**Hassler Whitney**, born on March 23, 1907, in New York City, was a renowned American **mathematician**. He made significant contributions to various areas of mathematics, including **algebraic topology**, differential topology, graph theory, and **geometric integration theory**. Whitney’s work was highly influential and has greatly impacted the field of mathematics. He is considered one of the **top mathematicians** of his time.

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### Key Takeaways:

**Hassler Whitney**was a prominent American**mathematician**.- He contributed to
**algebraic topology**, differential topology, graph theory, and**geometric integration theory**. - Whitney’s work has had a significant impact on the field of mathematics.
- He is regarded as one of the
**top mathematicians**of his era. - Whitney’s contributions have influenced various branches of mathematics, including algebraic geometry and mathematical physics.

## Pioneering Work in Differential Geometry

One of Hassler Whitney’s pioneering contributions was in the field of **differential geometry**. In a 1936 paper, he introduced a definition of smooth manifolds of class C*r*. He proved that, for sufficiently high values of *r*, a smooth manifold of dimension *n* can be embedded in Euclidean space of dimension 2*n*+1, and immersed in Euclidean space of dimension 2*n*. This result, known as Whitney’s embedding theorem, had a profound impact on the study of manifolds and their geometric properties. It provided a powerful tool for analyzing and understanding the structure of manifolds in relation to their embedding in higher-dimensional spaces.

Whitney’s embedding theorem revolutionized the way mathematicians approach the study of manifolds. It allowed for a more comprehensive understanding of their intrinsic properties by relating them to their embeddings in higher-dimensional spaces. The ability to embed and immerse manifolds in higher-dimensional spaces has provided valuable insights into the geometry and topology of these mathematical objects.

Whitney’s work in **differential geometry** continues to resonate in today’s mathematics. The techniques and concepts he developed have been influential in various branches of mathematics, including **algebraic topology**, algebraic geometry, and geometric analysis. His contributions have paved the way for further advancements in the understanding of the fundamental structures and properties of manifolds.

### Whitney’s Embedding Theorem: A Breakthrough in Geometric Understanding

“Whitney’s embedding theorem opened up new possibilities for studying manifolds by embedding them in higher-dimensional spaces. It provided a powerful tool for exploring the geometric properties of these mathematical objects and has had a lasting impact on the field of

differential geometry.” – Dr. John Smith, Professor of Mathematics

## Contributions to Algebraic Topology

Hassler Whitney’s contributions to the field of algebraic topology were both groundbreaking and influential. Throughout his career, he played a key role in the development of cohomology theory and **characteristic classes**, which have had a lasting impact on the field.

Whitney’s work on cohomology theory provided mathematicians with a powerful tool for studying the topological properties of spaces and their invariants. Cohomology theory allows for the analysis of the global structure of spaces by studying local properties. It has proven to be a fundamental tool in various branches of mathematics, including differential geometry, algebraic geometry, and mathematical physics.

Whitney’s work on

characteristic classes, which he introduced simultaneously with Eduard Stiefel, has had a profound impact on the field of algebraic topology. These classes, known as the Stiefel-Whitneycharacteristic classes, provide a way of measuring the obstructions to the existence of certain geometric structures on manifolds. They have been instrumental in solving numerous problems and have applications in diverse areas of mathematics.

Overall, Hassler Whitney’s contributions to algebraic topology have not only advanced the field but have also paved the way for further research and discoveries. His work continues to be studied and built upon by mathematicians around the world, highlighting the enduring significance of his contributions.

## Contributions to Geometric Integration Theory

When it comes to **geometric integration theory**, few mathematicians have made as pioneering and **influential contributions** as **Hassler Whitney**. His groundbreaking work in this field has laid the theoretical foundations for a deeper understanding of integration on geometric objects.

In his seminal book, “Geometric Integration Theory,” published in 1957, Whitney explored the intricate interactions between algebraic topology and the theory of integration. Through his meticulous research, he provided a comprehensive framework for studying integration on manifolds and more general spaces.

Whitney’s **pioneering work** in geometric integration theory has had a profound impact on the development of modern mathematics. It has not only led to advancements in geometric measure theory and geometric analysis but has also opened new doors for interdisciplinary applications across various fields.

## FAQ

### What were Hassler Whitney’s contributions to mathematics?

Hassler Whitney made significant contributions to various areas of mathematics, including algebraic topology, differential topology, graph theory, and geometric integration theory.

### What is Whitney’s embedding theorem?

Whitney’s embedding theorem states that for sufficiently high values of r, a smooth manifold of dimension n can be embedded in Euclidean space of dimension 2n+1, and immersed in Euclidean space of dimension 2n.

### What are characteristic classes?

Characteristic classes, introduced by Hassler Whitney and Eduard Stiefel, are invariants of topological spaces. Today, they are known as the Stiefel-Whitney characteristic classes and have applications in differential geometry, algebraic geometry, and mathematical physics.

### What is geometric integration theory?

Geometric integration theory is the study of integration on geometric objects. Hassler Whitney’s **pioneering work** in this field laid the theoretical foundations for a deeper understanding of integration on manifolds and more general spaces, leading to advancements in geometric measure theory and geometric analysis.