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Maryam Mirzakhani MARYAM MIRZAKHANI (Persian : مریم میرزاخانی, pronounced ; 3 May 1977 – 14 July 2017) was an Iranian mathematician and a professor of mathematics at Stanford University Stanford University . Her research topics included Teichmüller theory , hyperbolic geometry , ergodic theory , and symplectic geometry . On 13 August 2014, Mirzakhani was honored with the Fields Medal Fields Medal , the most prestigious award in mathematics. Thus, she became both the first woman and the first Iranian to be honored with the award. The award committee cited her work in "the dynamics and geometry of Riemann surfaces and their moduli spaces ". On 14 July 2017, Mirzakhani died of breast cancer at the age of 40 [...More...]  "Maryam Mirzakhani" on: Wikipedia Yahoo 

Closed Geodesic In differential geometry and dynamical systems , a CLOSED GEODESIC on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic flow on the tangent space of the manifold. CONTENTS * 1 Definition * 2 Examples * 3 See also * 4 References DEFINITIONIn a Riemannian manifold (M,g), a closed geodesic is a curve : R M {displaystyle gamma :mathbb {R} rightarrow M} that is a geodesic for the metric g and is periodic. Closed geodesics can be characterized by means of a variational principle. Denoting by M {displaystyle Lambda M} the space of smooth 1periodic curves on M, closed geodesics of period 1 are precisely the critical points of the energy function E : M R {displaystyle E:Lambda Mrightarrow mathbb {R} } , defined by E ( ) = 0 1 g ( t ) ( ( t ) , ( t ) ) d t [...More...]  "Closed Geodesic" on: Wikipedia Yahoo 

Earthquake Map In hyperbolic geometry , an EARTHQUAKE MAP is a method of changing one hyperbolic manifold into another, introduced by William Thurston (1986 ). EARTHQUAKE MAPSGiven a simple closed geodesic on an oriented hyperbolic surface and a real number t, one can cut the manifold along the geodesic, slide the edges a distance t to the left, and glue them back. This gives a new hyperbolic surface, and the (possibly discontinuous) map between them is an example of a left earthquake. More generally one can do the same construction with a finite number of disjoint simple geodesics, each with a real number attached to it. The result is called a simple earthquake. An earthquake is roughly a sort of limit of simple earthquakes, where one has an infinite number of geodesics, and instead of attaching a positive real number to each geodesic one puts a measure on them. A GEODESIC LAMINATION of a hyperbolic surface is a closed subset with a foliation by geodesics [...More...]  "Earthquake Map" on: Wikipedia Yahoo 

Ergodicity In probability theory , an ERGODIC dynamical system is one that, broadly speaking, has the same behavior averaged over time as averaged over the space of all the system's states in its phase space . In physics the term implies that a system satisfies the ergodic hypothesis of thermodynamics . A random process is ergodic if its time average is the same as its average over the probability space, known in the field of thermodynamics as its ensemble average . The state of an ergodic process after a long time is nearly independent of its initial state. The term "ergodic" was derived from the Greek words έργον (ergon: "work") and οδός (odos: "path," "way"). It was chosen by Ludwig Boltzmann Ludwig Boltzmann while he was working on a problem in statistical mechanics [...More...]  "Ergodicity" on: Wikipedia Yahoo 

Alex Eskin ALEX ESKIN (born May 19, 1965) is an American mathematician , born in the former USSR. He works on rational billiards and geometric group theory . For his contribution to joint work with David Fisher and Kevin Whyte establishing the quasiisometric rigidity of sol, he was awarded the 2007 Clay Research Award . Eskin was born in Moscow. He is the son of a RussianJewish mathematician Gregory I. Eskin (b. 1936, Kiev Kiev ), professor of UCLA. The family emigrated to Israel in 1974 and in 1982 to the United States. Eskin earned his doctorate from Princeton University Princeton University in 1993, under supervision of Peter Sarnak Peter Sarnak . He has been a professor at University of Chicago since 1999. In 2012 he became a fellow of the American Mathematical Society . In April 2015 Eskin was elected a member of the U.S. National Academy of Sciences [...More...]  "Alex Eskin" on: Wikipedia Yahoo 

Breast Cancer BREAST CANCER is cancer that develops from breast tissue. Signs of breast cancer may include a lump in the breast, a change in breast shape, dimpling of the skin, fluid coming from the nipple, or a red scaly patch of skin. In those with distant spread of the disease , there may be bone pain , swollen lymph nodes , shortness of breath , or yellow skin . Risk factors for developing breast cancer include being female, obesity , lack of physical exercise, drinking alcohol , hormone replacement therapy during menopause , ionizing radiation , early age at first menstruation , having children late or not at all, older age, and family history. About 5–10% of cases are due to genes inherited from a person's parents, including BRCA1 and BRCA2 among others. Breast cancer most commonly develops in cells from the lining of milk ducts and the lobules that supply the ducts with milk [...More...]  "Breast Cancer" on: Wikipedia Yahoo 

Moduli Space In algebraic geometry , a MODULI SPACE is a geometric space (usually a scheme or an algebraic stack ) whose points represent algebrogeometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus ) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects [...More...]  "Moduli Space" on: Wikipedia Yahoo 

Ergodic Theory ERGODIC THEORY (Greek : έργον ergon "work", όδος hodos "way") is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. 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 phase space eventually revisit the set. More precise information is provided by various ERGODIC THEOREMS which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. Two of the most important theorems are those of Birkhoff (1931) and von Neumann which assert the existence of a time average along each trajectory [...More...]  "Ergodic Theory" on: Wikipedia Yahoo 

