This year is a milestone for us because we’re going to finally get married. Besides that, we’re also going to move in together and potentially even more cities! This year is full of change, so I think the biggest thing on my mind is to enjoy the time for what it is.
This year is a bit hard to figure out. On one hand, I’m still very much interested in having a career in research and potentially academia. On the other hand, I understand the challenges of finding a geographical location both my partner and I would live at, with sufficient pay with a fulfilling career. With that in mind, I set a few goals assuming my postdoc will come to an end towards the later half of this year.
Wow this year was crazy! I got married to my partner, moved in together, and started a new job at Amazon as a Senior Applied Scientist. I am super grateful for the amount of positive changes that have come my way, and so excited for my life with my partner. I think this year I want to focus on a few things:
This year was a big personal year for me. I got engaged to my best friend. I ran a half marathon. I picked up playing golf (thanks to my fiance and her family). It was also a great year for my professional growth. I got a paper accepted into NeurIPS 2023, the premier machine learning conference venue.
I turned 31 years old, which was my first year into my 30’s. It did not feel that momentous to me, but when I look back, this year was just the start of an amazing new decade. I started playing golf right after new years and found that I really enjoyed it! It’s such an intricate sport that’s challenging no matter how much better you think you’re getting. It is a great way for me to spend time with friends and family in an outdoor setting. Drinking a few beers on the course doesn’t hurt! Besides golf, I started enjoying more time outdoors. I picked up pickleball and running more. I also went on loads of long walks during the spring, summer and fall. These outdoor activities really made me love NYC in these prime weather months.
Besides NYC living, some other notable mentions for this year are travel / big-events.
I started off this year going to Cancun and watched a time-share presentation (lol), and got amazing tacos and mexican street food with my partner. She took her first board exam. Then we went off to Hawaii and visited Oahu for the first time together. We rented a tesla and realized how crappy having an EV rental on an island with one charger is. My partner and I drove around the island and saw how beautiful things were, and also stayed in a very nice 5-star resort. We also woke up extremely early each day and saw the sun rise because we were so jet lagged. It’s interesting how small things like that got me excited this year.
I went to see Dabin in Red Rocks, but unfortunately it was canceled. Still had a blast with friends from college in Denver. Haha can’t forget a wild night at Temple. My partner then finished her second board exam and we went off to Sedona with her family! I was super bad at golf, but really enjoyed playing a course there and also hiking the beautiful red rock mountains there.
I went on to plan an engagement for my partner. It was one hell of a ride to coordinate family, friends, her schedule and also different moving parts like getting helium filled balloons from Party City (lol). Even though it was raining, it was an emotion-packed day and I was extremely grateful for all the friends and family that put up with such a hectic day.
We then went to loads of places: Los Angeles, Baltimore, Los Angeles again, New Orleans for a lit trip, Atlanta and New Orleans again. We ended the year by driving down a 20-hour road trip to Fort Lauderdale, Florida to play golf and spend time with my partner’s family. I don’t know if I could do another road trip like that without more stops. Finally, we ended the year in Los Angeles and Sunny San Diego.
I think this year, I made it a goal to be even more health-conscious. I noticed that I had harder hangovers and less energy for activities I used to be enthusiastic about when I was in my 20s. On the other hand, I’ve really gotten into running and managed to finally run a half-marathon (something that’s been on my goal list for years). Besides getting into golf, I also surprisingly started really enjoying hot-yoga! It’s an efficient excercise that improves my dexterity and mobility, while also being very good for getting a sweat in.
In my professional life, I think this year is going to bring a lot of changes.
I donated again whole blood 5 times this year (one time it was double red), so I managed to meet my goal. I’ve also been trying to re-use and reduce my waste more as before. I’m still looking for ways to improve, so hopefully this year brings about more personal committments. I ended this year by selling my dad’s Apple Watch 3 and my old 2015 MBP! Also brought my ancient 2011 MBP into Apple for them to “recycle”. Hopefully they go to good use.
pywhy-graphs and dodiscover packages.scikit-tree package. In this package, I will guide development of modern scikit-learn compatible tree algorithms. The result of this would be a paper on conditional mutual information estimation using unsupervised trees.This year is a milestone for us because we’re going to finally get married. Besides that, we’re also going to move in together and potentially even more cities! This year is full of change, so I think the biggest thing on my mind is to enjoy the time for what it is.
This year is a bit hard to figure out. On one hand, I’m still very much interested in having a career in research and potentially academia. On the other hand, I understand the challenges of finding a geographical location both my partner and I would live at, with sufficient pay with a fulfilling career. With that in mind, I set a few goals assuming my postdoc will come to an end towards the later half of this year.
Currently, a work-in-progress. If you feel that I’m missing anything, please let me know! I’m hoping this will be a light-weight resource for my friends and family and network.
