Fields Medal Research Articles

Global optimization and structural analysis of Coulomb and logarithmic potentials on the unit sphere using a population-based heuristic approach

Global optimization of high-dimensional non-convex functions arises in many important applications and researches. In this paper, we focus on the minimum energy configuration problem on the unit sphere, whose goal is to determine the minimum-energy configuration of N particles interacting via the standard Coulomb potential or the discrete logarithmic potential on the unit sphere. This is a classic global optimization problem with many important applications in mathematics, physics, biology and chemistry, and is one of the 18 great mathematical problems for the twenty-first century listed by the Fields Medal winner (Smale, 1998). To determine the minimum-energy configuration for these two potential functions, we propose a population-based global optimization algorithm that combines a two-phase local optimization method with early-stopping strategy, a random perturbation operator, a distance-based population update strategy using a novel distance function to measure the difference between solutions, and a population rebuilding method. Computational results show that the proposed algorithm is very competitive compared to the best known results in the literature. In particular, it improves the best-known result for one instance of the classic Thomson problem (i.e., N=360) widely studied in the literature. The configurations of instances with up to N=500 particles are systematically optimized for the first time by the proposed algorithm for the discrete logarithmic potential. The comparative study shows that the putative optimal configurations have a high similarity for the Coulomb and logarithmic potentials and that the defects play an important role in reducing the energy of configuration. The energies of particles on the putative optimal configurations are analyzed for the studied logarithmic potential.

Uniwersytet w imadle rankingów

The author of the article focuses on a critical assessment of university rankings existing in general, not only in academic, space. Three different rankings with an international reach: (1) The Academic Ranking of World Universities (ARWU) also known as the Shanghai Ranking, (2) Times Higher Education World University Rankings (THE), (3) CWTS Leiden Ranking and one Polish ranking: The Perspektywy University Ranking have been subjected to a detailed critical evaluation. The main objection raised against the discussed international rankings is disregarding by them scientific works published in national languages of the authors. This decision of rankings’ creators is particularly detrimental to universities with scientifically strong faculties in the field of humanities and social sciences. The fact that monographs are ignored is difficult to comprehend (e.g. CWTS Leiden Ranking). Moreover, favouring publications in two journals: “Science” and “Nature” as well as the Nobel Prize and the Fields Medal (mathematics) has no deeper justification. The author calls for recognition of monographs and also academic textbooks which shape the culture of university. The percentage proportion of particular criteria in the total assessment leading to ranking results – which, in fact, boils down to marginalising scientific achievements of representatives of humanities and, partly, social sciences – raises the author’s reservations. Finally, if we look at our Polish local “backyard” where universities were fragmented after the Second World War and rebuilt according to the Soviet model of higher education with smaller higher education entities of a narrower profile (medical, agricultural, economic, sports) functioning alongside standard universities which had been stripped of these sections, we will notice that these classic, “full” universities have no chance of competing for a scientifically decent position in world rankings.

Editorial

Editorial

The Birth of a Review or (How Villani Both Reinforces and Challenges Our Beliefs About Mathematics) A Review of Cédric Villani’s Birth of a Theorem: A Mathematical Adventure

In the book Birth of a Theorem: A Mathematical Adventure, Cédric Villani paints a portrait of himself, from the (dare I say) obsessed mathematician who works until the wee hours of the night, to the ordinary citizen, enjoying an impressive variety of music, to the slightly odd man sporting an ornamental spider daily. From Lyon to Paris, through the despair of not being able to eat quality cheese, Villani takes us on a journey, revisiting the path that led him to win the Fields Medal. He takes us back in time and punctuates his account with anecdotes or stories of mathematicians. We recognize well-known names (Nash, Poincaré, etc.) through stories that are less well-known. He lets us in his adventure, and we jump in with both feet (e.g., see Figure 1).

