So, I will not talk about standard parabolic functions, the most standard of it is y equal to x square, again you should immediately stop me, say sir please write domain and the function, first of all I am not sure whether it is actually a function, I should call it as a relation, so you should say sir stop using the word F, you should call it as a relation, you should say I get it as f of x, are you sure that this is a function, so from my experience I humbly say yes I am sure, but from your side you should always challenge the fact how are you using the word F, F for functions again do not go to some other trajectory altogether, so here F should be used only when you know it is a function, else use R with a single line, is that fine, but since I know I have been teaching this for 11 years now it is a function, does not mean for that, I will never give you wrong information on this, now assuming it is a function, can we see the graph for it, I am sure you would have seen this empty number of times in the chapter quadratic equations, and even by closing your eyes you can actually draw the graph for this, yes or no, by the way couple of things which I want to know, what does this arrow signify, goes up and up never stops, now if it never stops how much is the span of this function along x axis, I have only mentioned it, so you should say sir you have only stated this span is going to be R, but if I say this is not R, but this is only a chart, there is some intervals that I would be denoting in our maths, so I will be giving you just a quick idea about that, this is also a part of the bridge because these small small things you should be doing, intervals are basically from their subsets of the real numbers from where my function can be spanning, for example if I see my function only spans between minus one to one, so I will not draw this part of the function, so I will not draw an arrow, it will only stop like this, then that interval would be called as minus one to one closed, minus one and one are included in your, understood whats the good thing, arpita, arpita you understand the meaning of the word closed means I include minus one and one point in my domains, then what is the meaning of open then, this is called closed, open means, what does it mean, it means not they themselves at the corner position, yes I am repeating this again, when you say open interval minus one to one, everything is point zero, minus point nine nine nine, that is minus one and one they are, are you getting it, so that is it, don't get confused with this, what is the difference between these three expressions, this is a set, this will only contain two elements, what are they, minus one and one, this contains all the set of points which lie between minus one and one, all the x value which lie between minus one and one on the real number line, including minus one and one itself, when you write this what does it mean, all the set of points between minus one and one are that then minus one and one themselves, so for this if I have to plot this graph, dear students please note I have to put a whole like this, what does this whole signify, that exactly at one this value here is missing, it is just ending before one, now how before, it is a matter of limits, that is a concept in calculus, so how before somewhere point nine nine nine nine nine nine, how much nine you want to add it, it is not attaining one, that is what I want to see over here, here also this whole signifies the same thing, it is just starting after minus one, are you getting this point, so if I write this minus one to one and I ask you, if I ask you what is the domain of this function, one to one set interval that is the domain of the function, I just know I am going to talk about the entire real number, so I have been raising this, I am putting the entire real number, so this graph is now extended in both the directions like this, now the next question is what is the range of this function, what is the range of this function, for domain look at the span of the function along x axis, for domain look at the span of the function along y axis, domain, range, domain, range, understood, so domain you said all real numbers, I agree with you, what about range, this is not the range by the way, this is what, its range is actually, I will write it as a short form like this, r f equals r f to signify range of a function, d f to domain of a function, don't ask me what is r f, so r f is going to be all positive real numbers, can I write it like this, hope you are aware of the symbols which we use r and w, z, what is z used for, integers, i is also used for integers, i plus means positive integers, r plus means positive real numbers, is that fine, what is this minimum, are you going to calculate something, in case of a parabola, is that fine, okay, it is called the vertex of the parabola, again you have sections coming up in class 11 for you, I will speak in detail about this, okay, now I have certain questions for you to answer, so we just now saw the graph of y is equal to x square and it was like this, assuming its domain is all real numbers and cotomain is also all real numbers, I would like, if I have to plot this graph, how would it look like, what do you think Shishant, it would become somewhat, is this correct, okay so we have another answer, it will shoot down, Shishant is outrightly rejecting your claim, what do you think, Shishant is correct, just because it comes from that one, correct, how do you think, I don't think so, right now, especially in maths class, I love challenges, please challenge somebody's answer, if you accept the answer, tell me why it is correct, if you are not, tell me why it is not correct, Ananya, you can't think, okay, at this point at least this guy is saying that the vertex will come, okay, so initially when it, correct, the vertex position initially was suggested, he is saying that when x is, so when x is 1 is y is 0, at least he is correct with respect to that, yes sir, but that doesn't mean the answer that he is giving is completely correct, only that one point satisfied it, may not be the other case, now Foreman sense says that, if I had, in the previous graph I had to put x value also as 1 or minus 1, but to get the same, the second graph as 1, I have to put either my x as 0 or my x as 2, so to get a value of y is 1, either I have to put x as 0 or y as 2, so 0 or x as 2, right, this is a value of y is equal to 1, right, so what he said is absolutely correct, the blue one would be the graph of y is equal to x minus 1 the whole square, now no points for guessing the graph of y equal to, this is the y equal to x plus 1 the whole square, how would be the graph of this, now the same logic, now your graph will actually which the ratio I choose there, the graph will shift on this side, that means the vertex will now come at minus 1, 0, understand, now here comes the rule for you, what is the rule, basically says in any function, when I say any function means any function, not only with a parabola, not only for a parabola, but even for a line, even for a trigonometric function, even for a logarithmic function, even for an exponential function, even for any special function, whatever function you can think of in this word, the rule is this rule, it says that if you replace your x with x plus h, h being a positive quantity, then the graph will shift h units to the left, h units to the left, that's what happened over here, when I replace x with x plus 1, my graph shifted 1 units to the left position, so from your side left position, if you replace your x with x minus h, again h being a positive quantity, your graph will shift h units to the right, your graph will shift h units to the right, is this rule understood, and you can actually apply it to any graph, let's say can I move to the next slide, let's say you know the graph of y is equal to x, yes everybody knows it, how does it look like, it looks like a straight line like this, correct, I will signify it, it indifferently keeps on going, the moment I say hey draw the graph of x minus 4, won't it appear like this point, and this point the shift will be of, that's why it ends, as you can see minus 4 is your y into 7, so my point here is that any function in the world, that rule is going to be holding 2, understood, but let's return back to the same graph, now my question is 5 minus 4, x square equals 1, I mean the word h, this graph, original graph black mark will move where, 1 unit half, 1 unit x will move, okay Krishna, Krishna is not part of the world today, so shares is everything, saying that the same graph will go 1 unit up, is it true, is it correct, why, because if you see this is another way, we know that x plus 1, the graph of x square plus 1 is the graph of y equal to x square shifted 1, is correct because if you put, initially when you have, but now when your x is 0, y will now become a 1, so for the same x, y will now assume a value as 1, so it's vertex will shift to which position, 0 comma 1, correct, so now the graph, something like this hanging in the air, okay, this point we have for, understood, what will happen if I ask you to plot the graph of 1 equal to x square, can I say that, can I append this rule and say that, if you replace, what's the quantity, the graph shifts the graph sideways, now understood, can I combine these two rules and give you a problem, will you be able to solve it, okay, sure, so this is the handout which I am going to, anyways give it to you, okay.


