Life has been quite hectic lately. It’s increasingly difficult to find time to write on the blog. I should have more time to write a few weeks from now, and once that time comes I should be able to make progress on a lot of the ideas I’ve been wanting to develop here. For the time being, I thought it would be fun to write about what my plans are going forward and to offer an opportunity for my readers to make suggestions of areas that I can write on in the future.

**What’s Going on Now**

*My Mathematical Studies*

There is quite a lot going on right now in my mathematical work. I’ll summarize the many converging areas in which I am involved right now.

The most natural place to begin is with my own “education” in the sense that most people would use the word – my coursework. Since it is now the summer, I do not have normal semester courses going on, but I do have a four week summer class. The reason? In graduate schools (I believe all of them, but I can only directly speak for mathematics), there isn’t exactly a clean organization in terms of years. That isn’t the best way to think about it. You might think of graduate school in terms of three “barriers”: the qualifying exams, the research proposal, and the thesis. To give a very brief metaphor of these stages, ‘Stage 1’ might be like learning about the world at large, ‘Stage 2’ might be going deeper into continent or large chunk of a continent, and ‘Stage 3’ might be like learning more about a particular city or state than anyone has ever done before. I am currently on the cusp of finishing up ‘Stage 1’ – at my school we call the end of this stage of learning the qualifying exams. Without passing this stage, you cannot get a Ph.D. At my particular institution, three categories of qualifying exam are offered twice a year, and we must pass two of the three to pass into ‘Stage 2’. I have already passed one exam – the Algebra Exam – and my summer course is a preparatory course for what is called the Analysis Exam. So, I am spending a lot of my time preparing for this exam.

I am also working part-time as a teaching assistant at an REU, which stands for ‘Research Experience for Undergraduates’ at the University of Virginia (being held virtually, since we are still in quarantine). This is a 6 week program at which are assembled many of the brightest mathematical minds of college or high school age. The goal of the program is to provide mentorship for these bright young men and women and to guide them towards publishing cutting-edge academic papers. Working with these young mathematicians (who really aren’t that much younger than I and all of whom have much more natural talent than I) is such a pleasure. They move so quickly that I sometimes can’t quite keep up with them, and it is an absolutely wonderful learning opportunity for me – both learning more advanced mathematical concepts and in learning more about education.

Finally, there is a research component to my job. I have been write papers to publish in academic journals fairly regularly since entering into graduate school (if you want to see my publications, you can check the link at the end of the article, though this link might be changing sometime in the near future). All of my recent work is related closely to the theory of modular forms – which is an area of study primarily within number theory but which also has applications in the study of black holes and many other areas in academia. My writing so far has been focused on studying partition functions – which can be thought of as studying the various ways in which numbers can be written as sums of smaller numbers, though I also helped write some papers relating to Ramanujan’s tau function pioneered by my advisor Ken Ono that are very exciting.

There is also a networking aspect to my job – sadly, this can’t really happen during a global quarantine, so I cannot comment much on any of this.

**Critical Thinking Series**

I have also recently begun a series of posts about critical thinking that will go on to be quite a long series. Sadly, as far as I can tell we don’t really teach good critical thinking anymore – and if we do, there is no evidence whatsoever of this in the public sphere. Almost every time I see commentary on some important public event, I see that at least some important people and very many “regular” people speaking about it make fairly blatant logical fallacies or major miscues in some philosophical area like ethics, epistemology, metaphysics, or theology. And these miscues, when there are a great many of them, contribute to a culture that no longer knows how to use proper critical thinking.

Therefore, I have decided to devote a lot of time on this series, because if critical thinking and logical skills are mastered, then a great many blunders can be avoided immediately. I recognize, to be fair, that I also fall prey to bad thinking sometimes – but regardless of your political party, religion, race, or anything else, you will fall prey to way, way more fallacies in your thinking when you don’t know they are fallacious. Because I know sometimes my own thinking will fall short, I want to be surrounded by people who will call me out when it does. This is how growth works and is I believe an essential component of what we need to do in order to solve many of the largest problems we face as a society.

I hope that series will be helpful and educational for all who read it.

**What’s Coming**

*Database Series*

This is a project that has been in the works for a long, long time, and still is not quite ready yet. But I look forward to the day that it is ready.

