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I argue that since technology is already changing the way we do mathematics, we can use this technology to allow mathematics to be more congressive without human mathematicians becoming redundant in the face of technological advances. In thinking about what it means to "do mathematics," I examine the following aspects of technology in mathematics: teaching and learning, asking questions, collaboration, dissemination, and the act of doing research. This is not intended to be a rigorous analysis but an informed reflection based on my experience as a mathematician. [ABSTRACT FROM AUTHOR]

The extant letters exchanged between Kervaire and Milnor during their collaboration from 1958-1961 concerned their work on the classification of exotic spheres, culminating in their 1963 Annals of Mathematics paper. Michel Kervaire died in 2007; for an account of his life, see the obituary by Shalom Eliahou, Pierre de la Harpe, Jean-Claude Hausmann, and Claude Weber in the September 2008 issue of the Notices of the American Mathematical Society. The letters were made public at the 2009 Kervaire Memorial Conference in Geneva. Their publication in this issue of the Bulletin of the American Mathematical Society is preceded by our commentary on these letters, providing some historical background. [ABSTRACT FROM AUTHOR]

In this article we discuss how abstraction boundaries can help tame complexity in mathematical research with the help of an interactive theorem prover. While many of the ideas we present here have been used implicitly by mathematicians for some time, we argue that the use of an interactive theorem prover introduces additional qualitative benefits in the implementation of these ideas. [ABSTRACT FROM AUTHOR]

The article discusses the 1908 12-foot high bronze statue of mathematician Niels Henrik Abel (1802-1829) by sculptor Gustav Vigeland at the Royal Park in Oslo, Norway as of 2015. Topics covered include the object's depiction of Abel as a heroic figure with windblown hair astride two vanquished prone figures, and his stature as one of the great romantic figures of mathematics. Also noted is Abel's work on proof of the general insolvability of the fifth-degree polynomial equation by radicals.

Neural networks have demonstrated impressive success in various domains, raising the question of what fundamental principles underlie the effectiveness of the best AI systems and quite possibly of human intelligence. This perspective argues that compositional sparsity, or the property that a compositional function have "few" constituent functions, each depending on only a small subset of inputs, is a key principle underlying successful learning architectures. Surprisingly, all functions that are efficiently Turing computable have a compositional sparse representation. Furthermore, deep networks that are also sparse can exploit this general property to avoid the "curse of dimensionality". This framework suggests interesting implications about the role that machine learning may play in mathematics. [ABSTRACT FROM AUTHOR]

The author takes colleagues to task for failing to "live deliberately," and specifically for failing to pay attention to debates over technology and artificial intelligence in the wider society, when contemplating a mechanical future for mathematics. [ABSTRACT FROM AUTHOR]

Matroids are combinatorial abstractions of independence, a ubiquitous notion that pervades many branches of mathematics. June Huh and his collaborators recently made spectacular breakthroughs by developing a Hodge theory of matroids that resolved several long-standing conjectures in matroid theory. We survey the main results in this development and ideas behind them. [ABSTRACT FROM AUTHOR]

The article offers information on theexotic fusion systems which is related to finite groups. It mentions that the classifying space BDI(4) is homotopy equivalent to the 2-completion of the nerve of a union of subcategories. It mentions that the construction for the exotic fusion systems FSol(q). It is based on the notion of "representation" of a p-local finite group as the p-local group of a not necessarily finite group G.

The article focuses on the concept of homotopy-theoretic perspective which is a class of spaces that includes the p-completions of classifying spaces of compact Lie groups G. It mentions hat the BLO2 offers an analogous context for the p-completions of classifying spaces of finite groups, and offers a codification of the local study of group theory.

The article offers information on the theory of p-compact groups that has been developed as a homotopy-theoretical version, or extension, of the theory of compact Lie groups. It mentions that the A p-compact group is related to the p-complete space whose loop space has finite mod p homology and algebra.

