Hello everyone, my name is Sam Vaknin and I'm the author of Malignan self-love narcissism revisited sake So today we are going to discuss yet another disagreement between Freud and his erstwhile disciple Yung And I'm in a bit of a pickle here because I'm a Jew and I'm a narcissist as a Jew Obviously, I should support Zygmunt Freud against the German Yung And as a narcissist, obviously, I should support the psychopathic Yung against the relatively middle-class bourgeoisie Freud So I don't know what to do. I will let you be the judges I'm going to present to you the two views of Freud and of Yung regarding narcissism They are not diametrically opposed, but they are like 80% or 70% different and Yung's view Did not prevail Freud's view of pathological narcissism is the current orthodoxy. It's what we teach at school Universities, colleges, what you can read in textbooks and of course The prevailing view online. It was also my point of departure when I started the whole narcissism craze in 1995 But today, 25 years later, I'm older More handsome and consequently I think more clearly and I am not quite true that Freud had been a hundred percent right about Pathological narcissism. I'm going to present a more subtle and nuanced variant nuanced view So be prepared because Um Yung is not easy. Yung is not easy. He was a Strange bird to use another statement And when it came to narcissism, he had equally strange notions coupled with amazing diamond-like incisive insights Generally speaking in the narcissist the false self usurps the role of the ego So narcissists don't have an ego don't have a functional ego exactly the opposite people people say that narcissists ego teases They don't have an ego the false self fulfills these functions the mediation between the individual and the world in a sense of personal continuity and identity These are these reside in the false self and because the false self is false. It's a piece of fiction It needs constant input from other people which we call narcissistic supply Yung used a totally different language. He used terms like shadow complexes images archetypes repressed material And he used this terminology to describe the narcissist's early childhood So the video is divided into two parts. The first part is Yung and I assume that all of you know the basics of psychoanalysis The trilateral model what is ego? What is superego? What is Eid? Um drives urges pleasure principle reality principle, etc. If you do not Please fast forward to the second half of the video And watch that second half first There's a second half describes Freud's view of all these issues Freud preceded Yung of course and we're still using his language. So you can if you know The basics the rudiments of psychoanalysis watch the video as it unfolds If you don't watch the first the second part first the first part later So the classical Freudian concept of the ego is that it's partly conscious Partly pre-conscious and partly unconscious It operates on what Freud called the reality principle as opposed to the Eid Which operates on the pleasure principle The ego maintains an inner equilibrium between the onerous and unrealistic or ideal demands of the superego And the almost irresistible and unrealistic drives of the Eid So there are like three guys inside everyone The Eid is very basic very primitive very disorganized It's wants to eat it wants to drink and of course, it's one it wants to have sex Actually, it mostly wants to have sex Then there is the uh, and it reminds me of some some people. I know Then there's the superego and the superego is all the The inner critic voices the intro checks the chastising Diminining debasing degrading humiliating something critical voices That we accumulate your life could be our parents teachers peers Other voices as on and they all call less into the superego and superego Contains a component which is known as conscience.   You're gonna end up spending 20 years in jail without sex And then it goes to the super ego and says could you could you chill a bit? Could you stop criticizing this criticizing this poor guy so horribly and could you give him a break? So the ego is like an arbiter or a mediator And it also has to fend off the unfavorable unfavorable consequences of comparisons between itself And another structure called the ego ideal. The ego ideal is what you want to be The what you want to be when you grow up when you mature when you have acquired skills If you were more talented, if you were more handsome or beautiful, whatever. So this is the ego ideals ideal The superego uses the ego ideal to torment and torture the person It tells the person see how far you are from the ego ideal, you know, here's the ego ideal Here's you you will never bridge his gap and the ego comes in and says listen Some parts of the ego ideal are realistic if you only put your mind to it You have to you know, you have to study you have to work hard This so in many respects the ego in Freudian psychoanalysis is the self But not in Jungian psychology So Jung was a psychoanalyst. He was Freud's disciple before they broke up And before he rebelled against him in an act of symbolic patricide And he was very controversial. He was also very unethical. He slept with his patients. He was in my view a psychopath But a brilliant psychopath I'm going to quote from his work and all these quotes are from his collected works These Collected works were edited by G. Adler, Fordham and Reid. It's 21 volumes. Yes Princeton University Press was published over 23 years between the 1960s and 1983 So here's the first quote from Jung. It's from the structure and dynamics of the psyche Volume 8, page 121 if you want He says complexes are psychic fragments which have split off Owing to traumatic influences or certain incompatible tendencies As the association experiments prove Complexes interfere with the intentions of the wheel and disturb the conscious performance They produce disturbances of memory and blockages in the flow of associations Complexes appear and disappear according to their own laws They can temporarily Obsess consciousness or influence speech and action in an unconscious way In a word complexes behave like independent beings a fact especially evident in abnormal states of mind In the voices heard by the insane Complexes even take take on a personal ego character like that of the spirits who might be in a state of mind Spirits or manifest themselves through automatic writing and similar techniques He continues in another work titled the archetypes and the collective unconscious Volume 9 page 275 He says I use the term individuation to denote the process by which a person Becomes a psychological individual that is a separate indivisible unity or whole And he further elaborates on individuation in two essays on analytical psychology volume 7 page 266 He says individuation means becoming a single Homogenous being and in so far as individuality embraces our innermost lust and incomparable uniqueness It also implies becoming one's own self We could therefore translate individuation as coming to selfhood or self realization Not to be confused with self actualization must laws Jung continues in the structure and dynamics of the psyche volume 8 page 226 But again and again I note He says that the individuation process is confused With the coming of the ego into consciousness and then the ego is in consequence identified with the self Which naturally produces a hopeless conceptual model.  It's it's the ego plus. It's an archetype It's actually the archetype It is the archetype of order as manifested in the totality of the personality and as symbolized by a circle a square Or the famous quaternity Sometimes Jung uses other symbols. He was very very pot and strong and symbols He would have been a symbologist like, you know, Dan Brown's character so He uses the child uses mandala the mandala and so on he says the self is a quantity that is Supraordinate to the conscious ego It embraces not only the conscious but also the unconscious psyche And it's therefore so to speak a personality Which we also are There is little hope of our ever being able to reach even approximate consciousness of the self Since however much we may make conscious They will always exist an indeterminate An indeterminable amount of unconscious material which belongs to the totality of the self This is what we feel in consciousness is not the self. It's part of it In psychology and alchemy He was also Very big on the occult. He studied ufo's and alchemy and he came up with Nonsensical concepts such as synchronicity and collective consciousness and and so on. He is not by far my favorite psychologist or Example of a rational person. I think he's a throwback to the middle ages in many respects. He is he is an anti Profit He acts against enlightenment and he's a very irrational person in many of his of his writings, which I find rank nonsense But there are diamonds in there Here's one of the diamonds. He said the self is not only the center But also the whole circumference which embraces both conscious and unconscious It is the center of this totality Just as the ego is the center of consciousness and that's A diamond observation an insight that escaped actually Freud In two s's of analytical psychology young Kind of ends with an aphorism The self is our life's goal for it is the completest expression of that fateful combination that we call individuality Okay So it's clear by now Ego is the ego self is self ego is a component of the self. They are not coterminous. They're not the same The same thing as Freud suggested And so now we go to a Jungian another Jungian. His name is Jeffrey Satinover and he wrote a brilliant essay Where are items the narcissistic relation to the self? That essay has deals with another issue That's the fact that that narcissists are immature that they are eternal children or eternal adolescents And so he tries to explain why but he sums up At the beginning of the essay which is 18 or 20 pages long He sums sums up the differences between Freud and Jung and he says and um, um, it's an I'm quoting him extensively he says Freud considered that all people begin life in a blissful state that he called primary narcissism In this state no distinction between self and world exists Hence no painful tensions in the form of as yet unfulfilled desires of the subject for any object And therefore no conscious experience of drives and frustrations He says just to summarize this segment.  