Hello and welcome to this session. Now in level 1 under topic similarity and transformation we have discussed the various types of transformations, translation, reflection, rotation and dilation. In this session we will see that the transformation acts as a function that takes points in the plane as input and give other points as output. Now let us recall the three types of transformations. Now translation moves the object so that every point on the object moves in same direction at same distance. Here only the object size, shape and size remain same. Now this figure represents translation of image where every point has moved in same direction and at same distance. Now let us see what is rotation. Now rotation is a transformation in which the object is rotated about a fixed point. The direction of rotation can be clockwise or empty clockwise and here also shape and size remain same that is shape and size of the figure remains same. Now here you can see this figure represents rotation of image about center of rotation A that is the point A by some angle. Now let us see the third type of transformation which is reflection in reflection. An object is split across a line of reflection so shape and size remain same. Now this figure represents a reflection of image in vertical axis. The image is split in the line of reflection. Now let us discuss transformation as a function. Now transformation is an operation that moves, fits or changes a figure to create a new figure and here the new figure created by a transformation is called image and original figure is called the P image. Now if the P image is given by A then its image will be given by A dash and image of A dash will be given by A dash. Now a transformation is one to one correspondence between two sets of points to only one point in set A dash called its image and every point in set A dash is image of one and only one point in set A called its P image. Thus a transformation maps an original figure sample let image of point with coordinates x y in set A be a point with coordinates 2 minus x y which is a point in set A that is transforms to now and with this transformation a point with coordinates 2 minus 1 transforms to a point with coordinates 2 minus x y now where x is 2 y is minus 1 so coordinates of new point will be 2 minus 2 that is 0 and y that is minus 1 thus this point with coordinates 0 minus 1 in set A dash is the image point with coordinates 2 minus 1 in set A under this transformation here the point 2 minus 1 that is the point with coordinates 2 minus 1 is called input point and the point with coordinates 0 minus 1 is called so here we can see that whenever we are given the coordinates of any figure along with the transformation rule we can easily find the coordinates of the new image after the transformation and hence write the coordinates of the image of the given figure now let us discuss transformation rules for reflection first of all let us discuss reflection in x axis now under reflection in x axis the image of point A with coordinates x y the point A dash with coordinates x minus y so here we write x y transforms in y under reflection in x axis now let us discuss an example here when we reflect the point A with coordinates 3 4 in x axis we get its image point as the point A dash with coordinates 3 minus 4 now let us discuss reflection in y axis now under reflection in y axis the image of the point A with coordinates x y is A dash with coordinates minus x y and we write x y transforms to minus x y under reflection in y axis now here we can see under reflection in y axis the image of point A with coordinates 3 4 is the point A dash with coordinates minus 3 4 now let us see reflection in the line y is equal to x now under reflection in the line y is equal to x the image of the point A with coordinates x y is the point A dash with coordinates y x y transforms to y x now here we can see under reflection in the line y is equal to x the image of point A with coordinates 3 5 is the point A dash with coordinates 5 3 now let us see reflection in original now under reflection in original the image of the point A with coordinates x y is point A dash with coordinates minus x minus y and we write x y transforms to minus x minus y now we know that the point where the x axis and y axis intersect each other is called original now under reflection in original the image of point A with coordinates 3 3 is point A dash with coordinates minus 3 minus 3 now let us discuss transformation rules for translation now a translation of A units in horizontal direction and B units in vertical direction is the transformation of the plane such that the image of point A with coordinates x y is the point A dash with coordinates x plus A y plus B and we can write it as x y transforms to x plus A y plus B now let us discuss an example here and the translation where x y transforms to x minus 3 y plus 2 the image of the point 4 minus 3 will be 3 y plus 2 now where x is 4 y is minus 3 so image of this point will be 4 minus 3 that is 1 minus 3 plus 2 that is minus 1 also we must note that in a translation if a point is moved to the right then A is positive and if it is moved to the left then A is negative similarly when point is moved up then B is positive if it is moved down then B is negative now let us discuss transformation rules for rotation now let us discuss rotation at angle of 90 degrees about original now when image is rotated at 90 degrees angle about original in clockwise image of point A with coordinates A dash with coordinates y minus and where we write transforms to y minus and when image is rotated at 90 degrees angle about original in anti clockwise direction then the image on A with coordinates x y with coordinates minus y transforms to minus y x now on rotation of 90 degrees angle about original in clockwise direction the image of point A with coordinates minus 3 minus 2 will be the point A dash now where y is minus 3 so minus x will be minus of minus 3 which is 3 dash with coordinates minus 2 3 the image of point A with coordinates minus 3 minus 2 on rotation of 90 degrees about original in anti clockwise direction is the point A double dash with coordinates where y is minus 2 minus y will be minus of minus 2 which is 2 so image point is A double dash with coordinates 2 now here we can see when we have rotated the point A with coordinates minus 3 minus 2 at an angle of 90 degrees about original in clockwise direction then we get its image point as A dash with coordinates minus 2 3 and when we rotated it at an angle of 90 degrees about original in anti clockwise direction then its image point is the point A double dash with coordinates 2 minus 1 just discuss rotation at angle of 180 degrees about original now when image is rotated at 180 degrees angle the direction does not matter in this rotation then image of point A with coordinates x y is the point A dash with coordinates minus x minus y and we write x y transforms to minus x minus y for example image of point A with coordinates when rotated at 180 degrees angle about original is point A dash with coordinates minus 2 minus 1 now here we can see when we rotate the point A with coordinates of 180 degrees about original in anti clockwise direction we get the image point as A dash with coordinates minus 2 minus 1 and when we rotate at an angle of 180 degrees about original in clockwise direction then again we get the same point that is the image point A dash with coordinates minus 2 minus 1 so we can see in this rotation direction does not matter and now let us discuss rotation at angle of 270 degrees about original now then image is rotated at 270 degrees angle about original in clockwise direction then image of the point A with coordinates x y is the point A dash with coordinates minus y x and we write x y transforms to minus y x and when image is rotated at 270 degrees angle about original in anti clockwise direction then of the point A with coordinates x y is the point A dash with coordinates y minus x y transforms to y minus for example image of point A with coordinates 2 minus 3 when rotated at an angle of 270 degrees about original in clockwise direction then image will be A dash that is the point A dash with coordinates 3 2 and when it is rotated at an angle of 270 degrees about original in anti clockwise direction then its image point is the point A dash with coordinates minus 3 minus 2 so in this session we have learnt that transformation x as a function that takes points in the plane as input and gives other points as output and this completes our session hope you all have enjoyed the session


Although attraction is not the same as love, it is often the beginning of a love story. Attraction temporarily gives you the bond needed to know your partner and express your love interest. So, would it not be pleasant to know some details about this beautiful feeling? I bet you want to know. In this video, I will share with you 15 psychological facts about attraction you probably did not know. If you are new here, consider subscribing to this channel for more awesome videos such as this one. 1. Women with older fathers often get attracted to older men and vice versa. Research has pointed out that the popular conception that women are attracted to older men because they have underlying daddy issues is not precisely true. According to recent studies, women who were born to older fathers are more likely to find older men attractive. Notably, they are more likely to consider men with more facial creases and less hair to be beautiful. The opposite of this fact is true as well. Women who were born to younger fathers are more likely to be attracted to younger men. 2. Smell determines levels of attraction. It is known used at SANS play a significant role in the psychology of attraction, but which smells are considered desirable and which are not? A survey published by the Social Issues Research Center found out that women are attracted to Androsthenon, a natural chemical found in fresh male sweat. However, while Androsthenon acts as a pheromone, the scent of Androsthenon, which is produced when there is too much sweat, acts as a natural turn-off. On the other hand, men may be attracted to the smell of perfume, but turned off by the scent of tears. In a 2011 Signs Magazine article, it was published that when men sniffed tears, they felt a decrease in sexual arousal, whereas there was no change when they perceived a natural cell in solution. 3. When a woman is flirting, the tone of her voice naturally increases. In a survey carried out by Canada's McMaster University, it was observed that the style of a woman's voice rises a beat when she is teasing. Even more interesting is the fact that the intensity of a woman's voice varies on different occasions. For instance, a woman's highest chord occurs during her most fertile time, around a violation. Amazingly, the same study found that not only do men prefer these higher-peached voices, but they rated the highest speech as the most attractive of all. 4. I Love You has more effect on the brain when whispered in the left ear. Being in the ear of someone you love is no doubt a romantic gesture. But do you know that the ear you whisper in two matters? Well, if you are trying to impress a potential partner with romantic whispers, be sure to speak into his or her left ear. According to the findings of a survey carried out by MIT Cognit, when emotion stimuli enters the left ear, accuracy of the recall is more than 6% higher than when the same stimuli enters into the right ear. So, the chance of winning the heart of your crush is higher when you whisper in the left ear. 5. Opposites Attract You have likely heard the maxim that opposites attract. If you are a science student, you certainly have. A recent study conducted at the University of Dresden found that both men and women are naturally attracted to those with a different human leucoside antigen, HLA complex, basically a genetic blueprint than their own. As a person's HLA complex is crucial to immune process, this exciting fact infers that this subconscious attraction has to do with the survival of the species. So how do we know which potential mates have genetic blueprints different than our own without having multiple discussions with them? Interestingly, further study reveals our brain can tell whether or not two people are opposite based on sand alone. 