The overall goal of every single human being (that can afford to do so) should be to drastically reduce their consumption of materials and energy. With that in mind, I won’t post things like “recycling plastic water bottles” because most plastic is unable to be recycled more than a few cycles and the goal there should actually be to “reduce any plastic water bottle consumption”.
I won’t mention commonly recyclable metals, such as soda cans, which typically can still be “recycled for-profit”.
Electronics incur an environmental toll when done at scale. For example, silicon may deplete sand from an ecosystem. The lithium batteries may exact very harsh consequences on the natural surroundings. Overall, electronics like plastic water-bottles should be used as long as possible. insert reference
Another year has gone by and it was definitely a jam-packed year. This year, I’ve decided to begin organizing my yearly summaries into sections for personal, notable events, travel, health, professional-related and societal impact. Societal impact is a new one for me, where I’m trying to focus on making a net positive impact on society with my life. There is an emphasis on environmental awareness and education that I will try to abide by.
I turned 30 years old, which seemed daunting at first, but in my opinion the best years are still ahead. I moved out of Harlem and started living in Midtown, NYC, which was honestly one of the best decisions. The apartment has made me really love living in NYC and enjoy my time here in the city. In the city, I’ve been fortunate to try loads of new restaurants from Hyun (all you can eat Japanese A5 Wagyu), to Kono (a Michelin star worthy Yakitori) and many more. This year, I bought tickets to different comedy clubs and shows. NYC is definitely the best city in America and lives up to its name.
Besides NYC living, some other notable mentions for this year are travel / big-events, professional developments and health.
I traveled a lot this year. I attended a few big friend weddings, such as my friends (BC and AS) to going to The Yacht Week with 30 other friends. I did a lot of traveling with the K family from Atlanta all the way to Costa Rica for the holidays. I also went to LA and San Diego a few times to catch up with old friends and see Gryffin in a super-crowded LA historic park.
I went to Mammoth with my family to ski and see my newphew “try skiing” for the first time. I ended the year by playing golf for the first time with friends and the K family. I definitely sucked, but enjoyed it immensely, which will be a motivation for an updated health goal this year.
I started becoming more active and even more health-conscious. I bought an air purifier, which I always keep on when I’m sleeping now because the air we breathe is something we take for granted. I installed water faucet filter (which has a higher filter efficacy compared to e.g. Brita) because water even in NYC is something we ingest constantly. I refined some of my vegetarian cooking dishes, such as lentils. I began running consistently thanks to my roommate and achieved a VO2 max of more than 47 (as measured by my Apple Watch). According to Apple, I’ve excercised an average of 72 minutes per day, with a daily walking/running distance of 6.2 miles. I also slept an average of 7 hrs and 16 minutes per day in 2022. Disclaimer: although I wear my Apple Watch consistently, I do not always have it on. Moreover, the Apple Watch series 3 and SE versions are not perfect and thus these measurements are all rough estimates.
In my professional life, I feel that this year I have been finding a balance between my personal and professional lives. I started my postdoc at Columbia University studying something completely different from my PhD. I delved more into “theory” and rigorous proofs, compared to analyzing clinical data and working with clinicians on solving applied problems. This has been a great learning experience (many thanks to one of my mentors, Adele Ribeiro) for thinking about notions of causality and how they play a role in data analysis and how they might be applied in the biomedical sciences. I’m slightly disappointed in myself for not being able to take a project to completion this year, but I think that was somewhat expected considering I pivoted my research. I am very glad I learned so much from repeatedly reading and implementing different causal inference papers. I felt that I have a very holistic understanding of causal discovery now because of the time spent rigorously reading the literature. I finished up the year being accepted as a KP Fellows 2022 Program (deferred to 2023) and being a Machine Learning Engineering Intern at Uber. I learned a lot by working on an applied project within Uber that combined causal estimation with combinatorial optimization. It gave me a glimpse of the problems that industry and academia need to tackle and solve in-order for us to have an improved causal model of the world. I also made various commits to the dodiscover and pywhy-graphs packages in py-why organization. Moreover, I’ve made several Cython PRs into scikit-learn. I’m happy I’m still heavily involved in open-source software and plan on continuing to do so in my spare time.
This year, I began donating blood on a consistent basis. I’ve donated whole blood 4 times this year. Besides the benefits of saving a life, getting a regular iron and blood pressure checkup, donating blood consistently has been shown to reduce the levels of microplastic in our blood stream [1]. Moreover, according to the NYBC, every month, we need over 10,000 blood donations. Assuming you donate whole blood, you can only donate ~6 times per year, so this is quite a limiting factor. Finally, donating blood as an Asian increases the likelihood that another Asian who has sickle-cell disease would have a match.