You don't want for anything. You don't lack anything By the way, that was Spinoza's attribute of God. He said God Spinoza said God cannot want anything because he includes everything How can he want anything?  He continues Satinova As the infant develops it separates itself from its surroundings and begins to experience needs for other things As the infant grows these needs put pressure on the developing ego to acquire the skills necessary to fulfill these needs And so the ego adapts to object reality All the energy which an infancy was bound to the subject In this way Slowly extends out and becomes bound out bound up in the subject's pursuit of objects So at first the energy was inward and then it's externalized to obtain objects. This process is normal development Satinova continues Freud originally described the essence of neurosis as an interruption in this smooth transition from subject bound to object bound libido The childhood libido reaches out fascinated by the objects of its desire But being as yet insufficiently adapted to succeed It fails to attain its goal To compensate for this failure in adaptation and for the consequent lack of gratification An alternate easier form of gratification is sought One with which the ego is already familiar The libido regresses and reactivates an earlier form of adaptation It reactivates the blissful state of narcissism now called secondary narcissism In this view a narcissistic neurosis consists of the habitual seeking of gratification Through self-stimulation and the consistent refusal to take the more difficult path of adaptation or work The grandiose fantasy is preferred to the modest accomplishment The brief idealized affair or masturbation is preferred to the rocky long-term commitment Jung's modification of this idea is that says Satinova Is that the retreat to earlier forms of psychic life and behavior to secondary narcissism Is not only or even primarily an alternate means of gratification It is rather the necessary way that has yet unused Instinctive modes of adaptation latent within the psyche are released Thus the retreat to the narcissistic state releases archetype archetypal fantasies These fantasies are the representations in consciousness Of inherited but as yet unused adaptive behaviors I beg to differ with the word inherited because it implies a kind of Collective unconscious unconsciousness that we all tap into One of Jung's ideas, which I find to is a british understatement unacceptable, but still The regression according to Jung is not in itself neurotic says Satinova But rather it is the sign of a compensatory process of the psyche whose purpose is enhanced adaptation Early in his career Jung equated narcissism with introversion The general notion that introversion per se is pathological Stems from the early Freudian idea that narcissism is a substitute employed Where adaptation to object reality or extraversion has failed In consequence of his expansion of Freud's conception Jung separated the two terms And the general turning inward of libido Introversion was recognized as a servant of psychological development rather than an enemy to it In his book psychological types Jung suggested that introversion does not occur only in response to failures of extraversion But that the habitual turning of attention inwards to the self Is a normal function of the psyche Which in some individuals actually predominates in degree over the habitual turning of attention towards objects to summarize According to Jung narcissism or introversion can be one the pathological state And here he agrees with Freud But more often A compensatory response regression in the service of the ego and to further personal development And three a normal form of personal of psychological development And this last idea of of Jung means that there is There is such a thing as normal narcissism Jung says that there is such a thing as healthy and normal narcissism It implies that to some extent narcissism or introversion is actually a necessary aspect of all individuals And that like adaptation to the outer world. There is such a thing as better or worse thoughts of adaptation to the inner world We adapt to the outer world. We adapt to the inner world Sometimes we succeed to adapt to the outer world. Sometimes we don't Sometimes we succeed to adapt to the inner world. Sometimes we don't and when we don't Narcissism becomes pathological, but when we do it's novel and healthy That neurosis can develop which are narcissistic Not in the sense that narcissism per se is a neurotic response to failures of external adaptation In other words, the narcissistic neurosis is not because of the narcissism But they are they are narcissistic in the sense that they are failures to develop healthy introversion Failures to develop a proper form and degree of narcissism Jung postulated the existence. So this was Satinova this excellent overview and expose of Jung's attitude to introversion and narcissism Jung postulated the existence of two personalities actually two selves One of them being the shadow Technically the shadow is a part or an inferior part of the overreaching overarching personality one's chosen conscious attitude The shadow develops in a peculiar way Inevitably some personal and collective psychic elements are found wanting or incompatible with one's personality personal narrative to use today's phrase And so the expression of these psychic elements, which I remind you are wanting and incompatible their expression is suppressed And they coalesce into an almost autonomous splinter personality The second personality is contrarian It negates the official chosen personality. So it is totally relegated to the unconscious Jung believes therefore in a system of checks and balances The shadow balances the ego consciousness And this is not necessarily negative as most people think The behavioral and attitudinal compensation offered by the shadow can and usually is actually positive Shadow our dark side is a positive element Jung says The shadow personifies everything that the subject refuses to acknowledge about himself And yet is always thrusting itself upon the individual directly or indirectly For instance inferior traits of character and other incompatible tendencies are aspects of the shadow And this is in his in his essay the archetypes and the collective unconscious He continues Jung the shadow is the hidden that hidden repressed for the most part inferior and guilty personality Whose ultimate ramifications reach back into the realm of our animal ancestors And so comprise the whole historical aspect of the unconscious. That's the part I strongly disagree with If it has been believed here that too that the human shadow was the source of all evil It can now be ascertained on closer investigation That the unconscious man that is his shadow does not consist only of morally reprehensible tendencies But also displays a number of good qualities such as normal instincts Appropriate reactions realistic insights creative impulses and so on It would seem fair to conclude that there is a close affinity between the complexes Split of materials remember and the shadow Perhaps the complexes also the result of incompatibility with the conscious personality Yes, they're also the result of dissonance attention Perhaps the complexes are the negative part of the shadow. Perhaps it just reside in the shadow Or I don't know closely collaborate with the shadow in a feedback mechanism Perhaps perhaps whenever the shadow manifests itself in a manner which is obstructive destructive or destructive to the ego We call it complex Complexes may really be one and the same the result of a massive split off of material and its relegation To the realm of the unconscious and this is part of past and parcel of the Individuation separation phase of our early childhood development Prior to this phase the infant begins to differentiate between self and everything that is not self The infant tentatively Explose the world and these excursions bring about a differentiated worldview suddenly the world is populated with objects suddenly The unity Cracks and breaks. It's a break of the world. It's cartesium. It's like the carp Suggested that as an observer and observed us and nature We and the world the child goes through this early on child begins to form And store images of himself and of the world initially of the primary object in his life normally his mother And these images are distinct First time it's very traumatic mother is not me She is separate and if she's separate I can lose her She can abandon me to the infant. This is revolutionary stuff Nothing short of a breakdown of an erstwhile unitary universe and his substitution with fragmented unconnected entities. And as I said, it's very traumatic Moreover, these images in themselves are split The child has separate images of a good mother and a bad mother Respectively associated with the gratification of his needs and desires and then she's good Or with the frustration of his needs and desires and then she's bad mother The child also constructs separate images of a good self and a bad self linked to the ensuing states of being gratified By the good mother and being frustrated by the bad mother Everything fractures at this stage the child is unable to see That people are both good and bad that an entity with a single identity Can both gratify him and frustrate him at times. He has no concept of time And so he splits he derives his own sense of being good or bad from the outside The good mother in Inevitably and invariably leads to a good satisfied self And the bad frustrating mother always generates a bad frustrated self But the image of the bad mother is very very threatening Imagine imagine a rich conclusion that the person you depend on for your life Is evil. It's it's frightening. It's a horror movie.  