6. Attraction is more than sight It is easy to assume that the crux of the attraction comes down to the view. It may be a few brain chemicals thrown in. Even though sight certainly is crucial in the psychology of attraction, sound and smell have also been discovered to play a big role. A study published in Frontiers of Psychology says that people can distinguish several characteristics about a potential partner, such as weight, emotional state, and dominance level, only by listening to them talk. The same research also established that people could determine traits through smell, although sight is necessary. Those brain chemicals that deal with physical attraction are spiked not just by sight, but also by sounds as well. 7. The quickest way to a person's heart is through the eyes. The eyes linking to the heart may not be the traditional ending to the famous adage which says the fastest way to a man's heart is through his belly. But scientifically speaking, it is a proven fact. A study published in the Journal of Research and Personality stated that people who share prolonged eye contact form stronger attachments with, as well as increase affection for, their staring partner, as opposed to those who are introduced with a bare handshake. Locked eyes can quickly become locked lips. 8. Beards are attractive. Some beards are not. According to research published in the Journal of Evolutionary Biology, heterosexual women are naturally drawn to men with beards, however, not every beard is attractive enough. In the study, it was deducted that men with subtle or some small degree of facial hair are considered more beautiful by the female participants. The men who were regarded least attractive were those who were clean shaven and those with large, bushy beards. Thus, the final note is that many women subconsciously view men with stubble as active, healthy, and more likely to be a good parent. 9. A woman's cycle determines the kind of man she will be attracted to. Do you think it is abnormal for a lady to crush on another man, even though she is in a blissful relationship? Well, research has proven that while a heterosexual woman may be in a happy, committed relationship, she is likely to find a range of other men attractive throughout her menstrual cycle. According to several different studies, ordinarily, a few leading women are attracted to several things depending on their hormone levels during menstruation. During a woman's fertile stage, which usually lasts for 3 to 6 days, she is most likely to prefer men with deeper voices, competitive natures, and other general musculine features. At peak fertility, the day of a volation, a woman is most likely to seek out a male stronger than her usual partner. Psychologists attribute this to the genetic handwriting in mammals that drives the need to find a stronger mate. 10. Red is the most attractive color. While some individuals believe that the color, red, is a color of love, others conclude that it spells danger. However, regardless of how you feel about the color, red, studies have shown that both men and women are perceived as more attractive and more sexual when they are wearing red or at least another color that's both bright and warm, like the color, red. Furthermore, one study from the University of Manchester showed that men spent 7.3 seconds staring at images of red lips, 6.7 seconds on those with pink lips, and 2.2 seconds on those who wore no lipstick. Many scientists say that the reason for this love for red is because red triggers a primal sexual reaction in both men and women. 11. Dog owners have more chances of having a date. Dogs are lovely creatures and, truthfully, there are a million good reasons to get a dog. But did you ever think that finding a partner may be one of them? In the words of a health writer, Cecily Tilbridge, men are three times more likely to get a woman's digits when a pup was accompanying him versus when he was riding solo. Of course, the same would be right for women too. 12. In most cases, women make the first moves. So, you probably believe in the theory that men are always the first to chase after women, right? Well, that fallacy is wrong. Although many traditions of the world forbid a woman making the first signal, most times, the women are the first to give the green light. 90% of the time, a romantic encounter is prompted by a woman's solo flirting, like prolonged eye contact, small smiles and open body language. So, while it is true that a man might initiate discussion, 9 out of 10 times, it is because he's been indicated that discussion is desired. But this is not on all cases though. The fact that a woman shows that she finds you attractive does not mean you should heed on her. 13. Facial symmetry plays a vital role in attraction. Do you find a certain someone particularly fetching, but can't seem to put your finger in the reason why? Well, aside from just thinking that a person is beautiful, what you might be drawn to is the balance of their face. Research shows that men and women find symmetrical faces more attractive than non-symmetrical ones. In relationship therapist, Rhonda Milrad's LCSW voice, the more a person's face is roughly the same on both sides, the more you are going to want to approach them. 14. Blushing is a signal of attraction. If your date is blushing a lot to your compliments, take it as a sign you're winning them over. People only blush at the words of people they know. 15. The blood vessels in your face dilate, leading to high color in your cheeks, full lips and sparkling eyes. 15. The butterflies you feel in your stomach when you see someone you like is a stress response caused by adrenaline. Suppose you have ever been anxious about something that is about to happen. In that case, you may have felt the sensations of nausea and fluttering. The recognizable and unusual feeling deep in your heart, generally known as having butterflies in your stomach. For instance, you are about to meet a potential love interest. Instead of actual butterflies bouncing around your large intestine, there is of course something more scientific going on and it's all down to your nervous system. As the old saying goes, beauty is in the eyes of the beholder. What you find attractive, another person may not. Even though there are several general features that defines attractiveness, it is still for you to determine.


So, I'm here today to talk to you about empowering early career developers. We're going to talk about why it's important, what we've been doing on my team, and some strategies that you can bring to your team to empower your early career devs. Like Megan said, my name is Mercedes Bernard, and I'm a senior software engineer with Tandem, formerly known as DevMind, so you guys all get to be the first ones to hear that we're changing our name and rebranding. So, yeah, let's get started. There we go. Okay. So, I want you all to just take a second and reflect back on your career progression. How did you get to where you are today? How did you grow from an intern or an apprentice or an entry level to your current role? Did it feel a little bit like trial by fire? Or what about like drinking from a waterfall? My experience definitely did, and I can think of a few instances in particular. Like at my first job while I was still in school, and I had to explain to a very angry client at 1 a.m. why deploying untested code to production was a bad idea. Or another time at my first job out of school when a large payment transaction project fell in my lap that had been poorly estimated by someone else, and then I was on my own to manage unrealistic client expectations. In both of these scenarios, I did not have the support of someone in a senior or lead role, and unfortunately, I don't think my experience is uncommon. There has to be a better way. On my current team, I lead a team of eight developers. This isn't my first time leading a project, but it is the first time I've led such a large team of varying experience levels and that I've shared responsibility for their professional growth. And like Megan said, I've also started a community in Chicago called DevTogether that connects people starting their development careers with technical mentors. Both of these experiences have opened my eyes to the tremendous need in our industry for mentorship and coaching for those early in their career, which brings me to the problem with our current technology market. Companies are scared to invest in junior talent. If you look at any of the job boards right now, everybody is trying to hire teams of rock star senior devs, and I don't mean to call out anybody here at the conference or any of our sponsors, but I took a look at that job board, senior, principal, lead, senior-ish, that was my favorite. And in my opinion, this is just a losing strategy. It costs so much money to hire a team of all seniors. There's the recruiting costs, their salaries, and the cost of attrition that comes from lack of career growth opportunity, because when everybody's a senior, how do you grow and improve? Also, not every task on a project requires a senior's experience. Companies are overpaying for work that people early in their career could be challenged by and learn from. In my opinion, it costs less to mentor early career devs than it does to hire a team of all seniors. And the argument that the seniors that you do have don't have time to do this mentorship isn't valid. Your mid-level and senior-level engineers need to practice these mentorship and coaching skills if you want to grow managers and leads who teams want to work with. It's an unfortunate reality in our industry that job-hopping is often the quickest and easiest way to level up, so I can understand why that would make some companies hesitant to invest in their early career talent. They fear that after they spend all of this time, money, and energy, their early career devs are just going to go leave to get that mid- or senior title. But where do you think senior developers come from? I was talking to somebody recently who made the joke of the senior developer's stork. It just doesn't exist. People who feel valued, who are challenged, and who have the support of their leads and management are more likely to stay and grow with your company than those who are just given the intern work or the busy work, and they'll just leave to find variety or something different. So I've been reflecting on how to improve this mentorship and coaching that our industry so desperately needs. And we need to empower our early career talent if we want to retain them and grow them into the senior devs that we're all looking for. So I want to go back to the title of this talk for just a second, Empowering Early Career Devs, and I want to unpack what I mean for just a minute. This is the definition of empower that I particularly like and use, to make someone stronger and more confident through encouragement and support of their ability. I want to highlight just two pieces of this definition. The first is more confident. Of course, we want to grow strong developers and strong development skills, but we should also want our developers to own their growth. We should want confident, independent devs. And the last four words of this definition are my favorite. Support of their ability. Everyone has innate strengths, and everyone is capable. By encouraging the talents that the members of your team already have, you can grow them into the skills needed to make amazing team leads. So you may have noticed by now that I'm trying really hard to avoid using the word junior, and I'm replacing it instead with early career devs. So let me just make explicit what I mean when I say early career. All I mean is somebody who's beginning their development career. So why is this reframing from junior to early career so important? Let's just be blunt. Junior has a negative connotation. I don't think it's warranted, and I don't think it's