Proof, Semiotics, and the Computer: On the Relevance and Limitation of Thought Experiment in Mathematics

This contribution defends two claims. The first is about why thought experiments are so relevant and powerful in mathematics. Heuristics and proof are not strictly and, therefore, the relevance of thought experiments is not contained to heuristics. The main argument is based on a semiotic analysis of how mathematics works with signs. Seen in this way, formal symbols do not eliminate thought experiments (replacing them by something rigorous), but rather provide a new stage for them. The formal world resembles the empirical world in that it calls for exploration and offers surprises. This presents a major reason why thought experiments occur both in empirical sciences and in mathematics. The second claim is about a looming aporia that signals the limitation of thought experiments. This aporia arises when mathematical arguments cease to be fully accessible, thus violating a precondition for experimenting in thought. The contribution focuses on the work of Vladimir Voevodsky (1966–2017, Fields medalist in 2002) who argued that even very pure branches of mathematics cannot avoid inaccessibility of proof. Furthermore, he suggested that computer verification is a feasible path forward, but only if proof is not modeled in terms of formal logic.

Sir Vaughan Jones. 31 December 1952—6 September 2020

Vaughan Jones made fundamental contributions to mathematics and mathematical physics, bringing together disparate areas of operator algebras, knots, links and low-dimensional topology in mathematics, and statistical mechanics, quantum field theory and quantum information in physics, while opening up the new field of quantum topology. The key that unlocked all this was his seminal work on subfactor theory in von Neumann algebras of operators, which led to a new invariant of links: the Jones polynomial. For this he was awarded the Fields Medal in 1990.

Operation STEM fatale - how an equity, diversity and inclusion initiative has brought us to reflect on the current challenges in cell biology and science as a whole.

Operation STEM fatale - how an equity, diversity and inclusion initiative has brought us to reflect on the current challenges in cell biology and science as a whole.

Making the most of world talent for science? The Nobel Prize and Fields Medal experience

Opportunities in science largely affect the accumulation of scientific knowledge and, therefore, technological change. However, there is little evidence of how much of people’s talent is actually wasted. Here we focus on scientists with the highest performance, the recipients of the Nobel Prize and Fields Medal. We found that the average age of scientists at the time of the breakthrough was higher for researchers from less developed countries. Moreover, individual opportunities in the world were extremely unequal by country of birth, gender significantly conditioned any participation in research, and the probability of becoming a top researcher more than doubled for individuals with parents belonging to the most favoured occupational categories. Thus, inequality of opportunity in science at the highest level was higher than in sports excellence (Olympic medals) and educational attainment. These findings would not be so negative if opportunities in science at the highest level had increased over time. Contrary to the expectations, our results show that opportunities in science, in contrast with humanities, have stagnated.

Recollection of Maryam Mirzakhani's Love for Mathematics: Igniting Female Students' Passion in Mathematics

Many of the famous women mathematicians experienced a spark. Something happened, and a love of mathematics began. This paper is on Maryam Mirzakhani who was the only woman and Iranian to be awarded with the Fields Medal, the most prestigious award in mathematics. Recollecting her love for mathematics may ignite female students' passion in mathematics. Perhaps, through her story, teachers can provide that inspiration. Mathematics teachers can present the beauty of mathematics to the students, helping engage them in a lifelong passion.

Voevodsky's unfinished project: Filling the gap between pure and applied mathematics

In a series of lectures given in 2003, soon after receiving the Fields Medal for his results in the Algebraic Geometry, Vladimir Voevodsky (1966–2017) identifies two strategic goals for mathematics, which he plans to pursue in his further research. The first goal is to develop a ‘‘computerised library of mathematical knowledge,’’ which supports an automated proof-verification. The second goal is to ‘‘bridge pure and applied mathematics.’’ Voevodsky's research towards the first goal brought about the new Univalent foundations of mathematics. In view of the second goal Voevodsky in 2004 started to develop a mathematical theory of Population Dynamics, which involved the Categorical Probability theory. This latter project did not bring published results and was abandoned by Voevodsky in 2009 when he decided to focus his efforts on the Univalent foundations and closely related topics. In the present paper, which is based on Voevodsky's archival sources, I present Voevodsky's views of mathematics and its relationships with natural sciences, critically discuss these views, and suggest how Voevodsky's ideas and approaches in the applied mathematics can be further developed and pursued. A special attention is given to Voevodsky's original strategy to bridge the persisting gap between pure and applied mathematics where computers and the computer-assisted mathematics play a major role.