It's like a bad bank There's a kind of reservoir where he puts all the bad feelings bad emotions frustrations Bad images of mommy and everything this process is called object splitting It is the most primitive defense mechanism When still used by adults For example in borderline personality disorder narcissistic personality disorder. It is an indication of pathology And this is followed by the phases of separation and individuation between 18 and 36 months of life The child no longer splits his objects He no longer puts put bad objects to one repress one repress side and good objects to another conscious side He learns to relate to objects to people as integrated holes Holes with w1 As with nuances with subtleties with the good and bad aspects coalesce Gray shades instead of black and white and integrated self-concept also follows inevitably Because if you if you merge the good and bad aspects and you begin to see people as they are You begin to see yourself as you are partly good partly bad It's integration the child internalizes the mother. He memorizes her roles He becomes his own parent becomes his own mother and he performs a function by functions by himself He becomes way less independent personal autonomy He develops boundaries. He acquires object constancy. He learns that the existence of objects does not depend on his presence or on his vigilance Mother always comes back To him and for him after she had disappeared from sight A major reduction in anxiety follows and this permits the child to dedicate his energy to the development of stable Consistent and independent senses of self and interjects internalize images of others And this is the junction at which personality disorders form between the ages of 15 months and 22 months A sub phase in this stage of separation individuation is known as raposhmon Raposhma the child at this stage is exploring the world This is a terrifying and anxiety inducing process Terai kognita here be dragons the child needs to know that he is protected That he's doing the right thing and it is gaining the approval of his mother for this trip of exploration voyage of exploration The child peer periodically returns to his mother for reassurance affirmation admiration and unconditional love as if making sure that his mother endorses Excepts his newfound autonomy and independence is not angry at him and and accepts his um separate individuality What happens when the mother is immature narcissistic suffers from a mental pathology has her own abandonment anxiety She withholds from the child what he needs She doesn't give him approval admiration and reassurance On the contrary she feels threatened by the child's independence. She feels abandoned.  This is quiddity. This is This is the the core that it creeds founds and and vortex vortexes Vortices and tsunamis of rage the child isn't rage because he's frustrated in his quest for himself He's anxious fearful of losing mother if it's guilty for being angry at mother. He's attracted to mother repelled by mother In short, he's in a chaotic state of mind Whereas healthy people experience such eroding dilemmas now and then To the personality disorders these dilemmas are constant These dissonances are characteristic emotional states To defend himself against this intolerable vortex of emotions the child keeps them out of his consciousness Slices them off The bad mother and the bad self plus all the negative feelings of abandonment anxiety rage. He splits them off But the child's over reliance on this primitive defense mechanism obstructs His orderly development. He fails to integrate the split images The bad parts are so laden with negative emotions that they remain Virtually untouched throughout life in the shadow as complexes It proves impossible to integrate such as such explosive material With the more benign good parts and so the adult remains fixated at this earlier stage of development He's unable to integrate and to see people as whole objects People are either all good or all bad and he goes through idealization and evaluation cycles This kind of adult deformed partly developed arrested developed adult is terrified unconsciously of abandonment He actually feels abandoned Or at least on the threat of being abandoned and he subtly plays it out in his or her interpersonal relationships This is the core of borderline narcissism And so is the reintroduction of split-off material in any way helpful Is it likely to lead to an integrated ego or self? The question itself is wrong Because it confuses two issues With the exception of schizophrenics and some types of psychotics the ego or the self is always integrated There's not such thing as non-integrated ego or self That the patient cannot integrate the images of objects both libidinal and non-libidinal Doesn't mean that he has a non-integrated or disintegrative ego The inability to integrate the world this is the case in borderline or narcissistic personality disorders This inability relates to the patient's choice of defense mechanisms. It's a secondary layer The crux of the matter is not what state the self is in integrated or not But what is the state of one's perception of his self? So personality disorders are secondary layers They are like shrink wraps Pardon the pun They are like Self They're like introspective things. They're like self observation self referential That's why auto eroticism is very strong there Who is From the outside to the inside. So it's like the narcissist or borderline Has a self perception or self image or That is divorced from the integrated ego because it cannot It cannot put together every all the content Some of it is hidden shelved in the darkest resources of the unconscious And thus from a theoretical point of view The reintroduction of split-off material does nothing to increase the ego's integration And this is especially true if we adopt the Freudian concept of the ego as inclusive of all split-off material And of course, it's right if we are Jungians You


All right, so I guess we can start the talk for the sessions. Thank you very much for Professor Wan for hosting us here. This is like a brief experience sharing on teaching our caption course. So my background is in chemical engineering departments. And to set you up for the course conditions, we are teaching a final year students. So that's why we are working with these capstone course where all of the contents are the aggregations of learning experience of the students from the first year, second year, third year. Up to their final year. Now, we have the four outlines here we first introduce you about the our process design course. And we also have also introduced our MOOCs development on on on the on the course itself. And we also also briefly introduced you about the our flipped learning experience. And so we also bring you our key takeaways on on these talks. So, first thing first, I would like to briefly introduce about the chemical engineering departments. We basically try to teach about the process engineering. So it's mainly focuses on the design of a process plant, the operational, the control, and also the optimization of the chemical plant based on the chemical physical and biological processes. Secondly, we want to develop a develop talents that are able to work with the designs of process design chemical designs and try to come up with a solution that will potentially be quicker cheaper and efficient. And that's why we use our process design as our departments captain course and then this is a course that mainly tries to develop the skill of the students to work with the process departments so that they can organize they can design chemical plant and the organization part is that they need to work in groups so that they can organize themselves. And then this is an open problem course so there is no single solutions to be to be taught but this is an open problem. Development for the students so this is our syllabus for for the process design so you can see that this is a required course so every students in our department use this but not only in our departments this is like for the students of all the universities in Taiwan and moreover is also required course by universities all around the world so process design that the only single course that is required by almost every chemical engineering departments worldwide. So students are required to apply their knowledge on the unit operations chemical plant suspensions and etc we also include economic analysis and everything so this is a group work so we we use minimal supervision so they can use their time to get together with their with their peers to come up with a solutions for a process plant that is already given in the in the beginning of the course and we try to provide the resources for the students but as we know this is a very tough course so they need to implement everything from their first year to the final year of learning experience so by only having a semester of this course is just the workload is quite enormous so the power the objective of this course is to develop the students understanding on on the process synthesis the design and so this is the the main objective for our course and then we want to enhance the students skills on process and this is the key part when students want to design their their process plan they need to simulate the process to verify if the process itself is visible is workable and also fulfill the fundamental fundamental laws for them fundamental physics that is already given by the by the previous courses so they need to propose a