Just out of curiosity

It was 2004 when I started mathematics at University.  In 2004 studies were not divided between Bachelor and Master. It was just one degree and it lasted 5 years. It is my understanding that that was pretty much the case in every European country, and things changed with the "Bologna plan". So, back in the day, in Spain, to become a mathematician, one had to study for 5 years, from which 3 years and a half were filled with compulsory subjects. Now, of course, things are very different in the study plans.  But some things do not change: the transition from high school to university was tough. Very tough. And we had to pass all the subjects of the first year (a total of 8) to be allowed to go into the second year (and you had to pay for every course that you registered to, and you could not de-register - you were given a fail -, and every time you had to repeat that subject, it became more expensive). Exams were written, at the end of each semester, and they had a fixed date. S...

I am enjoying doing the little exercises for my exercise class. It is very nice to have concrete, well-defined, definite problems. It is also nice when there is more than one way of proving the same thing (though it is always tougher when it comes to correcting the exercises).  Honestly, maths may be tough, but they are so nice. A bit like what happens with sports. I can get so absorbed trying to solve a problem, or trying to find a shorter, clearer proof. I can spend hours. I used to spend hours as a student and then hours as a PhD student on just one question. It is hard to describe, I get so immersed in it, that I forget about myself. At points, I get frustrated if I feel I am taking "too much time" to solve it, but yeah, I like the process. In maths the answer is not just the answer: it is the answer and how you get to it.

The impostor syndrome is widespread. I know it is especially the case in academia. Many people suffer from it. They learn to live with it. I think it is not so easy to fight, it is a bit like that feeling of not "being good enough": it is a mental pattern that is very hard to get rid of. Actually, resisting it makes it stronger. In any case, I found myself doing more and more the things that I like, the way I like them, and this includes things that I am not asked to do (like writing a blog, or doing coaching sessions for the first year students, or trying to improve how we deal with diversity at the Faculty).  Yes, I am not "supposed" to be doing all these things. Nobody has asked for them. But I want to do them, I care for them, so I have the drive. This means that I am being myself: doing what I want, the way I want it. I do not feel like an impostor. I only feel like an impostor when I am trying to force myself to fit some pre-established mold or fixed idea. I ...

Let's apply what we learned about logic to some particular scenarios: Definition: we say that a sequence \((x_n)_{n\in \mathbb{N}}\), \(x_n\in \mathbb{R}\) converges to \(x\in \mathbb{R}\) when for all \(\varepsilon>0\) there exists a \(N\in \mathbb{N}\) such that for all \(n\geq N\) it holds that \(|x_n-x|<\varepsilon\) Definition: we say that a function \(f: \mathbb{R}\to \mathbb{R}\) is continuous at \(x\in \mathbb{R}\) when for all \(\varepsilon>0\) there exists a \(\delta>0\) such that if \(0<|y-x|<\delta \), then \(|f(y)-f(x)|<\varepsilon\).  Quiz: Write the negation of these definitions, i.e., what does it mean that a sequence \((x_n)_{n\in \mathbb{N}}\) DOES NOT converge to \(x\in \mathbb{R}\), and what does it mean that a function \(f: \mathbb{R}\to \mathbb{R}\) is NOT continuous at \(x\in \mathbb{R}\).  Cognitive bias We have many cognitive biases, which include biases on how we think about things. For example, there is a well-kno...

I will try to make this short. Here are some common mistakes made during presentations (here I am assuming a slide presentation). The title page says nothing: the title is full of jargon that is not explained. The second slide is a table of content: before putting a table of content one should motivate the talk, i.e., why should we listen to the talk. The table of content does not provide new information (either it has a standard structure or again it is full of jargon). The truth is that a table of content is not needed for short talks. Skip it if possible. The slides contain too much information or too much text, and you do not give time to read the slides. Not defining well the mathematical notation or the concepts. Not giving motivation for the problem or topic that you are presenting. Not explaining why you are explaining what you are explaining (i.e., where are taking us?).  Not drawing conclusions or giving a clear message at the end (what is the take-home message?). Wanting...

