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A Note on the Origin of the Twin Prime Conjecture
A Note on the Origin of the Twin Prime Conjecture by William Dunham Department of Mathematics & Computer Science, Muhlenberg College; Department of Mathematics, Harvard University (visiting 2012-2013) More than any other branch of mathematics, number As the title suggests, de Polignac’s paper was about theory features a collection of famous problems that took prime numbers, and on p. 400 he stated two “theorems.” centuries to be proved or that remain unresolved to the It should be noted that he was not using the word in its present day. modern sense. Rather, de Polignac confirmed the truth of Among the former is Fermat’s Last Theorem. This these results by checking a number of cases and thereby states that it is impossible to find a whole number n ≥ 3 called them “theorems.” Of course, we would call them, at and whole numbers x , y , and z for which xn + yn = zn . best, “conjectures.” The true nature of these statements is suggested by the fact that de Polignac introduced them In 1640, as all mathematicians know, Fermat wrote this in under the heading “Induction and remarks” as seen below, a marginal note and claimed to have “an admirable proof” in a facsimile of his original text. that, alas, did not fit within the space available. The result remained unproved for 350 years until Andrew Wiles, with assistance from Richard Taylor, showed that Fermat was indeed correct. On the other hand, there is the Goldbach conjecture, which asserts that every even number greater than 2 can be expressed as the sum of two primes. -
Karen Keskulla Uhlenbeck
2019 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2019 to Karen Keskulla Uhlenbeck University of Texas at Austin “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” Karen Keskulla Uhlenbeck is a founder of modern by earlier work of Morse, guarantees existence of Geometric Analysis. Her perspective has permeated minimisers of geometric functionals and is successful the field and led to some of the most dramatic in the case of 1-dimensional domains, such as advances in mathematics in the last 40 years. closed geodesics. Geometric analysis is a field of mathematics where Uhlenbeck realised that the condition of Palais— techniques of analysis and differential equations are Smale fails in the case of surfaces due to topological interwoven with the study of geometrical and reasons. The papers of Uhlenbeck, co-authored with topological problems. Specifically, one studies Sacks, on the energy functional for maps of surfaces objects such as curves, surfaces, connections and into a Riemannian manifold, have been extremely fields which are critical points of functionals influential and describe in detail what happens when representing geometric quantities such as energy the Palais-Smale condition is violated. A minimising and volume. For example, minimal surfaces are sequence of mappings converges outside a finite set critical points of the area and harmonic maps are of singular points and by using rescaling arguments, critical points of the Dirichlet energy. Uhlenbeck’s they describe the behaviour near the singularities major contributions include foundational results on as bubbles or instantons, which are the standard minimal surfaces and harmonic maps, Yang-Mills solutions of the minimising map from the 2-sphere to theory, and integrable systems. -
Karen Uhlenbeck Awarded Abel Prize
COMMUNICATION Karen Uhlenbeck Awarded Abel Prize The Norwegian Academy of Science and Letters has awarded the Abel Prize for 2019 to Karen Keskulla Uhlenbeck of the University of Texas at Austin, “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” The Abel Prize recognizes contributions of extraordinary depth and influence in the mathematical sciences and has been awarded annually since 2003. It carries a cash award of six million Norwegian krone (approximately US$700,000). Citation detail what happens when the Palais–Smale condition is Karen Keskulla Uhlenbeck is a violated. A minimizing sequence of mappings converges founder of modern geometric outside a finite set of singular points, and, by using resca- analysis. Her perspective has ling arguments, they describe the behavior near the singu- permeated the field and led larities as bubbles or instantons, which are the standard to some of the most dramatic solutions of the minimizing map from the 2-sphere to the advances in mathematics in the target manifold. last forty years. In higher dimensions, Uhlenbeck in collaboration with Geometric analysis is a field Schoen wrote two foundational papers on minimizing of mathematics where tech- harmonic maps. They gave a profound understanding niques of analysis and differ- of singularities of solutions of nonlinear elliptic partial ential equations are interwoven differential equations. The singular set, which in the case Karen Keskulla Uhlenbeck with the study of geometrical of surfaces consists only of isolated points, is in higher and topological problems. -
