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Bernd Sturmfels
Bernd Sturmfels (born March 28, 1962 in Kassel, West Germany) is a Professor of Mathematics and Computer Science at the University of California, Berkeley and is a director of the Max Planck Institute for Mathematics in the Sciences in Leipzig since 2017. Education and career He received his PhD in 1987 from the University of Washington and the Technische Universität Darmstadt. After two postdoctoral years at the Institute for Mathematics and its Applications in Minneapolis, Minnesota, and the Research Institute for Symbolic Computation in Linz, Austria, he taught at Cornell University, before joining University of California, Berkeley in 1995. His Ph.D. students include Melody Chan, Jesús A. De Loera, Mike Develin, Diane Maclagan, Rekha R. Thomas, Caroline Uhler, and Cynthia Vinzant. Contributions Bernd Sturmfels has made contributions to a variety of areas of mathematics, including algebraic geometry, commutative algebra, discrete geometry, Gröbner bases, toric varietie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kassel
Kassel (; in Germany, spelled Cassel until 1926) is a city on the Fulda River in northern Hesse, Germany. It is the administrative seat of the Regierungsbezirk Kassel and the district of the same name and had 201,048 inhabitants in December 2020. The former capital of the state of Hesse-Kassel has many palaces and parks, including the Bergpark Wilhelmshöhe, which is a UNESCO World Heritage Site. Kassel is also known for the '' documenta'' exhibitions of contemporary art. Kassel has a public university with 25,000 students (2018) and a multicultural population (39% of the citizens in 2017 had a migration background). History Kassel was first mentioned in 913 AD, as the place where two deeds were signed by King Conrad I. The place was called ''Chasella'' or ''Chassalla'' and was a fortification at a bridge crossing the Fulda river. There are several yet unproven assumptions of the name's origin. It could be derived from the ancient ''Castellum Cattorum'', a castle of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minneapolis
Minneapolis () is the largest city in Minnesota, United States, and the county seat of Hennepin County. The city is abundant in water, with thirteen lakes, wetlands, the Mississippi River, creeks and waterfalls. Minneapolis has its origins in timber and as the flour milling capital of the world. It occupies both banks of the Mississippi River and adjoins Saint Paul, the state capital of Minnesota. Prior to European settlement, the site of Minneapolis was inhabited by Dakota people. The settlement was founded along Saint Anthony Falls on a section of land north of Fort Snelling; its growth is attributed to its proximity to the fort and the falls providing power for industrial activity. , the city has an estimated 425,336 inhabitants. It is the most populous city in the state and the 46th-most-populous city in the United States. Minneapolis, Saint Paul and the surrounding area are collectively known as the Twin Cities. Minneapolis has one of the most extensive public par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Introduction To Tropical Geometry
''Introduction to Tropical Geometry'' is a book on tropical geometry, by Diane Maclagan and Bernd Sturmfels. It was published by the American Mathematical Society in 2015 as volume 161 of Graduate Studies in Mathematics. Topics The tropical semiring is an algebraic structure on the real numbers in which addition takes the usual place of multiplication, and minimization takes the usual place of addition. This combination of the two operations of addition and minimization comes up naturally, for instance, in the shortest path problem, where concatenating paths causes their distances to be added and where the shortest of two parallel paths is the one with minimum length, and where some shortest path algorithms can be interpreted as tropical matrix multiplication. Tropical geometry applies the machinery of algebraic geometry to this system by defining polynomials using addition and minimization in place of multiplication and addition (yielding piecewise linear functions), and studying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dave Bayer
David Allen Bayer (born November 29, 1955) is an American mathematician known for his contributions in algebra and symbolic computation and for his consulting work in the movie industry. He is a professor of mathematics at Barnard College, Columbia University. Education and career Bayer was educated at Swarthmore College as an undergraduate, where he attended a course on combinatorial algorithms given by Herbert Wilf. During that semester, Bayer related several original ideas to Wilf on the subject. These contributions were later incorporated into the second edition of Wilf and Albert Nijenhuis' influential book ''Combinatorial Algorithms'', with a detailed acknowledgement by its authors. Bayer subsequently earned his Ph.D. at Harvard University in 1982 under the direction of Heisuke Hironaka with a dissertation entitled ''The Division Algorithm and the Hilbert Scheme''. He joined Columbia University thereafter. Bayer is the son of Bryce Bayer, the inventor of the Bayer filter. C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Biology
Computational biology refers to the use of data analysis, mathematical modeling and computational simulations to understand biological systems and relationships. An intersection of computer science, biology, and big data, the field also has foundations in applied mathematics, chemistry, and genetics. It differs from biological computing, a subfield of computer engineering which uses bioengineering to build computers. History Bioinformatics, the analysis of informatics processes in biological systems, began in the early 1970s. At this time, research in artificial intelligence was using network models of the human brain in order to generate new algorithms. This use of biological data pushed biological researchers to use computers to evaluate and compare large data sets in their own field. By 1982, researchers shared information via punch cards. The amount of data grew exponentially by the end of the 1980s, requiring new computational methods for quickly interpreting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Statistics
Algebraic statistics is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing. Traditionally, algebraic statistics has been associated with the design of experiments and multivariate analysis (especially time series). In recent years, the term "algebraic statistics" has been sometimes restricted, sometimes being used to label the use of algebraic geometry and commutative algebra in statistics. The tradition of algebraic statistics In the past, statisticians have used algebra to advance research in statistics. Some algebraic statistics led to the development of new topics in algebra and combinatorics, such as association schemes. Design of experiments For example, Ronald A. Fisher, Henry B. Mann, and Rosemary A. Bailey applied Abelian groups to the design of experiments. Experimental designs were also studied with affine geometry over finite fields and then with the introduction of association sche ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tropical Geometry
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: : x \oplus y = \min\, : x \otimes y = x + y. So for example, the classical polynomial x^3 + 2xy + y^4 would become \min\. Such polynomials and their solutions have important applications in optimization problems, for example the problem of optimizing departure times for a network of trains. Tropical geometry is a variant of algebraic geometry in which polynomial graphs resemble piecewise linear meshes, and in which numbers belong to the tropical semiring instead of a field. Because classical and tropical geometry are closely related, results and methods can be converted between them. Algebraic varieties can be mapped to a tropical counterpart and, since this process still retains some geometric information about the original variety, it can be used to help prove and generalize classica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toric Varieties
In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be normal. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special, but still quite general class of toric varieties, this information is also encoded in a polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space. Toric varieties from tori The original motivation to study toric varieties was to study torus embeddings. Given the algebrai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gröbner Basis
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps. Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems. Gröbner bases were introduced in 1965, together with an algorithm to compute them (Buchberger's algorithm), by Bruno Buchberger in his Ph.D. thesis. He named them after h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology. History Although polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb; and ''p''-adic integers. Commutative algebra is the main technical tool in the local study of schemes. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have led to the no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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