mathematics

Louis Billera

Louis Billera 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 Louis Billera rather than just read about it. In short: Louis Joseph Billera is a Professor of Mathematics at Cornell University. Career Billera completed his B.S. at the Rensselaer Polytechnic Institute in 1964.

Key takeaways

Louis Billera belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Louis Billera to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Louis Billera from memory before moving on to harder problems.

Reference excerpt

Louis Joseph Billera is a Professor of Mathematics at Cornell University.

Career Billera completed his B.S. at the Rensselaer Polytechnic Institute in 1964. He earned his Ph.D. from the Graduate Center of the City University of New York in 1968, under the joint supervision of Moses Richardson and Michel Balinski. Louis Billera served as the first Associate Director of the National Science Foundation Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) at Rutgers University. In 2010 he gave the invited lecture, "Flag enumeration in polytopes, Eulerian partially ordered sets and Coxeter groups" at the International Congress of Mathematicians in Hyderabad.

Contributions The common thread through much of his research is to study problems motivated by discrete and convex geometry. A sampling includes constructing polytopes to prove the sufficiency condition for the g-theorem (with Carl Lee), discovering fiber polytopes (with Bernd Sturmfels), and studying the space of phylogenetic trees (with Susan Holmes and Karen Vogtmann).

Awards and honors In 1994 Billera won the Fulkerson Prize for his paper, Homology of smooth splines. This prize is given every three years to the best paper in discrete mathematics. In 2012 he became a fellow of the American Mathematical Society.

Selected publications Louis Billera, "Homology of smooth splines: Generic triangulations and a conjecture of Strang", Transactions of the American Mathematical Society 310 (1988). 325–340. Louis Billera, Anders Björner, Curtis Greene, Rodica Simion, Richard P. Stanley (eds.): New Perspectives in Algebraic Combinatorics, MSRI Publications, Cambridge University Press 1999, http://library.msri.org/books/Book38/

See also Simplicial complex Combinatorial commutative algebra Quasisymmetric function

References

External links Billerafest 2008, Conference in Honor of Louis Billera's 65th Birthday. https://www.math.cornell.edu/event/conf/billera65/ Louis Billera's Homepage profile at Cornell University archived

Illustrations

Worked examples

Example 1 — a first encounter with Louis Billera

Start with the simplest possible case. Write down what Louis Billera 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 Louis Billera 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 Louis Billera 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 Louis Billera

In research: Louis Billera 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 Louis Billera 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: Louis Billera is common in secondary-school and first-year university syllabi. It links to neighbouring topics 20th-century American mathematicians, 21st-century American mathematicians, CUNY Graduate Center alumni, so understanding it makes those chapters shorter.
In everyday life: Look for Louis Billera 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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How to study Louis Billera in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Louis Billera means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Louis Billera out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Louis Billera in simple terms?

Louis Joseph Billera is a Professor of Mathematics at Cornell University. Career Billera completed his B.S. at the Rensselaer Polytechnic Institute in 1964.

Why does Louis Billera 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 Louis Billera?

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 Louis Billera.