Another theme is that I’ve been mentally tracking and trying to reduce my material consumption, such as reducing clothing purchases, reducing water bottle purchases/consumption, reducing usage of low-efficiency products (e.g. air purifiers without a first-stage washable filter), and attempting to re-use plastic bags, boxes, paper bags, and other materials more before finally throwing them away. In general, it was quite an eye-opener when I first learned a few years ago that recycling is essentially putting a bandaid on a gigantic wound. While it does not hurt, it certainly is not the main effort that people should focus on. Moreover, I recently learned that adding non-recyclable materials, or dirty materials (e.g. with food scraps, or unwashed containers) can actually increase the cost of recycling. So nowadays, I definitely try to be more conscious of what/how I am recylcling and overall reducing my consumption as much as possible.
Overall, I’m excited for this year because I have a positive outlook and exciting plans for the upcoming years. I am eager to execute on the ideas I have for both personal and professional development.
Most of the goals that I set out to accomplish were in the personal development and health category.
This year, Sarah and I decided to set a few goals together that we can work towards both on our own, but also collectively.
pywhy-graphs and dodiscover packages.scikit-tree package. In this package, I will guide development of modern scikit-learn compatible tree algorithms. The result of this would be a paper on conditional mutual information estimation using unsupervised trees.[1] Gasiorowski R, Forbes MK, Silver G, et al. Effect of Plasma and Blood Donations on Levels of Perfluoroalkyl and Polyfluoroalkyl Substances in Firefighters in Australia: A Randomized Clinical Trial. JAMA Netw Open. 2022;5(4):e226257. doi:10.1001/jamanetworkopen.2022.6257. https://jamanetwork.com/journals/jamanetworkopen/fullarticle/2790905.
]]>Maybe it’s because I’m getting older, or maybe it’s because of Covid, but the years have been passing by faster and faster. There’s a lot to be thankful for this year, especially given the amount of different transitions happening.
In terms of trips, I started out with a Vermont skiing trip, which turned into quite a memory when the power went out for approximately 8 hours. We had to burn firewood to keep warm and melt snow to water in order to flush a toilet. Sarah took me on a cabin trip in Severna Park, Maryland, where we stayed at a beautiful AirBnB. During the summer, we also spent a lot of time traveling. First off, we went to Yosemite with Sarah’s family, which was awesome. We hiked upper Yosemite Falls, a grueling 3000 feet elevation over 7 miles. We then went to Portland for Sarah’s birthday. I went to Miami to celebrate my buddy’s Bachelor party and then went back to Baltimore. Afterwards, Sarah and I went back to California, going up the 101 from Thousand Oaks, Santa Barbara, Morro Bay, Monterey all the way to the Bay Area for my friend’s wedding. Going back to the east coast, I made the final move into NYC. Throughout the rest of the year, Sarah and I went on a few more trips: Chicago to see Porter Robinson, and then Portugal, going through Lisbon and Porto. To top off the new years, we celebrated in New Orleans.
Professionally, I had a lot to celebrate this year. My main PhD study was officially published in Nature Neuroscience, and moreover it was featured as the cover of the October Issue. I took a final course in constrained optimization and received my Master’s degree in Applied Mathematics & Statistics. I applied for and got the NSF Computing Innovation Fellowship, which secured my postdoctoral research funding for two years. I also made the move to Columbia Unviersity to begin working with Dr. Elias Bareinboim on causal inference.
I’ve also been an active contributor in open-source, specifically MNE-BIDS, MNE-Python and MNE-Connectivity. I was accepted into the Google Summer of Code 2021 to work on MNE-Connectivity, which ported over the connectivity analysis functions from MNE-Python. As a result of my contributions, the core developers invited me to join the core dev team and the steering council. I also initiated conversations with scikit-learn core developer team to try to bring oblique trees into the scikit-learn codebase, which will probably lead to a multi-year effort due to the complexity and size of the scikit-learn codebase.
I’m excited for this year because I’ll be turning 30 and starting a new chapter in NYC and Columbia University. I’ll be attempting to obtain a tenure-track faculty position, while studying causal inference as a postdoc.
This year, Sarah and I decided to set a few goals together that we can work towards both on our own, but also collectively.
Academic journal papers (preprint and peer-reviewed published) are the lifeblood of scientific researchers. These samples of writing describe pieces of scientific work in a certain sub-field. For example, it might describe experiments demonstrating the mechanisms of action to generate an action potential, or a cohort study of epilepsy patients that underwent surgery or drug treatment. These are generally characterized by hypotheses, prior art motivation and a rigorous methodology that leads to falsifiable results.
Here, I assume that you have generated all your results already! I mainly discuss optimal workflows for writing papers and dealing with revisions.
These are recommended tools for writing a paper.
Really, overleaf has an awesome interface that lets you look at latex and PDF and table of contents simultaneously.

