Dr. Hilda Geiringer was an Austrian mathematician whose life is closer to the plot of *Indiana Jones and the Last Crusade* than almost any other mathematician. She was the first woman to serve as a university instructor in applied mathematics in Germany and, completely against the current of the times, was up for promotion to *extraordinary professor* when the Nazi laws kicking anyone Jewish out of any worthwhile job kicked in. She fled through Turkey, eventually coming to the United States where her general excellence triumphed — partially — over the open discrimination against women in mathematics that was usual at the time. She was extraordinary and made remarkable contributions.

# Math Love

# The Integers (mod n)

The whole numbers, the first numbers we learn, are infinite. If you take a number and add one to it, you get a new and bigger number — but not always. Keep counting up from one and you can get to 213 — but start at one o’clock in the afternoon and you never get to two-hundred and thirteen o’clock. In this post we are going to look at a type of number where we cannot keep going forever. Instead, as we keep adding one, the numbers cycle around, like the numbers on a clock. It turns out that there is an infinite family of these number systems, and some of them obey almost all the normal rules of arithmetic.

When you are first learning to add and multiply, you might make tables. Memorizing the multiplication tables is somewhat controversial in math education (Occupy Math comes down firmly in favor of memorizing your one-digit multiplication facts). At the top of the post are addition and multiplication tables, but not the normal ones. This post is about a whole collection of number systems based on the integers, each of which has only a finite number of numbers. The addition and multiplication tables above have five numbers, for example. As we will see, these number systems show up in several applications. They obey many of the same laws of arithmetic as the integers. Mathematicians think of these number systems as smaller images of the integers that preserve some of their properties.

# What do mathematicians do all day? Part VI

This post is the next in our series on what mathematicians do with their time. The other posts in the series are indexed at the end of this post. Since everyone knows about the teaching mathematicians do, the series focuses on research. Today we are going to look at the *walking triangle representation* for optimizing functions. If you have finished a calculus course, it’s likely you already have optimized a function; this means finding the place where the function is highest (maximizing) or lowest (minimizing). We call the high points and the low points the *optima* of the function. To demonstrate the walking triangles, we will be maximizing the function shown at the top of the post. This function is really easy to maximize — you just head uphill.

# What are complex numbers?

A post on calculus and its higher level forms recently appeared in Occupy Math as part of the Islands of Mathematics series. One key piece of feedback from Occupy Math’s editors was substantial confusion about complex numbers. Since complex numbers are key to a number of the types of fractals that Occupy Math likes, including the one at the top of the post, there are now several reasons to have a post which explains complex and imaginary numbers.

There is also a rather startling mathematical technique to be explained: *if, in a system you are using, some object does not exist, then declare it to exist by fiat, and see what impact that has*. Here is a key point the post tries to make. The object that “does not exist” that we are going to investigate is the square root of a negative number. Many people “know” that negative numbers do not have square roots, but this is false. Something much narrower is true — those square roots only exist if we enlarge our concept of what a number is. Interestingly, the new notion of number actually lets us explain laws of nature and do engineering tasks that are either impossible or much more difficult without these new numbers.

# The Island of Calculus, the Mountains of Analysis

Today’s post is the next in our tour of the Islands of Mathematics. The island *du jour* is one that some people think of as Skull Island, the home of calculus and its higher level form, analysis. Even professional mathematicians — outside of the field of analysis — find the mental adjustments needed to do analysis to be challenging, hence the analogy to King Kong’s home. If you find yourself lost or confused, you are not stupid. Rather, you should probably not take up mathematical analysis as a profession. The name *analysis* is very poorly chosen, as there are many other fields of endeavor with exactly the same name. The linked article on Wikipedia resolves this issue by calling the field *Mathematical Analysis* which is enough to be going on with. As a first crack at explaining what it is, analysis is the mathematics of moving things, continuous functions and processes. This includes many of the laws of physics, e.g., gravity and planetary orbits. It also covers everything done with calculus, from calculating volumes to finding optima (most favorable points in a formula).

Without actually going through the calculations, the kinds of things you can compute with calculus include the following.

- A cone with height
**h**and a circular base of radius**r**has a volume of**π r**. This lets you figure out the volume of sand in a pile — which is usually roughly cone-shaped. This uses integral calculus to find the volume formula of a cone.^{2}h/3 - If you want a cylinder of a specified volume to have the smallest possible surface area, then the height of the cylinder must be twice the radius. This is an example of an
*optima*and lets you make a can, of a specified volume, with the smallest possible amount of materials. This uses differential calculus. - As a rocket takes off, it burns fuel, which means it get lighter. This means that the acceleration of the rocket increases as it travels. The motor is pushing at a fixed power level, but the rocket gets lighter and lighter. Calculus lets you deal with the changing acceleration and compute the rocket’s final speed. This is another integral calculus problem.

