In the real numbers, we say that exactly when , that is, is *positive. *So since we’re trying to generalize the idea of “ordering”, one way is to do that is to figure out how to generalize the idea of “positive” numbers. So (and I’m being a little loose here), let’s say we only have the idea of “adding”, which we’ll denote with +, and multiplication, which we’ll denote by “*”. This is sort of what it means to be a “general field” (You might want to think about which properties of positive numbers can be defined only with +, *, or if this is too vague, just keep reading.)

Here are the two that are most important about the positive numbers–one is that if you add two positive numbers, you get a positive number again. The other is that if you multiply two positive numbers, you get a positive number again. These properties say that whatever set of positive numbers we have, it must be *closed under addition* and *closed under multiplication* resepectively.

Oh–and one other point. If we pick a generic number , we want to say that either is positive, negative, or zero. Also, has to be exactly one of them (so 10 can’t also be negative, for example). It turns out that’s all we need to make our definition of an ordered field:

**Definition: **We say that a field is an *ordered field *if it has a set (of “positive numbers”) such that:

- ( is closed under addition) If we have two elements and , then their sum is also in , that is, .
- ( is closed under multiplication) If we have two elements and , then their product is also in , that is, .
- (All nonzero numbers are positive or negative) For all in our field, exactly one of the following holds: or .

Now we’ll show something pretty cool.

**Proposition: **The complex numbers is *not *an ordered field.

Proof: To show this, we’re going to use a method called *proof by contradiction. *We’re essentially going to show that if *was *an ordered field, something bad will happen. So let’s assume *was *an ordered field and see if we can find anything weird happening.

Well one special element in that’s not in the real numbers is , where . So since , either or is positive, according to (3) above.

If was positive, then is a positive number, by (2). But again by (2), this says that is positive, so and -1 are both positive. This violates (3).

Okay, so what if was positive instead? Well, a pretty similar thing happens, since will still be positive, so we’ll get the same contradiction that 1 and -1 are both positive.

So there’s no way to order the complex numbers, at least as a field. Woah! That’s pretty neat. The mathematicians reading this may argue that if you loosen up your definition of just a set ordering, instead of a field ordering, you could put an ordering on . But instead of doing that and arguing with your computer screen, you should try to prove to yourself that any finite field can’t be ordered (as a field). It’s more fun that way.

]]>This kind of phenomenon seems to happen to people in every stage of their mathematical career after meeting new people, but it appears to be especially prevalent among graduate students. I think it’s due (at least in part) to the fact that the math that students learn in high school and early college tends to be somewhat standardized, but there’s a lot you can learn in your later undergraduate years and early graduate years that other people just don’t come across.

I’ve pieced together advice I’ve gotten from various people and compiled it here. Being a math person, I have decided to divide it into two cases (although half of the problem with imposter syndrome is actually not knowing which case you fall into).

*Case 1: You are not the person that knows the least in the room.*

In this case (which, according to anyone I’ve spoken to—professors, other graduate students, etc.—is extremely likely), getting over imposter syndrome simply comes down to recognizing the assertion that it is tremendously unlikely that you are the smartest person in the whole world so eventually you are going to run into someone smarter than you. Moreover, graduate school is the bridge into real mathematics where knowledge is much less “well ordered” like it is in undergraduate mathematics and much more “partially ordered.” For example, I had a professor for real analysis in my undergraduate career who had forgotten all of his algebra knowledge. I mean, all of it. We knew this because some number theory actually came up in an analysis problem (believe it or not), and he had marked it as a “Challenge” problem in our homework when really it was the first lemma in our algebra textbooks. Now, am I smarter than this person? No way! I won’t mention the person by name, but there is exactly a 0% chance that I am anywhere near his level of intelligence. The point is that honestly, there’s a mistaken belief people seem to have that the concept of “intelligence” is well ordered.

*Case 2: You really are the person that knows the least in the room.*

Okay, so you know the least amount of mathematics in the room. Woe is you. Who cares? Sometimes it can actually be an advantage to be the person that knows the least in the room. That way, everyone has something to teach you. Flipping that around, you can learn something from everybody!