Comments on Vladimir Voevodsky's biologically motivated works

Vladimir Voevodsky's contribution to Algebraic K-Theory is well known and was recognised by awarding him the Fields Medal. He became widely known in mathematical circles for his daring project of reforming the Foundations of Mathematics, a project he pursued until his untimely death. His ideas on the so-called “univalent” foundations of mathematics were taken up by a broad community of mathematicians and continue to be developed today. Less well known is that he devoted several years of his life to a biologically motivated project aimed at developing a mathematical theory of population dynamics. He never published anything about it. The aim of the article is to summarise his ideas on this project and to draw them to the attention of the scientific community.

Why U.S. Immigration Matters for the Global Advancement of Science

This paper studies the impact of U.S. immigration barriers on global knowledge production. We present four key findings. First, among Nobel Prize winners and Fields Medalists, migrants to the U.S. play a central role in the global knowledge network—representing 20-33% of the frontier knowledge producers. Second, using novel survey data and hand-curated life-histories of International Math Olympiad (IMO) medalists, we show that migrants to the U.S. are up to six times more productive than migrants to other countries—even after accounting for talent during one’s teenage years. Third, financing costs are a key factor preventing foreign talent from migrating abroad to pursue their dream careers, particularly for talent from developing countries. Fourth, certain ‘push’ incentives that reduce immigration barriers—by addressing financing constraints for top foreign talent—could increase the global scientific output of future cohorts by 42 percent. We concludeby discussing policy options for the U.S. and the global scientific community.

Why U.S. Immigration Matters for the Global Advancement of Science

This paper studies the impact of U.S. immigration barriers on global knowledge production. We present four key findings. First, among Nobel Prize winners and Fields Medalists, migrants to the U.S. play a central role in the global knowledge network—representing 20-33% of the frontier knowledge producers. Second, using novel survey data and hand-curated life-histories of International Math Olympiad (IMO) medalists, we show that migrants to the U.S. are up to six times more productive than migrants to other countries—even after accounting for talent during one’s teenage years. Third, financing costs are a key factor preventing foreign talent from migrating abroad to pursue their dream careers, particularly for talent from developing countries. Fourth, certain ‘push’ incentives that reduce immigration barriers—by addressing financing constraints for top foreign talent—could increase the global scientific output of future cohorts by 42 percent. We concludeby discussing policy options for the U.S. and the global scientific community.

Elitism in mathematics and inequality

The Fields Medal, often referred as the Nobel Prize of mathematics, is awarded to no more than four mathematicians under the age of 40, every 4 years. In recent years, its conferral has come under scrutiny of math historians, for rewarding the existing elite rather than its original goal of elevating under-represented mathematicians. Prior studies of elitism focus on citational practices while a characterization of the structural forces that prevent access remain unclear. Here we show the flow of elite mathematicians between countries and lingo-ethnic identity, using network analysis and natural language processing on 240,000 mathematicians and their advisor–advisee relationships. We present quantitative evidence of how the Fields Medal helped integrate Japan after WWII, through analysis of the elite circle formed around Fields Medalists. We show increases in pluralism among major countries, though Arabic, African, and East Asian identities remain under-represented at the elite level. Our results demonstrate concerted efforts by academic committees, such as prize giving, can either reinforce the existing elite or reshape its definition. We anticipate our methodology of academic genealogical analysis can serve as a useful diagnostic for equity and systemic bias within academic fields.