design concept for our process design competitions and for the timeline of the course this is the week timeline so the course itself is on Monday the first four hours of the of the of the week and we spend almost four hours each week to teach the content of the course and you can see that for these four hours of content it's not enough to deliver everything that's why we want to develop this kind of most teaching content so that the students can access their teaching materials outside of the four hours so basically they can use their own time for the other for the other days so as we know that this is a capstone course then we try to introduce the process simulation try to verify the their their exercise through software and the main software that we are using and this is a worldwide well-known software this is the so-called asperd plus so this is a market leading chemical process of demonstration software for the chemical industries and because this software itself is a licensed software so we are bound to install the software in our servers so we don't we cannot install the software to their students laptops so yeah so we need to think on how the students have the access to this to that particular software so this is the I guess the framework of our course so we the process this time itself is the aggregation of the previous courses the previous required course like we have fluid mechanics thermodynamics and mass balance process control process safety and everything and then we because this is an open problem course we try to collect the problem itself from our from our plan visits and also from the process design competitions problems this is the national Taiwan process design competition competition problem that we are usually provided and in the previous years we only provide students of course by learning by doing that we provide them servers we have them we give them the virtual PCs access points so that they can access to the software they have they have also access to the the main software asperd plus and also we have the team fee were as our remote access platform and also mentor for the I guess the feedback platform from from the students but now we develop a new way to do this because the students have their opinion that by only giving them the software giving them the hardware they don't have the resources on getting used to the software itself that's why we did we develop the so called on demand online materials are the most online tutorials and also we have the certification so we we use this so called the assistive learning so that the students can yeah I guess have more time and more resources to learn on the aspen plus software itself so that when they have the higher understanding on the software itself they can implement that process simulation software to their main process design course that is designing their own process plant and then after they they have the simulation results they can show and also yeah submit their their final reports to as their as their final project of their in their final year so what we have now is the hardware part is that we have the fully updated servers with these memories so we have a very big ram because there will be a lot of students accessing the same PCs at one time and to to serve these demands so we need to have a lot of memories and then we also provide a virtual PC servers that can serve 50 students at once so you can see the scale of our students one class is almost 50 to 60 students and at one time they will access the same platform and this will be a trouble trouble some if if we don't have enough resources to do this to do that so but lately we also introduced a Windows server 2019 and we also developed a remote desktop connection platform so that the students can access the software and we can solve the problem on how we can eliminate the problem where the students need to go to the to the classroom need to be to go to our computer center to access to access the software they can work at home you they don't have to really worry on the their PCs their laptops that is not adequate for the for the server itself so they can just use the remote desktop connection and then they can get the access through our servers and we use a team viewer this is for the students access previously but now we use the remote desktop connection which is internally installed from their laptop so they don't have to really install another software but so we eliminate the team viewer but previously we use the this platform for the students and we when we do the classroom face to face meetings we use the mentor so that students can also ask questions directly and we can also give them some get some feedback from from the from the students right away and this is our some of the feedback from the students when you use the mentor so they can you can have these interactive classroom and as we explained before we have to plan visits or students can get a real live experience from the when they decided to plan the their their equipment they know exactly what kind of equipment they're they're trying to design and we also have the the chemical engineering competitions and then this is the results from our previous years that the students can have some performance in the international process design competition this is our recent result from students performance after having these the most the most platform and also the online on-demand platform is introduced so they can perform well on the national process design competition and this is one of the the the the advantage that we developed the MOOCs course they can get the the completion certificates and by doing so the students can have the the certificate itself and they want to apply for jobs they can use it as their additional values to their to their credits and it's also mentioned about the aspen process is very very related software by the industry so basically students when they show these the certificate they can get the job easily because of these qualifications so now work with the MOOCs itself we have several platforms first is from the iLearning platform this is the the internal platform by our university national junction university they have their iLearning this is like an internal website internal servers that students can have access to and we open a whole year videos whole year to whole year tutorials on the aspen plus so that if the students require some information aspen plus even though they are not in the final year they are in the first second year and they can directly get some access to the tutorials and we also developed the three more MOOCs courses this is put these courses are put in the open edu platform and this is our MOOCs course we also try to promote this course to our international collaborators so we have all over the world worldwide exposure on this course and students from chemical engineering department can have access to these the MOOCs content so the concept itself for the MOOCs is that for this process design course we have the objective that we want to design the process plan and on top of the theory on top of the fundamentals of the chemical engineering we try to introduce the process simulation software by first we have the introduction of aspen plus course and also we work with the more intermediate introduction to the aspen plus and then we have the more advanced courses so that we can have the final objective that we want to fulfill our process design teaching yeah I guess we will have this platform so that we can achieve our teaching objective we also work with these international universities and that's why we will follow the new software policy by the Taiwan government and we have the beginner intermediate advanced course on the MOOCs platform and of course because we are sharing with our international students we have the multilingual set up so we have the subtitles and also we also introduce some kind of AI and VR content to our MOOCs parts so the other thing is that as you can see we have four hours of time slot teaching the process design for the first three hours we work with the process design itself but in the final hours we have an hour on letting the students get their hands into the process simulation software and we do a flip learning settings in this kind of course so that we I guess we put some problems and we put the video so that the students can see and learn they can see the videos in the previous parts and then they can implement the knowledge that they get from the videos on the same day we have on the same week that we do the flip learning so I would like to share our videos on how the students have the interaction so we have the computers and setups and they have discussions and they make the group and then share the knowledge that they have got from the videos that we given in the previous weeks and this is a very intensive discussion that they have in our course so from these introduction we can give you the several key takeaways that in chemical engineering the process design course is very very important so that they can aggregate all everything that they have learned in the first years, second years and the third year and we also emphasize the use of the process monitors and by doing so we can develop the students understanding on their skill on this content and we know that the students can integrate the chemical process designs and by having these processes and course and we increase the student exposure by not only in the classroom but we have them at the yeah I guess we have them outside the classroom by using this remote desktop connections we also have developed the teaching materials from the beginner intermediate and advanced material for this part and we also show you how we develop the remote infrastructure and then we also have the potential integration for the courses on their first, second and third year and we also give you the experience on teaching these capstone course and we also able to engage international interest by these multilingual subtitles and we use the MOOCs as the cross utilization from the flip learning and the problem based learning settings so that's that from me, thank you