Additionally, you don’t need to be the cleverest to do mathematical research. I (and many of my peers) fall victim to this idea in research of “Why am I even trying this when [name of prominent famous mathematician] could have this done in a week?” Well, the real answer to that question is, your prominent famous mathematician doesn’t have the time or the interest to work this out. There are other things that person is tackling. Further, just because you’re the (hypothetical) least smart person in the room, doesn’t mean you can’t or shouldn’t solve a problem. It is well documented that math is about persistence, not “genius.” (See this post by Terence Tao for an interesting expansion on this subject). Plus, once you solve this problem, you become the sole person in the world who best understands your problem and maybe that will yield problems in the future that you are the best person to solve!

Everyone needs to cross the bridge at some point in their life where they learn that they’re not the best. Hopefully these tips lead you to recognize and start to get over your imposter syndrome just a little more easily. (But really, I don’t know what I’m talking about. Sometimes I feel like I’m the least qualified person to talk about imposter syndrome.)

]]>When most people think of basketball, they picture the tall players, the fast-paced plays, and the seemingly impossible shooting skills. However, spatiotemporal pattern recognition does not come to most people’s minds when discussing the game. In his Ted Talk titled The Math Behind Basketball’s Wildest Moves, Rajiv Maheswaran discusses the use of spatiotemporal pattern recognition in analyzing the players’ movements and using this analysis to help coaches and players create effective game strategies. This up-and-coming science aims to understand and to find patterns, meaning, and insight in all of the movement in our world today.

What is spatiotemporal pattern recognition? In layman terms, it is the analyzation of “moving dots.” For those more interested in the technical aspects behind this pattern recognition, the University of North Carolina released an analysis of a number of papers that are good examples of spatial-temporal modeling, a source which is readily available online. This very new kind of data is becoming more and more popular, especially with the popularization of devices such as cellphones and GPS. Because of its newness, data scientists have the challenge of finding patterns within the data. In an interview with writer Ben Lorica, Maheswaran explained these difficulties, “There’s no language of moving dots, at least not that computers understand…There’s no computational language of moving dots that are interacting. We wanted to build that up.”

Here is one example of how this data can be used. To the left is a bubble chart with each bubble representing an NBA player. On the X-axis is their shot probability, and on the Y-axis is their shooting ability. If you take a player who generally made 47% of his shots, before, that was all you knew about him. Now, scientists can tell he would take shots that an average NBA player would make 49 % of the time (shooting probability), and they are 2% worse at their shots (shooting ability). This is significant for teams because it allows scouts to distinguish between all of the 47% shooters and determine their relative shooting ability and probability to the other 47% shooters.

What does this new science have to do with sports such as basketball? Maheswaran explained to Lorica that sports is one of the areas in which there is really great data available. Maheswaran said that “in sports in the last year, there have been tracking technologies placed in all major sports where they’re tracking all the players and the ball at a very, very high frame rate.” This availability of data as well as the large amount of people interested in finding patterns in this data, such as coaches and front offices, makes sports one of the best places to start building this science.

An important tool in developing this science is the use of machine learning, which allows scientists to go beyond their own ability to describe the things that they know. By giving the machine specific examples of movement and specific examples of non-movement, these scientists can teach the machine to see the game through the eyes of a coach. The machine is able to find features that enable it to separate particular movements and to discover the relationships between these movements. With this new information, new game strategies are being formed that are helping teams win games. In the near future, Maheswaran believes that real-time data will not only become a game changer, but also will help us to move better, move smarter, and move forward.

Maheswaran is also the CEO and founder of Second Spectrum, a company that applies analytics to sports tracking data. Those at Second Spectrum specialize in creating products that “fuse cutting-edge design with spatiotemporal pattern recognition, machine learning, and computer vision to enable the next generation of sports insights and experiences.” The company’s main goal is to revolutionize the way that people play, coach, and watch sports.

As Maheswaran mentions in his TED Talk, spatiotemporal pattern recognition can be used for much more than just sports analyzation. Writer Maureen Dowd fears other ways in which this type of data tracking can be used. In her *New York Times* article titled “Walk This Way”, Dowd discusses the Pentagon’s attempt at creating a grand database that can be used to track Americans’ every move, both literally and virtually. The Pentagon has been developing this technology as an antiterrorist surveillance system. The report outlining this research and development states that the “goal of this program is to identify humans as unique individuals (not necessarily by name) at a distance, at any time of the day or night, during all weather conditions, with noncooperative subjects, possibly disguised.” Though the reasoning for this data collection is for the good of the general public, many people like Dowd may find it to be an invasion of their privacy. Despite this, most, like Maheswaran, still have hope that this science will help to revolutionize how we think about movement in our world today.