The database series will not exactly be articles in the way that I usually write. Basically, the goal of this series is to compile large lists of resources and data that would be useful in understanding various problems. So far, I have begun compiling two such databases, neither of which is quite ready yet. In one of them, I have used a “Top 100 Mathematicians of All-Time” list and tried my best to determine the religious beliefs of these men and women. The reason for this is that there is a fairly prevalent theme in a large chunk of modern society that religious people are anti-science and always have been, and so I thought that such a review might be at least one piece of data that would be relevant in evaluating whether such a claim is accurate or not.

I am also working on a database of apologetics ministries through which those who want to investigate religions intellectually can look. I do my best to only list a ministry that I feel is both intellectually honest and helpful. The list is primarily of Christian ministries, but I will also include at least one Muslim speaker and one atheist speaker who I have gained a lot of respect for as I have listened to them.

The database series will also be updated as I come across more information. So, unlike my regular posts, these posts will be updated over time when I find it appropriate to do so. I also am hoping that these will be helpful – though I expect they won’t be as helpful at first, when there are only one or two of them – but I hope that over time they can help people who are curious to learn more.

*Book/Article Summary Series*

I’m not quite sure what I want to do with this yet, but I want to provide an ongoing series in which I summarize some of my favorite intellectually significant books and books that have had a large influence on culture. I expect it will be quite a while before this series really gets going, but probably before the end of 2020 I will have done one or two of these.

*What Is Mathematics All About?*

This is a series about mathematics that I have wanted to write every since the blog began, but which I haven’t quite yet figured out how to write. What I want to do with this is to attempt to explain how a mathematician thinks about math. I want to answer questions like the following:

- What does it mean for mathematics to be new?
- What are the different fields of math (analogous to physics, chemistry, biology, etc. in science) – and why do we think of them in that way?
- What does it mean for an idea to be an ‘example’ of another idea?
- How do we shift between different mathematical fields and ideas?

These, and many more, are the sorts of questions I hope to address in that series. Once I feel satisfied that I have done a good enough job with my goal for the series, I will begin writing it.

*Understanding Calculus*

This coming semester at UVA, I am going to be an instructor for Calculus 1. Although I have tutored dozens of people in these topics and have been a teaching assistant in this class before, this semester will be the first time I have taken the lead instructor role, and it’s an exciting step up.

It is, however, a lot of work. Calculus is a different sort of conceptual entity than much of the math that comes before it. By that I mean that it isn’t like calculus is just “algebra 3” or “geometry 2” or “trigonometry 2”. It uses algebra, geometry, and trigonometry in some important places, but it has its own unique elements.

As an instructor, this “newness” means that an important part of what I need to do for my students is to ground these new concepts clearly for them and lay out good, clear examples of how to use these new tools to solve new problems.

In light of this responsibility of mine as a calculus teacher, I want to take this sort of education to the blog. I’m still working on how exactly I want to present the material, but my goal is to write a series of discussions that enable someone with essentially no background in mathematics to understand what it is that calculus is all about and what we can use it for. For example, I will go through how calculus is used to discussing important concepts in physics like areas, speeds, acceleration, and smooth surfaces.

*The Kalam Cosmological Argument*

I’ve also been thinking a lot about how to begin writing about apologetic arguments, and I decided to soon begin a fairly long reading list that will enable me to discuss in detail what is known as the Kalam cosmological argument for God’s existence.

I’ve discussed this argument briefly elsewhere in the blog, but I think for a few reasons that this is a good place to start. In discussing the Kalam, we get a good example of both a rigorous deductive argument and of an inference to the best explanation. Going through a full discussion of the Kalam is also intellectually quite rich – it involves discussing metaphysics, philosophy of time, philosophy of science, developments in modern physics, and theology. Off the top of my head, no single line of reasoning in natural theology has such a diversity of deep discussions readily accessible on the surface. This is probably why the Kalam is my personal favorite area of study in natural theology.

So, you can expect that, a few months from now, I will be writing up a series making an effort to be as careful as possible providing a thorough discussion of the various ins and outs of the Kalam. It will take me a while to get there, since I want to bone up on my reading on the subject before I write on it, but it should be a very interesting discussion.

*Q&A Series*

This one isn’t really up to me, but I have wanted to start doing Q&A type posts. Of course, I need Q’s to do this, and I do not yet have anyone asking me good questions directly that I can address in Q&A’s. Therefore, I would love for my current readers to direct questions you’d like to see me address to my email address, mathematicalapologist@gmail.com.

*Closing Comments*

I’d love to hear what my readers would like to see discussed, because one of my main objectives in my writing is to help others learn about interesting and important ideas. Feel free to contact me over email any time, I have an email address that I use for the blog and would be happy to have discussions and take suggestions from anyone.

**Link to my Academic Page:**