To pa počuje, da se napravila, da se najboj nekaj inčen očel tega modulača, in to je početje, ko je nekaj inčen, če je izgleda, če je to nešto, če je je početje, če je? So, kaj smo zelo izgleda? Tukaj sem tudi bila na mgn. Zato ne bi se vzelo, da je to modulaj spas, očeč genus G kerz, z njim njim vzelo, Markzpojnje. Zato, da je bilo, da je zelo, da je to modulaj spas, zelo, ki je za njim, na vse relacij, kaj je zelo Markzpojnje in genus, ko je zelo. tako, če je to? To je modulaj spas, n plus 1 tapas, c p1 up to pn, where c is an unsingular projective curve of gsg, and p1 up to pn at distinct marked points. So, we will consider it not so much as a modular space, like a collectionisomorphism process, but rather as a stack. So, we will consider the fact that it parameterizes families over a given base, in particular it has a universal family. So, for the kind of invariance I am interested in the fact that this is a stack rather than the scheme is not particularly important. There is something to do with the fact that if you fix a curve and the number of marked points on it, there may be automorphism. And this is the kind of additional structures, which is encoded if you look at the stack rather than the associated course modulaj space, which is just parameterizing the isomorphism classes. The only nice thing of working in this set up is that it ensures that there is an existence of a universal curve. And this gives some kind of canonical way to reconstruct to the point of the modular space, the curve with its marked points, and even to fit it into a family. So, if we fix a base scheme s, can you read here, then the close points of mgn over s are flat families of gsg curves and to be able to give n marked points over each element of the family, we need to have n distributed sections. And I have, of course, to be at least still. So, if there are no marked points, there is no need to find the sections, but this is the way to look at the points of our space. And as I said, if one prefers to look at the scheme, one can look at the associated course modulaj space and modulaj, I mean, it does not fit well in this more formal definition of what the stack was, but modulaj space of isomorphism classes and pointed genus g curves. I guess I forgot to write smooth here. And, of course, then one should do it over s and then it could be curves over s. Well, this is just a scheme of the same dimension. And as I said, if you don't worry about how many marked points you have, you don't have to worry too much about this scheme versus stack business, because if you have sufficiently many marked points, then there are no automovils on your curve, so these two objects are going to become eventually the same. Moreover, this is not a particularly scary space to look with, because it can be actually realized quite nicely as a quotient. Exactly in the way it was originally constructing, using geometric invariant theory, and then realizing that there is a global space, paramethizing pluricanonically embedded curves and then dividing by the appropriated automovils group in the objective space in which the curves are embedded. So, what I was trying to convince you, yes. So, something I forgot to write is, when this guy, M.G.N., one of the advantages of looking at it as a stack, it is a smooth stack, because of the fact that all information of curves with automovils is coming together here, this scheme may well have singularities. But anyway, because of the fact that it comes from this nice smooth stack, this guy just has locally quotient singularities. And then the course module is based. So, this is all a very good theory, but we have already discussed today that, of course, smooth curves can degenerate to singular curves. There is just no way to prevent it, because if you have a curve, there are families that can't be filled by adding a smooth curve in the missing points. So, if one wants to be able to complete families, one needs to allow singular curves. The other problem is, of course, that we have distinct marked points. So, even if the curve does not degenerate, we have to find some kind of replacement for the phenomenon in which two points are coming together. So, if two points are coming together, let's say that p1, we have p1 and p2 in the family approaches p1, so we have a family parameterized by c itself, we need to be able to decide what happens in the case in which p1 and p2 coincide. And we don't want exactly to say, well, but we could as a first approximation say, well, we don't care that the points are distinct, because if we want to have a good modular space, we always want the automorphism group of the objects we are considering to be finite. And this was exactly the condition that was ensuring finateness, but then if the points are coming together, then the fact that we have n points when we actually have less, so this part of the definition is not going to be satisfying anymore. So, we are lucky. For curves, there is a nature that way to find a nice class of singular curves that can be added to compactify our modular space, our modular part. This leads to the linear manifold compactification. I guess it has not been in existence for 50 years yet, but the closure. So perhaps we should track down the first paper and organize a birthday party at some point. And this is the modular space and pointed of genus G. So, what are now the elements of m bar g n s, a family, a flat family of curves with n sections. Well, we have to look at flat families, because of course we want things to be invariant under the formation, so this is our guarantee that there are no strange jumps in the kind of objects we are taking, but this also is giving us an indication of what kind of genus we need to consider, because if you remember, one can define the genus in several ways. There is a geometric genus, which is the genus of the normalization, and this is what you find by looking at global differential forms, generalizing the definition for smooth curves in that way. But anyway, what we need to look at if we want to have a flat family, would be a flat family of arithmetic genus G. And because of the fact that we want the genetic objects to be as easy as possible, we are lacking in this case, we can just, it's enough to assume that we are working with nodal curves, so we simply need to allow the easiest type of singularities. Now the genus we have to consider is the arithmetic genus, which in itself already ensures that the family is flat, and the curves we consider may be reducible, but they have to be so possible. What I want to say, they may possible be reduced, but they anyway have to be connected. Then there is a condition on the section, they still have to be disjoint sections, but what we want them is that they can't hit the singular loci of the curves. So they have to stay inside the singular locus, and then we want to have a stability condition that ensures that the automorphism group of the curve is finite. So when is the automorphism of the group finite? Here is that this kind of condition, which is actually the Euler characteristic of a complex smooth, what I want to say, this is actually the Euler characteristic of a smooth curve of genus G with n points removed and then multiplied by minus 1, but the point is we need to have, this is exactly the condition that ensures the finiteness of the automorphism group, so for each reducible component we have the genus for the component 2 minus 2 plus the number of special points, so we take the preimages of the marked points and the singular points in the normalization of the component, this should be positive.  Or the equivalent one could take the Euler characteristic of the component, remove from it the set of marked points, not all of them may lie on it, and also all singular, and this should be negative, because if you try to understand why I'm writing this, oh, this is not minus. The reason why I'm writing this, so this is just the same formula we have here multiplied by minus 1, the reason I'm writing this is because it has some more intuitive interpretation if we think over the field of complex numbers, then our curves, then all components, or at least their normalizations, give Riemann surfaces, and the idea is that the automorphism group of the Riemann surface with some marked points is finite if and only if we remove from the surface all marked points, we get a hyperbolic curve, so a curve of a negative genus. The idea is that the compact Riemann surface has an untrivial automorphism group when there is some group acting on it, and this sort of flattens the metric on it, and this cannot happen if the Euler characteristic of the curve is negative. If we look at this formula, just 2g minus 2 plus n is bigger than 0, we see that actually this can become negative, well, the only thing that can be negative is minus 2, so this is only significant, if we have g equal to 1, in which case we need n to be at least 1, or if we have g equal to 0, in which case we need n at least 3, so actually the only additional condition if we assume this to start with is that every time we have a rational component, there are at least 3 special points on it. So each rational component, so each component of a genus zero, and the points should be counted on the normalization, so if we have something like a self-intersection, then it counts as 2 special points. And again, this compactifying space has all good properties that the previous one had, so it should again be reducible, the linear manifold style, but now it's complete. And if we look back at the history of the subject, this is the space that may realize from the construction, from the geometry invariant theory, and this ensures that we can actually define this directly over z, so