The tools are recommended for the following reasons:
Note that many of these tools require some bit of a learning curve. Notably, overleaf for latex, and Mendeley/Zotero for references. If you have collaborators that simply cannot use these tools, then that’s an unfortunate circumstance… I have yet to discover an optimal workflow besides: render pdf / convert to word using Adobe, and then incorporate changes/notes afterwards.
Here’s a good comparison on word vs latex.
If you intend on using these tools, possibly a similar workflow applies, but these are not directly discussed.
I recommend using Latex to write papers because they are modular and support version-control, whereas Word documents inherently lack these features. In order to use latex, one can setup a local environment, but this becomes difficult to share. Therefore, we highly recommend Overleaf, which is an online latex editor that allows for tracking changes and simultaneous editing across many people. In addition, if you pay for the subscription, it allows integration with Github/Dropbox/Mendeley/Zotero.
Many journal papers are moving towards reproducability, since this has become a major
issue in recent years. If you setup your code to do so, it helps facilitate this process,
but also has the added benefit that people will more likely cite your study, since they
can confirm results for themselves! In order to facilitate trivial reproduction of your
main figures in the paper, I suggest using a github repository with the following:
README file describing installation process, main instructions to generate figures, instructions for downloading
additional data and any misc. notes that users need to be aware of.data directory containing all the source data needed to reproduce figures and or run the experiment. These should
be relatively small files, since they are housed in a Github repository.*.ipynb Jupyter notebooks that contain self-contained code to generate all main
figures and/or experiments described in the paper.One should also include scripts (e.g. .py files) needed to support the generation
of figures and/or experiments as well.
Many journal papers before publishing your manuscript will want the raw high-resolution figures.
This generally means using .pdf, or .svg format, so that they do not lose resolution during
zooming in/out operations. In order to prepare for these later on, it is helpful to have each
figure/subfigure in a file already! Inkscape (and/or Photoshop) can load SVG/PDF files and
modify them. If you edit figures in these programs, then if you need to perform downstream
edits (due to revisions), it becomes easy because you just pull up the file and perform edits.
Due to latex modularizing figures, you simply have to re-save the figures with the same filepath as where your latex file is pulling from. An example with Overleaf and Dropbox integration would be:
figures/figure-1-blah.svg.figures/figure-1-blah.svg.When writing a manuscript, you’ll need to keep track of all your references! Managing this manually is unwieldy and extremely prone to error. Use Mendeley (and/or Zotero)! These tools all have Overleaf integration, Desktop application and a Chrome browser extension. Thus one’s workflow looks like:
.bib) file to directory for your Latex file to useThis has the advantage that one can easily change the citation formats, edit metadata and easily port the bibliography file to other manuscripts.
When dealing with revisions, one generally will need to generate a tracked-diff between the submission file and the revision file. This helps facilitate editor/reviewer feedback and if you do not do this, you will not be liked! This is obviously trivial with Microsoft Word, but it is an incredible headache to convert Latex/PDF files to Word.
To handle a tracked-diff, save a copy of the Latex version upon initial manuscript submission.
Then, when you have your final revised manuscript after revision, use latexdiff to generate
a new Latex file, which will now render a nice PDF with tracked changes inside the PDF file itself!
latexdiff-so old_version.tex new_version.tex > differences.tex
Here’s a great reference: http://www.peteryu.ca/tutorials/publishing/latexdiff
The only tool I know of currently that even relatively works is Adobe’s export PDF to word
functionality. Even there, it does not work great when you have mathematical equations,
complex figures, and custom formatting! However, some journals for some reason still require
a word document for initial submission, so there you have it.
Great! Now you have your manuscript version 1 ready to go. Well, unfortunately the submission process for many academic journals is more involved. Fortunately, you can still make your work publicly accessible via preprint servers!
The preprint servers (bioarxiv and arxiv) are extremely easy to release a preprint. Bioarxiv is for generally more biological related sciences and arxiv is for the physical and mathematical sciences. There are also other preprint servers, like Medarxiv.
One must have:
and you are ready to go! Just make an account with the preprint server you are using and go through their submission process!
Nowadays, there is a deluge of preprints. To “advertise” your study, it is recommended to send this out to 1) collaborators and 2) post on Twitter!
Last but not least, remember that preprints are version controlled! You can always submit a new version when work has been improved and updated.
