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Hillel Furstenberg
Hillel (Harry) Furstenberg ( he, הלל (הארי) פורסטנברג) (born September 29, 1935) is a German-born American-Israeli mathematician and professor emeritus at the Hebrew University of Jerusalem. He is a member of the Israel Academy of Sciences and Humanities and U.S. National Academy of Sciences and a laureate of the Abel Prize and the Wolf Prize in Mathematics. He is known for his application of probability theory and ergodic theory methods to other areas of mathematics, including number theory and Lie groups. Biography Furstenberg was born to German Jews in Nazi Germany, in 1935 (originally named "Fürstenberg"). In 1939, shortly after Kristallnacht, his family escaped to the United States and settled in the Washington Heights neighborhood of New York City, escaping the Holocaust. He attended Marsha Stern Talmudical Academy and then Yeshiva University, where he concluded his BA and MSc studies at the age of 20 in 1955. Furstenberg published several papers as an u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constituent states, Berlin is surrounded by the State of Brandenburg and contiguous with Potsdam, Brandenburg's capital. Berlin's urban area, which has a population of around 4.5 million, is the second most populous urban area in Germany after the Ruhr. The Berlin-Brandenburg capital region has around 6.2 million inhabitants and is Germany's third-largest metropolitan region after the Rhine-Ruhr and Rhine-Main regions. Berlin straddles the banks of the Spree, which flows into the Havel (a tributary of the Elbe) in the western borough of Spandau. Among the city's main topographical features are the many lakes in the western and southeastern boroughs formed by the Spree, Havel and Dahme, the largest of which is Lake Müggelsee. Due to its l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Furstenberg's Proof Of The Infinitude Of Primes
In mathematics, particularly in number theory, Hillel Furstenberg's proof of the infinitude of primes is a topological proof that the integers contain infinitely many prime numbers. When examined closely, the proof is less a statement about topology than a statement about certain properties of arithmetic sequences. See discussion immediately prior to Lemma 3.2 or see Section 3.5. Unlike Euclid's classical proof, Furstenberg's proof is a proof by contradiction. The proof was published in 1955 in the ''American Mathematical Monthly'' while Furstenberg was still an undergraduate student at Yeshiva University. Furstenberg's proof Define a topology on the integers \mathbb, called the evenly spaced integer topology, by declaring a subset ''U'' ⊆ \mathbb to be an open set if and only if it is a union of arithmetic sequences ''S''(''a'', ''b'') for ''a'' ≠ 0, or is empty (which can be seen as a nullary union (empty union) of arithmetic sequences), where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Washington Heights, Manhattan
Washington Heights is a neighborhood in the uppermost part of the New York City borough of Manhattan. It is named for Fort Washington, a fortification constructed at the highest natural point on Manhattan by Continental Army troops to defend the area from the British forces during the American Revolutionary War. Washington Heights is bordered by Inwood to the north along Dyckman Street, by Harlem to the south along 155th Street, by the Harlem River and Coogan's Bluff to the east, and by the Hudson River to the west. Washington Heights, which before the 20th century was sparsely populated by luxurious mansions and single-family homes, rapidly developed during the early 1900s as it became connected to the rest of Manhattan via the A, C, and 1 subway lines. Beginning as a middle-class neighborhood with many Irish and Eastern European immigrants, the neighborhood has at various points been home to communities of German Jews, Greek Americans, Puerto Ricans, Cuban Ameri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kristallnacht
() or the Night of Broken Glass, also called the November pogrom(s) (german: Novemberpogrome, ), was a pogrom against Jews carried out by the Nazi Party's (SA) paramilitary and (SS) paramilitary forces along with some participation from the Hitler Youth and German civilians throughout Nazi Germany on 9–10 November 1938. The German authorities looked on without intervening.German Mobs' Vengeance on Jews", ''The Daily Telegraph'', 11 November 1938, cited in The name (literally 'Crystal Night') comes from the shards of broken glass that littered the streets after the windows of Jewish-owned stores, buildings and synagogues were smashed. The pretext for the attacks was the assassination of the German diplomat Ernst vom Rath by Herschel Grynszpan, a 17-year-old German-born Polish Jew living in Paris. Jewish homes, hospitals and schools were ransacked as attackers demolished buildings with sledgehammers. Rioters destroyed 267 synagogues throughout Germany, Austria and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
German Jews
The history of the Jews in Germany goes back at least to the year 321, and continued through the Early Middle Ages (5th to 10th centuries CE) and High Middle Ages (''circa'' 1000–1299 CE) when Jewish immigrants founded the Ashkenazi Jewish community. The community survived under Charlemagne, but suffered during the Crusades. Accusations of well poisoning during the Black Death (1346–53) led to mass slaughter of German Jews and they fled in large numbers to Poland. The Jewish communities of the cities of Mainz, Speyer and Worms became the center of Jewish life during medieval times. "This was a golden age as area bishops protected the Jews resulting in increased trade and prosperity." The First Crusade began an era of persecution of Jews in Germany. Entire communities, like those of Trier, Worms, Mainz and Cologne, were slaughtered. The Hussite Wars became the signal for renewed persecution of Jews. The end of the 15th century was a period of religious hatred that ascribed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Academy Of Sciences And Humanities
Israel Academy of Sciences and Humanities, based in Jerusalem, was established in 1961 by the State of Israel to foster contact between Israeli scholars in the sciences and humanities and create a think tank for advising the government on research projects of national importance. Its members include many of Israel's most distinguished scholars. The offices of the Israel Academy of Sciences and Humanities are located next door to the official residence of the President of Israel and the Council for Higher Education in Israel in Albert Einstein Square in Jerusalem. In the sciences, the Academy funds projects on the geology, flora, and fauna of Israel, and facilitates the participation of Israeli scientists in research at international projects, such as high-energy physics at CERN and synchrotron radiation at the European Synchrotron Radiation Facility. Israel has the highest concentration of scientists and engineers in the world. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hebrew University Of Jerusalem
The Hebrew University of Jerusalem (HUJI; he, הַאוּנִיבֶרְסִיטָה הַעִבְרִית בִּירוּשָׁלַיִם) is a public research university based in Jerusalem, Israel. Co-founded by Albert Einstein and Dr. Chaim Weizmann in July 1918, the public university officially opened in April 1925. It is the second-oldest Israeli university, having been founded 30 years before the establishment of the State of Israel but six years after the older Technion university. The HUJI has three campuses in Jerusalem and one in Rehovot. The world's largest library for Jewish studies—the National Library of Israel—is located on its Edmond J. Safra campus in the Givat Ram neighbourhood of Jerusalem. The university has five affiliated teaching hospitals (including the Hadassah Medical Center), seven faculties, more than 100 research centers, and 315 academic departments. , one-third of all the doctoral candidates in Israel were studying at the HUJI. Among its first ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolf Prize In Mathematics
The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. According to a reputation survey conducted in 2013 and 2014, the Wolf Prize in Mathematics is the third most prestigious international academic award in mathematics, after the Abel Prize and the Fields Medal. Until the establishment of the Abel Prize, it was probably the closest equivalent of a "Nobel Prize in Mathematics", since the Fields Medal is awarded every four years only to mathematicians under the age of 40. Laureates Laureates per country Below is a chart of all laureates per country (updated to 2022 laureates). Some laureates are counted more than once if have multiple citizenship. Notes See also * List of mathematics awards References External links * * * Israel-Wolf-Prizes 2015Jerusalempost Wolf Prizes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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