earned, but unfortunately there's a lot of people in our industry who view junior as probationary and other forms of less than, rather than just as somebody who's learning. And because of that negative connotation, it causes imposter syndrome to a bound. If the person you're talking to thinks you mean junior as less than, they're going to feel less than when you use it. And just because this is somebody's first development job does not mean that they're junior. Career switchers have so many transferable skills, and often it's the communication and soft skills that are the hardest to learn on the job. Early career devs bring fresh perspectives and no bad habits to unlearn. And in my experience, they bring all of the enthusiasm and none of the arrogance that comes from a lot of senior technologists. So when we're talking about this group of people, shouldn't we want to use positive language when we're talking about them instead of labeling them as junior? So I want to switch gears for just a second and talk about the opposite end of the spectrum and talk about the senior developers. So what makes a senior developer? I'm sure everyone in this room has some ideas about the defining characteristics. But for me, I've broken the traits into two main categories of skills and experience. So in my opinion, a senior dev is somebody who has really strong technical communication, design management, meeting facilitation, consulting, and client engagement. But they also have experience, making decisions, being held accountable to those decisions, looking at the big picture, and leading a team. Now you might look at my list and go, well, that skews really heavy towards the consulting side, and it's totally true it does. That's where all my experience is. So if you focus on product dev or other forms of dev, your list might look different and that's totally cool. So how do we foster growth in someone who's early in their career into a team-leading senior dev who has all of these traits? We need to give our early career devs the opportunity to practice these skills and to gain this experience before they step into a lead role.  We don't want it to feel like a trial-by-fire experience. On my project, we've been putting some processes in place to facilitate these kinds of opportunities. So I'm going to talk about what we've done, what we've learned, and then specific strategies that you can bring to your teams to empower your early career devs. So looking back on what my team's done over the last few months, it kind of follows this framework. So this is what I'm going to be using for this talk. There's three phases to what we did. So in the first phase, prepare, my manager and I had a lot of conversations about ways that we could coach and mentor our team to help their professional growth. We wanted to find ways in which they could practice skills that they weren't getting a chance to practice. We also wanted the whole team to share more ownership over our project. So I started by thinking, how am I a silo? How am I, as a lead, kind of hoarding all of these responsibilities and not letting my team get insight into them and not letting them get the experience that they need to level up? For our project and my team, the answer was client communication. I owned almost all of the technical communication with the client, which left my team with almost no space in which to practice those skills and grow. I was also responsible for leading a lot of our meetings and for making a lot of our decisions. And quite frankly, my team was ready to share that responsibility with me. So after reflecting on that, I tried to come up with a purpose of what I wanted to do. So for me, I wanted all members of our team to gain consulting, client engagement and leadership experience. Ultimately, what we ended up doing is having every member of our team own a feature area of our application. And then the last step in prepare is I had to prepare myself to delegate. This was the biggest challenge for me personally, and I'm seeing some head nods in the room, so I think it's a really common challenge for anyone in a leadership role. But it's really important to share that responsibility and that accountability with your team. The way that I've been trying to reframe how I think about it is like the grounds crew at an airport. So I'm sure a lot of you flew in over the last couple days, and when the pilot needs to connect the airplane to the jetway bridge, it's vital that they perfectly center themselves on a white J-Line that's painted on the tarmac. And the FAA actually requires three grounds crew members to direct the pilot because the door is behind the cockpit and that J-Line needs to be under the plane, so there's no way that the pilot can see everything they need to in order to make that happen. And so as a lead, that should be your role on a project, is paying attention to the things that your team might not see or might not be aware of and ultimately directing them towards their success. You're not jumping in the cockpit and doing it for them. So in the first phase, it was mostly my manager and I just talking and trying to come up with ways to put this plan into place. But I wanted to make sure that I introduced the plan to the team, that we created a shared understanding and that they were able to give me feedback on what they wanted out of it. So if we go back to the purpose, ours was client engagement, consulting and leadership, and so these were the responsibilities that I presented to my team. So for us, our feature owner is the point of contact for everything related to their feature. They own all of the client communication, all of the questions, answers, everything related to their feature. They get to practice asking questions of the client, holding the client accountable to communication deliverables. But they also get to practice answering questions from the client, like feature requests, priority, and timeline. The feature owner is also responsible for documenting all the conversations they had with the client and everything they learned. Plus things like system architecture, user stories, access control requirements, and pretty much everything. Documentation is not a flashy part of consulting and we all know this, but I think that written communication skills are vastly undervalued and really, really difficult to learn. A natural extension of documentation is knowledge sharing. So we don't want all of this consulting work to happen in a vacuum and we don't want there to just be instead of one silo on the project, me, now eight silos. So it's up to the feature owner to make sure that everything that they've learned from the client and that they've documented gets disseminated out to the rest of the team. The feature owner also gets to practice their decision making skills. So they're responsible and they're a key stakeholder in all decisions related to their feature. They get to practice both making the decision and then being held accountable to it later. They also have to advocate for whatever implementation decisions they make when they present them to the team. And finally, meetings. The feature owner is responsible for leading all of the meetings related to their feature. This builds domain expertise, meeting facilitation skills, and confidence. And on our team, it had the added benefit of having everyone learn when a meeting is valuable and when they could have just sent an email. So when my team and I were talking about what our new process would look like, I wanted to make sure that we all understood what it was not. The feature owner was not going to be the sole owner and decision maker. They'd still have the support of me, my manager, and the other seniors on the project. And they would be responsible for making decisions collaboratively with me, the design lead, and the product owner. And in the same way that a team lead doesn't write all the code for their project, the feature owner was not going to be writing all the code for their feature. They'd still have the support of their entire team members to make sure that all the tickets got picked up and worked on in a sprint. So now it was time for us to put our new process into practice. We have an ongoing rotation of Epic ownership, where a few sprints before we're scheduled to start working on a feature. I assign it to one of our team members. It's up to them to go initiate the communication with the client and gather technical requirements, understand their current workflow, and to raise or identify any risks that may happen before we start working. So you can see here that Shamile, one of my team members, had two really closely related features in the first sprint. And then his next one was in sprint four. Where Sasha had a feature that spans sprints one and two. And so then her next feature was in sprint five. And like I mentioned before, the feature owner was responsible for all of the documentation for their feature. So what my team has done is created a folder in our Google Drive for each feature where they put any and all documentation that they come across. So you can see here, it might be legacy whistles, the results of data modeling exercises, or just an ongoing list of questions and answers that we receive from the client. Chris, one of my team members has actually started just keeping a Google Doc where he labels every time he initiates questions with our client and then the answers that we get back. This has been really helpful in increasing transparency on the team, because our client doesn't always respond to emails. Sometimes they slack us, sometimes they just pop into the office to answer something. And it can be really challenging for us to know which questions are still outstanding and which ones we have answers for. We also need to document decisions that we make. So Sasha, another one of my team members, had a conversation where we made some architecture decisions about what legacy infrastructure we would need to keep around for backwards compatibility, and what we could sunset, because we didn't require it anymore. So for her, all she had to do is take a picture of the whiteboard, add a few bullets about the decisions that were made, and then send an email to the team that's like, hey, if you have any questions, like let me know, but just take a look. But sometimes documentation is not enough. So in this case, Sasha sent a meeting request to the entire team to say, hey, I wanna review these two classes and how we're supposed to be using them, not like what we're actually using them for, which was really great to hold the whole team accountable to the responsibilities of our code. The feature owner also has to write the user stories for their feature. In the last few months of doing this, I have seen incredible growth. Our user stories have way more detail. The estimates are way better because we actually know what work we're gonna be doing, and anyone on the team can pick them up and just hit the ground running. Rather than before, it was almost always the feature owner had to do all the work. So here, you can see that Shamile wrote a user story where he gave us like a high level description, some implementation notes, and he also attached a sample request file. So the person who is ready to work on it wouldn't have like any questions. And throughout the feature implementation, our feature owner is also responsible for continuing that dialogue with the client as questions come up. So like I said before, it can be as simple as just, hey, these are the questions that are still outstanding or like this one came up. But sometimes it can be a little bit more challenging, like reminding the client that they promised us some deliverables and that they need to get those to us on time. So what are we learning from using a process like this? Well, I'm learning that delegating is really hard. It can be really challenging to clearly explain what it is that you're looking for and to fight that feeling of, well, just let me do it, it'll be faster. But it's really important as a lead and as a senior engineer on your team that you're taking a step back and giving your team the