Works Cited:

Dowd, Maureen. “Walk This Way.” *New York Times* 21 May 2003. *National Newspapers Expanded*. Web. 31 March 2016.

Lorica, Ben. “The science of moving dots: the O’Reilly Data Show Podcast.” *O’Reilly*. O’Reilly, 20 November 2014. Web. 31 March 2016.

Maheswaran, Rajiv. “The Math Behind Basketball’s Wildest Moves.” TED 2015. March 2015. *TED*. Web. 30 March 2016.

*Revolutionize Sports Through Intelligence*. Second Spectrum. Web. 31 March 2016.

“Spatial-Temporal Models.” *University of North Carolina, *University of North Carolina, n.d. Web. 24 April 2016.

Images:

http://tamilculture.com/category/videos/general

http://www.sportsgrid.com/nba/ted-talk-how-computers-are-replacing-coaches-in-the-nba/

The plane with the new metric is, in effect, exactly the 2-d sphere (with the north pole removed). The way I had come up with the appropriate scalings was by looking the stereographic projection from the north pole of a sphere and by requiring lengths of curves on the plane to be equal to the lengths of their corresponding curves on the sphere. (For instance, a line through the origin corresponds to a great circle through the two poles.) Other interesting geometric objects which can be obtained from the usual Euclidean plane by modifying its geometry include the hyperbolic plane.

Now let’s see if this example provides enough intuition to arrive at the definition of a 2-d manifold. Imagine the shell of an ovoid in . This is a very geometric object: one can measure lengths of curves which live on its surface, calculate the areas of different regions, and so on. In fact, developing tools to solve these problems are the subject of most introductory calculus courses. However, our goal is to give these notions meaning without any explicit reference to . We want to endow the ovoid, our manifold, with an identity of its own, independent of the ambient space .

Thanks to our extended examples, we have a clue. From far away, the ovoid does not look like a plane at all. But if we fly very close to its surface, it does look very similar to parts of a plane. Mathematically speaking, we can bijectively map a piece of the manifold onto a domain in the plane:

$$\phi : V \longrightarrow U \subset \mathbb{R}^2. $$

To cover the whole manifold, we need many such mappings.

Thus, we have disassembled our shape into (probably many) separate tabs of data, each of which involves a region in the plane along with a new inner product – a way of measuring angles and distances in that region.

Now, the question is: given only the tabs, how much of the shape can we reconstruct? Can one differentiate an ovoid from a sphere by only looking at their disassembled versions? This, of course, depends on how much geometric information we record in those tabs.

Before proceeding any further, however, there is a crucial issue we have to deal with. Assume we have a curve contained in one of the regions from our tabs. Part of the corresponding curve on the manifold may cross into another region as well. Now, the image of this segment in the other tab may look nothing like a curve. (Compare the images of the orange curve under two tabs in the above picture.) The way out of this is to put requirements for the *transition maps *that describe the correspondence between regions from different tabs. In the above picture, the transition map that sends the orange curve in to the one in is . If we require this map to be then the image of the orange curve must be in as well.

These conditions on the transition maps will be key to our definition of a manifold. We will be able speak of “derivatives of real functions on the manifold” only if the transition maps are differentiable. We can talk about smooth maps or curves on the manifold only if the transition maps are smooth. (Notice that smoothness and differentiability do make sense for transition maps, because they are maps between open subsets of . However, for the same notions on manifolds, we need definitions.)

Next post, we will see the definition of a (Riemannian) 2-d manifold. Meanwhile, please comment your thoughts!

]]>That said, until now we have never taken a specific political stance on an issue, nor called for our audience to do the same. However, in light of President Trump’s recent executive order placing immigration and travel restrictions on individuals from Iran, Iraq, Syria, Sudan, Yemen, Libya, and Somalia, we feel that there are certain issues that are too important not to take a vocal and principled stand against.

