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Caroline Klivans
Caroline Jane (Carly) Klivans is an American mathematician specializing in algebraic combinatorics, including work on cell complexes associated with matroids and on chip-firing games. She is an associate professor of applied mathematics at Brown University, and associate director of the Institute for Computational and Experimental Research in Mathematics at Brown. Education and career As an undergraduate at Cornell University, Klivans was the 1999 winner of the Alice T. Schafer Prize of the Association for Women in Mathematics for excellence in mathematics by an undergraduate woman, for an undergraduate research project involving robot navigation algorithms. She graduated from Cornell in 1999, and completed her Ph.D. at the Massachusetts Institute of Technology in 2003. Her dissertation, ''Combinatorial Properties of Shifted Complexes'', was supervised by Richard P. Stanley. After postdoctoral research at the Mathematical Sciences Research Institute and the University of Chic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Combinatorics
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. History The term "algebraic combinatorics" was introduced in the late 1970s. Through the early or mid-1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association schemes, strongly regular graphs, posets with a group action) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric functions, Young tableaux). This period is reflected in the area 05E, ''Algebraic combinatorics'', of the AMS Mathematics Subject Classification, introduced in 1991. Scope Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematics Of Chip-Firing
''The Mathematics of Chip-Firing'' is a textbook in mathematics on chip-firing games and abelian sandpile models. It was written by Caroline Klivans, and published in 2018 by the CRC Press. Topics A chip-firing game, in its most basic form, is a process on an undirected graph, with each vertex of the graph containing some number of chips. At each step, a vertex with more chips than incident edges is selected, and one of its chips is sent to each of its neighbors. If a single vertex is designated as a "black hole", meaning that chips sent to it vanish, then the result of the process is the same no matter what order the other vertices are selected. The stable states of this process are the ones in which no vertex has enough chips to be selected; two stable states can be added by combining their chips and then stabilizing the result. A subset of these states, the so-called critical states, form an abelian group under this addition operation. The abelian sandpile model applies this m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Massachusetts Institute Of Technology Alumni
Massachusetts (Massachusett: ''Muhsachuweesut Massachusett_writing_systems.html" ;"title="nowiki/> məhswatʃəwiːsət.html" ;"title="Massachusett writing systems">məhswatʃəwiːsət">Massachusett writing systems">məhswatʃəwiːsət'' English: , ), officially the Commonwealth of Massachusetts, is the most populous state in the New England region of the Northeastern United States. It borders on the Atlantic Ocean and Gulf of Maine to the east, Connecticut and Rhode Island to the south, New Hampshire and Vermont to the north, and New York to the west. The state's capital and most populous city, as well as its cultural and financial center, is Boston. Massachusetts is also home to the urban core of Greater Boston, the largest metropolitan area in New England and a region profoundly influential upon American history, academia, and the research economy. Originally dependent on agriculture, fishing, and trade. Massachusetts was transformed into a manufacturing center during th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cornell University Alumni
Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to teach and make contributions in all fields of knowledge—from the classics to the sciences, and from the theoretical to the applied. These ideals, unconventional for the time, are captured in Cornell's founding principle, a popular 1868 quotation from founder Ezra Cornell: "I would found an institution where any person can find instruction in any study." Cornell is ranked among the top global universities. The university is organized into seven undergraduate colleges and seven graduate divisions at its main Ithaca campus, with each college and division defining its specific admission standards and academic programs in near autonomy. The university also administers three satellite campuses, two in New York City and one in Education City, Qatar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Women Mathematicians
American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States of America ** American ancestry, people who self-identify their ancestry as "American" ** American English, the set of varieties of the English language native to the United States ** Native Americans in the United States, indigenous peoples of the United States * American, something of, from, or related to the Americas, also known as "America" ** Indigenous peoples of the Americas * American (word), for analysis and history of the meanings in various contexts Organizations * American Airlines, U.S.-based airline headquartered in Fort Worth, Texas * American Athletic Conference, an American college athletic conference * American Recordings (record label), a record label previously known as Def American * American University, in Washington, D.C. Sports teams Soccer * B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
21st-century American Mathematicians