space they need to practice these skills. We're also learning how important feedback is. So it's super important that you as the senior engineer are giving positive feedback to the team about the things that they're doing amazing. Point out the ways in which you're seeing them grow. You wanna fight that imposter syndrome. But it's also really important that you make the cycle of giving and receiving feedback so normal that constructive feedback isn't scary. It can be as simple as, hey, don't be afraid to redirect the client when they go on a tangent. Some strategies that I like to use are XYZ. You wanna make sure that you're giving your team members the tools that they need to succeed. And similarly, make sure that you are asking them for constructive feedback on how you're doing. I got some feedback really early in this process that I needed to spend more time with my team prepping with them before they let a meeting. That was super valuable feedback and it raised my awareness so now I can try to make more of an effort to put some time on the calendar so that they feel prepared to succeed in these meetings. We're also learning when to offer support. This can be as simple as explicitly asking, hey, how much support do you need on this and anyone on the team can do it. We're gonna be coming back to this idea of support in a minute, but it's really important that you're there for your team members. And finally, remember to check in with your team. It can be really easy to make assumptions about what challenges you think they are or are not ready for, but they know themselves best. So make sure you just check in, ask them how they're feeling and what skills they wanna work on so that you can incorporate that as you continue to iterate and improve this process. So how can you bring a framework like this into your team? If we think about the three phases, I'm gonna use those again while we talk about this. So in the first phase, prepare. Identify the skills that you, your team, and your company value in a senior engineer. Some ideas to get you started might include technical ability. Maybe it's learning React or Kotlin or whatever the new hot new framework might be. Or maybe it's getting better at database design. Or knowing when to do unit testing or integration testing. Or maybe it's just, hey, we're doing really well, but like our object oriented skills aren't so great. So let's like have some fun and learn algorithms. Technical communication, both written and verbal. Time management. I don't think this one gets talked about enough and I don't think people think of it as a skill that can be practiced and learned. But I hope that any of my senior engineers have really good time management skills since they're gonna be juggling a lot of responsibilities. Maybe you want your senior engineers to be able to look at the big picture and be able to make decisions based on the product and not just the short term feature timeline. Mentorship, this is especially important if you have a lot of mids on your team who are gonna be stepping into the senior role soon. Consulting and client engagement. I know my list skews very consulting focused. This is just meant to get you started. So once you've identified the skills that you and your company value in your senior engineers, narrow it down and identify what are the gaps on my team specifically. The whole goal of this process is to create something that's tailored to your current team as it stands today. So some questions to ask to narrow this down might be, how are you as a lead, a silo on the team? What skills do you wish your team got to practice? And what skills are needed for individuals to level up? And while you're going through these thought exercises, make sure you're considering the scope of your project. You wanna make sure that your project is big enough to accommodate all of this new shared responsibility. You wanna avoid the too many cooks in the kitchen problem. So on my project, it's 18 months long. There's lots of distinct application areas. By spreading this responsibility around, it actually decreased overhead and lightened everybody's load so that they could focus on their personal career goals. So now you have the skills you value, the ones you specifically wanna encourage on your team. Make a clearly defined purpose that you want to achieve with that. You want your purpose to be narrow enough that it sets crystal clear expectations, but broad enough that there's a variety of ways that you can achieve it and that it gives your team a variety of opportunities to practice these skills. So as a reminder, my team's purpose for this new process was for everybody to gain consulting, client engagement, and leadership experience. You'll notice that I didn't list every skill that I value in a senior engineer. And I specifically stayed away from the hard technical skills because that just wasn't important for my team. But you'll also notice that my purpose isn't for people to get better at leading meetings. Because that's just too narrow and that doesn't give everyone enough space they need to grow. So how do you achieve this purpose? It's time to identify the responsibilities that would help you do that. Again, some questions that I asked that helped me identify them were what responsibilities satisfy one or more of the skill areas you chose? What types of tasks serve the purpose you identified? What responsibilities do you have as a lead that everyone on your team can share? And what responsibilities do you as an individual find challenging to keep up with? So these were the responsibilities that we outlined on my team. And I find it super helpful to identify three to seven responsibilities. Anything less than that and it becomes really easy to ignore and anything more than that is just unmanageable and you can't track it. I've also found it helps to write it down or visualize it in some way. Because if you can find responsibilities that satisfy more than one of your skill areas, that's a really good indicator that it's a good candidate for you to focus on. And by making them as clear as you can, it also makes it easier to track and talk about in one on ones or other types of reviews that you may do on your team. We've actually had one team member get a promotion in the last few months since we've done this and it was super helpful for him to have the vocabulary of all of the things that he's been working on and all of the experience he's gained during his yearly review. And I know I've said it a million times, but really, really you have to prepare yourself to delegate. It is so challenging, but it is vital to making this process work. So in the proposed step, schedule some time with your team to introduce this plan. I know that having a framework like prepare, propose and practice can feel really formal, but the only goal is to tailor it to your team. So make it as formal or as informal as you want. When I introduced it to my team, I used a Wizard of Oz themed deck, where every responsibility and every feature was a step on the yellow brick road until we hit Oz, which was the project's over, we've all grown and we're done. And make sure that you're using this opportunity to solicit feedback. You should be asking your team or trying to understand if there's skills that they would like to practice that you may be missed. Do they feel comfortable but challenged by the responsibilities that you've outlined? Is there anything they want to add or clarify? And do they feel that this implementation will be sustainable? This last piece is really important. You don't want the responsibilities to be so overwhelming that your team's going to get burned out, but you also don't want it to feel so easy that they can be ignored and put on the back burner. So now it's time to put your plan into practice. Again, follow through and delegate. Don't jump into that cockpit. Take the time to explain what you need your team to do and to support them while they do it. So there's lots of different ways that you can offer support. Some can be really formal. You can have regularly scheduled one-on-ones with all the individuals on your team. That way they have a time and a place to talk about their specific challenges and their specific questions. But it might be as simple as reading an email before they send it, just to make sure that their ask is clear or that their request is actionable. One of the challenges we've had on our team is getting responses from the client. So it can be really helpful to have another set of eyes on that email.  And for those who haven't had the responsibility of writing stories for their team, pairing can be really helpful. We all know it's hard to know how big to write the story or how small to write it or what details to include. And so having somebody there with you can make it a lot easier. And related to that, if somebody doesn't have a lot of experience doing data model or other architecture exercises, it can be helpful to pair with them on those as well. You can be there to ask questions that help guide their thinking, so that ultimately it is them that's doing it. But you're just kind of there as training wheels until they get used to those types of decisions. And related to the story writing, data modeling, and architecture decisions, I found that it can be really helpful early on to send calendar invites just to set expectations for how long activities like that should take. You wanna make sure that you don't have people on your team who are spending three days on something that should have just taken an afternoon. Or that they spent more than a cursory 15 minutes on something that they should have given their full day to. And just be kind and give them reminders about deadlines. They're taking on a lot of new responsibility and things can get lost in the shuffle. So it's as simple as saying, hey, I know you promised that documentation to the client by Friday. It's Wednesday, I'm just checking in to see how it's going. But above all, you wanna make sure that the support that you're giving to your team is individualized. Every member of your team is gonna have different strengths and challenges, and you wanna make sure that you're giving them the support that works for them. Our industry is missing out on so much talent because we're not doing enough to support those early in our career. We all need to do our part to empower early career devs to grow them into the amazing seniors and team leads that we all know they can be. Thank you.