# What Do Mathematicians Do All Day? Part V

Today’s post is about *ring species* and a useful discovery that Occupy Math and his collaborators found while trying to understand them. A ring species is spread out around a major barrier that it cannot cross, so that the members of the species are in a long, thin ring-shaped domain. Examples of ring species include larus gull populations around the north polar ocean or greenish warblers spread around the Himalayan massif. A picture of the greenish warbler’s distribution appears at the top of the post. The funny thing about ring species is that they are arguably one or several species.

The biological species concept says that a group that form a species must be able to breed with one another. In a ring species, adjacent groups in the ring can breed, but on the far side of the ring where the groups meet up again, they cannot. Mostly this shows that the biological species concept is a little too simple. We thought that wolves and coyotes were different species, for example, but there is a difference between “cannot breed” and “choose not to breed”. The hybrid coywolves show that coyotes and wolves can breed — when humans thin out the wolf population to where they cannot find mates that are wolves. Below the fold, we will get to how this led to mathematical research.

# Rediet Abebe: A Rising Star in AI

This is Rediet Abebe, an AI researcher and new faculty member at the University of California at Berkeley (picture by Anoushnajarian). Berkeley seems very proud of their new assistant professor, and with good reason. Her insights about the way inputs to artificial intelligence are chosen have huge implications for social justice and discrimination, and are foundational to progress in the application of AI. Occupy Math has chronicled a number of people from under-represented groups that made huge contributions in math, but most of those posts were obituaries. Professor Abebe is a current phenom. She is a co-founder of Black in AI, an organization that is trying to increase the number of black people working in artificial intelligence. The post will look at how this remarkable woman got where she is today and discuss some of the reasons why increasing the representation of underrepresented groups in AI is both important and a really good idea.

# The Island of Numbers

This post is the next in our tour of the islands of mathematics, the Island of Numbers. Almost the first thing that every civilization that discovered math found was the whole numbers. Leopold Kronecker observed that “God made the integers, all else is the work of man.” This is why number theory is one of the oldest fields of math — as soon as you find the whole numbers, you start seeing strange patterns in them. We are still finding strange patterns, new ones, even in the present day.

The field of number theory shares with set theory the quality that it can be used as a basis for the rest of math. To serve as the foundation of mathematics, number theory uses the Peano axioms for arithmetic instead of the axioms of set theory. There is a conflict, within mathematics, over which of these two fields is the most natural foundation for the discipline. Occupy Math notes some things are easier in set theory and others are easier in number theory, which leads him to suspect that this conflict tells you more about the interests of parties to the conflict than anything else. The place where number theory starts is the set of *prime numbers*, numbers that lack whole number divisors, other than one or themself. The prime numbers were one of the first mathematical mysteries humanity noticed, going all the way back to ancient Greece. The prime numbers also are at the core of many problems that we have not, in thousands of years, solved. Some of them appear in the post.

# Another Hidden Figure: Raye Montague

Occupy Math sometimes chronicles the trials and tribulations of women in mathematics. Katherine Johnson, Margret Hamilton, and Elizabeth Williams are examples from earlier posts. Today’s post is the next in this sequence, concerning naval architect Raye Montague, pictured on the left working at someone else’s desk. In addition to having a huge talent for mathematics and computer science, Raye Montague was a black woman from the American south, born in Arkansas in 1935, quite a load to carry while revolutionizing ship design and winning awards. Ms. Montague was clearly a genius level talent, and while this is wonderful, genius is too rare to carry civilization. The system that Ms. Montague bucked to become the remarkable person she was must have turned back many excellent and many competent persons simply for being women or having the wrong color of skin.

# Set theory contains real and unreal worlds.

This post is part of the series on visiting the islands of mathematics. A **set** is a collection of objects in which no object appears twice. **Set theory** is the mathematics you get by thinking about sets really hard. Occupy Math has already introduced a standard method of diagramming the relationship between sets, the Venn diagram, a humorous example of which appears at the top of the post. Set theory turns out to be a way to establish an intellectual foundation for the rest of mathematics — and this leads to a mildly awful anecdote about the interaction between mathematical research and mathematics education, near the middle of the post. Occupy Math has published a book on set theory and regularly teaches a course in the subject for second-year students.