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Study of geometric properties of sets through measure theory
mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows
Geometric_measure_theory
Area formula from geometric measure theory
In geometric measure theory the area formula relates the Hausdorff measure of the image of a Lipschitz map, while accounting for multiplicity, to the
Area formula (geometric measure theory)
Area_formula_(geometric_measure_theory)
Generalization of mass, length, area and volume
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions
Measure_(mathematics)
Chinese mathematician (born 1991)
Chinese mathematician known for her research in Fourier analysis and geometric measure theory. She was awarded the Fields Medal in 2026 for her contributions
Hong_Wang
Polish-Canadian mathematician
Columbia. Her main research specialties are harmonic analysis, geometric measure theory, and additive combinatorics. Łaba earned a master's degree in 1986
Izabella_Łaba
Mathematical function
now called Hausdorff dimension. Outer measures are commonly used in the field of geometric measure theory. Measures are generalizations of length, area
Outer_measure
Generalization of volume to non-integer number of dimensions
d-dimensional Hausdorff measures for any d ≥ 0, which is not necessarily an integer. These measures are fundamental in geometric measure theory. They appear naturally
Hausdorff_measure
American mathematician
2010) was an American mathematician. He is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical
Herbert_Federer
Italian mathematician
calculus of variations, hyperbolic systems of conservation laws, geometric measure theory and fluid dynamics. He is a permanent faculty member in the School
Camillo_De_Lellis
Shape containing unit line segments in all directions
dimension n; it remains open for n>3. These questions belong to geometric measure theory. In 1920, while considering a problem of integration, Besicovitch
Kakeya_set
Italian mathematician
His main fields of research are the calculus of variations and geometric measure theory. Ambrosio entered the Scuola Normale Superiore di Pisa in 1981
Luigi_Ambrosio
and Fleming in 1960. It forms a fundamental part of the field of geometric measure theory. The notion was applied to find solutions to Plateau's problem
Flat_convergence
Measure of a two-dimensional surface
of the twentieth century. Their work led to the development of geometric measure theory, which studies various notions of surface area for irregular objects
Surface_area
Distributions on spaces of differential forms
particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on
Current_(mathematics)
Expression that may be integrated over a region
succinct proof may be found in Herbert Federer's classic text Geometric Measure Theory. The Wirtinger inequality is also a key ingredient in Gromov's
Differential_form
specifically, in geometric measure theory — spherical measure σn is the "natural" Borel measure on the n-sphere Sn. Spherical measure is often normalized
Spherical_measure
American mathematician (1933–1997)
1933 – February 5, 1997) was an American mathematician working in geometric measure theory. He was born in Birmingham, Alabama. Almgren received a Guggenheim
Frederick_J._Almgren_Jr.
Concept in measure theory
generalize the idea of a rectifiable current, and are studied in geometric measure theory. Varifolds were first introduced by Laurence Chisholm Young in
Varifold
To find the minimal surface with a given boundary
variations. The existence and regularity problems are part of geometric measure theory. It is named after Joseph Plateau. The problem was first raised
Plateau's_problem
Italian mathematician (1928–1996)
the foundations of mathematics. De Giorgi's first work was in geometric measure theory, on the topic of the sets of finite perimeters which he called
Ennio_De_Giorgi
Italian mathematician (born 1985)
the calculus of variations, partial differential equations and geometric measure theory. In 2016, he was awarded the EMS Prize, "for his outstanding contributions
Guido_De_Philippis
Branch of mathematics
understood as geometric objects since Klein's Erlangen programme. Geometric group theory studies group actions on objects that are regarded as geometric (significantly
Geometry
Australian mathematician (born 1945)
mathematician, known for deep contributions to the fields of geometric analysis, geometric measure theory, and partial differential equations. He is currently
Leon_Simon
Mathematics concept
everywhere. Rectifiable sets are the underlying object of study in geometric measure theory. A Borel subset E {\displaystyle E} of Euclidean space R n {\displaystyle
Rectifiable_set
Mathematical concept in measure theory
into measurable functions with applications in real analysis and geometric measure theory. Let E ⊆ R n {\displaystyle E\subseteq \mathbb {R} ^{n}} be a Lebesgue
Approximately continuous function
Approximately_continuous_function
Finnish mathematician (born 1948)
Mattila (born 28 March 1948) is a Finnish mathematician working in geometric measure theory, complex analysis and harmonic analysis. He is Professor of Mathematics