The title of this paper is inspired by the work of Poincaré [1890, 1892], who introduced many key dynamical systems methods during his research on celestial mechanics and especially the three-body problem. Since then, many researchers have contributed to his legacy by developing and applying these methods to problems in celestial mechanics and, more recently, with the design of space missions. This paper will give a survey of some of these exciting ideas, and we would especially like to acknowledge the work of Michael Dellnitz, Frederic Gabern, Katalin Crubits, Oliver Junge, Wang-Sang Koon, François Lekien, Martin Lo, Sina Ober-Blöbaum, Kathrin Padberg, Robert Preis, and Bianca Thiere. One of the purposes of the AMS Current Events session is to discuss work of others. Even though we were involved in the research reported on here, this short paper is intended to survey many ideas due to our collaborators and others. This survey is by no means complete, and we apologize for not having time or space to do justice to many important and fundamental works. In fact, the results reported on here rely on and were inspired by important preceding work of many others in celestial mechanics, mission design and in dynamical systems. We mention just a few whose work had a positive influence on what is reported here: Brian Barden, Ed Belbruno, Robert Farquhar, Gerard Gómez, George Hailer, Charles Jaffé, Kathleen Howell, Linda Petzold, Josep Masdemont, Vered Rom-Kedar, Radu Serban, Caries Simó, Thrgay Uzer, Steve Wiggins, and Roby Wilson. In an upcoming monograph (see Koon, Lo, Mars- den, and Ross [2005]), the dynamical systems and computational approach and its application to mission design are discussed in detail. One of the key ideas is that the competing gravitational pull between celestial bodies creates a vast array of passageways that wind around the Sun, planets and moons. The boundaries of these passageways are realized geometrically as invariant manifolds attached to equilibrium points and periodic orbits in interlinked three-body problems. In particular, tube-like structures form an interplanetary transport network which will facilitate the exploration of Mercury, the Moon, the asteroids, and the outer solar system, including a mission to assess the possibility of life on Jupiter's icy moons. The use of these methods in problems in molecular dynamics of interest in chemistry is also briefly discussed. [ABSTRACT FROM AUTHOR]

This note will provide a lightning tour through the centuries, concentrating on the study of manifolds of dimension 2, 3, and 4. Further comments and more technical details about many of the sections may be found in the Appendix. [ABSTRACT FROM AUTHOR]

The article focuses on the uniquness of centric linking systems for saturated fusion systems. It mentions that the open question in the theory of fusion systems with increasingly vibrant period of research. It offers information on the fusion system FS(G) of a finite group G with Sylow p-subgroup S.

The American mathematical research community celebrated, symbolically at least, its fiftieth anniversary in 1938. Many of those fifty years had marked a period of consolidation and growth at home of programs in mathematics at institutions of higher education supportive of high-level research as well as of a corps of talented researchers capable of making seminal contributions in a variety of mathematical areas. By the middle decades of the twentieth century—the 1930s, 1940s, and 1950s—members of that community, like members of the broader American public, began increasingly to look outward beyond the national boundaries of the United States and toward a larger international arena. This paper explores the contexts within which the American mathematical research community, in general, and the American mathematician Marshall Stone, in particular, deliberately worked in the decades around mid-century to effect the transformation from a national community to one actively participating in an internationalizing mathematical world. [ABSTRACT FROM AUTHOR]

In this paper, using the ideas of Bessi and Mather, we present a simple mechanical system exhibiting Arnold diffusion. This system of a particle in a small periodic potential can be also interpreted as ray propagation in a periodic optical medium with a near-constant index of refraction. Arnold diffusion in this context manifests itself as an arbitrary finite change of direction for nearly constant index of refraction. [ABSTRACT FROM AUTHOR]

In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated Fuchsian groups. Another method, of arithmetic nature, uses continued fraction expansions of the end points of the geodesic at infinity and is even older—it comes from the Gauss reduction theory. Introduced to dynamics by E. Artin in a 1924 paper, this method was used to exhibit dense geodesics on the modular surface. For 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. We present some of the ideas, results (old and recent), and interpretations that illustrate the multiple facets of the subject. [ABSTRACT FROM AUTHOR]

The article discusses the history of graph theory which was introduced by Leonhard Euler on August 26,1735 after solving the Konigsberg Bridge puzzle which he presented to the Petersburg Academy. His work was published on academy's proceedings called "Commentarii" with the title "The Solution to a Problem Relating to the Geometry of Position." This paper can be considered as the earliest work in topology.

In this paper, we describe some central mathematical problems in medical imaging. The subject has been undergoing rapid changes driven by better hardware and software. Much of the software is based on novel methods utilizing geometric partial differential equations in conjunction with standard signal/image processing techniques as well as computer graphics facilitating man/machine interactions. As part of this enterprise, researchers have been trying to base biomedical engineering principles on rigorous mathematical foundations for the development of software methods to be integrated into complete therapy delivery systems. These systems support the more effective delivery of many image-guided procedures such as radiation therapy, biopsy, and minimally invasive surgery. We will show how mathematics may impact some of the main problems in this area, including image enhancement, registration, and segmentation. [ABSTRACT FROM AUTHOR]

The article represents a major contribution to the literature surrounding E. Szemerédi's theorem on arithmetic progressions. The paper gives a new proof of the theorem and the same time provides a significant improvement of the upper bound to the function. The ergodic-theoretic approach depends essentially on passing to a limit. There is an argument about the idea that a sufficiently random subset of given density will be expected to contain many term of arithmetic progressions.

The article offers information on the theory of modular characters of a finite group G with modular representation.