If you want to learn maths, you need to learn how to write maths and how to explain maths. The reason is simple: the process of communicating is a process of assimilation. If you cannot explain something clearly, then something is not clear in your thinking, so practice explaining things to other fellow students and let others explain things to you (and reflect on how they communicate). So far, so good. But there is an aspect that is often ignored: writing maths. And this is crucial. I did not learn to write maths properly until very late (and, actually, to write them really properly, until my post-doc). The reason? Two-fold: I did not give it so much importance (I was thinking that the important thing is to have the right answer), and nobody taught me (nor I knew how to go around learning it). Writing and thinking are inseparable. Fix that in your mind. Writing is the most powerful tool that you have to empower your thinking. Mathematical thinking is an iterative process of refining i...

After today's coaching session, I decided that I will do various blog entries on these topics: - It is normal that it is tough to study mathematics (and especially struggling to do proofs). What is different now from mathematics in high-school. - Writing mathematics, doing proofs, the way of communicating maths, and how important it is to write properly to refine your thinking. The entry is now here . - Make sure you understand the exercises (point 1 of Polya's problem-solving). Does it sound like ridiculous advice? Well, it tends to be a common mistake. - How was it for me when I started mathematics at University (and the main reason why I am doing this coaching). There is quite a lot to say about all these points.

It was just two Sundays ago that I wrote a blog  entry about me working on a Sunday because I had a lot to do. And yes, I like my work, so I am happy working beyond normal hours. However, I always try to keep the weekends work-free because I need to recharge my batteries. It is important for everything to function properly. Guess what? Today, Sunday, I worked a full day again.  I know many researchers that work basically all the time. Others do not do it. But I always tell people that being a researcher is like running your own company (at times): if you want things to roll, you cannot just say I will work 8 to 5, Monday to Friday. There are times when one needs to walk that extra mile.  I know, it is not very family-friendly. But let's be honest, society is not very family-friendly right now. It is hard for women to keep up with their careers in general. Women going to work has created a vacuum in childcare that kindergarten and schools do not fill. In my case, it also ...

It was only when I was a teenager that people started to have cell phones, and when I say cell phones, I do not mean "smart" phones. These were big heavy phones that could only be used for calling and texting. It was only when I was going to university that we got internet at home. There was no point in having it before. And it was also at university when I got my first email account, a Gmail account (and I was so eager at the time to get an email..., that time is long gone). Also, until my teenage years, things were very expensive. Any possession was extremely valued, even a watch (I mean, a normal one, not a "smart" one, those did not exist), because everything was so expensive. Traveling was expensive too. Having music was expensive. Normally I would listen to the radio, and I was lucky one to have a radio set that could record what was being broadcasted on a cassette tape. Then I could listen to that song that I liked again. Sometimes I would record just in case...

This has been a tough week. Actually, it has been the third tough week in a row, and I have held up pretty well, just until now... That's it! Negativity is winning the battle, and I know it.  Prescription: keep distance from people (to avoid chewing someone's head off), take some rest, and sleep a lot. My source of stress is typically that I want to do too many things, too many projects. But that is also what gives me the drive, it is just that I can lose the balance.  Anyway, I am just saying this because when negativity is winning the battle I know that I need to retreat. Keep fighting just pulls me deeper into the mud. Doing something from a state of negativity or anger just makes things worse. I know that I will even probably do things that I regret or that will give me a headache later. It is like going for a run in the middle of a terrible flu: you are just digging your grave. So, grudgingly, grumpily, I retire and give myself time to restore, and stay away from my beaut...

when you are stressed, obsessed, everything seems insurmountable, and you have tried everything (meditating, running, journaling), and it is not enough... ... there is just one thing left to do: dance! (and if you can, also, sing, and if you can, do it with others, the more you are, the more intense the effect) You don't believe me? Just follow the guy: Ah! The power of music!

The renovation at home has thrown away most of my habits, which makes everyday life a bit more difficult. But one of the losses I have suffered the most is journaling. In the morning, one of the first things I would do is sit down and write. Typically I would write about the day before, going through it. This process helps me to assimilate emotions that had gone under the rack or how decisions were taken or if priorities were met or, even, if there are some patterns in my actions or hidden intentions in my choices. This process helps me to understand myself better and also to improve in all aspects of life. It is a way of keeping the mind and the emotions clean. It is also a way of learning from the experiences rather than just being swept by them. I did do journaling during the last 2 weeks and, combined with the fact that teaching has started, my level of stress has increased. Funny thing, when things get a bit overwhelming, I stress, and then, I lose clarity of mind. Clarity of mind...