Pertti_Mattila
Colombian–American mathematician (born 1964)
University of Washington. Her research is "at the interface of geometric measure theory, harmonic analysis and partial differential equations". Toro was
Tatiana_Toro
Hungarian mathematician
real analysis, geometric measure theory, and geometric nonlinear functional analysis. She proved the equivalence of the zero measure notions of infinite
Marianna_Csörnyei
Mathematics award
work in harmonic analysis, partial differential equations, and geometric measure theory, including the local smoothing conjecture, Furstenberg set conjecture
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
Italian mathematician
active in the fields of calculus of variations, real analysis and geometric measure theory. Alberti has studied at Scuola Normale Superiore under the guide
Giovanni Alberti (mathematician)
Giovanni_Alberti_(mathematician)
Mathematic formula
In the mathematical field of geometric measure theory, the coarea formula expresses the integral of a function over an open set in Euclidean space in terms
Coarea_formula
function and the measure of the domain. The Eilenberg's inequality has applications in geometric measure theory and manifold theory. It is also a key
Eilenberg's_inequality
Mathematics award
études scientifiques, France "For her work in harmonic analysis and geometric measure theory, including applications of multiscale and decoupling techniques
Fields_Medal
Hungarian mathematician
Hungarian mathematician mainly noted for his work on real analysis and geometric measure theory. His most famous result is the solution of Tarski's circle-squaring
Miklós_Laczkovich
Topics referred to by the same term
may also refer to: Generic Mapping Tools in graphical computing Geometric measure theory Giant Magellan Telescope, a telescope under construction in Chile
GMT_(disambiguation)
American mathematician (1928–2023)
emeritus. Together with Herbert Federer, Fleming was a pioneer of geometric measure theory. Later in his career, he worked on stochastic processes, stochastic
Wendell_Fleming
problems on the border between the theory of partial differential equations, calculus of variations and geometric measure theory". He received an ERC Starting
Filip_Rindler
manifolds. Tangent measures (introduced by David Preiss in his study of rectifiable sets) are a useful tool in geometric measure theory. For example, they
Tangent_measure
Mathematics concept
mathematical fields of differential geometry and geometric measure theory, homological integration or geometric integration is a method for extending the notion
Homological_integration
Computationally feasible measure in real-valued number of dimensions
is equal to the m-dimensional Hausdorff measure of A. Gaussian isoperimetric inequality Geometric measure theory Isoperimetric inequality in higher dimensions
Minkowski_content
Geometric measure theory the study of geometric properties of sets (typically in Euclidean space) through measure theory. Geometric number theory Geometric
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Finnish mathematician
Järvenpää is a Finnish mathematician specializing in fractal geometry, geometric measure theory, and dynamical systems. She is a professor of mathematics and dean
Maarit_Järvenpää
Professor of mathematics
to partial differential equations, calculus of variations and geometric measure theory, with specific emphasis on free boundary problems. She received
Donatella_Danielli
Area in mathematics devoted to the study of finitely generated groups
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups by exploring the connections between algebraic properties
Geometric_group_theory
Geometric minimal hypersurface
In geometry and geometric measure theory, the Simons cone refers to a specific minimal hypersurface in R 8 {\displaystyle \mathbb {R} ^{8}} that plays
Simons_cone
American mathematician
American mathematician who specializes in differential geometry and geometric measure theory. He is a professor of mathematics and a former chair of the mathematics
Brian_White_(mathematician)
On distance sets of high-dimensional sets
In geometric measure theory, Falconer's conjecture, named after Kenneth Falconer, is an unsolved problem concerning the sets of Euclidean distances between
Falconer's_conjecture
Italian mathematician (born 1966)
mathematical analysis, including the calculus of variations, geometric measure theory, potential theory, and applications to the mathematical modeling of materials
Adriana_Garroni
mathematics — specifically, in geometric measure theory — a uniformly distributed measure on a metric space is one for which the measure of an open ball depends
Uniformly_distributed_measure
Key result in general relativity
of differential geometry, partial differential equations, and geometric measure theory. Richard Schoen and Shing-Tung Yau, in 1979 and 1981, were the
Positive_energy_theorem
conjecture is a fundamental problem in Riemannian geometry and geometric measure theory which states that the classical isoperimetric inequality may be