The article profiles mathematician Yitang Zhang, a lecturer at the University of New Hampshire. He submitted a manuscript about the breakthrough on the twin prime question that there are infinitely many pairs of primes that differ by no more than seventy million. George Csicsery is editing a documentary about the life of Zhang.

The article provides an overview of the two published papers related to mathematics. The papers present important improvements in the analysis of low-degrees tests. The second paper presents two probabilistically checkable proof (PCP) systems which are amenable to proof composition and are aimed at achieving very probe-efficient PCP systems.

This article begins with a provocative question: Are there any genuine continuous multivariate real-valued functions? This may seem to be a silly question, but it is in essence what David Hilbert asked as one of the 23 problems he posed at the second International Congress of Mathematicians, held in Paris in 1900. These problems guided a large portion of the research in mathematics of the 20th century. Hilbert's 13th problem conjectured that there exists a continuous function ƒ : I3 → R, where I = [0,1], which cannot be expressed in terms of composition and addition of continuous functions from R2 → R, that is, as composition and addition of continuous real-valued functions of two variables. It took over 50 years to prove that Hilbert's conjecture is false. This article discusses the solution. [ABSTRACT FROM AUTHOR]

The five exceptional simple Lie algebras over the complex number are included one within the other as g2 ⊂ f4 ⊂ e6 ⊂ e7 ⊂ e8. The biggest one, e8, is in many ways the most mysterious. This article surveys what is known about it, including many recent results, and it focuses on the point of view of Lie algebras and algebraic groups over fields. [ABSTRACT FROM AUTHOR]

This is a survey article on the theory of fusion systems, a relatively new area of mathematics with connections to local finite group theory, algebraic topology, and modular representation theory. We first describe the general theory and then look separately at these connections. [ABSTRACT FROM AUTHOR]

The article surveys the progress in applying techniques from moduli spaces in mathematics and physics to studying the geometry and topology of three families of macromolecules of interest in biology, including RNAs, proteins and polysaccharides. It is intended for audience such as mathematicians whose expertise does not necessarily extend to either moduli spaces or macromolecules and who may be interested in the nascent application of geometry to problems in biology.

This is an introductory discussion of limit shapes, in particular for random partitions and stepped surfaces, and of their applications to supersymmetric gauge theories. [ABSTRACT FROM AUTHOR]

The article focuses on mathematics and Newton and Leibniz who developed the ideas of the calculus. It states that physicist Isaac Newton's Principia was an impressive mathematical and scientific talent during the first half of the 18 century and Newton's complicated notation is often cited as a reason that British mathematics lagged behind. It mentions that the mathematician Marquise Du Châtelet became a potent force in advancing the case for Newton on the Continent.

The article presents information on the cover which is about two theorems on geometric constructions. Jean-Victor Poncelet (1788-1867) was a professional engineer and sometimes mathematician who is probably best known for bringing back an interest in projective geometry which had long faded since the golden age of the subject in the 17th century.

The article features algebraist Emil Artin, and his work at several universities in the U.S. He established graduate education as one of two main priorities when he became president of the University of Notre Dame in 1934. He also thought mathematics at Indiana University. In 1946, he moved from Indiana University to Princeton University, where he thought advanced calculus, relativity, trigonometry and differential equations, among others.

We discuss a number of natural problems in arithmetic, arising in completely unrelated settings, which turn out to have a common formulation involving "thin" orbits. These include the local-global problem for integral Apollonian gaskets and Zaremba's Conjecture on finite continued fractions with absolutely bounded partial quotients. Though these problems could have been posed by the ancient Greeks, recent progress comes from a pleasant synthesis of modern techniques from a variety of fields, including harmonic analysis, algebra, geometry, combinatorics, and dynamics. We describe the problems, partial progress, and some of the tools alluded to above. [ABSTRACT FROM AUTHOR]

The article announces the retraction of a paper by Mohammad A. Asadi-Golmankhaneh on multiple point self-transverse immersion published in the journal "Bulletin of the American Mathematical Society" in the February 3, 2009 issue.

The article announces the retraction of a paper by Mohammad A. Asadi-Golmankhaneh on multiple point self-transverse immersion published in the journal "Bulletin of the American Mathematical Society" in the February 3, 2009 issue.

We expose a theorem of Pila and Wilkie on counting rational points in sets definable in o-minimal structures and some applications of this theorem to problems in diophantine geometry due to Masser, Peterzil, Pila, Starchenko, and Zannier. [ABSTRACT FROM AUTHOR]

The article discusses the cover of the "Bulletin of the American Mathematical Society" which was published on April 11, 2011 entitled "Early Images of Minimal Surfaces." The cover image, courtesy of David Hoffman, presents images of complete minimal surfaces in space and accompanied by mathematical descriptions. The authors pioneered in this setting a conversation between visual aspects of the minimal surfaces and advancing theory.