So, this topic on malicious software or malware, we're just really classifying the different types and gone through some examples of viruses and worms, so we're classified based upon how the malicious software spreads because it genuinely tries to infect not just one computer but multiple computers, so it spreads in different manners, so a virus attaches itself to another program, so you can think a virus is some code that does something malicious that attaches to some normal program and spreads by finding other programs to attach itself to. A worm, you can think more as an independent program that finds its own way to spread like using network connections to copy itself from one computer to another, whereas a virus attaches to other programs, a worm distributes itself, copying itself, for example, across a network or between computers. Another form of spreading which we just mentioned was towards the end of the lecture was social engineering, taking advantage of tricking someone into doing something malicious on their computer and malware can take different actions to do harm. It may try and corrupt the computer system, it's infected, delete files, for example, will mention it may be used to take control of the computer to do other malicious things, so treat the computers that it's infected as zombies that it can then use them to perform a different type of attack on other targets, steal the information, hide itself as some of the things that it tries to do and we'll finish with some touch upon antivirus very briefly. So we've looked at viruses, some different types of viruses, polymorphic, metamorphic, to hide from antivirus software, the virus tries to change itself because if we know of a virus in advance, what an antivirus software can do is scan looking for patterns or a signature of that virus. So if we know the virus contains these lines of code, the antivirus software can look for those lines of code in any file or any email the computer receives. So therefore it's easy to detect. So as a way to conceal, the virus tries to change itself. So it's harder for the antivirus software to find that signature. One way to change is to change how it looks, which what a polymorphic virus does it, changes some code inside itself. The virus still does the same thing, but if you look at the code of the virus, it's different from the previous version. A metamorphic virus does that as well as changing its behavior. So a polymorphic virus just changes the way it looks, whereas a metamorphic virus also changes what it does, maybe what the payload is, how it infects other programs, which is harder to do, but if you can do it from the virus' perspective, harder to detect. We had a few examples. We mentioned worms, where did we get to last week. Social engineering, I think you've seen, we mentioned that, okay, Spam email is used where people send unsolicited email with the hope that the target will read the email, think it's a real email, and take some action, like click on some link or run some executable file or open some document attached to that email. And then that's the starting of the attack. And we mentioned phishing attacks as one example there. Trojan horses are software that are from the target's perspective useful, but also contain some malicious payload. You download a program to convert word to PDF. It's useful for you, but actually that software includes some malicious functions as well. What can malware do? What actions does it take? So once it's infected your computer or your software, what does it do? Well, different things. It can do data destruction, delete data, overwrite data. So it can be worse in some cases than deleting data is to overwrite it because you think the file is still there, is still on the system, maybe still the same size as before, but the contents has changed. Encrypt data and a recent and current malware present is called CryptoLocker. What it does, and I don't remember the details of how it initially infects, but once it infects a computer, CryptoLocker encrypts or searches for a set of files on your computer, like JPEGs, thinking about photos or other documents. It searches for files and then encrypts those files using a strong encryption algorithm, using public key cryptography. So it encrypts the files and then sends you an email or contacts you and says, if you want to decrypt these files, you need to pay some money. So that's an example of ransomware that the malicious software holds your data to ransom. So in this case it encrypts CryptoLocker, it encrypts the files with public key cryptography. The person who created the malware knows the private key to decrypt. You don't know the private key, they encrypt it with a public key and therefore they say, if you pay this amount of money then they'll decrypt the files and you'll get your files back. Otherwise your files will stay encrypted and they say that they'll delete the key so you can never get them again. And currently there are no known ways to decrypt those files. So that's a current malware that's in use today. We mentioned Stuxnet, which was a most likely a government sponsored worm that infected a number of hosts, but mainly factories and facilities in Iran with the intention of shutting down some of their nuclear power plants. And that did real well damage. By infecting the computers that control industrial systems, then it made those systems operate outside of their normal intended behaviour and made them fail. So it did real well damage via the computer. And related to both of them is a logic bomb which is something that activates a payload that activates when certain conditions are met. So here's some payload that's for example, your computer's been infected, the payload's on your computer when some conditions are met on your computer that payload is activated, whatever it does, whether it's data destruction or some real well damage. So that may be as simple as thinking about does this file exist on your computer? If so, then activate the payload, date and time, whether some particular software is being used on your computer or a particular user logs in, then activate the payload. So that's the general concept of a logic bomb that it performs some logic at a particular date and time or when some conditions are met. We saw a very simple example in the Melissa virus that we showed. At the bottom it had some code that says if the current date equals the current hour, then display some message. So it was a particular set of conditions. If this time and date are met, then do something, otherwise do nothing. So there may be some conditions to activate the payload. So we're going to finish with the last few slides to finish this topic today. What else have we got? Zombies and bots. So another common thing today is that once a computer is infected, then it allows the person who created the virus or worm