**We, the editorial board of the AMS Graduate Student Blog, condemn—in the strongest possible terms—these actions by President Trump, and we ask that he repeal this executive order as soon as possible. Moreover, we implore our readers, our fellow graduate students, and the entire mathematical community to stand up with us and other protesters around the world in condemning the president’s actions and demanding change.**

Given the somewhat unprecedented nature of this statement, we owe it to our readers to provide some context surrounding the thought process that went into its making. (For background about the order we recommend this video by *Vox *or this article by *NPR.*)

First, as noted in a recent statement by AMS Board of Trustees, this executive order threatens to “do irreparable damage to the mathematical enterprise of the United States” by choking the flow mathematical ideas from around the world. Having a shared stake in the success of this enterprise, we are concerned by policies that threaten it. The flourishing of any mathematical or academic community is contingent on the free exchange of ideas and researchers from around the globe. Many leading scientists, professors, and students in the United States are either from the countries included in the above list or have family members living there; we are made poorer, not safer, by their exclusion. *In making this statement we stand up for our mathematical community.*

Moreover, this executive order poses more than just an abstract threat; it has a human cost, one that will be felt by our friends and our colleagues. Already some in our community have been faced with difficult questions: Can I travel to this conference? To this job interview? To visit my friends? My loved ones? Will I be able to get home? The fact that this executive order puts members of our community in these positions, based solely on their national origin, is something we find unacceptable. *In making this statement we stand up for and with our friends and our colleagues whose lives are being negatively affected.*

Finally, we recognize that this issue goes beyond the small corner in which our blog is situated. President Trump’s executive order—placed indiscriminately and in many cases falling on those fleeing war and persecution—will negatively affect the lives millions of people around the world, both by promoting intolerance and bigotry and turning away those in need. By targeting individuals based on their national origin, used as a proxy for religion, this order is incongruent with our commitment to fostering a diverse and inclusive community, country, and world. *In making this statement we stand up for our shared values of diversity, inclusion, and basic human **rights.*

For these reasons, we are compelled to join the chorus of protests and publicly express our condemnation and outrage at President Trump’s recent actions. For those looking to stand up and become involved on this issue, here are a few resources:

- Academics Against Immigration Executive Order: An online petition of academics from all fields opposing the recent Executive Order.
- WhoIsMyRepresentative: A tool for finding your representatives in the House and Senate, as well as how to contact them and voice your concerns.
- Statement of Inclusiveness: An online pledge committing to maintaining an inclusive scientific community. See our previous article on the importance of such statements, especially now more than ever. Created by Juan Souto and Kasra Rafi.

Signed with Solidarity,

AMS Grad Blog Editorial Board

**Sarah Salmon, Editor-in-Chief**

University of Colorado, Boulder

**DJ Bruce, Managing Editor**

University of Wisconsin

**Irving Dai, Managing Editor**

Princeton University

What makes for a good task? Rachel Lotan, a teacher educator at Stanford, coined the term *groupworthy task* to describe what we strive for in task design. In a book review (link: http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/240/164 ) that I wrote of Mathematics for Equity, I describe groupworthy tasks as follows:

Groupworthy tasks facilitate students’ interdependence by foregroundingmultiple abilities and multiple representations, requiring students to worktogether in solving complex mathematical problems. These tasks involvesufficient interdependence and challenge; even those students who areperceived as “advanced learners” often experience difficulty completingthe tasks on their own.

Let’s take a closer look at interdependence, multiple abilities, and multiple representations:

**Interdependence****:** If we want students to work together, we need to create a task that actually requires working together to be able to solve it. Moreover, we need to *convince* students that they need to work together. If the task is not sufficiently complex and mathematically rich (See: What is a mathematically rich task) then there will be no need to work together. The typical end of chapter exercises in most textbooks are not mathematically rich; merely teaching a skill and having students practice it (so-called “drill and practice” or more derisively, sometimes called “drill and kill”) is not sufficient to satisfy this criterion.

**Multiple abilities and multiple representations:** A groupworthy mathematical task also requires students to use a lot of different academic abilities (verbal, written, spatial, visual) along with intra and interpersonal skills. Going hand in hand with this, a good task also requires the use of multiple representations—the so-called Rule of Four suggests that we need to use graphical, numeric, linguistic, and symbolic ways of representing mathematics.

Let us take a look at a task that I have used with both high school and college algebra students. This task is adapted from a text called Discovering Algebra.

You have a sheet of paper and are folding it in half, and then inhalf again, and so on. You need to find out how many layers thereare total for a given number of folds. For example, with two foldsthere are 4 layers.In other words, you are searching for a formula that represents therelationship between the number of folds and the number of layers.