... about quantum thermodynamics. And given that, and that's certainly a pleasure in the privilege, but given that I'm not entirely sure that the term quantum thermodynamics is correct, or that I agree with that formulation, I'd rather talk about, say, a slightly rephrased title, no? So non-equilibrium thermodynamics of quantum processes, because I believe this is more faithful to what we actually mean when we discuss quantum thermodynamics. Of course, it's a shortcut in notation, so to say, but then it's fine to use it. And this is, so this series of three lectures should be understood as an opening invitation to get your hands dirty in this field with these problems, with the set of issues that we are going to go through together and start wondering about what are the thermodynamic implications of quantum dynamics. OK, so just to give you not the time to get used to my stupid accent, let me spend a couple of words about the place I come from, or at least the place where I work, which also provides some motivations, so to say, for being interested in questions of thermodynamic nature. OK, so we're here in Belfast. Belfast is, I don't know if it can be seen from that. So first of all, I need to check. Can you hear me OK at the back? OK, I usually shout, I'm not necessarily... I don't know if I need that microphone, but anyway. So Belfast is there, is this red spot. I hope it's visible to everyone. And so the green spot all around the red one is what is called Northern Ireland. Northern Ireland is geographically speaking Ireland. Politically speaking is UK, but it depends who you are talking to, right? And then, say, despite what the present climate, say political climate claims, UK is in Europe. This is a statement, and there stays. OK, so as far as I'm concerned, that's where it is, and that's where it remains. Now, in Belfast, say, my university is Queens. So the university where I work at is Queens University, so this is the building. And Queens was the house of a few of quite well-known, quite well-known physicists. Some of them you might have heard of. Some of them probably are new. Larmor, of the Larmor frequency, I'm sure you have heard of him, was a professor there for a bit of time. Then there are these two guys, which I'm not entirely sure. So well-known, at least not to these audience, Harry Massey and David Bates. They were very well-known computational atomic physicists. And in particular, David Bates single-handedly basically founded the department that I'm working at now. And then finally, this is a very old picture that puts together both Massey and Bates. Bates is here, and Massey is somewhere. I cannot really, yes, it's a young Massey here. And then there is the final, probably the most well-known, or one of the most well-known persons coming from Belfast in link to Queens, which is John Bell. So Bell was a student at Queens. He never actually worked in Queens, at least for, OK, unless for a very brief steent as a lab technician. Then he was, say, his line manager, as they would say right now, a supervisor, recognized his potential, so he recommended him for a PhD, and then he moved on. But Bell is from Belfast, actually, from a few hundred meters from the university. That's where his birth house is. This is, say, uncommon, good day in Belfast. So this is the campus of the university. And at the campus, you find this thing, no, a blue plaque named after Bell, and put there by the UK Institute of Physics. And if you have nothing better to do on the 4th of November of each year, then we celebrate Bell on the 4th of November. So that's institutionally the John Bell Day. Why the 4th of November? Because that was the day when he submitted these big paper. And the Royal Irish Academy, in collaboration, which is basically the equivalent of the Royal Society in Ireland. The Royal Irish Academy, in collaboration with Queens, celebrates every year John Bell. So if you fancy a little bit of physics discussions, some good beer and bad weather just come along on the 4th of November to celebrate the big man himself. Yet he's not the biggest, or at least not the only big figure that we have from that place. Lord Kelvin. Lord Kelvin was from Belfast. So I think his family moved to Scotland in 1804. No, later on. So he was born in Belfast in 1824, and he moved to Scotland when he was 80-year-old. So around 1832. And if you come and see Belfast, so for the John Bell Day, then you can enjoy a kneel in front of the statue that is in Botanic Garden at the back of the university. So basically the university is here, and this is the statue named after Kelvin. And somehow the geographical proximity of Kelvin to the office, no? I mean, it's a few hundred meters from my office. And the fact that Bell was born in Belfast put these two things together. And then you start understanding why. Migropy is so interested in working at the interface between quantum and thermodynamics.  And unfit slides. So I have a lot to leave up to. Martin gave a fantastic lecture on the blackboard. I'm going to use heavily my slides, but at some point we are going to stop and go through a few calculations together on the blackboard, hopefully without any mistake from my side. Now, why thermodynamics? And why should we be interested in exploring, which are the implications that thermodynamics has for quantum dynamics? Or vice versa, how quantum mechanics can influence the development of a thermodynamic framework for microscopic processes. Motivations of values, no, and somehow variogate. Historically, thermodynamics was born because of the Napoleonic Wars. So Karno wanted to understand why the army, the French army, got defeated. And he thought, or he could pinpoint, in the inefficiency of the French weapons, a reason for, or at least one of the reasons, for the Napoleonic army to be defeated. So if you want thermodynamic historically as a very hands-on sort of origin or motivations, these guys wanted to build better weapons and realized that what they knew about the steam engine could have helped. And then they developed a framework that was so important for the development of the Industrial Revolution. So very technology-oriented motivations for the development of a fundamental framework that has been denused until now in many, many different contexts. And that has evolved, has evolved in time, in a quite ample way, quite ample breadth of ways. So rather than focusing on the various facets that the evolution of thermodynamics has undergone so far, in these three lectures, starting from today, we are going to go through the framework for non-equilibrium quantum processes, so the framework for non-equilibrium thermodynamics at the quantum level. Why? Because when you deal with a microscopic system, so these days we are, say, the hype linked to quantum technologies is growing. I mean, the interest in that is growing as well. When you are interested in the development of a framework for quantum technology, you are interested in kicking your systems considerably out of equilibrium. You want to process information.  So to say, you are really interested in genuine non-equilibrium processes. And therefore, this is the natural scenario to consider when trying and build up an interface between thermodynamics and quantum dynamics. The second motivation, or at least the second direction along which one can go, is more fundamental. And this link to the following question, say, are the concepts of work, heat, entropy that we use in macroscopic standard thermodynamics, if you want. The one that you are used to when you open, or the one that you have seen, by opening the book, when opening the book by Kalen, or the book by Zemanski. I studied on Zemanski. My thermodynamics course was based on that book. Now, you open that, and you have definitions of what work heat entropy is from a macroscopic thermodynamics sense. Are these concepts still valid, or do we need to reformulate them when we are interested in generally quantum mechanical processes, when the objects, the media upon which we reform the transformations that we are used to implement in a thermodynamic sense, are implemented on microscopic systems. And even, say, digging even more, say, into the foundations, Martin has introduced the concept of quantum resource and resource theory. And we're going to that probably. I don't know if this is the case, but I think you are going into resource theory for coherences towards the end of your lecture. So can we study, can we use what we know of resource theory? What we know about quantum coherences, quantum correlations to understand and to expand the domain of thermodynamics and make use of them to design better devices, thermodynamically inspired devices for the processing of information. These are the, somehow, not the three directions along which you might want to go when you address thermodynamics of quantum processes. To me, what is, as any lectures, this is a highly subjective presentation of a field. So just take it as my own perspective. The study of thermodynamics and interplay with quantum processes is interesting for two reasons. On one hand, it provides a very natural framework for the assessment of what we call the quantum to classical transition. So the process, according to which, or due to which, any quantum system loses its quantumness, its quantum nature, and get, basically, much more mundane, becomes describable through the laws of classical physics. Now, this diagram here should be taken as highly qualitative, so that's why I don't have anything very clearly specified along the axis, but what I want to illustrate with that is the common belief that if you look at the quantumness in a given system, and I'm very fuzzy about that, so I'm not defining what I mean by quantumness, take it as your preferred indicator of deviation from classical dynamics. For instance, the amount of quantum correlations that you have in a given register. And on the horizontal axis, they have an equally undefined or not very well-defined concept of complexity, which you can take as the size of your device or the number of elements that pertain to it, the number of particles that you have in your box or the mass of the systems that you have. What is commonly believed is that if you increase the degree of complexity of your system, then seeding quantumness in that or maintaining the quantum features of such a system becomes a much more difficult task. I mean, we might agree or not about that, I mean, I think it's a highly case-dependent problem. But say, without any request for a canonical nature of this plot, just take it as it is. So thermodynamic is an inherently many-body theory. I'm going to contradict this statement in a minute by addressing the dynamics of a single, from the thermodynamic viewpoint, of a single particle. But if you think of thermodynamics, what you have in mind is a gas or a piece of material across which you want to let some heat go. So macroscopic systems, complex systems in a very loose sense. So it's a perfect scenario to, say, adopt if you're interested in precisely this point in characterizing how quantum features are lost due to the growing complexity of your device. And more technologically oriented, well, just like I said, at the beginning, thermodynamics was somehow originated to improve the working principles of weapons and then machines. So one can take inspiration from that and dream of building thermodynamic machines, cycles and genes that might make use of elements of quantumness to boost their performance. Now, whether or not this is the case, we still don't know. There are steps towards that that are currently being taken. So this is just to give you an idea that both at the foundational, say, level, the fundamental level, and more technology-oriented level, there are open questions when you address the interplay between quantum and thermodynamics. And that probably makes the study of this field worthwhile. Now, a long introduction to something that is structured as follows. So this is a rough schedule of the discussion that we are going through in these three lectures. So I will start with the redefinition or the introduction of concepts like work, and in the second lecture we go to heat when you deal with an explicit non-equilibrium framework for thermodynamics and when your process, the transformation that you are dealing with, is inherently quantum. The second lecture, so roughly speaking, tomorrow, if I stop talking too much about motivations and so on, we'll deal with Landauer principle. And our Landauer principle can be intertwined with quantum-open system dynamics. Thirdly, and this is where the crux of the discussion is, now we are going to talk about irreversibility and entropy production. And they stay here in closed quantum system, but we are going to have a, say, to visit also the open dynamics, the open system dynamics framework. And if I have enough time, I'll go to a very brief discussion of, say, the interplay between the theory of quantum correlations and the framework for the characterization of quantum coherences that maybe Martin will talk about, and thermodynamics. So this is a rough schedule, so don't take it as a contract. So I'm very much likely not going to stick to this detailed plan. I'll be happy if we address irreversibility at some point in this discussion. So, without further ado, we start with the introduction of the non-equilibrium thermodynamic framework at the quantum level. And the objective here, the goal, is to go to what are called fluctuation theorems. Now, before we start, say, getting, not warming up and discussing really the technical aspect of the framework, is there any question at this point? Anything unclear? Anything you want to ask? Okay, great. Feel free to stop me anytime. Let's see how it goes. So I recommend you to have a look at this paper, okay? If you want to somehow to get a flavor of the possible discrepancies, the differences between quantum formulations of non-equilibrium thermodynamics and classical ones, well, this paper provides a very nice introduction. And the title, although I have taken, say, the abstract and the title that I have taken from the archive version of the paper, I think it's faithful to the published version of the paper itself, which is in PRE 10 years ago now. So the title illustrates exactly, I mean in a sentence, what the crooks of the messages, no? Fluctuation theorems, work is not unobservable, full stop, right? So this is some form of death sentence, right? So don't even think about the possibility of going into the lab and devise an experiment that directly provides you, directly provides you the amount of work that you can perform on a given system as, say, for instance, the eigenvalue of a given observer, right? That's not the case. According to this paper, you have to be a little bit more inventive and devise smarter ways of assessing how much work you are doing on a system or how much work a system is doing when you address work from a dynamic work from a quantum mechanical viewpoint. It appears somewhere in the, already in the first column of the paper. Yeah, yeah, yeah, already in the first, in the first.  