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62-543: The VDC was formed by Jerry Rubin and Stephen Smale between May 21 and May 22, 1965 during a 35‑hour‑long anti-Vietnam war protest that took place inside and around the University of California, Berkeley and attracted over 35,000 people, including Paul Montauk and Stew Albert . The VDC laid out three main objectives: to achieve national and international solidarity and coordination on action, to take part in militant action, including civil disobedience and to work extensively in

124-416: A robot can be described by a manifold called configuration space . In the area of motion planning , one finds paths between two points in configuration space. These paths represent a motion of the robot's joints and other parts into the desired pose. Disentanglement puzzles are based on topological aspects of the puzzle's shapes and components. In order to create a continuous join of pieces in

186-497: A smooth structure on a manifold to be defined. Smooth manifolds are "softer" than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold – that is, one can smoothly "flatten out" certain manifolds, but it might require distorting

248-401: A C average as a graduate student. When the department chair, Hildebrandt , threatened to kick Smale out, he began to take his studies more seriously. Smale finally earned his PhD in 1957, under Raoul Bott , beginning his career as an instructor at the University of Chicago . Early in his career, Smale was involved in controversy over remarks he made regarding his work habits while proving

310-641: A Distinguished University Professor at the City University of Hong Kong . In 1988, Smale was the recipient of the Chauvenet Prize of the MAA . In 2007, Smale was awarded the Wolf Prize in mathematics. Smale proved that the oriented diffeomorphism group of the two-dimensional sphere has the same homotopy type as the special orthogonal group of 3 × 3 matrices. Smale's theorem has been reproved and extended

372-412: A convenient proof that any subgroup of a free group is again a free group. Differential topology is the field dealing with differentiable functions on differentiable manifolds . It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds. More specifically, differential topology considers the properties and structures that require only

434-512: A few times, notably to higher dimensions in the form of the Smale conjecture , as well as to other topological types. In another early work, he studied the immersions of the two-dimensional sphere into Euclidean space. By relating immersion theory to the algebraic topology of Stiefel manifolds , he was able to fully clarify when two immersions can be deformed into one another through a family of immersions. Directly from his results it followed that

496-426: A given space. Changing a topology consists of changing the collection of open sets. This changes which functions are continuous and which subsets are compact or connected. Metric spaces are an important class of topological spaces where the distance between any two points is defined by a function called a metric . In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x

558-420: A homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. However, the sphere is not homeomorphic to

620-403: A key tool, Smale resolved the generalized Poincaré conjecture in every dimension greater than four. Building on these works, he also established the more powerful h-cobordism theorem the following year, together with the full classification of simply-connected smooth five-dimensional manifolds. Smale also introduced the horseshoe map , inspiring much subsequent research. He also outlined

682-667: A number of guest speakers, including Simon Casady, a former president of the California Democratic Council , Dorothy Healy , the Southern California chairman of the Communist Party USA , and the British philosopher Bertrand Russell . Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology , dynamical systems and mathematical economics . He

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744-575: A professor emeritus at Berkeley in 1995 and took up a post as professor at the City University of Hong Kong . He also amassed over the years one of the finest private mineral collections in existence. Many of Smale's mineral specimens can be seen in the book The Smale Collection: Beauty in Natural Crystals . From 2003 to 2012, Smale was a professor at the Toyota Technological Institute at Chicago ; starting August 1, 2009, he became

806-455: A research program carried out by many others. Smale is also known for injecting Morse theory into mathematical economics , as well as recent explorations of various theories of computation . In 1998 he compiled a list of 18 problems in mathematics to be solved in the 21st century, known as Smale's problems . This list was compiled in the spirit of Hilbert 's famous list of problems produced in 1900. In fact, Smale's list contains some of

868-534: A set (for instance, determining if a cloud of points is spherical or toroidal ). The main method used by topological data analysis is to: Several branches of programming language semantics , such as domain theory , are formalized using topology. In this context, Steve Vickers , building on work by Samson Abramsky and Michael B. Smyth , characterizes topological spaces as Boolean or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties. Topology

930-457: Is a π -system . The members of τ are called open sets in X . A subset of X is said to be closed if its complement is in τ (that is, its complement is open). A subset of X may be open, closed, both (a clopen set ), or neither. The empty set and X itself are always both closed and open. An open subset of X which contains a point x is called an open neighborhood of x . A function or map from one topological space to another

992-409: Is a set endowed with a structure, called a topology , which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity . Euclidean spaces , and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies . A property that

1054-661: Is a quantum field theory that computes topological invariants . Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory , the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson , Jones , Witten , and Kontsevich have all won Fields Medals for work related to topological field theory. The topological classification of Calabi–Yau manifolds has important implications in string theory , as different manifolds can sustain different kinds of strings. In cosmology, topology can be used to describe

1116-406: Is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in calculus . If a continuous function is one-to-one and onto , and if the inverse of the function is also continuous, then the function is called

1178-444: Is invariant under such deformations is a topological property . The following are basic examples of topological properties: the dimension , which allows distinguishing between a line and a surface ; compactness , which allows distinguishing between a line and a circle; connectedness , which allows distinguishing a circle from two non-intersecting circles. The ideas underlying topology go back to Gottfried Wilhelm Leibniz , who in