to take control of that computer to get access to that computer via other means and then use that computer to perform attacks on other computers. And we get a set of bots from the perspective of the attacker. So if you can, for example, infect one computer and take control of that, take control may mean the attacker can trigger that computer to do something to other computers. For example, send packets to other computers. And if you can do that to many computers, then the attacker has control of many computers on the internet. That becomes their network of bots or botnet that they then use to attack some other target computer. And this is common in distributed denial of service attacks, where some malicious user takes control by infecting many other computers on the internet and then uses those many other computers to, for example, send many packets to one target computer, overloading that target computer so it's denied service. We will look at denial of service attacks in more detail in a later topic. So the general concept of a computer that you take control of is a zombie or a collection of them is referred to as a botnet. And there's thoughts that there are botnets that attackers have control of in the order of millions of computers. So attackers have compromised millions of computers via different means and can control them to do things that they'd like to do. For example, denial of service attacks, send spam. So instead of sending spam from the malicious user's computer, they take control of normal users' computers and get them to send spam to some target. Monitoring what's happening in terms of those computers, what they're sending and receiving, sniffing traffic, logging actions on those computers, spreading new malware and other things. So that's a common problem in network security today is that many computers are compromised and attackers can control those computers to do other more malicious things. We'll see more detailed examples of them when we look at denial of service. So another form of payload or an action that can do harm is information theft, that the attacker, they steal information from you. So one thing they can delete your files, but other thing they may try and steal information from you. For example, try and steal your password. And one form of doing that is monitoring the keys that you press. So if the attacker can install software on your computer that monitors the keys that are pressed by the keyboard and can keep a log of all those key presses and send that log back to the attacker's computer, then hopefully in that set of key presses will be your password to different accounts. And that's the general concept of a key logger. Some software that captures all the key strokes that you enter into your computer. So how to get the password out of it. For example, monitor what's happening on the computer when you log into another computer or when you log into a website. So you look for patterns of what may be a password. So assuming the attacker can see all the key presses on your computer, then they can search for specific combinations or specific patterns to try and identify when you entered in a password. And then they can discover your password for different systems. So a key logger fingers a piece of software that the attacker may install on your computer to steal information from your computer and send it back to them. That information normally being passwords. But maybe other things. Maybe messages that you send. Key loggers may be software on a computer that they may be done via extra hardware mechanisms. For example, software installed via USB keys on computers without your knowledge. Generally, it's not hard to do because the operating system always keeps track of the key presses. So you can get access to the key presses. So you just need some software to read that information from the operating system and you can get a complete log of everything that happens on the keyboard. And that can be extended to not just key presses but also mouse clicks. So if some websites as a security mechanism require you to not type in your password but to click on an on-boards keyboard, on-screen keyboard, that you click on the letter A, B and C instead of typing in the password in case there's a key logger there. But even then, if the software, the malicious software could monitor where you're clicking on your mouse, then although it's a little bit more complex and they can do the same thing that they can monitor that you're clicking on these locations on the screen and send that information back to the malicious user so that they know where you're clicking and then map that back to the on-board, on-screen keyboard. So key loggers monitor keystrokes but more generally we can monitor whatever activity is performed on the computer including mouse presses, mouse movement and clicks. Yes, so the idea of the on-screen keyboards is to avoid this problem of capturing keystrokes. So if you just type in the password and someone has a keylogger then it's quite easy for that keylogger to detect the password typed in. Therefore websites will have an on-screen keyboard. If that keyboard is fixed, that is the keyboard is like a QWERTY keyboard as an image on your screen, then again if the attacker can install software to detect where your mouse is and when the button is pressed then they can determine okay you clicked on the letter A then you clicked on the letter B and C based upon the position of the keys on the screen. So then other systems will each time you log in will have a different arrangement of the keys. It won't be the QWERTY or won't be ABCD but it may be a random arrangement of the keys making it harder for the malicious software to detect what you were clicking on even though it knows where you clicked it doesn't know which letter you clicked on. So that in that case the malicious software needs some other way to know the mapping from where you clicked to the letter you clicked on. That may be from getting a screen capture or it may be about predicting what the arrangement of the keys was how that website displays them. So it gets more complex making it more secure but making it more inconvenient for the user. Most users it's easy to type in a password it's a little bit slower to click on an on-screen keyboard it's even slower if that keyboard is changing the arrangement of letters all the time. So the first time you log in it's one arrangement of letters so you know where to click the next time you log in it's completely different arrangement so you need to look and spend a few seconds to find out where to click to type in your password. So again this is the trade-off between you can make it more secure but it becomes more inconvenient for the users. And this is moving on to almost the techniques used in spyware is that once a computer is compromised by some malicious software