Okay, so let's see what these guys had in mind when stating this, okay? So am I going to quantify the amount of work that you are performing on a system that evolves unit highly, okay? So this is the frameworks for now. And it will be that until the end of this discussion and probably for a little bit tomorrow's discussion, okay, tomorrow's lecture. So I have a quantum mechanical, I have a system, I implement a transformation on it that let it evolve unit highly. And then you might wonder, okay, I'm changing the energy of this system. How much work am I doing on it? Or how much work can I extract from the system itself? And this paper provides you with an operational way of actually going into it and providing an answer to that. So, let's start with a familiar scenario. You have your, say, your favorite quantum system characterized by its Hamiltonian H.I. And a lot earlier than the start of your experiment, you have put it in contact with a bath at a given temperature. And you waited long enough for the system to thermalize with the bath itself, okay? Such a way, so this T smaller than zero means that I have done it as a preparation stage. So before the actual experiment starts. So at T equal to zero, that is when the experiment starts, what you do is that you detach your system from the bath. So now your system is in a thermal state at the temperature of the bath. It was at equilibrium with the bath. So I've detached it from that, assuming that the touching process didn't alter the equilibrium, say, the state of the system itself. And what I do is that I perform a measurement on it, okay? So what I do is that I ask myself, okay, what is the chance, what is the probability that I find the system at this instant of time in the nth energy eigenstate of this Hamiltonian H.I., okay? So I have a well-defined energy. I'm asking, oh, I have a very well-defined Hamiltonian. And I'm asking what is the probability that my system is in the nth energy eigenstate of this Hamiltonian, immediately after I detached it from the contact with the environment, okay?  And the not here, so the zero here, is because this is the instant of time T equal to zero. And then the A is because I'm asking how much is the probability of finding the particle in the nth, or the system, in the nth energy eigenstate. Now it's the, so far, very, somehow very passive, right? So you observe. We now dig into the active bit of the protocol. Okay, so I'm now taking a wrench, I'm changing something in the system. So what I'm doing is that I'm changing the Hamiltonian of the system itself. So I have a parameter in principle a time-dependent parameter that I decide to change in time. And I do it, given that my system is completely now isolated from its environment, right? So besides the bath, there is no external word, so the system is completely isolated. And from this point on, so from step before onward, there will be no environment whatsoever. So the dynamics that is induced by my modification of the parameter characterizing the energy of the system induces a unitary evolution. Fair enough. So my system is evolving in time according to this time evolution operator U. For a time tau, that I decide arbitrarily. At instant of time tau, I have the end of my protocol. Okay, at the end of my experiment, which I conclude in the following manner. So as I said, I've changed the Hamiltonian of my system from h i to h f. So h f is the final Hamiltonian. And now I do again what I did at t equal to zero. That is, I measure, no? I project onto one of the energy eigenstates of the final Hamiltonian, so of h f. And I wonder how much is the probability? And this is a conditional probability. How much is the probability p m known n? So this is the conditional probability to find the system at time t equal to tau in the nth energy eigenstate of the final Hamiltonian. Given that t equal to zero, the system was found in the nth energy eigenstate of the initial Hamiltonian. Does it make sense, guys? It's more difficult to explain than to write down. It's easier to understand what that is in symbol than to explain it. Are we okay with that? I measure d t in nth energy eigenstate of the initial Hamiltonian. The guy evolved. Then I measure again energy in the nth. This time I have a different set of eigenstates because my Hamiltonian has changed. And this is the conditional probability that I can get. Fair enough? Now, question. Suppose that I repeat this very same experiment a zillion of times. And for now, let's concentrate on the first step, so on part b only. So I do it many, many times, right? Will I get always the same eigenvalue if I measure the energy at t equal to zero? Objuzno, no. Because my initial state, the state of my system was a thermal state, right? So these probabilities will follow a Boltzmann distribution. I have a thermal state, that's what I get. So every time I repeat the experiment, I get a different outcome of energy. And the probability to get the nth energy eigenstate will follow Boltzmann distribution. So I have an element of thermal randomness in this experiment that is of an entirely classical nature. No quantum mechanics is concerned at this point or is involved at this point. Then again, now I have my zillion copies of the same experiment. I'm repeating my experiment many, many times. And they perform my second measurement. By the way, this protocol is usually called the two measurement protocol and you understand why, right? So you are doing two measurements, right? So will I get the same, no? Will I get the same energy eigenstate? No, will I find the system all the time in the same energy eigenstates at the end of the protocol? No, because my system has evolved quantum mechanically. And we know that if I perform a measurement, right? In quantum mechanics, the outcome is only probabilistic. So I have a second element of randomness in this process, which is, somehow, enforced by quantum mechanics. So if you repeat your experiment the zillion times that I mentioned, what you end up with is a distribution of values. It's not a single valued experiment, no? You have a distribution of values. You have many values of PM, no men. You have many values of PM, not. So what do you do with this distribution, right? You build the work probability distribution. So you immediately find out that according to this framework, work becomes a stochastic variable, right? How much is the work that you are doing in this, following this process? Well, the system was isolated. I've changed its energy. So this change in energy must all go into work. What is the change in energy? Well, it will be the difference between the energy of the final eigenstate that I'm finding the system into minus the energy of the initial eigenstate that I found the system into when I'm performing the first measurement. Does it make sense? Yeah? Please. Okay. Very good question.  There is a bit of noise down here. No, no, I think I have heard it. So the question is, how about the cost of actually, not changing, right? Changing, not wrenching my system, not changing it. Yeah, we are going to ignore that cost for the discussion. You are absolutely right. I should account for all possible sources of work done on the system or possible welds for the work that I'm doing, right? On the system itself. The framework that I'm going to use at this stage is and set by that paper is such that basically all the thermodynamic costs or the thermodynamics relevant change in energy, energetics are all in this energy difference. In principle, I should reformulate in a way to put in the actual agent that changed the Hamiltonian itself. I might go farther and say, yeah, I mean, I'm performing two measurements here. Is this cost free?  This is a perfectly legitimate question that has been addressed. So there are papers working out or trying to work out the cost of general measurements, both projective and generalized. And it's a much more involved framework that I want to stay away from for now. Okay, I understand it's not a very satisfactory answer, but say, let's go step by step, if you allow me. Any other question? Costless at this stage? Costless, say, if you. Okay, so as I said, we have this bunch of numbers. What are we going to do with them? Well, I want to organize them in a distribution, in a probability distribution made out of delta peaks. These delta peaks are sanctioned precisely at the amount of work that I'm doing, by changing the energy of the system. Okay, so by changing the energy of the system from en to em dash, these are the two outcomes of my two measurements. How tall is each peak, right? It's delta peak that I have in this distribution. Well, the height of each peak is given by this product, by this product, right? By the probability to find the path, the system in the nth energy against data t equal to zero, and the condition of probability Pm known n, at time tau. W is the work that I'm doing, and this is simply the difference between. Average of hf? No, no, no, no, no, average. No, no, no, it's simply defined as the, if you want the point-like difference between em dash and em. So, there is no other, say, we are excluding all other possible costs, and we are excluding any environment, so the energy, the change in energy of my system all goes into the work that you are doing on the system, or that the system does. Sort of. Is there a construct of perpetual modeling? Aha, correct. I don't think you can build a perpetual motion machine from that, but, no, no, no, of course. Of course, say, a ratio should be there. A ratio should be there, so the reset step has to be included. And that costs, of course, that costs. I'm not resetting at this point, say. There is a, yeah, so the question was, whether I can, if any measurements, so if all measurements in this setting are cost free, can I build a, basically, a perpetual motion-like device through that? And the answer is that I think you escape from it by putting in your framework the necessity of resetting your machine. At this stage, we are not focusing on that. Okay, so cost of resetting will be introduced tomorrow. No, no, no, this is a single transformation. And if you want, even if you want to re-use the machine after the first transformation, then that is, you start a new, and you proceed from there. Say, at the start of my assumption is that the initial state of the system is a thermal state, so I have to re-initialize my system into such a thermal state. Okay. So we now have a probability distribution. We are now sort of agreeing on the fact that this work becomes a stochastic variable and is distributed according to this probability distribution. Now, there is a label here, F, which is going to be, so the presence of this label will be clarified in a bit, but this basically stands for forward. So this is the process according to which I'm changing the Hamiltonian of the system from HI to HF. Okay, at some point, possibly tomorrow, given the time, 12.30, right? I have to stop at 12.30, am I right? 