we can make sense of this over any possible field of ring we may wish to consider. And if we just wish if you don't, for some reason, don't wish to maintain all information about automorphism groups of the elements, so if we are just happy to work with the isomorphism classes of stable clouds parameterizing this way, then we can work directly with the coarse-modulated space, which is of course a scheme. So what are we going to do? Our lectures with these places. Well, the idea is that most of the time we work over the field C of complex numbers, so we are actually working with modular spaces of reman surfaces, or this kind of the generations that are created by patching together pieces of reman surfaces by identifying points and creating the nodal singularities. And we will look at the co-mology of the space. So this is the most flexible but significant invariant that is attached to this. So why is this interesting? Well, for some cases the constructions will be very close for what one can do with our groups. In other cases we will try to discuss classes that have nothing to do with cykers on the modular space. What is the motivation mainly? The first motivation is I guess how the one of the main motivations to study the modular space of curves to understand it's a numerative geometry. So to be able to count how many the generations and on which time they are in each family. And this of course is some kind of a numerative information which can be found by taking the intersection product in the charting but in many cases the relevant information survives to co-mology. So this is sufficiently interesting. And then there are two kind of motivations that are sort of autonomous. So because of the fact of the way in which the geometry of curves work co-mology groups in themselves may have interesting interpretations coming from number theory in spaces of modular forms. So in some cases studying co-mology group of these modular spaces is going to give us some kind of geometric interpretation of spaces of modular forms that are coming from number theory. And you know it doesn't matter whether the input is coming from number theory and we can give some kind of geometric interpretation for it or if the thing is going the other direction we can use geometry to prove something that people in number theory can see for themselves but the point is there is some kind of interaction which can be very useful for both sides. And then I must say I would say most of the application of results about modular spaces of curves so far in the last decades has been to mathematical physics because they are related to the construction of co-mological field so this is like when looking at Gromov written in writing. So as I hinted at before both the modular space of smooth curve and the modular space of stable curves are actually quotients. So the idea is that we don't want to make this so very explicitly but the idea is that anyway globally it has been constructed by saying well there is projective variety that parameterizes all possible embedding bendings of the stable curve in a sufficiently large projective space now I am not trying to add the marked points in this in this story but then of course one can add them as well by taking some kind of incidence correspondence and then reshenting it by the action of the automorphism group of the space in which it thinks we are embedding so the idea is that this is globally a quotient and this means that when we are looking at its homology we can reinterpret it as a equilariant homology of this space and this is the way to define it actually it would even work if we wanted to work with homology with integral coefficients and this kind of construction would work even with integral coefficients the problem is that everything becomes much more delicate and in that case the results just in very small genomes I think nothing further than genomes since we are working on the sea there is even a way to realize n, g and bar globally as a quotient of a smooth projective variety by a finite group so this group business can be done very one can even reduce to the case in which one is working with a finite group so the idea is that to do this one needs to put some extra structure which is called in this case a level structure we perhaps touch on the basic case of this which only works for smooth curves anyway and in previous I think it is too low and on the compactification for n equal to zero and then to the students in the beginning of this century so where is this nice well if we have that our mgn but it actually works with any stack the idea is that I didn't want to write it in too much detail but the idea is that if you want to parameterize curves you want to find some kind of intrinsic way to embed them into a projective space so natural choice is to look at the canonical embedding and then you look again and you say but this is not going to work for hyper elliptic curves because those are the exception so one needs to look at least at the canonical embeddings but actually if one wants to be able to embed all smooth curves without any trouble one needs three canonical embeddings ok, that's good but then one wants to apply geometric invariant theory and make sure that the kind of the degenerations that one takes is independent of how high the embedding is because one says well this is actually a procedure that stabilizes so we need to take a pluric canonical embedding but we want to be able to be free to take a higher power without changing the kind of stable curves because of course there are clear theoretical advantages in doing this also the other point is that if you take canonical embedding which is to otherwise we will not get our stable curves we will get something slightly slightly different than with curves which are not stable but just pre-stable so I think that one needs to take a five canonical embedding at least to make it work so it will not be nice to write the equations for all possible five canonical embeddings of stable curves but it's a problem that one can approach theoretically and look at which kind of stability conditions that are there for the action of the of PGL and this is the kind of embeddings first if one wants to take marked points into the story one needs to add them into the projective varieties one needs to take again an n plus one tamper in which the first part is the embedding of the curve and then one needs to take all the curves and I must say if you are interested in the GIT construction with marked points actually this is very not explicitly done in any of the historical references so one can find much more recent work perhaps 10 to 15 years old can give you reference about this so what I was trying to say is if we are lucky and what we get is simply the quotient of a variety by a group as I said already a story of a sea so if we want to look at the chronology of the quotient with rational coefficients this is actually just the same thing as looking at the chronology of the variety and then take the G invariant part so if one is working with a finite group one does not even need to look for the for the equivalent chronology it's just the invariant part classical chronology this is the reason why the situation is much nicer there but this only works when taking rational coefficients because the map going from here to there has something to do with the fact that you have to take an average I'm afraid it's only over the sea I should have to look it up but I mean it's made using level structures so I do think that they are truly taking structures on the chronology for classical level structures if you want to frame chronology so this is one of the reasons why it's nice to work over Q the other reason is that from the point of view of geometry it's very nice to be able to work with the stack because we can make many more constructions here because of the fact that as I said we have a universal curve over this so it's not just that we can we have some kind of isometric classes below it's just over each point we have some kind of natural way to recover the curve but for many constructions it's also very difficult to keep track exactly that the kind of construction we are using is respecting the stack structure so it's very practical that if we work with rational coefficients we don't need to make a distinction between the chronology of the stack and one of the cross-modulate space why is that? the point is that the information which we are forgetting has something to do with the automotives groups which are finite, they are torsion so they can't produce anything interesting at the level of chronology as soon as we take q coefficients this is the definition of the rational coefficient so I know I am being vague here so actually what I wanted to say here I am just taking the scheme related quotient of a scheme by a group and I am not worrying about the stack structure but the point is that in any case it's not the definition if you want to take a definition you need to work with equivalent chronology or if you want to have charrings with equivalent charrings that's the good way to define the chronology otherwise you could go back there are some alternatives but mainly the best choice is to work directly with equivalent chronology with these kinds of spaces what I wrote here was already thinking I was already thinking with a variety in mind because this would not be the standard notation for the quotient stack quotient stacks are written with square brackets but I don't want to make a point of this because as I said if we are not behaving so nicely with respect to the stack structure actually for what we are doing it will not matter so I try to think about the stack as often as possible because the geometry here is much better but if we replace by mistake our stack with a different one with the same coarse modulate space we are not in time we can do it all the time and if one is topologically oriented our way of