Why U.S. Immigration Barriers Matter for the Global Advancement of Science

This paper studies the impact of U.S. immigration barriers on global knowledge production. We present four key findings. First, among Nobel Prize winners and Fields Medalists, migrants to the U.S. play a central role in the global knowledge network— representing 20-33% of the frontier knowledge producers. Second, using novel survey data and hand-curated life-histories of International Math Olympiad (IMO) medalists, we show that migrants to the U.S. are up to six times more productive than migrants to other countries—even after accounting for talent during one's teenage years. Third, financing costs are a key factor preventing foreign talent from migrating abroad to pursue their dream careers, particularly talent from developing countries. Fourth, certain 'push' incentives that reduce immigration barriers – by addressing financing constraints for top foreign talent – could increase the global scientific output of future cohorts by 42% percent. We conclude by discussing policy options for the U.S. and the global scientific community.

Editorial: Biographical Memoirs, Volume 69

Editorial: Biographical Memoirs, Volume 69

Los rankings académicos y la distribución por género de las universidades

Este trabajo examina la distribución por género de las universidades, a través de un porcentaje de mujeres dentro del alumnado, y comprueba si tal distribución tiene alguna relación con la posición de las universidades en los rankings universitarios. Se han seleccionado siete rankings, globales y sintéticos, disponiendo de datos de la distribución por género del alumnado para 924 universidades. Los resultados confirman que las universidades con una equilibrada distribución entre hombres y mujeres presentan mayores puntuaciones, por tanto, mejores posiciones en los rankings. Las universidades con mayor presencia femenina en el alumnado son las de ratio más alto de estudiantes por profesor y las que alcanzan mayores valoraciones medias en los indicadores de internacionalización. En los indicadores de premios Nobel y medallas Fields, e ingresos procedentes de la industria, las universidades con mayor presencia masculina tienen mayores puntuaciones medias. Los resultados pueden contribuir a la mejora de las políticas de igualdad en las universidades, así como a orientar las decisiones estratégicas y de gestión de las mismas.

A Personal Journey into the World of Mathematics

Caucher Birkar received the 2018 Fields medal for his work in algebraic geometry. This article is based on a talk given by Birkar during the 7th Heidelberg Laureate Forum in Heidelberg, Germany. Birkar chose his current name after he immigrated to the United Kingdom from Iran. The person who introduced this talk noted, “Caucher Birkar means ‘migrant mathematician.’ In that sense, I am also Caucher Birkar, as are many people in this room.”

Sir Michael Atiyah, a Knight Mathematician

Sir Michael Atiyah was considered as one of the world's foremost mathematicians, He is best known for his work in algebraic topology and the co-development of a branch of mathematics called topological K-theory together with the Atiyah-Singer index theorem for which he received Fields Medal (1966). He received also the Abel Prize (2004) along with Isadore M. Singer for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics. Indeed, his work has helped theoretical physicists to advance their understanding of quantum field theory and general relativity.

FACTORS OF HIGHER EDUCATION INSTITUTIONS RANKING (COMPARISON OF THREE WORLD UNIVERSITY RANKINGS)

Many universities have drawn attention to world university rankings, which reflect the international competition of universities and represent their relative statuses. This study does not radically contradict all types of global university rankings but calls for an examination of the effects of their indicators on the final ranking of universities. This study investigates the indicator contribution to the ranking of universities in world university ranking systems including the Academic Ranking of World Universities (ARWU), Times Higher Education (THE), and QS World University Rankings. Results showed that in the ARWU system, three indicators regarding faculty members who won Nobel Prizes and Fields Medals and papers published in Nature and Science and in the Science Citation Index and Social Science Citation Index journals predicted the ranking of universities. For the QS and THE systems, the more powerful contributors to the ranking of universities were expert-based reputation indicators.