# IRFAN ALAM

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Click here for Career Update; and a tribute to Peter Loeb (Written during January 11-20, 2024):

(I realize with this update that I am now using my homepage almost like an academic diary. While that idea feels awkward to some parts of me, I think it is an effective method of keeping record. This entry describes my experiences at Joint Mathematics Meetings 2024, and provides some context to my decision to apply to Philosophy PhD programs. For any entity interested in funding my planned thesis, generic versions of my "STATEMENT OF PURPOSE", "PERSONAL HISTORY STATEMENT", and "WRITING SAMPLE" used for these applications can be accessed by clicking on the yellow links. The document for "writing sample" up here may keep changing occasionally until I am done working on it to my heart's content.)

I recently came back from a very eventful Joint Mathematics Meetings (JMM) in San Francisco. It was also a bit emotional because about six months ago I had decided that I want to pursue a PhD in Philosophy once my appointment as Hans Rademacher Instructor at Penn ends later this year, which meant that this would be the last time that I would be attending a big mathematics conference solely in my capacity as a mathematician. I plan to still be involved in mathematics research to some extent once I transition to whichever Philosophy graduate program I get to begin, especially since a job description cannot stop me from still pursuing personally interesting mathematics on my own time, the same way I pursue abstract art or music now.

At JMM, I gave talks in the following sessions:

Special session on Loeb Measures After 50 Years.

Special session on Mathematics and the Arts.

The first of these was emotional for very different reasons. It was in honor of Peter Loeb, who had discovered in 1973 how to effectively use the existing tools of standard measure theory in the then-newly discovered nonstandard mathematical universes. Nonstandard mathematical universes were originally discovered in the works of Skolem in the 1930s, but it was Abraham Robinson who first recognized the wide scope of such universes in the 1960s when he developed the so-called Nonstandard Analysis (NSA). With NSA, Robinson was able to give rigor to Leibniz's original conceptions of Calculus, as infinitesimals were now legitimate objects in the nonstandard interpretation of what it means to be a real number. With Loeb's insight, tools from standard probability theory could, for the first time, be used to obtain mathematical results in these nonstandard universes, which would also lead to new results in the usual standard mathematical universes, as the nonstandard universes logically behave similarly to the standard ones in certain precise model theoretic senses. A mathematician working only in these standard mathematical universes (, which would include almost all contemporary mathematicians, unless you are a logician/model theorist, as our culture has standardized the idea of a "real" number in ways that no longer appeal to the infinitesimals of Leibniz,) may or may not believe in the existence of infinitesimals in any philosophical sense. Yet, they could not refute the myriad of new results about their standard mathematical universes that started snowballing with the use of Loeb measures, including in areas as diverse as probability theory, physics, economics, statistics, combinatorics and number theory. The session celebrating 50 years of Loeb measures started with an hour-long panel where many of these mathematical achievements of the last half-century were commemorated.

This session was personally emotional for me because Loeb measures have empowered me in my mathematical life in a manner that is difficult to express in words. Peter Loeb's 1973 discovery came into my life on September 21, 2015, when my then-future mathematics PhD co-advisor Karl Mahlburg gave a talk in the LSU Probability Seminar titled Loeb Measure and Additive Number Theory. At that time, I was a second year PhD student who had come to the United States to pursue a PhD in the area of probability theory. Inspired by the beauty of infinitesimals that I perceived from his talk, I started meeting Karl regularly as I started learning nonstandard analysis out of interest, with no intention of working with it on my thesis as well. (For any students reading this, I initially used Isaac Goldbring's excellent lecture notes, attached here, which I would recommend to anyone who wants to quickly get a good gist of the subject.) When it was later time to ask my other-future mathematics PhD co-advisor Ambar Sengupta whether he wants to be my advisor, he was the one who suggested that I should have two co-advisors instead, as in his words, I seemed not too focused on any particular area of mathematics, but that I seemed so interested in the intuitions provided by nonstandard analysis that I should keep learning it and finding out new ways to apply it broadly (including in standard probability theory). That was great advice and it led to my works on applications of nonstandard analysis in probability and measure theory in my PhD thesis.