Cartan–Hadamard_conjecture
Generalized function whose value is zero everywhere except at zero
compactly supported functions f. Using the coarea formula from geometric measure theory, one can also define the composition of the delta function with
Dirac_delta_function
Robbins lemma Factorization lemma Fatou's lemma Frostman's lemma (geometric measure theory) Malliavin's absolute continuity lemma Lindelöf's lemma Urysohn's
List_of_lemmas
Set of mathematical rules governing the structure of soap films
minimal surfaces was proved mathematically by Jean Taylor using geometric measure theory. Young–Laplace equation, governing the curvature of surfaces in
Plateau's_laws
Mathematical theorem
p(Ω). Rademacher's theorem is also significant in the study of geometric measure theory and rectifiable sets, as it allows the analysis of first-order
Rademacher's_theorem
American mathematician
emeritus, at Williams College. He is known for contributions to geometric measure theory, minimal surfaces, and differential geometry, including the resolution
Frank_Morgan_(mathematician)
Name list
Miklós Laczkovich, Hungarian mathematician noted for his work on geometric measure theory Miklós Nyiszli, Jewish prisoner doctor at Auschwitz Paulo Miklos
Miklós
American mathematician
Berkeley. Her research interests include geometric analysis and the intersection of algebra and geometric measure theory. Harrison grew up in Tuscaloosa, Alabama
Jenny_Harrison
Swiss mathematician
his theory of currents is basic to geometric measure theory and related fields. His work is particularly important for Hodge theory and sheaf theory. In
Georges_de_Rham
American mathematician (born 1945)
Pittsburgh) is an American mathematician. His research deals with geometric measure theory, partial differential equations, and continuum mechanics. He is
Robert_Miller_Hardt
Spanish mathematician
on harmonic analysis (Calderón-Zygmund theory), complex analysis, geometric measure theory, and potential theory. Specifically, he is known for his research
Xavier_Tolsa
Calculus of Variations. London: Imperial College Press. ISBN 1-86094-508-2. Federer, Herbert (1969). Geometric Measure Theory. New-York: Springer-Verlag.
Minkowski–Steiner_formula
Measure-theoretic generalization of the concept of a tangent space
In geometric measure theory an approximate tangent space is a measure theoretic generalization of the concept of a tangent space for a differentiable manifold
Approximate_tangent_space
Theorem about mass-minimizing surfaces
In geometric measure theory, a field of mathematics, the Almgren regularity theorem, proved by Almgren (1983, 2000), states that the singular set of a
Almgren_regularity_theorem
American mathematician
the Fields Medal in 2026 for her work in harmonic analysis and geometric measure theory. Guth won an Alfred P. Sloan Fellowship in 2010. He was an invited
Larry_Guth
French mathematician (born 1946)
systems theory, compact negatively curved manifolds and their abelian covers, linear actions and random walks on linear groups, geometric measure theory, and
François_Ledrappier
Israeli mathematician
particularly Banach space theory, finite- and infinite-dimensional convexity, geometric nonlinear functional analysis and geometric measure theory. He authored more
Joram_Lindenstrauss
Statistical parameter
In mathematics, concentration of measure (about a median) is a principle that is applied in measure theory, probability and combinatorics, and has consequences
Concentration_of_measure
Counterintuitive observation
numbers of sides (and decrease in the length of one side). In geometric measure theory, such a smooth curve as the circle that can be approximated by
Coastline_paradox
Australian and American mathematician (born 1975)
combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing, and analytic number theory. He is regarded by many as "the
Terence_Tao
Broadest definition of sizes in integer-dimensional spaces
everywhere," meaning except on a set whose Lebesgue measure is zero. Lebesgue measure extends ordinary geometric length (or volume) in a way that is compatible
Lebesgue_measure
Mathematics award
to the Fourier restriction problem, the Kakeya conjecture, and geometric measure theory". List of awards honoring women List of mathematics awards Pereyra
Sadosky_Prize
Romanian mathematician
interface of this field with partial differential equations, geometric measure theory, scattering theory, complex analysis and validated numerics. She is also
Irina_Mitrea
Branch of mathematics concerning probability
space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. This became the mostly undisputed
Probability_theory
Mathematical theorem related to real and functional analysis
(2011). "Kirszbraun's theorem" (PDF). Preprint. Federer, H. (1969). Geometric Measure Theory. Berlin: Springer. p. 202. McShane, E. J. (1934). "Extension of
Kirszbraun_theorem
Technique in integral evaluation
w(x)=(g\circ \varphi )(x)} for some Borel measurable function g on Y. In geometric measure theory, integration by substitution is used with Lipschitz functions.