I finally decided to read the short story that I started during my trip back from Konstanz.  It sucks. I was surprised. Not that I expected a masterpiece, but definitely something much better. However, I heard that professional writers with long careers always feel that what they wrote is not good. So it seems to be a widespread phenomenon, independently of the quality of the story. On the other hand, I also heard of this 1/3 "depression" when writing a novel, i.e., after having written around 1/3 of a novel, authors think that the novel sucks and want to throw it away. The only advice to overcome this seems to be: just accept it, and power through it. I wonder if this is what is happening here: I want to throw away the story, but I am going to try instead to finish it by putting it into shape (I will try to rewrite as little as possible). But it is hard to put myself to work on it because it feels like it is going to be a lot of work and I may not end up satisfied. But I gue...

These reflections came from a discussion about how long should be the blocks of time dedicated to solving problems. First, I said: - Allocate time to activities rather than to specific goals . For example, decide "today from 5-7 p.m. I will work on the problem set" rather than "today from 5-7 p.m I will solve 3 problems". The first goal is under your control, the second is not. If you do not fulfill a goal, you will feel bad about yourself. But then again, how long is a reasonable block of time?  That is a question without a right answer, in my opinion. But this question came from a student that is working and studying at the same time, so he does not have long blocks of time.  This reminded me of an old belief of mine: Belief : I need a long block of time to be able to work on my math homework. Do you believe that? Do you have 15 minutes. Can you do some homework with that time? What about 10 minutes? What about 5 minutes? I thought for a long time that I needed a...

I find it very hard to rest.  For me resting is not equivalent to sitting on the sofa watching Netflix, or browsing aimlessly on the internet, or some other form of modern "chill-out". For me resting is some form of regaining energy so I am able to continue whatever activity I want to do. Sometimes resting means going for a run, because my body has been inactive on a chair for a long time; other times it means stretching, because my body has been under a lot of tension; closing my eyes for a minute because they are in pain after so much screen-watching,... Resting for me is "bending to the other side" to find an equilibrium for all my activities. What I find hard about resting is understanding towards which other side I need to bend to. Actually, the body and the mind are very misleading. For example, you may feel tired and think "I should rest", meaning, "I should sit on the sofa and do nothing", but actually, what you may need to counteract tha...

On Wednesday I went to a talk by Georg Kaser (University of Innsbruck, Austria) on climate change. It is a public lecture and easy to understand. I found it very interesting. In particular, it is interesting to see why increasing more than 1 degree as we have done right now is so dramatic, and reaching 2 degrees can be catastrophic.

Finally, I managed to do more videos! Full of mistakes that I tried to correct somehow. It will get better with time, but I like this new set-up of writing directly on paper, rather than on the tablet. This is a series of 3 short videos explaining three basic tools from Ordinary Differential Equations that are extensively used in modelling:  - conservative Newtonian dynamics;  - gradient dynamics (overdamped dynamics);  - friction dynamics (damped dynamics). Here are all the videos:

Yesterday I stayed past my bedtime to finish the short fiction story I was writing. Finally, it is finished! At least the first version. I will try to edit as little as possible, but I will try to correct the bits that felt "less believable". Funny stuff, my brain located the story in a time when there were no cell phones and no internet. I was born during that time, and somehow my brain likes that set up for stories. The internet and the cell phones feel still a bit surreal to me when they appear in stories, also because they become outdated very soon. For example, I can make a story where MySpace appears, but now MySpace does not exist anymore (I think) and people would not know what is it when they read the story. But it is just so easy to mix things from the pre-internet/pre-phone era with the post-one.  The story is darker than I thought. I had the story clear in my mind to begin with, but it evolved quite a lot during the process of writing it.

I had a very enjoyable first class of kinetic theory today. Of course, I have always found the subject of kinetic theory fascinating, but it is easy in everyday life to forget about it. I was lucky enough to discover this topic by chance when I was doing my first year of PhD. I felt relaxed in class, probably because I was not thinking in terms of what "I should" or "I should not" do. I was just telling them things that I found important and interesting. Now that I think about it, I was telling them a story: the story of kinetic theory.