The 1st century was the century spanning AD 1 ( I) through AD 100 ( C) according to the Julian calendar. It is often written as the or to distinguish it from the 1st century BC (or BCE) which preceded it. The 1st century is considered part of the Classical era, epoch, or historical period. The 1st century also saw the appearance of Christianity. During this period, Europe, North Africa and the Near East fell under increasing domination by the Roman Empire, which continued expanding, most notably conquering Britain under the emperor Claudius ( AD 43). The reforms introduced by Augustus during his long reign stabilized the empire after the turmoil of the previous century's civil wars. Later in the century the Julio-Claudian dynasty, which had been founded by Augustus, came to an end with the suicide of Nero in AD 68. There followed the famous Year of Four Emperors, a brief period of civil war and instability, which was finally brought to an end by Vespasian, ninth Roman empero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Living People
Related categories * :Year of birth missing (living people) / :Year of birth unknown * :Date of birth missing (living people) / :Date of birth unknown * :Place of birth missing (living people) / :Place of birth unknown * :Year of death missing / :Year of death unknown * :Date of death missing / :Date of death unknown * :Place of death missing / :Place of death unknown * :Missing middle or first names See also * :Dead people * :Template:L, which generates this category or death years, and birth year and sort keys. : {{DEFAULTSORT:Living people 21st-century people People by status ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Year Of Birth Missing (living People)
A year or annus is the orbital period of a planetary body, for example, the Earth, moving in its orbit around the Sun. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by change in weather, the hours of daylight, and, consequently, vegetation and soil fertility. In temperate and subpolar regions around the planet, four seasons are generally recognized: spring, summer, autumn and winter. In tropical and subtropical regions, several geographical sectors do not present defined seasons; but in the seasonal tropics, the annual wet and dry seasons are recognized and tracked. A calendar year is an approximation of the number of days of the Earth's orbital period, as counted in a given calendar. The Gregorian calendar, or modern calendar, presents its calendar year to be either a common year of 365 days or a leap year of 366 days, as do the Julian calendars. For the Gregorian calendar, the average length of the calendar year (the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H-vector
In algebraic combinatorics, the ''h''-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions and allows one to express the Dehn–Sommerville equations in a particularly simple form. A characterization of the set of ''h''-vectors of simplicial polytopes was conjectured by Peter McMullen and proved by Lou Billera and Carl W. Lee and Richard Stanley ( ''g''-theorem). The definition of ''h''-vector applies to arbitrary abstract simplicial complexes. The ''g''-conjecture stated that for simplicial spheres, all possible ''h''-vectors occur already among the ''h''-vectors of the boundaries of convex simplicial polytopes. It was proven in December 2018 by Karim Adiprasito. Stanley introduced a generalization of the ''h''-vector, the toric ''h''-vector, which is defined for an arbitrary ranked poset, and proved that for the class of Eulerian posets, the Dehn–Sommerville equations continue to hold. A different ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shelling (topology)
In mathematics, a shelling of a simplicial complex is a way of gluing it together from its maximal simplices (simplices that are not a face of another simplex) in a well-behaved way. A complex admitting a shelling is called shellable. Definition A ''d''-dimensional simplicial complex is called pure if its maximal simplices all have dimension ''d''. Let \Delta be a finite or countably infinite simplicial complex. An ordering C_1,C_2,\ldots of the maximal simplices of \Delta is a shelling if the complex :B_k:=\Big(\bigcup_^C_i\Big)\cap C_k is pure and of dimension \dim C_k-1 for all k=2,3,\ldots. That is, the "new" simplex C_k meets the previous simplices along some union B_k of top-dimensional simplices of the boundary of C_k. If B_k is the entire boundary of C_k then C_k is called spanning. For \Delta not necessarily countable, one can define a shelling as a well-ordering of the maximal simplices of \Delta having analogous properties. Properties * A shellable complex is homotop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohen–Macaulay Ring
In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways. They are named for , who proved the unmixedness theorem for polynomial rings, and for , who proved the unmixedness theorem for formal power series rings. All Cohen–Macaulay rings have the unmixedness property. For Noetherian local rings, there is the following chain of inclusions. Definition For a commutative Noetherian local ring ''R'', a finite (i.e. finitely generated) ''R''-module M\neq 0 is a ''Cohen-Macaulay module'' if \mathrm(M) = \mathrm(M) (in general we have: \mathrm(M) \leq \mathrm(M), see Auslander–Buchsbaum formula for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Simplicial Complex
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. Lee, John M., Introduction to Topological Manifolds, Springer 2011, , p153 For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1). In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems. An abstract simplex can be studied algebraically by forming its Stanley–Reisner ring; this sets up a powerful relation between combinatorics and commutative algebra. Definitions A collection of non-empty finite subsets of a set ''S'' is called a set-family. A set-family is called an abstract simplicial c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