Loeb measures have been essential in almost all of my own mathematical discoveries so far. I feel honored that Professor Loeb not only (remotely) attended my mathematics PhD thesis defense in 2021, but has also given me unquantifiable encouragement and motivation in my career through personal conversations at numerous occasions by now, including most recently at this event celebrating his mathematical legacy. The feeling of someone akin to a mathematical idol being genuinely interested in your work gives a sense of validation that can hardly be expressed in words --- it is a feeling I will never forget. Some of my proudest moments as a mathematician have been in conversations with Professor Loeb. I was very excited to speak at this session in San Francisco, and to meet him for the first time in person.

Sometime during his opening speech, Loeb proclaimed (and I quote the first sentence below from memory, so that part is only roughly his quote, though what he said was very similar, if not the same): "Don't let the skeptics make you think any different. You know what you have!" The skeptics that Loeb is referring to are those who are skeptical of the use of infinitesimals in mathematics. The second sentence is referring to the epistemic value of nonstandard analysis, something Loeb is right in predicting that the audience at this session would definitely know very well. Indeed, in my case, the infinitesimals have empowered me mathematically as well as philosophically. My working paper titled "Ethnomathematical Reflections on Nonstandard Analysis" (click here to access a current draft) explores these themes, which I am also using as my "writing sample" for applications to Philosophy PhD programs. As I transition to philosophy, some of the mathematical directions that I still plan to follow on my own time going forward will be highly motivated by my personal interactions with Professor Loeb, and I am very grateful to him for his encouragement in this endeavor.

I am personally excited about becoming a student again because being expected to learn is the type of expectation I want to have from an academic culture. Too often, the prevalent "publish or perish" culture does not lend itself well to the need of some learners to take their own time while processing the presented material using intuitions that they tend to end up developing because they took the time to struggle with the original presentation of the material long enough. Parts of my philosophical thesis will be about certain forms of "intellectual ableism" and associated epistemic injustices in contemporary mathematical practice at all levels.

My transition to philosophy from academic mathematics is at least partly motivated by certain personal discoveries in "hidden" artistic abilities I found soon after my diagnosis of autism, much of which I have documented elsewhere on this site. These personal discoveries re-shaped my perspectives on infinitesimals in particular, and mathematics as an artistic endeavor in general. My talk in the special session on Mathematics and the Arts was on this theme, as I attempted to explain the "nonstandard" creative processes in my own abstract art (see my created abstract art folder), and I put into context how my proposed philosophical thesis, which grew out of my recognition of epistemic injustices in mathematical practice, is actually related to similar issues stemming largely from socio-cultural norms in education, as well as in society more broadly.

The attached Statement of Purpose contains a tentative abstract for this thesis. Also attached is a Personal History Statement in the same context. I have been using customized versions of these statements for individual applications to graduate schools.

Originally posted April 16, 2023 (updated periodically):

Is mathematics an art, or is it a science? Can it be both? What does it mean to be art? What does it mean to be science? Why does it even matter which way mathematics is classified? Well, it sometimes matters economically as the humanities departments tend to get less funding than STEM departments. Are we (,the mathematicians,) using the higher funding that we get responsibly? Is the mathematics that we create socially informed?

For mathematicians wondering about what I mean by the last question, there is a relatively new field of study in philosophy called "ethics in mathematics." Here is a link to Cambridge University's Ethics in Mathematics Project that may have some useful resources for people who are curious to learn more. If you are really short on time, but would still like to learn more about why there is a need for mathematicians to be ethically informed, I found this short article rather insightful and highly relevant.

Mathematics is pervasive in our society, whether it be at the socio-cultural level, where computer algorithms can help shape trends and more (tangential comment: I highly recommend this video called "The A.I. Dilemma" from the Center for Humane Technology for those who have not seen it); or at the very personal level for a student who might be considered "mathematically weak", something that could adversely impact such a student's life given how much more difficult our current models of education and employment are to such students. But what constitutes a "mathematically weak" student? When can we convincingly say that someone has no or few "mathematical talents"? Are our models of education underinclusive to students who might have mathematical talents that we are too close-minded in our methods to see?

