mathematics

Tobias Colding

Tobias Colding is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Tobias Colding rather than just read about it. In short: Tobias Holck Colding (born 1963) is a Danish mathematician working on geometric analysis, and low-dimensional topology. He is the great-grandchild of Ludwig August Colding.

Key takeaways

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Reference excerpt

Tobias Holck Colding (born 1963) is a Danish mathematician working on geometric analysis, and low-dimensional topology. He is the great-grandchild of Ludwig August Colding.

Biography He was born in Copenhagen, Denmark, to Torben Holck Colding and Benedicte Holck Colding. He received his Ph.D. in mathematics in 1992 at the University of Pennsylvania under Chris Croke. Since 2005 Colding has been a professor of mathematics at MIT. He was on the faculty at the Courant Institute of New York University in various positions from 1992 to 2008. He has also been a visiting professor at MIT (2000–01) and at Princeton University (2001–02) and a postdoctoral fellow at MSRI (1993–94). Colding lives in Cambridge, Massachusetts, with his wife and three children.

Work In the early stage of his career, Colding did impressive work on manifolds with bounds on Ricci curvature. In 1995 he presented this work at the Geometry Festival. He began working with Jeff Cheeger while at NYU. He gave a 45-minute invited address to the ICM on this work in 1998 in Berlin. He began coauthoring with William P. Minicozzi at this time: first on harmonic functions, later on minimal surfaces, and now on mean curvature flow.

Recognition He gave an AMS Lecture at University of Tennessee. He also gave an invited address at the first AMS-Scandinavian International meeting in Odense, Denmark, in 2000 and an invited address at the Germany Mathematics Meeting in 2003 in Rostock. He gave the 2008 Mordell Lecture at the University of Cambridge and gave the 2010 Cantrell Lectures at University of Georgia. Since 2008 he has been a Fellow of the American Academy of Arts and Sciences, and since 2006 a foreign member of the Royal Danish Academy of Sciences and Letters, and also since 2006 an honorary professor of University of Copenhagen, Denmark. In 2010 Tobias H. Colding received the Oswald Veblen Prize in Geometry together with William Minicozzi II for their work on minimal surfaces. In justification of the reward the American Mathematical Society wrote: "The 2010 Veblen Prize in Geometry is awarded to Tobias H. Colding and William P. Minicozzi II for their profound work on minimal surfaces. In a series of papers they have developed a structure theory for minimal surfaces with bounded genus in 3-manifolds, which yields a remarkable global picture for an arbitrary minimal surface of bounded genus. This contribution led to the resolution of long-standing conjectures of initiated a wave of new results. Specifically, they are cited for the following joint papers, of which the first four form a series of establishing the structure theory for embedded surfaces in 3-manifolds:

Colding, Tobias H.; Minicozzi, William P. (2002). "The space of embedded minimal surfaces of fixed genus in a 3-manifold I; Estimates off the axis for disks". arXiv:math/0210106. Colding, Tobias H.; Minicozzi, William P. (2002). "The space of embedded minimal surfaces of fixed genus in a 3-manifold II; Multi-valued graphs in disks". arXiv:math/0210086. Colding, Tobias H.; Minicozzi, William P. (2002). "The space of embedded minimal surfaces of fixed genus in a 3-manifold III; Planar domains". arXiv:math/0210141. Colding, Tobias H.; Minicozzi, William P. (2002). "The space of embedded minimal surfaces of fixed genus in a 3-manifold IV; Locally simply connected". arXiv:math/0210119. Colding, Tobias H.; Minicozzi, William P. (2004). "The Calabi-Yau conjectures for embedded surfaces". arXiv:math/0404197. In the final paper cited here, the authors show that a complete embedded minimal surface of finite genus is properly embedded, proving the embedded version of the Calabi-Yau conjectures."

Honors Sloan Fellowship ICM Speaker Oswald Veblen Prize in Geometry Carlsberg Foundation Research Prize, 2016 Rolf Schock Prize in Mathematics, 2026

Major publications Cheeger, Jeff; Colding, Tobias H. (1996). "Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products". The Annals of Mathematics. 144 (1): 189. doi:10.2307/2118589. JSTOR 2118589. Retrieved 28 July 2026. Colding, Tobias H. (1997). "Ricci Curvature and Volume Convergence". The Annals of Mathematics. 145 (3): 477–501. doi:10.2307/2951841. JSTOR 2951841. Retrieved 28 July 2026. Cheeger, Jeff; Colding, Tobias H. (1 January 1997). "On the structure of spaces with Ricci curvature bounded below. I". Journal of Differential Geometry. 46 (3). Bibcode:1997JDGeo..4659974C. doi:10.4310/jdg/1214459974. ISSN 0022-040X. Retrieved 28 July 2026. Cheeger, Jeff; Colding, Tobias H. (1 January 2000). "On the structure of spaces with Ricci curvature bounded below. II". Journal of Differential Geometry. 54 (1) 42145. Bibcode:2000JDGeo..5442145C. doi:10.4310/jdg/1214342145. ISSN 0022-040X. Retrieved 28 July 2026. Cheeger, Jeff; Colding, Tobias H. (1 January 2000). "On the structure of spaces with Ricci curvature bounded below. III". Journal of Differential Geometry. 54 (1) 42146. Bibcode:2000JDGeo..5442146C. doi:10.4310/jdg/1214342146. ISSN 0022-040X. Retrieved 28 July 2026. Colding, Tobias; Minicozzi, William (1 March 2012). "Generic mean curvature flow I; generic singularities". Annals of Mathematics. 175 (2): 755–833. doi:10.4007/annals.2012.175.2.7. ISSN 0003-486X. Retrieved 28 July 2026.

References

External links Tobias Colding at the Mathematics Genealogy Project

Worked examples

Example 1 — a first encounter with Tobias Colding

Start with the simplest possible case. Write down what Tobias Colding claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Tobias Colding before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Tobias Colding ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Tobias Colding

In research: Tobias Colding appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Tobias Colding in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Tobias Colding is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1963 births, 20th-century Danish mathematicians, 21st-century Danish mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Tobias Colding outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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Frequently asked questions

What is Tobias Colding in simple terms?

Tobias Holck Colding (born 1963) is a Danish mathematician working on geometric analysis, and low-dimensional topology. He is the great-grandchild of Ludwig August Colding.

Why does Tobias Colding matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Tobias Colding?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Tobias Colding.