35, okay, so 10 minutes, 45. Oh wow, okay, so we have time. So at some point we are going to introduce a label B, which will stand for backward process, and that will be the process that takes the Hamiltonian of the system from HF to HI. Okay, so it's the time reversed, it's the time reversed process. Now, I'm acquainted with T-Shum by training, so you give me a probability. Fully Hamiltonian process. Fully Hamiltonian process. Completely reversed. Completely reversed. But again, so you can understand what I'm going to clarify, but say, take two different sets of experiments. One where I initially give you HI, and you are asked to change it into HF, and one where I initially give you HF, a completely set different set of processes, and you change it all the way down to HI, okay? Mind you guys, I've not been prescriptive about how you are changing the Hamiltonian of the system. So, I didn't say anything about, say, first of all, how quickly you are changing this Hamiltonian, right? So, it might be extremely slow in time, right? So, you can be very close to the paradigm of quasi-static processes, which is what you read in standard thermodynamics book. You have to be quasi-static to be able to define at every instant of time thermodynamically meaningful quantities, heat work. Or you can be very quick, no? You are impatient like unusually, so you change the energy very quickly, no? Non quasi-statically. I didn't put any constraint on that, so you are free to change, or actually Fabrizio is able to, is free to choose any way of implementing such transformation. This, of course, has consequences, but we are going to see that for things like fluctuation theorem, it's actually highly immaterial, so to say, the way you implement the transformation itself. Now, what I was saying is that, if you give me a probability distribution, given my background, I'm naturally inclined to take the Fourier transform of it and look at what is called the characteristic function of it, okay? So, let me introduce the characteristic function for work distribution, which is what I'm going to call chi from now on. U is the conjugate variable for work there. And this is defined as the Fourier transform, the complex Fourier transform of my probability distribution for work. Now, why am I making it more complicated than what it was already? Why am I introducing this characteristic function? Because sometimes, sometimes it's easier to work with the characteristic function than with the probability distribution itself. So, there are problems that are more easily addressed by means of the characteristic function than the probability distribution. And vice versa, okay? So, it's, again, a very problem dependent, case dependent problem, okay? But now, and this is when I want to, I want to do the first calculation of this set of lectures. Let's try and work out together this nice expression here, which is the general form that the characteristic function for work distribution takes when you implement precisely the protocol that I have explained so far.  U dagger is obviously, no? The emission conjugate of it. Lambda is the parameter that I'm changing in time. So, it's the guy that defines basically the process itself. Lambda goes from lambda naught to lambda tau. And if you take h of lambda naught, that is my, I had some chalk here. So, h of lambda naught, you can assume, is your h i, and h of lambda tau, on the other hand, is your final Hamiltonian hf. Is my handwriting historically bad. People complain every time. Is it okay? Can you, yeah? Whenever you don't understand what I'm writing, it's my fault, and you tell me, okay? So, just to warn you, this is my x, right? So, when you don't understand what I write, that's very much likely an x, yeah? So, what is a rho g here? Rho g of lambda naught is my initial thermal state. So, this is the initial state of my system, which you can write like that in terms of the initial Hamiltonian of the system itself. And lambda and z, z will be used to label the partition function. So, the trace of this quantity of e to minus beta h of lambda naught. The last parameter that I need to define is beta. Beta is one over kT. So, it's the inverse temperature of the bath. Okay, so, I'm throwing this expression on this slide. Let's see how we get it, yeah? Let's see how we get it starting from the actual definition of the work probability distribution that I've given you here, okay? So, again, so, you will have to suffer my handwriting. Let's see how it goes. First of all, so, let's write down what the probability distribution for the forward processes. So, as I said, these are sum over n, and then, right? The two labels for initial and final eigenstates, so, initial and final energy eigenvalues of my measurement, p of m, known n, at time tau, times pn, at time zero, times my delta peak, centered at the difference between the final energy, em dash, and the initial energy, em. Okay, makes sense. Okay, what is this guy, right? We define the initial and Newtonian hi as having eigenstates and eigenvalues, en, and n, okay? And I'm going to say that hf, the final and Newtonian, has energy eigenvalues, em dash, and energy eigenstates, m, okay, right? Right, so, what is pn naught in this notation? Pn naught is e to minus beta, en over z, right? Z of lambda naught. So, I'm going to set this notation, z of lambda naught. So, the partition function associated with the initial state of my system, z naught. And I'm going to call the partition function associated with lambda tau at z tau. So, pn naught, easy peasy, that's what we have. No, ga, see, Boltzmann distributed. No, okay, let's call them em dash. These are two different sets of eigenstates. If you are commuting a Newtonian, becomes very, very boring, you can do it, of course, but it becomes extremely boring. I'm assuming that these two sets, no, are different sets of eigenstates, okay? So, the two Hamiltonians don't commute. Okay, what is pm known n at time tau? Well, no, let's follow the prescription. This is the probability to find the system in m, given that it was in n at time t equal to zero. So, I found the system in n, I've implemented my process, my transformation. So, I've implemented my time evolution operator u, until time tau, and then at that time, I've performed my measurement, my second measurement, getting my state em dash. So, let's call these guys em dash, dash, dash. Well, there is already a dash there, so I don't need to dash anything else, okay? And this must be square modulus, to get my probability. Okay? So, I'm now going to put these two guys into my probability distribution for work, and I'm also using the definition for the characteristic function for work distribution. Are you okay if I delete these two guys? Is it clear what h i and h f are? Okay, so I'm going to get rid of them, and I write explicitly that chi f of u is equal to the integral over all possible values of work, of my forward probability distribution for work, e to the i u w. No, I'm not, and this is extremely, no, and this is actually, so basically giving up the, so, discovering who the mother is, okay? Because we are going to, we are going actually to, I'm going to stress the fact that in principle, your process doesn't take you to equilibrium, and this has very, very strong implications at the thermodynamic level, okay? So, no, by no means, I'm at equilibrium. Okay, so I now put in my definition for probability distribution of work into this integral, and I'm using these two definitions there. So, I'm going to get a sum over n and m dash of an integral over the work w of e to minus beta em en over z naught times this guy, no, the square modulus of m dash u tau n square modulus. Then I have a delta of w minus em dash plus en. And then I have the final guy, which is e to i u w, which is set, no, is enforced by the fact that I want to do this Fourier transform. Okay, now I use the, no, the nice properties of Dirac delta sum, the integration. So, this gives me, this gives me the sum over n and m dash of e to minus beta en over z naught. Now, I'm going to split this square modulus into one term, and it's complex conjugate, right?  And then here, in this exponential, I have w, so I'm going to replace w with em dash minus cn. So, e to di u times em dash minus cn. Okay, now a little bit, we are basically done, because what follows, simply a rearrangement of things, a trivial rearrangement of things, okay? So, I'm going to rewrite this expert, everyone okay with that? I've just opened up the definition of the probability, the conditional probability, and use the property of Dirac delta under integration. So, now I'm doing a sum over m dash of, well, I can take this guy here, and I can take this guy up there, right? And there are c numbers, so I can play with them as I want, right? So, I have m dash u of tau n, then I have e to minus, I have a sum, sorry, the sum is over m and n, okay? M dash and n, then I have e to minus beta en minus i u en over z naught, then I have an m u tau dagger m dash, and then e to the i u em dash. Did I forget any term? I think I didn't, should be okay, yeah? Now, guys, I can do another trick, right? So, I can actually, and I'm doing it because I can, I'm doing it on the blackboard, it should say, should this be an undergraduate lecture people would have killed me, so I'm doing it right now, you forgive me for that. So, what I'm doing is that I'm taking e to minus i u en, which is a c number, and I'm plugging it here, okay? I can do that, so that is perfectly legitimate. So, I just move a little bit my time evolution operator u, and rewrite here e to minus i u en, and delete it from there. And I remind myself, I remind myself that this object here, right? This is an eigenstate of my initial Hamiltonian, with an associated eigenvalue em, yeah? So, I can reinterpret this object, just these two guys as e to minus i u hi, and make sense, yeah? Okay. So, now I have, I want to continue in this direction, so now I have a sum over n, and m dash of m dash, u of tau, e to minus i u hi, staj 10, e to minus beta en, over z naught, staj 10, then I have u of tau dagger, staj 10 dash, and I'm doing now the very same thing I did here, this time, on these two terms, okay? So, I'm taking this object here, okay? So, I have too much space, u tau dagger e to di u em dash, and I remind myself again, that this guy was an eigenstate of the final Hamiltonian, with this eigenvalue. So, I can take these up, these two guys, and reinterpret them as e to di u h final m dash. Make sense? They can do something else, no? Even more unfriendly for you that writing, right? So, I can say, well, I have a double sum over n and m dash, this guy does not depend on n, this guy does not depend on n, the only guys that, this guy does not depend on n, the only guys that depend on n are these three terms, so let me move this summation inside, and I want to isolate these three guys together. Why? Because this is simply, the spectral decomposition of my initial state, rho g of lambda naught. I want to follow the same notation that is there. No, this is my initial state, my initial thermal state, nothing fancier than that. So, now, I'm getting things that look a little less unfriendly, no?  And this is a trace, nothing else, right? I'm taking the trace of this object. Make sense? Which is a representation of my characteristic function for work distribution. So, I think this is a good point to stop. We go back to what we need, so which sort of use we make of this object tomorrow, and we extend the framework from closed system dynamics, so from unitary processes, all the way down to open system dynamics, characterizing in the very same way we did for work, the distribution for a new stochastic variable, which will become heat. And that will give us the chance to illustrate a bit, the interplay between open system dynamics and thermodynamics in light of Landauer principle. Okay, any question? Over lunch, thank you so much.