1240-592: Is on community organization and education as well as on direct action against the war. Similar groups began to form outside California, notably in Mexico City and Tokyo . In California, the International Days of Protest were to culminate with a peace march toward the Oakland Army Terminal , where men and materials were sent to Vietnam . On October 15, 1965, the protests took place across the country, with

1302-776: Is point-set topology. The basic object of study is topological spaces , which are sets equipped with a topology , that is, a family of subsets , called open sets , which is closed under finite intersections and (finite or infinite) unions . The fundamental concepts of topology, such as continuity , compactness , and connectedness , can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words nearby , arbitrarily small , and far apart can all be made precise by using open sets. Several topologies can be defined on

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1364-453: Is relevant to physics in areas such as condensed matter physics , quantum field theory and physical cosmology . The topological dependence of mechanical properties in solids is of interest in disciplines of mechanical engineering and materials science . Electrical and mechanical properties depend on the arrangement and network structures of molecules and elementary units in materials. The compressive strength of crumpled topologies

1426-459: Is studied in attempts to understand the high strength to weight of such structures that are mostly empty space. Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics. A topological quantum field theory (or topological field theory or TQFT)

1488-437: Is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside. In one of the first papers in topology, Leonhard Euler demonstrated that it

1550-428: Is the set of all points whose distance to x is less than r . Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the real line , the complex plane , real and complex vector spaces and Euclidean spaces . Having a metric simplifies many proofs. Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces. The basic goal

1612-437: Is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. The most important of these invariants are homotopy groups , homology, and cohomology . Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for

1674-525: The National Guardian : Preparations are being made in about two dozen American cities for coordinated mass protests Oct. 15-16 in opposition to U.S. aggression in Vietnam. Advance information indicates that demonstrations may surpass previous anti-war protests not only in total numbers and intensity of action, but in long-range benefit to the peace movement, for the emphasis of the "national days of protest"

1736-445: The Greek words τόπος , 'place, location', and λόγος , 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations , such as stretching , twisting , crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space

1798-429: The geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries. 2-dimensional topology can be studied as complex geometry in one variable ( Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from

1860-460: The hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick ." This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg , the result does not depend on

1922-595: The plane , the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the Klein bottle and real projective plane , which cannot (that is, all their realizations are surfaces that are not manifolds). General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology

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1984-501: The real line , the complex plane , and the Cantor set can be thought of as the same set with different topologies. Formally, let X be a set and let τ be a family of subsets of X . Then τ is called a topology on X if: If τ is a topology on X , then the pair ( X , τ ) is called a topological space. The notation X τ may be used to denote a set X endowed with the particular topology τ . By definition, every topology

2046-443: The 17th century envisioned the geometria situs and analysis situs . Leonhard Euler 's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although, it was not until the first decades of the 20th century that the idea of a topological space was developed. The motivating insight behind topology

2108-582: The US, he was unable to renew the grant. At one time he was subpoenaed by the House Un-American Activities Committee . In 1960, Smale received a Sloan Research Fellowship and was appointed to the Berkeley mathematics faculty, moving to a professorship at Columbia the following year. In 1964 he returned to a professorship at Berkeley, where he has spent the main part of his career. He became

2170-558: The VDC itself organising a sit-in at San Francisco State College , which saw a performance by Country Joe and the Fish . The VDC organized another peace march which took place on November 21, 1965, and saw over 10,000 people marching through Oakland. The march was the first of its kind in California and was one of many orchestrated by the VDC from 1965 through 1972; a number of pro-war protesters lined

2232-590: The community to develop the movement outside of the university campus. Attending the event were several notable anti-war activists, including Dr. Benjamin Spock , however the State Department declined to send a representative, despite the burning of an effigy of president Lyndon Johnson . On May 5, 1965, the VDC was involved in a march of several hundred students from campus to the Berkeley Draft Board, where

2294-509: The concepts now known as homotopy and homology , which are now considered part of algebraic topology . Unifying the work on function spaces of Georg Cantor , Vito Volterra , Cesare Arzelà , Jacques Hadamard , Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space potentially giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined

2356-458: The definition of sheaves on those categories, and with that the definition of general cohomology theories. Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular, circuit topology and knot theory have been extensively applied to classify and compare the topology of folded proteins and nucleic acids. Circuit topology classifies folded molecular chains based on

2418-461: The dimensions of the (un)stable manifolds . Part of the significance of these results is from Smale's theorem asserting that the gradient flow of any Morse function can be arbitrarily well approximated by a Morse–Smale system without closed orbits. Using these tools, Smale was able to construct self-indexing Morse functions, where the value of the function equals its Morse index at any critical point. Using these self-indexing Morse functions as

2480-638: The doughnut. While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n -dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n . Lines and circles , but not figure eights , are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces , although not all surfaces are manifolds. Examples include

2542-407: The hairy ball theorem applies to any space homeomorphic to a sphere. Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking

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2604-782: The higher-dimensional Poincaré conjecture. He said that his best work had been done "on the beaches of Rio." He has been politically active in various movements in the past, such as the Free Speech movement and member of the Fair Play for Cuba Committee . In 1966, having travelled to Moscow under an NSF grant to accept the Fields Medal, he held a press conference there to denounce the American position in Vietnam , Soviet intervention in Hungary and Soviet maltreatment of intellectuals. After his return to

2666-402: The hole into a handle. Homeomorphism can be considered the most basic topological equivalence . Another is homotopy equivalence . This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object. Topology, as a well-defined mathematical discipline, originates in the early part of

2728-476: The number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signaling the birth of topology. Further contributions were made by Augustin-Louis Cauchy , Ludwig Schläfli , Johann Benedict Listing , Bernhard Riemann and Enrico Betti . Listing introduced the term "Topologie" in Vorstudien zur Topologie , written in his native German, in 1847, having used

2790-686: The original Hilbert problems, including the Riemann hypothesis and the second half of Hilbert's sixteenth problem , both of which are still unsolved. Other famous problems on his list include the Poincaré conjecture (now a theorem, proved by Grigori Perelman ), the P = NP problem , and the Navier–Stokes equations , all of which have been designated Millennium Prize Problems by the Clay Mathematics Institute . Topology Topology (from

2852-475: The overall shape of the universe . This area of research is commonly known as spacetime topology . In condensed matter a relevant application to topological physics comes from the possibility to obtain one-way current, which is a current protected from backscattering. It was first discovered in electronics with the famous quantum Hall effect , and then generalized in other areas of physics, for instance in photonics by F.D.M Haldane . The possible positions of

2914-462: The pairwise arrangement of their intra-chain contacts and chain crossings. Knot theory , a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis . Topological data analysis uses techniques from algebraic topology to determine the large scale structure of

2976-481: The planar and higher-dimensional Schönflies theorem . In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory. Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature /spherical, zero curvature/flat, and negative curvature/hyperbolic – and

3038-426: The point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure. Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are structures defined on arbitrary categories that allow

3100-639: The route holding signs that said, "Stamp out VDC". By this time, the activities of the VDC had attracted the interest of the California Senate Factfinding Subcommittee on Un-American Activities . On March 25, 1966, the UCLA VDC, a group not organizationally tied to the Berkeley VDC, sponsored a well-attended, 12-hour 'teach-in' at UCLA . This clashed with a small rally that supported America's involvement in Vietnam. The antiwar event had

3162-484: The shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes. To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism . The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and

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3224-408: The space and affecting the curvature or volume. Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology. Some examples of topics in geometric topology are orientability , handle decompositions , local flatness , crumpling and

3286-740: The staff was given a black coffin, and students burned their draft cards. Later that year, the VDC planned a nationwide protest, the International Days of Protest Against American Military Intervention, which was scheduled between October 15 and October 16. In arranging and coordinating the protest movement, the VDC headquarters in Berkeley communicated with anti-war groups in New York City , Boston , New Haven , Philadelphia , Pittsburgh , Detroit , Ann Arbor , Chicago , Madison , Milwaukee , Minneapolis , Los Angeles , Portland and Atlanta . The planned movement attracted attention from some newspapers like

3348-474: The standard immersion of the sphere into three-dimensional space can be deformed (through immersions) into its negation, which is now known as sphere eversion . He also extended his results to higher-dimensional spheres, and his doctoral student Morris Hirsch extended his work to immersions of general smooth manifolds . Along with John Nash 's work on isometric immersions , the Hirsch–Smale immersion theory

3410-646: The term "topological space" and gave the definition for what is now called a Hausdorff space . Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski . Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series . For further developments, see point-set topology and algebraic topology. The 2022 Abel Prize

3472-565: The twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler . His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realized the importance of the edges of a polyhedron . This led to his polyhedron formula , V − E + F = 2 (where V , E , and F respectively indicate

3534-512: The word for ten years in correspondence before its first appearance in print. The English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". Their work was corrected, consolidated and greatly extended by Henri Poincaré . In 1895, he published his ground-breaking paper on Analysis Situs , which introduced

3596-541: Was a good student, placing into an honors calculus sequence taught by Bob Thrall and earning himself A's. However, his sophomore and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in nuclear physics . Smale obtained his Bachelor of Science degree in 1952. Despite his grades, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale performed poorly in his first years, earning

3658-504: Was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty of the University of California, Berkeley (1960–1961 and 1964–1995), where he currently is Professor Emeritus, with research interests in algorithms , numerical analysis and global analysis . Smale was born in Flint, Michigan and entered the University of Michigan in 1948. Initially, he

3720-436: Was awarded to Dennis Sullivan "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects". The term topology also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology describes how elements of a set relate spatially to each other. The same set can have different topologies. For instance,

3782-462: Was highly influential in Mikhael Gromov 's early work on development of the h-principle , which abstracted and applied their ideas to contexts other than that of immersions. In the study of dynamical systems , Smale introduced what is now known as a Morse–Smale system . For these dynamical systems, Smale was able to prove Morse inequalities relating the cohomology of the underlying space to

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3844-439: Was impossible to find a route through the town of Königsberg (now Kaliningrad ) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory . Similarly,

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