran a column on mathematical typesetting in the AMS Notices, and he invited me to be guest columnist. This is what I wrote–a short exposition of my ideas about producing mathematical documents.
TL;DR: More Postdocs and Fewer Jobs: The employment structure for PhD mathematicians is undergoing challenging changes, as reported by the Joint Data Committee over the last few years as discussed by the authors, which addresses two of these challenges in light of results from the AMS annual survey and from the Conference Board of the Mathematical Sciences [2].
Abstract: NOtices Of the AMs 1057 The employment structure for PhD mathematicians is undergoing challenging changes, as reported by the Joint Data Committee over the last few years. This article addresses two of these challenges in light of results from the AMS annual survey [1] and from the Conference Board of the Mathematical Sciences [2]. (1) More Postdocs and Fewer Jobs. The number of postdocs in mathematics is increasing while the number of academic jobs potentially leading to tenure in doctoral departments of mathematics is stagnating. Therefore, research-focused departments should educate graduate students and postdoctoral fellows for a wider range of career opportunities and responsibilities. Departments with postdocs should record where postdocs find their next—and, if possible, subsequent—employment. (2) Recent Hiring Off Tenure Track. Over half of recent hiring of full-time PhD mathematicians by all US universities and 4-year colleges has been off the tenure track: 811 out of 1,551 for fall 2014 [1a]. The mathematical community should discuss the roles of PhDs in full-time non-tenure-track positions. In particular, an ad hoc committee from the AMS and the MAA should discuss, articulate, and disseminate responsible practices. Details will surely vary across various sectors of higher education. Broad awareness of the issues by employers and by job seekers will be at least as important as details of the report.