This task allows for multiple representations; I generally provide physical paper for students to fold and some find the physical folding helps to make things more concrete. The task asks for a formula, but many students record their observations/answers in a table as an intermediate step. There are also multiple abilities needed to solve the task; students need to be able to count, to notice patterns (such as the doubling relationship), to understand operations conceptually (such as how repeated multiplication becomes exponentiation), and to communicate their ideas with each other. The task requires interdependence; there is not a simple procedure for finding this answer and students have to be able to explore and test different ideas. Moreover, the task is mathematically rich; the concepts of multiplication, doubling, exponentiation, geometric series, exponential functions, and recursive functions are all things that have come up when my students have worked on this problem.

In the next part of this series of posts, I will discuss how to adapt problems that you might already have access to in order to make them more groupworthy.

]]>Some of the posts are done live so attendees can jump in on events while they happen. Other posts are done after the event which allows the authors to give more of an overview of everything that occurred.

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Anytime we have a problem, what do we do? We use an app! Here to save you at JMM is the 2017 JMM Mobile App! You can search for events, personalize your schedule (which can add things to your calendar!), and even give updates/announcements about the conference. The app also has a networking feature that can help you connect to others attending JMM.

Okay, so maybe it’s not a “hidden gem” since it’s the first links on the JMM website and there’s a description in the registration packet but I’m still excited to check it out and take some of the bookkeeping out of my brain and onto my phone. I didn’t get a chance to utilize the app last year so if you give it a shot or have any tips or tricks, let us know!

]]>One outlet for such support that I recently had the opportunity to help implement here at the UW–Madison (sorry for the humblebrag), is a statement of community commitment to, and value of, inclusivity. Based on this experience, I would like to talk about what such a statement is, why I think they are meaningful, and to (not so) secretly encourage others to do similar things within their own departments.

To start, such a statement simply outlines what inclusivity and diversity mean to you/your department, how this group values these concepts, and how they will fight for and support them. For example, here at the UW–Madison, the department adopted the following statement,

“As a diverse group, the Mathematics Department strives to foster an open and supportive community in which to conduct research, to teach, and to learn. In accordance with these beliefs and § 36.12 of the Wisconsin Statutes, the Mathematics Department affirms that all community members are to be treated with dignity and respect and that discrimination and harassment will not be tolerated. We further commit ourselves to making the department a supportive, inclusive, and safe environment for all students, faculty, staff, and visitors, regardless of race, religion, national origin, sexual orientation, gender identity, disability, age, parental status, or any other aspect of identity.

To all members of our community, we, the members of the Department of Mathematics, welcome you.“

Of course what precisely such a statement should say, and how it should say it varies person-to-person and group-to-group. I see power in these statements having unique, personal voices—serving as honest expressions of one’s beliefs and commitments, laid bare for the reader to hear. So if you are thinking about drafting such a statement, consider trying to make it personal. Be honest and speak to those who you hope eventually read it.

As a second example of such a statement, and to highlight a slightly different voice, here is (in some sense) my own statement. (Note my statement was influenced by Federico Ardila-Mantilla’s amazing recent article in the Notices, “Todos Cuentan: Cultivating Diversity in Combinatorics”.)

No student may be denied admission to, participation in or the benefits of, or discriminated against in any service, program, course or facility of the {UW} system or its institutions or centers because of the student’s race, color, creed, religion, sex, national origin, disability, ancestry, age, sexual orientation, pregnancy, marital status or parental status.” ~ § 36.12, Wisconsin Statutes

I recognize the importance of a diverse, inclusive, and supportive community. In accordance with these beliefs, as well as § 36.12 of the Wisconsin Statutes, I am committed fully to the following axioms:

Axiom 1: All members of the department/university community should be treated with dignity and respect.

Axiom 2: I strive to promote a supportive, inclusive, and safe environment for all students, faculty, and staff, regardless of race, religion, national origin, sexual orientation, gender identity, disability, age, parental status, or any other aspect of identity.

Axiom 3: Incidences of hate, bias, discrimination, or violence have no place in a department/university community, and will not be tolerated.

Axiom 4: I will, to the best of my abilities, aid those facing instances of bias, discrimination, hate, or violence; directly and by helping individuals find the appropriate campus and community resources.

These axioms serve as guideposts in teaching, research, outreach, and all aspects of my career/life. Moreover I am dedicated to working/partnering with other campus organizations (Multicultural Student Center, University Health Service, LGBT Campus Center, End Violence on Campus, McBurney Center, Campus Women’s Center, etc.) to promote these axioms.

Finally, to all members of our campus community, I would like to say:

I seeyou, I acceptyou, and I affirmyou.