Dr. Yitang Zhang Professor Department of Mathematics University of California, Santa Barbara
Distinguished Lifetime Achievement Award Dr. Yitang Zhang Professor Department of Mathematics University of California, Santa Barbara Citation of Accomplishments Establishing the first finite bound on the gaps of prime numbers and thus solving a centuries- old problem in number theory, and his unsubdued passion for mathematics and science. Dr. Yitang Zhang was born on February 5, 1955 in Shanghai, China, and lived there until he was 13 years old. During the Cultural Revolution, he and his parents were sent to the countryside to work in the fields. He worked as a laborer for ten years and was unable to attend high school. After the Cultural Revolution ended, he entered Peking University in 1978 as an undergraduate student and graduated in 1982. Then he became a graduate student of Professor Pan Chengbiao, a number theorist at Peking University, and obtained his master’s degree in mathematics in 1984. With recommendations from Professor Ding Shisun, the president of Peking University, Yitang was granted a full scholarship at Purdue University. He arrived at Purdue in June 1985, studied there for seven years, and obtained his Ph.D. in mathematics in December 1991. After graduation, he had a hard time finding an academic position. After several years, he managed to find a position as a lecturer at the University of New Hampshire (UNH), where he was hired by Professor Kenneth Appel in 1999. He served as a lecturer at UNH until January 2014, when UNH appointed him to a full professorship as a result of his breakthrough on the distribution of prime numbers. In Fall 2015, he accepted an offer of full professorship at the University of California, Santa Barbara. -
President's Report
Volume 38, Number 4 NEWSLETTER July–August 2008 President’s Report Dear Colleagues: I am delighted to announce that our new executive director is Maeve Lewis McCarthy. I am very excited about what AWM will be able to accomplish now that she is in place. (For more about Maeve, see the press release on page 7.) Welcome, Maeve! Thanks are due to the search committee for its thought and energy. These were definitely required because we had some fabulous candidates. Thanks also to Murray State University, Professor McCarthy’s home institution, for its coopera- tion as we worked out the details of her employment with AWM. The AWM Executive Committee has voted to give honorary lifetime mem- IN THIS ISSUE berships to our founding presidents, Mary Gray and Alice T. Schafer. In my role as president, I am continually discovering just how extraordinary AWM is 7 McCarthy Named as an organization. Looking back at its early history, I find it hard to imagine AWM Executive Director how AWM could have come into existence without the vision, work, and persist- ence of these two women. 10 AWM Essay Contest Among newly elected members of the National Academy of Sciences in the physical and mathematical sciences are: 12 AWM Teacher Partnerships 16 MIT woMen In maTH Emily Ann Carter Department of Mechanical and Aerospace Engineering and the Program in 19 Girls’ Angle Applied and Computational Mathematics, Princeton University Lisa Randal Professor of theoretical physics, Department of Physics, Harvard University Elizabeth Thompson Department of Statistics, University of Washington, Seattle A W M The American Academy of Arts and Sciences has also announced its new members. -
Non-Self-Dual Yang-Mills Connections with Nonzero Chern Number
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 24, Number 1, January 1991 NON-SELF-DUAL YANG-MILLS CONNECTIONS WITH NONZERO CHERN NUMBER LORENZO SADUN AND JAN SEGERT We prove the existence of non-self-dual Yang-Mills connections on SU(2) bundles over the standard four-sphere, specifically on all bundles with second Chern number not equal to ±1. A Yang- Mills (YM) connection A is a critical point of the YM action S(A)= f \F\2dVol= f -Tr(*FAF), Js4 Js4 where F is the curvature of the connection A and * is the Hodge dual. The YM equations D * F = 0, where D denotes the covariant exterior derivative, are the variational equations of this functional, and constitute a system of second-order PDE's in A. Absolute minima of the YM action, in addition to satisfy ing the YM equations, also satisfy a first-order system of PDE's, the (anti)self-duality equations *F = ±F. We call a connection non-self-dual (NSD) if it is neither self-dual ( *F = F ) nor anti- self-dual (*F = —F), i.e., if it is not a minimum of the YM action. (Anti) self-dual connections on *S4 have been well-understood for some time. The first nontrivial example, the BPST instanton [BPST], was found in 1975, and three years later all self-dual so lutions on S4 were classified [ADHM], not only for SU(2) but for all classical groups. The study of self-dual SU(2) connections on other four-manifolds led to spectacular progress in topology, including the discovery of fake R4 (see [FU] for an overview). -