constructing mgn is not particularly natural actually there is some kind of topological or analytic construction of mgn so how does one construct mgn when one is not in algebraic geometry depending exactly in the way in fact your modulate space is going to give you some kind of topological stack or just manifold or before perhaps so if you are topologically oriented and you want to parametize all possible compact Riemann surfaces of genus G when you look at them and you say well but actually topologically they are all the same because there is just one there is a morphin's glass of orientable surfaces of genus G compact orientable surfaces so the idea is that we can fix a surface of genus G and then mg parametizes all possible choices of a complex structure on it so for instance we can denote it by sg everyone obtains in this case it's a very large space the so called Taich-Müller space the space of complex structures and it can be realized itself as a topological as a smooth manifold and the important thing is that this is contractible so to take a choice of a complex structure on a smooth surface maybe a complicated business but anyway we can there is a contraction of this guy to a point so from a topological point of view it's not a complicated thing the problem is that of course some complex structures can give isomorphic reman surfaces so we have to divide it for some group this is what is called the mapping class group so the idea is that we have to divide by diffeomorphisms which are orientation preserving so this is just part of the definition the point is of course this would out on tg and the quotient will give the isomorphic classes we want to have but two diffeomorphisms may define the same structure they do this exactly when they lie in the same connected component of the group I've just written so one does not need to take this one needs to take the group of connected component of the group of diffeomorphisms on a fixed reman on a fixed orientation so this means that one needs to divide by the component of this group that contains the identity the idea is that you can deform a diffeomorphism to the identity then it's actually also trivial acting trivial on the trivial on the complex structure so if one takes the stack quotient here one gets again our mg in a different incarnation we want to stress that this is now a discrete group because it's a group of connected it's a group of connected components of a larger group and this and this gamma g is called the mapping class group this is true but let's say as topological stacks or if you want to take the quotient and then you take the course modeling space so if you just look at the isomorphism classes so at the quotient then you get the course modeling space and gamma g is called the mapping class group so the reason why this is done is that if you have fixed your orientable surface then you are free to make many to put a lot of structure on it and to find very explicit parametrizations so actually depending on the problem which one is interested this may be a very a very explicit description it can be useful but it has nothing to do of course with algebraic geometry and of course one can produce a variation of this idea that works with marked points so it simply one needs to add more data here and to be preserved there and even with with boundaries which is something that does not make sense either in algebraic geometry but makes sense in a topological setting so I will just show a picture so we all know what the marked point is it's just a fixed point so that's not the problem but something which one can do if one is working just on surfaces is to make a hole in them so instead of marking a point to cut a smaller circle into it one can use this kind of circle to attach together other reman surfaces or other compact surfaces with the boundary removed so this means we can take another one it works more nicely if we parameterize the boundary so the orientation is giving us in which direction we can move around the removed boundary and then we give an explicit parametrization of it so we can take another one and then sort of attach them together so in this kind of world there are natural ways to transform a surface with boundary of certain genus to a surface with boundary of a boundary with a higher genus if you start with a say a rational curve and we mark a point we can't simply say well now we have a standard way genus one with a mark point but in this kind of world one can say well if we fix the thing we attach to it perhaps we need to and we have boundary here we can always attach it so it's a different way to look at it but it's also show us where this kind of approach is good for studying the dependency on the homology on the genus homologists have been more successful recently in comparison with the algebraic geometry yeah you can I can look up a reference about this for you if you wish but to me it looks very unnatural so to me it looks like they want to have mg and bar so they are sort of artificially gluing things on the boundary to produce exactly mg and bar but still the procedure of the generations is working well because somehow degenerating a curve from a topological point of view how do you create a node on a linear surface the idea is that you fix some kind of loop if you are truly into the business of course you will fix a geodesics not just anything because of course they are hyperbolic surfaces with a flat metric making the kind of things and the idea is that you can fix a parameterization and make this in another way so in this way you get the stable ribbon surface you expect as a limit so somehow if you want to produce mg and bar out of this construction for mg and what you need to do is to parameterize to add to the data the length of the inappropriate geodesics and make it go to zero and this explains to you how to put together the elements in the boundary but anyway when I submitted my proposal for these lectures I thought that I decided on the aim and my aim was to try to illustrate to you as many ways as possible to construct homology classes on these modular spaces to illustrate how we reach the homology theory but also which kind of meaning these classes may have for the geometry of the modular space so of course if you want to answer this question and you have a fixed candidate you may be in good shape as long as you have some nice explicit description of all curves that occur then you can try to parameterize and so on like in the case of m2 where there was just this polynomial of degree 6 and somehow all information on the isomorphism class of the curve was like that but if one wants to be able to construct homology classes in general one needs to have something that works independently of the genus anyway the first one to work on this problem in the 80s in 1982 where it actually published 1983 paper about the homology m2 bar the idea is that if one wants to have a universal construction one looks at the prototypical example of a modular space which is the grasmanian of vector subspaces in a fixed vector space so in considering it as a grasmanian matter of what then let's see k-dimensional subspaces in cn so it can be considered as the modular space of such subspaces and it has again a universal family and since it's parameterizing vector subbanders the universal family is just that universal subbander as for subbanders the idea is that s lies in the trivial vector bundle of rank n over g and it has of course a quotient so s has rank k and the quotient has rank n minus k so then from this construction where first of course we are not worrying too much about what the homology of the cell groups are because g has a nice cellular decomposition as you know of affine spaces you can stratify it by affine spaces so in this case the charting is the same thing as the homology ring it's just generated linearly by the classes of the cells and this leads to the theory of Schubert cycles and so on but in this case what I wanted to say well the generators are just 10 classes of the subbander so the thing to look at is the 10 classes of the universal bundle in this case of course we have no universal bundle in our case and then if one wonders about how to get relations when we can use the sequence to say because we know that the rank of q is n minus k then it's 10 classes up to in degree larger than the dimension have to vanish and actually this is enough also to give all relations so if we look here then we have to take what we have for s and invert because of the sequence so take the sun then we invert and then we will say well if we look at the part of this which has degree l which is at least n minus k plus 1 then this has to vanish and this can be an entity that's giving relations here so this is a completely explicit description of the cohomology of our Grassmanian so this is what we would like to copy the part about creating classes defining them can be copied very easily the business relations is actually much harder and of course there is no guarantee that what we get with this kind of constructions gives us the whole cohomology because m, g and bar is a space with an interesting geometry so we can't startify it by putting together affine spaces so for sure there is no reason why cohomology and Chow agree and no reason why it's generated linearly by classes that are easy to but anyway this was the idea so we look at the universal family n bar but actually the universal family m bar g n is the same thing as taking m bar n plus 1 because the idea is that well if we start with a curve with n mark points if we take a new point so the additional in the universal family we simply need to take an additional point and let it move on the curve so as long as it's distinct from the mark points but we have nothing to have if they hit so let's say that p hits the point n this is actually the image so we keep track of where the point n was and here we take the genus zero component and