Integration_by_substitution
Almgren isomorphism theorem is a result in geometric measure theory and algebraic topology about the topology of the space of flat cycles in a Riemannian
Almgren's_isomorphism_theorem
Geometric inequality applicable to any closed curve
In mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the
Isoperimetric_inequality
Cartan–Hadamard conjecture – an open problem in Riemannian geometry and geometric measure theory concerning the generalization of the classical isoperimetric inequality
List of things named after Jacques Hadamard
List_of_things_named_after_Jacques_Hadamard
Branch of mathematics studying (smooth) functions of manifolds
not homeomorphic. This was the origin of simple homotopy theory. The use of the term geometric topology to describe these seems to have originated rather
Geometric_topology
NP-hard problem in combinatorial optimization
our example, A → C → B → A). There is an analogous problem in geometric measure theory which asks the following: under what conditions may a subset E
Travelling_salesman_problem
Mathematician
space: from category theory to geometric measure theory by Tom Leinster and Mark W. Meckes". In Nicola Gigli (ed.). Measure Theory in Non-Smooth Spaces
Tom_Leinster
Theory of irregularities of distribution
discrepancy theory, namely distributing points in some space such that they are evenly distributed with respect to some (mostly geometrically defined) subsets
Discrepancy_theory
Italian mathematician (born 1984)
Alberto; Valdinoci, Enrico (2018). Partial differential equations and geometric measure theory : Cetraro, Italy 2014. Cham. ISBN 978-3-319-74042-3. OCLC 1038015728
Alessio_Figalli
American mathematician
bounded mean oscillation, geometric measure theory, sets of positive reach, the implicit function theorem, approximation theory, real analytic functions
Steven_G._Krantz
Chinese-American mathematician (born 1949)
made novel use of the Almgren–Pitts min-max theory of the area functional from geometric measure theory; Li and Yau's approach depended on their new
Shing-Tung_Yau
Statistical measure
theory and statistics, the geometric standard deviation (GSD) describes how spread out are a set of numbers whose preferred average is the geometric mean
Geometric_standard_deviation
Mathematical foam of equal-volume bubbles
Frank (2009), "Chapter 14. Proof of Double Bubble Conjecture", Geometric Measure Theory: A Beginner's Guide (4th ed.), Academic Press. Conway, John H.;
Weaire–Phelan_structure
Point minimizing sum of distances to given points
In geometry, the geometric median of a discrete point set in a Euclidean space is the point minimizing the sum of distances to the sample points. This
Geometric_median
Italian mathematician
outside a singular set (i.e. a closed subset of null measure) both in geometric measure theory and for variational systems of partial differential equations
Giuseppe_Mingione
American mathematician (born 1949)
parametric area minimizing surfaces. He derived an existence and regularity theory for a class of constrained variational problems. Parks has discovered, and
Harold_R._Parks
Real-valued number of spatial dimensions
In geometric measure theory, fractal dimensions enable consistent statistical indexes of complexity in patterns. Since fractal patterns can be scale-variant
Fractal_dimension
Theorem in probability and statistics
ISBN 0-321-50046-6. MR 0373075. Zbl 0619.62001. Federer, Herbert (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153.
Law of the unconscious statistician
Law_of_the_unconscious_statistician
Willmore conjecture. Almgren isomorphism theorem Varifold Geometric measure theory Geometric analysis Minimal surface Freedman–He–Wang conjecture Willmore
Almgren–Pitts_min-max_theory
Irish journalist and author (born 1968)
London. She got a PhD in geometric measure theory at University College London (1995) with the dissertation "Packing measures, packing dimensions, and
Helen_Joyce
mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if it is concentrated
Discrete_measure
sequence Surgery theory Fields Medal David Milman Krein–Milman theorem Vitali Milman Work in geometric measure theory Concentration of measure Israel Prize
List of second-generation mathematicians
List_of_second-generation_mathematicians
Branch of mathematics
measure spaces, and function spaces. Its major areas include complex analysis, functional analysis, measure theory, harmonic analysis, and the theory
Mathematical_analysis
German mathematician
Martina Zähle (born 1950) is a German mathematician specializing in geometric measure theory and stochastic geometry. She is a professor of mathematics at the
Martina_Zähle
Concept in mathematics
In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular
Invariant_measure
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