]]>This year has been pretty blessed considering what’s happening around the world with Covid-19. I did a cabin trip with Sarah and friends early in the year and did New Years Eve at a club seeing Kaskade(?). Definitely safe to say, I can’t do “long” nights out anymore. I went to Cancun with my family and Sarah at the beginning of the year before Covid hit and then spent 3-4 months at the Kim family house, which was definitely… longer then expected. Throughout quarantine, really got to know myself better. Found an interest in gardening and cooking and really had to explicitly focus on mental health. In the summer, got to make a trip home and even go down to San Diego to see some old friends from the BodyRock/SD crew. Was very memorable even with Covid restrictions, and happy I went. In the midst of Covid, got to experience July 4th still with some NYC friends up near Westpoint. Closer to Halloween, saw K-Town in Manhattan with all its outdoor seating! It was actually quite awesome, and something I hope they keep during all the non-winter months in the future.
AAMPLIFY managed to get through the pandemic by shifting its gears to a virtual summer program. It actually was quite successful and I was glad to introduce a statistical and data-driven thinking portion to the leadership curriculum. EverydayBME managed to shift gears in the face of the pandemic as well. We partnered with BMESDiversity and offered Patagonia 1/4 zips to entice a social media display of the diversity in BME science and research. It was actually quite awesome to see how many minority groups participated because it speaks to some of the successes at least we’ve had in the BME community in terms of diversity.
With the start of the Fall semester, my paper was accepted for review at a high-profile journal, which was a really nice touch (we’ll see if it gets accepted though heh). In addition, I managed to work with a small team (many talented individuals) to get an EEG analytics software in its MVP version approved by the FDA via a 510k approval process. Learned a lot about regulatory and how to work with the FDA and consultants to get a software approved. Finally, I took a very interesting course, called Real Variables with Professor Sogge at JHU. Learned a lot about Lebesgue measure, integration and differentiation theory. In addition, got a very nice introduction to Hilbert spaces and Fourier series, integrals, transforms. Wish I took the course 3 years earlier. I’ve also been an active contributor in open-source and was later added to the BIDS-standard team by Stefan. I’ve really become amazed by the open-source community. I’ve had lengthy Github conversations with many talented individuals that I’ve never even met, let alone know their face, or voice.
I think it’s important to reflect on how “blessed” this year was for myself, but also reflect on some of the bad that came with it. This year marked the onset of the Covid-19 pandemic, and with it, many many people were forced to stay home. An aggregation of poor response, and misinformation led to the pandemic bringing the USA to its knees. When we look at countries like South Korea, Taiwan, New Zealand, Vietnam and more, it’s quite apparent what differences there were to anyone that is scientifically trained. In addition, at the same time, we saw the very public unnecessary deaths of many Black Americans (Ahmaud Arbery, Breonna Taylor and George Floyd to name a few). For people like me that live in a world of first-world privilege, it probably “shocked” us. I know for me at least, it really started making me question the society we live in and what sort of biases I’ve been engraved with since childhood. It also was hard to come to terms with the fact that this kind of thing has been honestly happening far too frequently since before I was even born. It is a flaw in how we are taught history, and a flaw in our society. With the poor pandemic response, and racial problems of the USA, it is apparent that voting strong local public officials that are knowledgeable, scientifically driven and socially aware is more important then we realize. Going to be on the lookout for more ways I can actively, or passively help as time progresses. I’m glad that I was also able to at least do a bit more research in my local, state and federal elections then I normally would in the hope that I become a responsible and socially aware voter beyond just this year. A side effect of the pandemic actually made me do more reading and research into climate change and being a “green citizen”. It took me by surprise that recycling is more nuanced then just use, recycle, reuse. Projecting forward decades, it was became more apparent to me that we can’t keep living the way we do - taking from the earth without thought. It actually will lead to more racial inequality and potentially cause more social issues that we can’t even imagine. Part of my personal goals for this new year is to become a more “green citizen”, as well as being a “responsible socially aware citizen”.
13.1 miles. Longer then I've ever ran in my life, but it's going to be the goal of this year, and the entrance into adulthood. In order to get to this point, say for the Boston half-marathon on October 13, 2020, then there will be a decent amount of training involved. I should set a goal to:
- [X] run 1-2 times per week with 4+ miles by March.
- [ ] run 1-2 times per week with 6+ miles by June. Run the Boston 10k in June.
- [ ] and then start training in July for the half-marathon.
Learn another few signature dishes to maintain cooking health :).
- [X] pancetta wrapped stuffed chicken w/ fetta cheese, spinach, mushrooms and garlic
- [X] a red wine dish
- [X] a fresh pasta dish
I definitely could have ran more this year, but because of Covid, I’ve “exercised” less, but have been running significantly more. I’ve gone up to ~4 mile runs every now and then, but am nowhere close to a half-marathon level. Cooking-wise, Sarah and I have explored many many dishes, such as: eggplant lasagna, medium-rare rib-eye steak, indian marinated chicken, thigh curry chicken, fresh-made linguini with seafood and much more.
Internship, or open-source development with at least 10 closed Issues and Pull Requests.