These are the types of questions that can keep me up at night sometimes these days when I am able to work on academic stuff. I am not always able to work on academic stuff, and often some of the reasons can be surprisingly social. This website has threefold goals:

It is a repository of my academic work, most of which is mathematical but a non-trivial amount of which is not. I hope the people only interested in the mathematical aspects of my work do not get discouraged by all the "other stuff" on this page and still look for my mathematical work by clicking on the relevant sections. I also hope that they still see the other stuff at some point because mathematics (as a community) really needs to be more aware of the other stuff in order to be inclusive and ethical. This leads us to the second goal.

It is my small contribution in spreading awareness about some of the lesser known socio-cultural aspects of doing mathematics.

And closely related to the second item above in a way that I shall attempt to explain in some of my upcoming work --- it is also my small contribution in spreading awareness about disability advocacy, a topic that is criminally under-understood by a mathematical community for whom it is actually highly relevant.

(For some more context on why disability advocacy is relevant in mathematics, here is a link to Sines Of Disability, a group of mathematicians who are also helping spread similar messages that I hope to post here from time to time.)

Most of my own understanding of disability is shaped through the lens of my personal experiences as an autistic mathematician with a dissociative identity disorder (DID) . Naturally, a lot of what I hope to write about disability advocacy will also have the dual-goal of de-stigmatizing mental health "disorders" --- something that often goes hand in hand in the social fight against ableism. In particular, the socio-cultural context in which I aim to present my own mathematics here will hopefully be indicative of how neurodivergence might impact how one does mathematics.

These are not small goals, and therefore this website itself is a massive undertaking that I do not hope/expect to build in one day. Most of the other sections on the website are thus still under construction and will be updated from time to time.

About Me

My name is Irfan Alam (click for pronunciation: <Irfan> <Alam>). I am a Hans Rademacher Instructor of Mathematics at the University of Pennsylvania (Penn). Besides mathematics, I am also formally trained in Philosophy, which I pursued for a separate Master's degree while I was enrolled in a mathematics PhD program. I obtained both my PhD in mathematics and MA in Philosophy from Louisiana State University (LSU) in 2021. Besides mathematics and philosophy, I also work on topics related to disability advocacy, especially in the context of autism in particular and neurodivergence in general. As of Fall 2022, I serve on the Research Advisory Board for Autism in Context, a research lab in the Education department of University of Delaware.

I used to like to say that I am a mathematician. But then I discovered I was autistic at the age of 30 (less than a year after I had finished my PhD), and soon after I discovered I had a dissociative identity disorder (DID), both being events that can significantly change one's life and perception of self. (This essay that I recentlly wrote elaborates on this more. I will periodically post more details about Autism, DID, and related topics via similar blog-style essays in the Autism/DID page.) Since then, I have also discovered hidden "abilities" I never knew I had in "more conventional art", including in the types of art that I actually hopelessly struggled with when I was taught them earlier in educational settings, in a way not very dissimilar to how "mathematically weak" students struggle with mathematics in our educational settings.

These days, I would instead fancy identifying as an artist who happened to have found mathematics as their first introduction to art. I have had the good fortune of being impacted by short encounters with several "more conventional" artists in my personal life. If I ever talked about mathematics, most of those artists would repeat the same old expression of how they "suck" at math. However, I have an impression that almost every single one of the conventional artists I have met actually thinks highly mathematically in their artistry, although they might not be aware of it; which is not surprising as a good portion of our society, in general, is not always aware of what mathematical thinking means. When I made my "first abstract art" not too long ago, all I could think of was how similar it was to doing mathematics.

Someone once told me that "good mathematics is about making good definitions", and I am lately realizing that that is what making abstract art is about as well. When a more conventional artist is creating abstract art, they are often identifying new abstract objects (say, with a stroke of a drawing instrument, or with a sequence of notes in a tune) with a general hope to identify patterns among such objects that produce a coherent and interesting structure as the end-product (say, a painting or a song). Replace the word "artist" by "mathematician", the word "art" by "mathematics", and identify the word "end-product" with "theory" or "manuscript" in the case of the mathematician, and the last sentence still makes sense!