At this point, some of you reading this might be wondering what the point of these statements is. You, or well the conveniently constructed strawperson I so often use rhetorically at this point in my writing, may be thinking, “Surely people know that our math department is a nice friendly place, which is inclusive of all people.”

Well, a couple of points:

First, be careful in thinking your department and university are friendly places free from bias, hate, discrimination, and violence. The fact of the mater is that most are not. (Some privileges checklists!) So taking a moment to write this, and honestly evaluating your, and your department’s, commitment to inclusion and diversity can be eye opening. You may begin to see incidences of bias, hate, and violence you were previously blind to.

Secondly, even if your department is a magical land free of bias and hatred—it’s not, but regardless—not everyone knows this. Sure you might, but I am certain that there is someone, another grad student, a visitor, and a calculus student, who doesn’t. Someone who is unsure of whether the department will respect their identity, or whether there is someone who they can turn to for support. Letting this person know that you recognize them and will do your best to support them is meaningful. There is power in being an ally and power in recognizing and affirming the identities and rights of others, especially those who often face bias, hate, discrimination, and violence. This can even more true in turbulent times like these where many people feel uncertain, unwelcome, and unsafe.

All this is to say that creating a statement of commitment to inclusivity can be both meaningful and powerful. However, they are not ends in themselves. **They will not end bias, hatred, and discrimination within your department, university, or the mathematical community more generally. So while I encourage everyone to the time to write such a statement, I also think it is important to recognize that such a statement is best seen as a starting point, a guidepost that can direct your actions going forward.**

*Did you say your department is a safe place free from hate?*Well how will you, or others, handle a situation where someone creates a hostile environment in the classroom? Moreover is there training to help instructors identify and handle incidences of bias and hate in their classrooms?*Did you commit yourself to being an inclusive place for people of all gender identities?*Well are the health plans your department offers inclusive of trans-persons? Do you have gender-neutral washrooms in your department? Do instructors ask students what pronouns they use?*Did you commit yourself to support victims of violence and harassment?*Well do students know who they should report incidences of violence and harassment to within the department? Are your department’s policies on these matters clearly articulated and publicized? Do instructors know what campus resources exists to help victims of violence?

I could go on—in fact, that might make an interesting future post—but to keep it short: **creating a inclusive, supportive, and safe community takes dedicated, deliberate, and thoughtful action; writing a statement of commitment is just a first step.** (I know that this is my goal here at UW–Madison.)

Finally to all those reading this—especially those who may be spinning, either a bit or a lot—let me reiterate what I said above:

*I see you, I accept you, and I affirm you.*

My story of surfaces starts in a beautifully weird morning when I got up to realize that life in the usual Euclidean plane had changed dramatically. Vectors had shortened, areas had shrunk, and infinity was just a few feet away!

Here is the aftermath (why the term “after math?!”):

What had happened was this: A vector of unit length in the usual sense was now much shorter, of length where is the distance (in usual sense) of its base (=initial point) from the origin. Yes, here it does matter where the base of a vector is.

The vectors that used to be perpendicular are still perpendicular. By looking at parallelograms, we see that areas have shrunk by a factor of . So, in the above diagram, the square feet of the two shaded areas are in fact almost equal.

Let me do some measurements.

**Length of an infinite straight ray.**

Consider (see picture.) To calculate lengths we must add up lengths of tangent vectors. Tangent vectors are all , unit length in ordinary plane at each point, so their new length must be taken into account. The result is:

Hmm! Seems that the infinity is only feet away!

**The area of the whole plane!** (But wait, isn’t that infinite?)

We add up areas of infinitesimal rectangles . As discussed above, in our new *metric *(Ok, I had to finally use the term!) this rectangle’s area is shrunk by our factor. So, we get

Maybe my whole success in posting these series depends on my ability to convince you now to experiment with this new plane I have created. Be bold about it. Ask questions, do calculations for yourself, get a feel of the structure of this space. Ask what could be asked about this space? What other familiar constructions are possible on it? Could we talk about spaces of functions? What is the measure? How will we integrate a function? What is the shortest path, say between and ? Notice that going on the straight line isn’t the best, because we are better off bending away from the origin so that lengths are shorter. But how much to bend?!

And finally answer this: the plane with this tampered metric is a copy (in almost all aspects) of a familiar 2-d shape, can you guess what shape?!

To be continued with more fun…

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