Today was the first coaching session (with 3 interested people out of around 25). One student shared that during the theoretical class, he was copying everything written on the blackboard and had no time to follow what the teacher was saying. I did this too. Many people do this too. But unless you have no other resource (like no lecture notes or no books), this is a very ineffective way of making the most of the theoretical class. You will learn little, understand not much, and be unmotivated. And then you need to spend a lot of time at home assimilating your transcribed notes. If you have lecture notes the best is to prepare the theoretical class by: (1) Revising the content of previous classes; (2) Reading in the lecture notes what is coming in the next class. Try to understand as much as possible, and make note of your questions or any unclear points. Then in class do not copy the blackboard. Copy only when new material, information, or examples are given that are not on the lecture...

Yesterday I traveled all day long by train from Konstanz to Vienna. Today I finished preparing the exercise class, the seminar class, and some of my master class. I also wrote more than 1,000 words of a short story. I know. It is Sunday. But it is ok. However, the only reason my day went like this is because my husband is not with me, otherwise, he would have taken priority. My work was my priority number one for a long time, because it was my identity. But I got a healthier relationship with my job now. If my husband had been here, my time would have been for him (ok, except for the bits of the courses that I needed to prepare, but even that I would have organized myself to have done it earlier).

Something that I like about working on short spurs (or better said, keep changing tasks rather than devoting a big chunk of time to just one task) is that my brain keeps working "on the background" and keeps giving me ideas from time to time, and I just need to make a note on them.  For example, I started to prepare the exercise class for the first-year students and the difficulty of these classes is on how to explain the exercises, or rather, how to guide the students on how to think about these exercises without telling them the solution. My brain gave me very good ideas. It finally also gave me another idea for another story. I thought that maybe I was not keen on working on more than one story at a time. But today I had a full new story while I was showering. The pity is that my brain goes faster than I can write, and dictation just does not work for me. Blame my Spanish accent. I have discovered that I can work in very small spurs on tasks that I thought it was just not ...

The conference in Konstanz is over. It has been an intense week and I am running out of energy. But I have kept up with the good job, making the most of every second - which was necessary, since the teaching semester is starting and I had a lot of queries from students. I am still going with the attitude of "making the most of every minute" that I learned from the book 'Someday is today' by Matthew Dicks. It seems that I have learned to use the cracks of time to advance work, and indeed I manage to get much more work done. I am also going with the attitude of "doing what I want" (and, yes, sometimes I trick myself to want to do something in particular that is a bit pressing).  I was thinking today that being able to do what you want at every moment (or convince yourself that that is what you want to do) is what means to be successful. And now, what do I want to do? Ah, yes! Fall asleep...

This text in Dean Wesley Smith's blog on writer's fears inspired me: So if you are having fear issues, move the fear over to not writing and not finishing. I can tell you this for a fact: The idea of not writing and not finishing what I write scares the hell out of me. Get help with your fears, move the fear to a fear of not writing. And move the fear to a fear of not finishing. Because not writing and not finishing are true failures. I hate to tell you this folks: Every time you claim you want to write and then don’t write or don’t finish, everyone around you knows you are failing. That should scare you more than anything   https://deanwesleysmith.com/heinleins-rules-chapter-four/ I like this technique of moving the center of the fear to something else. It is applicable to many contexts when studying mathematics: Fear to present math exercises in front of your peers? Move the fear to actually dropping because it feels unpleasant. You know that if you drop this means that you ...

Socrates said "Know yourself". To me, this does not mean to make a judgment of yourself or form an opinion about yourself, it does not mean to make any type of statement about yourself. It is about knowing yourself "moment to moment". Right now, at this moment, know yourself: know your thoughts, know your emotions, know your reactions, know the physical sensations, know your conditioning, know everything that is going on in  you. And all that changes from moment to moment. So know yourself moment to moment, because whatever applied later, does not apply now, thinking that it does is judgment and judgment is limiting: you try to fit your life into your mental view of it; you do not allow for new things to happen, for change to take place. Judgement is negative, negative in the sense that it "negates" the possibility. The possibility of something different to your pre-conceived idea. I connect this also to the other famous sentence of Socrates "I only k...