Although I haven’t done an internship as I was hoping, I have been significantly more active in open-source. I’ve contributed to mne-bids, mne-python, the BIDS specification, bids-validator, and many more. Open source science rocks! I believe I’ve definitely gone beyond 10 closed Issues/PRs.
Think before speaking more, patience and focus on my friends, family and significant other more then myself. I've focused on my self-growth over the last 10 years, and it's time to focus on my external-growth over these next 10 years.
I should check up on my family more often via WeChat, and give better presents for the birthday and holidays.
I’ve done this relatively well, considering we grew up not “giving presents” normally. Since I’m very “calendar-oriented”, it helps that I have all my birthdays and important events on my google calendar.
I should check up on my friends that don't live in Baltimore every few weeks or so. Put things on the schedule for hanging out. I should invite a few friends to do Montreal with me in the summer.
I haven’t really had an opportunity to do this as much as I would’ve liked. However, due to Covid, I’ve definitely had a few Zoom get-togethers ranging from catchups, to board games to playing Among Us.
I should focus on listening to Sarah more this year. She's more right then I am in more instances then I would prefer to admit. The fallacy of trying to believe in yourself is that you convince yourself out of hubris that you are right even when you are wrong.
I could probably do better here still. I think I’ve been marginally improving, but there can be more instances where I reflect more on my thought patterns and actions.
This year, I’ve found that mental and physical health are intricately linked. In addition, they have consequences on professional productivity. Sarah introduced me to yoga this year, and I actually quite enjoy it since it involves a lot of stretching, deep breaths and movements that I otherwise wouldn’t do. It’s a great way to be active in the morning and after runs.
Other activities I’ve found helpful are running and cooking healthy. Cooking isn’t just healthy, but it is a relaxing activity. In addition, now that I’m more comfortable cooking, I want to explore more by cooking healthy. Cooking healthy means cooking with just vegetables and ingredients that are not currently being stressed by human activity (e.g. red meat, fish, fat).
This upcoming year, I’ll be graduating with my PhD and MS from Johns Hopkins. I’m looking forward to taking courses in constrained optimization, stochastic optimization and graph theory. I’ll also probably be starting a transition to a post doc, or industry position. In order to feel like my PhD is fulfilled, I would like to publish a few more projects:
This year I would also like to dedicate more time to outreach. I think there’s too often change that could be done at the local volunteer level that isn’t because of passiveness.
Real analysis is the study of real numbers in $\mathbb{R}^d$, with
d being the dimensionality. Although it is a pretty abstract topic
that does not seem to have applications, it is a very fundamental
in understanding other branches of applied mathematics. For example, in order
to define optimization algorithms, one is very interested in the ideas of convergence.
One then often uses theory of continuity. In computational neuroscience and
control theory, one often deals with integrals when defining theory. In
statistics, it is absolutely necessary to understand integrals (expectation
is defined with an integral).
In this blog post, I’ll review some of the basic concepts that one needs to know in general. A great course sequence I took was both at UCSD with Professor Dragos Oprea (Math 140AB) and with Professor Sogge (Math 605) at Johns Hopkins.
Excellent references are:
Measure is an abstract notion of “volume” of a set. In all “regular”
examples, one can assume the sets in consideration are measurable. However,
using the Axiom of Choice, one
can construct “unmeasurable” sets. This motivates the need to define Borel sets (sigma-algebras)
in probability theory, which denote measurable family of subsets of a set. This
is beyond the scope of this post though. It suffices to know that unmeasurable
sets do exist.
Measure from first principles is defined in Stein starting from outer-measure. Outer-measure is simply the measure obtained by covering a set with a countable almost-disjoint union of rectangles (circles, squares also suffice). Outer-measure is well-defined for any possible set. Lebesgue measure then is defined by the infimum over the possible outer-measures of a set.
Some important properties of measure of a set is non-negativity and countable sub-additivity. Countable additivity is obtained when they are pairwise disjoint sets in consideration.
For many interesting functions, and numbers, we are interested in
the notion of convergence. That is, if we define a sequence enumerated
by n of numbers, or functions using some rule that depends on n,
then what are conditions that these sequences converge? Generally,
if a sequence does not converge, then it goes to $\infty$.
In real analysis, we generally consider the Euclidean metric space. In fact, these metric spaces are complete (i.e. all Cauchy sequences converge inside the metric space).
When defining convergence for integrals, there are three fundamental results that are useful:
I won’t talk about them here because the above links actually provide a very nice reference (that is also accessible to beginners).
The main way of proving convergence relies on showing that a sequence is Cauchy. That is for every $\epsilon > 0$, we have:
\[|x_m - x_n| < \epsilon\]for $m, n \ge N$, for some large number $N_\epsilon$. The “|…|” sign
denotes the distance metric defined. This implies that even if
the sequence ${x_k}$ potentially oscillates, the range of its oscillation
is decreasing as you take values further and further down the sequence.