Karen Keskulla Uhlenbeck
2019 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2019 to Karen Keskulla Uhlenbeck University of Texas at Austin “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” Karen Keskulla Uhlenbeck is a founder of modern by earlier work of Morse, guarantees existence of Geometric Analysis. Her perspective has permeated minimisers of geometric functionals and is successful the field and led to some of the most dramatic in the case of 1-dimensional domains, such as advances in mathematics in the last 40 years. closed geodesics. Geometric analysis is a field of mathematics where Uhlenbeck realised that the condition of Palais— techniques of analysis and differential equations are Smale fails in the case of surfaces due to topological weaved with the study of geometrical and topological reasons. The papers of Uhlenbeck, co-authored with problems. Specifically, one studies objects such as Sacks, on the energy functional for maps of surfaces curves, surfaces, connections and fields which are into a Riemannian manifold, have been extremely critical points of functionals representing geometric influential and describe in detail what happens when quantities such as energy and volume. For example, the Palais-Smale condition is violated. A minimising minimal surfaces are critical points of the area and sequence of mappings converges outside a finite set harmonic maps are critical points of the Dirichlet of singular points and by using rescaling arguments, energy. Uhlenbeck’s major contributions include they describe the behaviour near the singularities foundational results on minimal surfaces and as bubbles or instantons, which are the standard harmonic maps, Yang-Mills theory, and integrable solutions of the minimising map from the 2-sphere to systems. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Optimal Execution of Portfolio Transactions∗
Optimal Execution of Portfolio Transactions¤ Robert Almgreny and Neil Chrissz December 2000 Abstract We consider the execution of portfolio transactions with the aim of minimizing a combination of volatility risk and transaction costs aris- ing from permanent and temporary market impact. For a simple lin- ear cost model, we explicitly construct the efficient frontier in the space of time-dependent liquidation strategies, which have minimum expected cost for a given level of uncertainty. We may then select op- timal strategies either by minimizing a quadratic utility function, or by minimizing Value at Risk. The latter choice leads to the concept of Liquidity-adjusted VAR, or L-VaR, that explicitly considers the best tradeoff between volatility risk and liquidation costs. ¤We thank Andrew Alford, Alix Baudin, Mark Carhart, Ray Iwanowski, and Giorgio De Santis (Goldman Sachs Asset Management), Robert Ferstenberg (ITG), Michael Weber (Merrill Lynch), Andrew Lo (Sloan School, MIT), and George Constaninides (Graduate School of Business, University of Chicago) for helpful conversations. This paper was begun while the first author was at the University of Chicago, and the second author was first at Morgan Stanley Dean Witter and then at Goldman Sachs Asset Management. yUniversity of Toronto, Departments of Mathematics and Computer Science; [email protected] zICor Brokerage and Courant Institute of Mathematical Sciences; [email protected] 1 December 2000 Almgren/Chriss: Optimal Execution 2 Contents Introduction 3 1 The Trading Model 6 1.1 The Definition of a Trading Strategy . 7 1.2 Price Dynamics . 7 1.3 Temporary market impact . 8 1.4 Capture and cost of trading trajectories . -
Ten Mathematical Landmarks, 1967–2017