One of my current projects that I am very excited about is an upcoming paper that intersects a significant number of topics that I have hand-wavingly introduced so far---disability advocacy, mathematics education, ethics in mathematics, philosophy of mathematics, and the impact of neurodivergence in all of them. It is influenced by my own personal experiences of surviving in the mathematics academia system with several hardships (and several points where I could have easily "failed"), and is hence a very personal project. I would perhaps not have the courage to broach such an interdisciplinary topic (in a way that may be "risky" for my conventional/mainstream mathematics research career) if Robert Ely did not pave the way with his important paper "Nonstandard student conceptions about infinitesimals". The subject of Ely's paper was one of his Calculus students named Sarah, who had invented certain mathematical intuitions about "infinitesimals" that should have empowered her to overcome her mathematical difficulties if our mathematics education system was built in a more intellectually inclusive way --- yet she struggled in mathematics because our system is not built in such an inclusive way. Guided by that case study and further motivated by my own experiences of isolation as a mathematician who primarily thinks nonstandardly in a system not built for such thinking, I aim to shed light on how our typical current models of mathematics education (and research) are ableist against certain forms of intellectual thinking, ironically making mathematics potentially underinclusive to the more artistic and imaginative students.

I was very fortunate to have found Ely's paper in which I felt surprisingly seen in a way that I could not have imagined before. I like to think of this current project of mine as a spiritual successor to Ely's paper, but one in which the student is now an actual mathematician who has different but still similar difficulties in being accepted at an intellectual level. An important theme in this project is to view these difficulties within the context of neurodivergence, and to study what is needed for the socio-ocultural framework of mathematics education and research (and academia in general) to be more inclusive to the intellectual needs of neurodivergent people who might have academic abilities, but not in the conventional sense. This inevitably leads to discussions about philosophy of mathematics---about what "mathematical ability" means, and how it affects society.

(Added June 20, 2023) As a first step toward the massive interdisciplinary project outlined above, I have just finished writing "A philosophically motivated introduction to nonstandard analysis". Instead of trying to publish it on its own, I have decided to add it as an appendix to my existing paper "Generalizing de Finetti--Hewitt--Savage Theorem", as that allows this paper to be essentially prerequisite-free now (only basic measure theoretic probability and basic point-set topology are assumed). This paper was originally completed and posted on the arXiv in 2020 when I was still a PhD student, but it has been updated in important ways this year, and remains my most substantive work so far.

Despite most of this introduction page being about "other stuff", I am still also a research mathematician, and most of the rest of this website (except for the sections <Autism>, <Art>, and, to some extent, <Travel>) is about that aspect of my academic life --- mathematics research and teaching, though most of it is still under construction.

My Mathematical Genealogy

My post-doc mentor at University of Pennsylvania (Penn) is Florian Pop.

Before Penn, I was a graduate student at Louisiana State University (LSU), where I was advised by Karl Mahlburg and Ambar Sengupta for my mathematics PhD, which I obtained in 2021.

Prior to LSU, I finished an M.Math from Indian Statistical Institute, Kolkata and a B.Math (Hons.) from Indian Statistical Institute, Bangalore.

Most of my mathematical research has intersected probability theory or nonstandard analysis (usually both). Nonstandard analysis is a theory based in mathematical logic that gives firm foundations to infinitesimals, among other things. Despite what may appear from the choice of topics I have worked on in my research, I do NOT identify myself as either a probabilist or a logician. I happen to have done tangible work at the interface of those two fields so far, but I also dabble in several other areas (such as combinatorics and functional analysis to name two) from time to time. I would hope to do more work across all of these topics that I am interested in. As such, instead of trying to present myself as an "expert" in a few areas when that is far from the truth, I would prefer to be identified as a life-long student of a variety of areas, including both probability theory and logic.

Please browse to <Mathematics Research> and <Articles> to learn more about my research work, and to <Teaching> where I compile my limited experiences as a mathematics educator.