Note that by construction any sequence that is Cauchy in a metric space
is in fact bounded.
The Bolzano-Weirstrass theorem, or sequential compactness theorem states that any bounded sequence in finite-dimensional Euclidean space has a convergent subsequence. Then a sequence is convergent if and only if it has a convergent subsequence, and so every Cauchy sequence is convergent.
Note that convergence of a sequence of real numbers requires a specific
component to emerge. In the space of integrable functions, $L^1(\mathbb{R}^d)$,
one can define convergence in norm. That is the sequence defined might not
converge exactly element by element to a fixed component, but their difference in norm
converges. That is for every $\epsilon > 0$, and ${f_k} \in L^1$, we have:
where the distance metric is now defined as $||.||{L^1} = \int{\mathbb{R}^d} |f_m(x) - f_n(x)| dx$. Convergence in norm can then be extended in general to $L^p$ spaces, since $L^1 \supset L^2 \supset … \supset L^\infty$, which follows from Holder’s inequality.
Note that there are other notions of convergences as well! Weak convergence, strong convergence, etc.
Integration with respect to Lebesgue measure is defined by considering the convergence of the integral of the absolute value of the function:
\[\int |f(x)| dx < \infty\]else, the function f is said to be not integrable. The space of Lebesgue integrable functions if formally known as the “L1” space. Note that by the Risz-Fisher theorem, we know that $L^p$ spaces for $1 \le p \le \infty$ is complete.
When considering differentiation in the context of integration, we are primarily interested in when the Fundamental Theorems of Calculus hold in the Lebesgue measure theory. We learned in high school calculus the following:
If $F(x) = \int_a^x f(t) dt$ for $x \in [a, b]$ interval, then
\[F'(x) = f(x)\]that is, we can reverse integration on f by taking the derivative of F.
This theorem holds in the space of Lebesgue integrable functions, when the
function is locally integrable (i.e. Lebesgue’s differentiation theorem).
The second part states that if we take the endpoints of the antiderivative, then we can obtain the integral of the function over those endpoints:
\[F(b) - F(a) = \int_a^b f(t) dt\]The result holds when F is absolutely continuous, which is a stronger
notion then continuity. It essentially bounds the degree of variation and is
actually introduced through the concept of Bounded Variation.
This blog post is meant to be brief, but in the future, I’ll probably try to cover Hilbert spaces (linear operators, orthogonality, compact operators, the Spectral Theorem, and Kernel integral operators), Fourier series and the Fourier transform, which are also broadly applicable across applied mathematics and engineering.
]]>This year was a great way to end what I like to call a decade of growth. These past 10 years, I went from high school to college. I found out the things I love, the things I hate and the things that I could care less about. Love came and went, along with different friends. Above all else, I am almost certain that I’ve found what I love to do. Science, and the ability to “make discoveries” that can help people are what I hope to do for the rest of my life. I came to graduate school without understanding why I wanted to be there, and where I wanted to go, but now I’m getting ready to leave… at least understanding why I came.
This year I am glad I found a partner that has been supportive of what I do, and patient enough to listen to my weird and obnoxious thoughts on things ranging from sleeping habits, to science. It has and should continue to be a thrilling experience to make mistakes, reflect and adapt while I face the next challenge of learning how to grow two minds at once.
I’m close to graduating from school where I will never have to “take courses” again, but rather then rejoicing that I never have to set foot in a classroom again, I’m realizing I’ll miss the lessons in logic you learn from challenging yourself through homework and exams. I took up the Masters in AMS at JHU this year, and I’m glad I went through with it. I am close to publishing most of my thesis work this year and then going through the final defense of the PhD. It didn’t seem so far away that I was struggling to understand the basic methodological papers my undergrad PI sent me, but now I’m confident to say that I at least understand 50% of things with a one time read. Time to grow and reflect on this year.
“Traveling to a Mandarin speaking country, Taiwan with my friends made me realize my Chinese is horrendous. I always went to China with my family, and I never had to get around by myself. This was the first time I wanted to rely on my Chinese to communicate with strangers, and get around the country, but I realized my speaking and reading ability is just not up to par. This year, I’m committed to speaking to my parents in as much Chinese as possible and I’ve even started a Wechat/Messenger/Line group chat with my Chinese friends to practice Chinese.”
Yeah this stopped after April when my friends Anthony and Justin stopped using Line chat lol.
I bought a bartending kit on Amazon and am aiming to learn how to make some basic drinks like an old fashioned, mojito and other drinks I can find online. Aiming to learn how to make 6 good drinks by the years end!
Definitely instead of doing this, I chose to not drink as much, … so I guess that’s an accomplishment.