Symplectic Geometry SYMPLECTIC GEOMETRY is a branch of differential geometry and differential topology that studies symplectic manifolds ; that is, differentiable manifolds equipped with a closed , nondegenerate 2form . Symplectic geometry Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. CONTENTS * 1 Introduction * 2 Comparison with Riemannian geometry Riemannian geometry * 3 Examples and structures * 4 Name * 5 See also * 6 Notes * 7 References * 8 External links INTRODUCTIONA symplectic geometry is defined on a smooth evendimensional space that is a differentiable manifold . On this space is defined a geometric object, the symplectic form , that allows for the measurement of sizes of twodimensional objects in the space [...More...]  "Symplectic Geometry" on: Wikipedia Yahoo 

Seoul SEOUL (/soʊl/ ; 서울; Korean: ( listen )), officially the SEOUL SPECIAL CITY – is the capital and largest metropolis of the Republic of Korea Korea (commonly known as South Korea). Seoul Seoul is the world's 16th largest city , and forms the heart of the Seoul Capital Area Seoul Capital Area , which includes the surrounding Incheon Incheon metropolis and Gyeonggi province. The Seoul Capital Area Seoul Capital Area houses about half of the country's population of 51.44 million people with 678,102 international residents. Situated on the Han River , Seoul's history stretches back more than two thousand years when it was founded in 18 BCE by Baekje Baekje , one of the Three Kingdoms of Korea Three Kingdoms of Korea . It continued as the capital of Korea under the Joseon Dynasty [...More...]  "Seoul" on: Wikipedia Yahoo 

Marina Ratner MARINA EVSEEVNA RATNER (Russian : Мари́на Евсе́евна Ра́тнер; October 30, 1938 – July 7, 2017 ) was a professor of mathematics at the University of California, Berkeley who worked in ergodic theory . Around 1990, she proved a group of major theorems concerning unipotent flows on homogeneous spaces , known as Ratner\'s theorems . Ratner was elected to the American Academy of Arts and Sciences in 1992, awarded the Ostrowski Prize in 1993 and elected to the National Academy of Sciences the same year. In 1994, she was awarded the John J. Carty Award from the National Academy of Sciences. BIOGRAPHICAL INFORMATIONBorn in Moscow , Russian SFSR to a Jewish family, her father was a plant physiologist and mother, a chemist. Ratner's mother was fired from work in the 1940s for writing to her mother in Israel, then considered an enemy of the Soviet state. Ratner gained an interest in mathematics in her fifth grade [...More...]  "Marina Ratner" on: Wikipedia Yahoo 

International Mathematical Union The INTERNATIONAL MATHEMATICAL UNION (IMU) is an international nongovernmental organisation devoted to international cooperation in the field of mathematics across the world. It is a member of the International Council for Science International Council for Science (ICSU) and supports the International Congress of Mathematicians International Congress of Mathematicians . Its members are national mathematics organizations from more than 80 countries. The objectives of the International Mathematical Union (IMU) are to promote international cooperation in mathematics. By supporting and assisting the International Congress of Mathematicians International Congress of Mathematicians (ICM) and other international scientific meetings/conferences [...More...]  "International Mathematical Union" on: Wikipedia Yahoo 

Applied Mathematics APPLIED MATHEMATICS is a branch of mathematics that deals with mathematical methods that find use in science , engineering , business , computer science , and industry . Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics [...More...]  "Applied Mathematics" on: Wikipedia Yahoo 

Palo Alto, California April 23, 1894 NAMED FOR El Palo Alto GOVERNMENT • TYPE CouncilManager • BODY City council City council members: * Mayor Patrick Burt * Vice Mayor Gregory Scharff * Marc Berm [...More...]  "Palo Alto, California" on: Wikipedia Yahoo 

Metastatic Breast Cancer METASTATIC BREAST CANCER, also referred to as metastases , advanced breast cancer, secondary tumours, secondaries or stage 4 breast cancer, is a stage of breast cancer where the disease has spread to distant sites beyond the axillary lymph nodes . There is no cure for metastatic breast cancer. There is no stage after IV. It usually occurs several years after the primary breast cancer, although it is sometimes diagnosed at the same time as the primary breast cancer or, rarely, before the primary breast cancer has been diagnosed. Metastatic breast cancer cells frequently differ from the preceding primary breast cancer in properties such as receptor status. The cells have often developed resistance to several lines of previous treatment and have acquired special properties that permit them to metastasize to distant sites. Metastatic breast cancer can be treated, sometimes for many years, but it cannot be cured [...More...]  "Metastatic Breast Cancer" on: Wikipedia Yahoo 

President Of Iran The PRESIDENT OF IRAN (Persian : رییسجمهور ایران Rayis Jomhure Irān) is the head of government of the Islamic Republic of Iran Iran . The President is the highest popularly elected official in Iran Iran (however, the president is still required to gain the Leader’s official approval before being sworn in before the Parliament, and the Leader also has the power to dismiss the elected president anytime). The President carries out the decrees, and answers to the Supreme Leader of Iran Iran , who functions as the country's head of state . Unlike the executive in other countries, the President of Iran Iran does not have full control over anything, as these are ultimately under the control of the Supreme Leader [...More...]  "President Of Iran" on: Wikipedia Yahoo 