we put on it the p and n and this is the kind of thing that happens the way we have interpreted the case in which the point that we left moved on the curve hit one of the mark points if the point hits a singular point then we can sort of blow up the node to give us two distinct branches then there will be an exceptional device of the blow up which will be something of genus zero and then we put p inside and if you remember that we already had in this lecture in the previous one when we have three points on the rational curve then they have no moduli so as this is just encoding the same information the same p has moved to the singular point so now if you are looking at the universal family we can create some kind of universal some kind of vector bundles using the data so the idea is that I want to think about there is something which happens on the universal family but we think we can mg n bar g n plus 1 is just some moduli space in which we have at least one mark point so there is something in simply a construction that makes sense at any point as long as we have some mark points and we can say well some way of constructive vector bundle over mg n bar is to take to look at where the mark point is or the point on the universal curve if you prefer and look at this cotangent space to the to the curve we have a smooth curve and we have some mark point pj unit then we can consider the tangent line through this and also take the cotangent space which is be having more positively so that is what we want to take and then if we let the move in the moduli space then this is going to give us a line bundle pj one of the mark points if you think about the universal family we can take the last one for instance and since this is a line bundle on m bar g n we can take its first-gen class and for sure this will give an element of the homology in degree 2 and if you prefer of course this would also work in the first chart group then of course this does not make sense if n is equal to zero but then as I said if n is equal to zero we simply go and look at the universal family and this gives the so-called kappa classes the morita non-fog classes so what is the idea here where we don't know what n is it may be equal to zero but then we look at the universal family we always have a side class there the last one and we can take any power of it and pushes down to m g n bar using the universal family looking at higher powers so we will have algebraic classes of higher degree whereas we already discussed in the previous lectures there is a natural bundle over m g n which actually extends also over m g n bar because it actually depends on the definition of the universal family the Hodge bundle so let's see the idea where this again works the same whether we have marked points or not so point c of the modular space we can take the space of regular differential so this of course works well if we are working over a smooth curve but it actually can be extended in general the idea is that one takes the shift of relative differential of the universal family I am not writing n here because it does not really play a role and this of course lives on the universal family so one takes the one push it down again and now we have rank g vector bundle over m g bar if you wish also over m g bar and one we can take yes the Hodge bundle has this nice natural bundle definition we can find the line bundles but you just find the term in terms of their fibers that is also feeding a little bit yes with the universal families so the idea is that if one truly understands how the thing is defined then there is a natural way to figure out how to give the global definition and personally I am specifically worried about this because you see the dimension of the space of differential drops if there is no component of genus g so the term classes of the Hodge bundle are going to vanish in which there is no on which there is no component of genus g so this makes much less intuitive to think about this bundle over the locus of m g n of curves of m g n bar of curves with only components of smaller genus this in general but of course since this is a rank g bundle one has to stop but if one wants to have something don't give so so how what Maffel thought about these classes is something one should keep in mind when defining them so the geometric intuition is that the lambda classes are natural because they have something to do with the sheaf of differential and this sheaf is geometrically related to the fact that curves have a Jacobian and abelian variety and so how the lambda classes come from the fact that they are the natural kormodric classes on the modular space of abelian varieties so they are there also for the modular space of curves because there is a very precise relation but they are not not necessarily so natural for the geometry of m g n bar on the other hand these kappa classes are new and at least this was what Maffel didn't expect but it turned out to be they should have some kind of specific meaning for the geometry of the modular space of curves and indeed a kappa one is the ample one one is used to prove the projectivity of the modular space so far we have a collection of classes we will actually see that the lambda classes are not independent one could actually forget them from the theory so they are useful because they are geometrically natural and they are an example of classes that can be expressed in terms of the other ones but which other natural classes do we want to consider? well the point is we want to consider the boundary strata classes the classes the components of the locus of curves that have at least a fixed number of nodes can't remember which notation it was but the idea is that one moves at curves that have at least a fixed number of points of singular points the number of singular points on the curve is at least and actually each one so these are of course the reducible components of this guy are of course closed subvarieties of the modular space of curves so one wants to be able to take a natural thing to take the fundamental classes if you remember we stated as a fact that if we impose the condition that there are at least k singular points there is something about the formation theory then the dimension of these locus the co-dimension of these locus of all components of these locus is k so they will give a collection of fundamental classes co-dimension k means in degree 2 k this one can do the same and these are what we call the boundary static classes of course we can deal better with them if we understand better what the combinatorics of the boundary is and usually we do this by using stable graphs so the idea is that each such reducible component corresponds to the topological type of the degeneration so smooth curves are all the same if we are working with the nodal curve then we need then there is some kind of combinatorial information attached to it we want to know for instance what the general of all components are so how many components there are what the general is which points have been identified if one wants to take the map from the normalization to the stable curve every time we have a stable curve of course if we have a constant map points we can extract from it the combinatorial information about this topological type and this is encoded in its dual graph so we try to sketch an example so let's say we have a curve a component which is a curve of genus 3 and then there are two other components that have genus 2 and here we have a component genus 0 where we said that this needs to be stable so we need at least two map points here but perhaps we can have 3 oh yes, from this picture there are no so I will turn this into a component of geometric genus 3 with a node so the genus I write is just the geometric genus of the component it's a genus of the normalization genus labor in this picture so let's say the points 1, 2 and 3 same this genus 0 component and then for instance on this one here we have the label 4 this is stable because I put sufficiently many points here so which is the information we want to keep in mind so we want to attach to each component a vertex of the graph labeled with the genus this will be actually a labeled graph with also some open leaves so here we have 1, 2, 3, 4 components 1 of genus 3 2 of genus 2 and 1 of genus 0 and components of genus 0 are often just written as black dots because of the fact that genus 0 inside without having it looking very strange also good for print so then each node should correspond to an edge of the graph and since a node joins to irreducible component it should be an edge that joins exactly the vertices which are associated to the components for instance so this genus 0 thing just means one of the two genus 2 components let's say it's this one then the genus 2 component which is here means also the genus 3 and the genus 2 component so it goes like that and then let's check this is everything so the other genus 2 component also means the genus 3 1 and then we have one additional node this self intersection of the genus 3 component which is going as an edge starting and arriving at the same component and then we need to keep in mind where the marked points are and the marked points correspond to marked leaves of the graph which start exactly the corresponding vertices so here 1, 2 and 3 genus 0 vertex and then we have an additional one which is on the genus 2 vertex node and this is the dual graph of this curve but this is actually a kind of construction that works in general we start with the curve we can reconstruct the graph but if we have the graph just by peaking the components we can construct the normalization of the of the stable curve so how many edges of leaves are starting so in this way we get the normalization and then the way in which we constructed the edges tells us what we need to identify to construct the stable curve so I guess that this is everything for today I hope it was not too long