Ten Mathematical Landmarks, 1967{2017 MD-DC-VA Spring Section Meeting Frostburg State University April 29, 2017 Ten Mathematical Landmarks, 1967{2017 Nine, Eight, Seven, and Six Nine: Chaos, Fractals, and the Resurgence of Dynamical Systems Eight: The Explosion of Discrete Mathematics Seven: The Technology Revolution Six: The Classification of Finite Simple Groups Ten Mathematical Landmarks, 1967{2017 Five: The Four-Color Theorem 1852-1976 1852: Francis Guthrie's conjecture: Four colors are sufficient to color any map on the plane in such a way that countries sharing a boundary edge (not merely a corner) must be colored differently. 1878: Arthur Cayley addresses the London Math Society, asks if the Four-Color Theorem has been proved. 1879: A. B. Kempe publishes a proof. 1890: P. J. Heawood notes that Kempe's proof has a serious gap in it, uses same method to prove the Five-Color Theorem for planar maps. The Four-Color Conjecture remains open. 1891: Heawood makes a conjecture about coloring maps on the surfaces of many-holed tori. 1900-1950's: Many attempts to prove the 4CC (Franklin, George Birkhoff, many others) give a jump-start to a certain branch of topology called Graph Theory. The latter becomes very popular. Ten Mathematical Landmarks, 1967{2017 Five: The Four-Color Theorem (4CT) The first proof By the early 1960s, the 4CC was proved for all maps with at most 38 regions. In 1969, H. Heesch developed the two main ingredients needed for the ultimate proof, namely reducibility and discharging. He also introduced the idea of an unavoidable configuration. -
Awards of ICCM 2013 by the Editors
Awards of ICCM 2013 by the Editors academies of France, Sweden and the United States. He is a recipient of the Fields Medal (1986), the Crafoord Prize Morningside Medal of Mathematics in Mathematics (1994), the King Faisal International Prize Selection Committee for Science (2006), and the Shaw Prize in Mathematical The Morningside Medal of Mathematics Selection Sciences (2009). Committee comprises a panel of world renowned mathematicians and is chaired by Professor Shing-Tung Björn Engquist Yau. A nomination committee of around 50 mathemati- Professor Engquist is the Computational and Applied cians from around the world nominates candidates based Mathematics Chair Professor at the University of Texas at on their research, qualifications, and curriculum vitae. Austin. His recent work includes homogenization theory, The Selection Committee reviews these nominations and multi-scale methods, and fast algorithms for wave recommends up to two recipients for the Morningside propagation. He is a member of the Royal Swedish Gold Medal of Mathematics, up to two recipients for the Morningside Gold Medal of Applied Mathematics, and up to four recipients for the Morningside Silver Medal of Mathematics. The Selection Committee members, with the exception of the committee chair, are all non-Chinese to ensure the independence, impartiality and integrity of the awards decision. Members of the 2013 Morningside Medal of Mathe- matics Selection Committee are: Richard E. Borcherds Professor Borcherds is Professor of Mathematics at the University of California at Berkeley. His research in- terests include Lie algebras, vertex algebras, and auto- morphic forms. He is best known for his work connecting the theory of finite groups with other areas in mathe- matics. -
Board of Retirement Regular Meeting Sacramento County Employees’ Retirement System
Board of Retirement Regular Meeting Sacramento County Employees’ Retirement System Agenda Item 6 MEETING DATE: September 20, 2017 SUBJECT: Travel Reimbursement for Participation at Global Absolute Return Congress (October 23-25, 2017 in Boston, Massachusetts) Deliberation Receive SUBMITTED FOR: X Consent and Action and File RECOMMENDATION Staff recommends that the Board accept the travel and lodging reimbursement offered by Global Absolute Return Congress (‘Global ARC’) for SCERS’ Chief Investment Officer Steve Davis to participate on three discussion panels at this conference. PURPOSE To inform the Board on upcoming out-of-state travel and obtain advance approval to accept the travel and lodging reimbursement offered by Global ARC for Steve Davis’ participation in compliance with Government Code section 89500. BACKGROUND Sacramento County Employees’ Retirement System (‘SCERS’) has received an invitation to have one of its employees participate on three panels at the Global ARC Congress, which will be held in Boston, Massachusetts, from October 23 through October 25, 2017. Global ARC has requested that Steve Davis, SCERS’ Chief Investment Officer, be the designated employee. Included with the invitation was the offer from Global ARC to reimburse SCERS for admission, travel expenses, and lodging for three nights (October 22, 23 and 24) for the conference (See Exhibit A). Staff recommends that your Board approve acceptance of the gift, permit Mr. Davis to use the gift and publicly report its use in the Minutes for this meeting. DISCUSSION Trustees have repeatedly been advised that they should not accept conference fees or travel payments to attend industry events and conferences because such offers fall under the definition of a gift as that term is defined in the ethics chapter of the Public Reform Act, California Government Code §89500 et seq.