“Cutting my own hair, benching 2 plates, reaching deadlift of 3 plates and running at least 5 miles per week. Been proud of myself for learning how to cut my own hair (thus saving time and money each month), and hitting a bench press of 2 plates, while still weighing 160 lbs. This year I hope to continue my fitness goals and potentially shave off 10-15 lbs to a flat 145-150 lb. This will be achieved with the help of my nifty Apple Watch Series 3.”
Still been cutting hair, but went all in on the high-rep, 1 plate for bench, deadlift and squat. I’ve been running a lot more recently and I think that will lead me to the goal-setting of 2020.
Managed to get a lot of things done this year :).
“This year, I’m going to try to post new photos to a new instagram blog page with more travel related photos. This is to sort of separate my personal instagram from my photography hobby. I have a lot more to learn about optimal lighting and photography of different scenes (e.g. portraits, landscapes, nightshots). I am aiming to post an additional 12 photos that I am proud of onto this IG page by the end of the year.
See link to keep up to dates with my adventure! https://www.instagram.com/ajltravels/ “
Didn’t happen as much as I wanted, but instead I worked with a few colleagues to get @EverydayBME off the ground, which is a super cool Instagram page off the ground.
13.1 miles. Longer then I’ve ever ran in my life, but it’s going to be the goal of this year, and the entrance into adulthood. In order to get to this point, say for the Boston half-marathon on October 13, 2020, then there will be a decent amount of training involved. I should set a goal to:
How can I measure side-effect success? Heart rate should ideally go to resting at 50 consistently. I should be stretching consistently and doing foam rolling.
Learn another few signature dishes to maintain cooking health :).
Internship, or open-source development with at least 10 closed Issues and Pull Requests.
Think before speaking more, patience and focus on my friends, family and significant other more then myself. I’ve focused on my self-growth over the last 10 years, and it’s time to focus on my external-growth over these next 10 years.
I should check up on my family more often via WeChat, and give better presents for the birthday and holidays.
I should check up on my friends that don’t live in Baltimore every few weeks or so. Put things on the schedule for hanging out. I should invite a few friends to do Montreal with me in the summer.
I should focus on listening to Sarah more this year. She’s more right then I am in more instances then I would prefer to admit. The fallacy of trying to believe in yourself is that you convince yourself out of hubris that you are right even when you are wrong.
]]>This is the main international airport of Iceland! It is about 40 minutes away from Reykjavik and is about 20 minutes away from Blue Lagoon! It’s quite a nice airport and has a great Joe & and the Juice.
-> You should just instantly go to Blue Lagoon if you’re thinking about it. It’s a nice place to relax after a long plane ride.
Pretty waterfalls southeast of Reykjavik about 2 hours. Skogafoss is this big waterfall that you can get very close to and then also hike up to the top of. Here, you can actually rent a campsite, or even a hotel room. After hiking up to the top, there is actually also a long hike of apparently 4 days. If it is in the springtime, feel free to walk along for 30-45 minutes. You’ll get a quiet area with some wild sheep and more mini waterfalls along the way.
Seljalandsfoss is on the way back to Reykjavik and is another waterfall that you can walk behind! It is a nice one.
Beware, if you get too close to either, you can get severely wet just from the mist.
Golden circle is this amazing ring of things to see and do. In Thingvellir national park, there are a bunch of hiking trails and overlooks. In addition, I would highly recommend booking a snorkeling tour at Silfra. You wear a dry suit and swim through a freshwater stream (glacier water), which you can drink from while swimming. It is very cold, so you’ll need to bring a pair of thermal underpants and wool socks. Totally worth it to swim in between two tectonic plates.
Geysir is this cool park that has a bunch of 80-100 degree C water geysers. One even shoots up every 5-10 minutes. Cool hike up.
Gulfoss is the granddaddy waterfall of them all! Very amazing sight.
Crater Kerid is a cool looking crater.
In Snaesfellsness, there is the Snaesfellsnessjokull National Park (a cool glacier with a lot of cliff views).
Kikjufell is a cool arrow-head looking mountain, and Kikjufellfoss is a nearby waterfall. Both very scenic and cool looking.
If you’re thinking about eating whale, or puffin, please don’t. These are both endangered, or in danger of being endangered animals because of how many people hunt for them. They are hunted for because the tourism industry’s craving to “try” these dishes cause massive demand.
Upscale tapas that is pretty nice!
Classic hotdog with 5 different sauces, raw onions and fried onions. Need I say more?
Deep fried hotdog with fries and cheese and paprika. Need I say more?
Great seafood restaurant near Kikjufell!
Nature.
You should probably rent a car. I rented one from Ace Rentals, which doesn’t have an office at the airport, so you will need to coordinate a pick up from them at the arrivals hall.
Pros:
Cons:
Tourbus: TBD
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