that malicious software spies on what that computer is doing or the user is doing with that computer it monitors what's happening which includes monitoring key presses mouse activity but other things like monitoring applications browsing history particular web page request documents opened to then learn about what you're doing so learn about your behavior and use that for other malicious things extracting data about websites you've visited things that you've done to use that for other purposes. Fishing we mentioned last week where we use someone uses social engineering they try to trick users into following some URL to some malicious website is it a common example they send you an email you think it's a real email you click on a link that link takes you to some fake website where you enter in your details and that malicious user now knows your details spear phishing is the specific case where it's targeted a particular person so the malicious user creates the email and that fake website especially for the person that are targeting so generally phishing target anyone but spear phishing is more more targeted to a specific user and therefore the website and email will have content that that specific user will think is real or more like likely to think is real any questions about the concept of phishing in a security context anyone seen examples anyone received emails spam emails which have links to other websites I receive them all the time yeah yes I think it's not limited to email email is the most common form of delivery of this this spam of this this social engineering technique but it's not limited to email so social networks chat software anything that makes it easy usually for the attacker to send the information to many users cheaply that that needs to be a part and most social networks email chat applications allow that it's about tricking in the user into usually visiting a website or executing some application that is malicious and there are many other types of malicious software so this topic is more about introducing several types you probably heard of a lot of them just to distinguish between several types but there are many others and there's some of them listed that you may come across and we may explain them as we go through some other topics when when needed just briefly to finish then well how do we stop them what countermeasures do we have so that the malicious software cannot take effect on our computer systems we'd like to prevent ideally prevent any malicious software into our computer prevent it from running but that's not always possible how do you prevent how do you prevent malicious software getting on your computer in an organization having some policy about what the user should do policy and awareness about don't follow links in emails or in unsolicited emails don't open attachments in emails which contain executables and awareness from the user's perspective will help vulnerability mitigation a lot of the malicious software takes advantage of that your computer has existing bugs vulnerabilities so make sure that those bugs or vulnerabilities are not present or as minimal as possible that includes making sure systems are computer systems up to date especially operating systems and software make sure that the the latest versions are installed and being used so in operating systems now have regular updates or patches so making sure that those patches are applied so that the software is up to date and there because many of those patches usually fix security bugs so if you don't update the software there still may be bugs with respect to security in that software that the malicious software can take advantage of make sure access controls are used and applied so that even if malicious software is executed on a computer that the access control can limit what it can do it cannot take control of all your files it cannot delete important files and so on so using access control to limit who can read write and execute different files can help limit the effectiveness of malicious software and an important part making sure users are aware of what malicious software is and how it may arrive detection is the main form used today as that as malicious software arrives at a computer system detect if it is present and then remove it detect identify what it is and remove it all the countermeasures to stop malicious software they have general or they have a set of requirements some of them listed here the countermeasures should work for any system and any type of malicious software be general you don't want a countermeasure that's tailored just to one type of malicious software they should work in a short amount of time if there are failures or if some malicious software gets in then the countermeasures should still work so if you think of antivirus software you have antivirus software as a countermeasure if some malicious software gets on your computer then resiliency means that that antivirus software will still work we shouldn't allow that malicious software to then take over the antivirus such that the antivirus no longer work it should be resilient to some forms of attack countermeasures like antivirus should not deny service too much at least that is if you install antivirus software on your computer and it takes up 50% of your CPU just scanning viruses all the time then that's effectively denying denying you service it denies you access to your the CPU to run your normal applications we want to minimize that the user should be transparent to the countermeasures so the user shouldn't have to deal too much with say the antivirus software should be transparent it should cover malware on your computer and across a larger set of computers not just across a single computer the three main approaches to provide countermeasures are to perform scanning on individual computers called host-based scanners for example antivirus on your own computer for an organization or a network perform perimeter scanning so think for SIT there's a gateway between SIT internal network and the rest of the world perimeter scanning would scan files as they pass through that gateway and try and detect and identify and remove malicious software at that gateway before they get into SIT's network so rather than relying on antivirus on all the computers inside SIT have one device on the perimeter of the network that performs the scanning and removal of malicious software that's better for larger organizations because you don't rely on the individual computers and more complex techniques use what's called distributed intelligence gathering which we'll not talk about but involves not just collecting information about your network but using information of what's happening in many networks to try and improve the ability to detect malicious software let's just look at some general concepts of scanners and you know them as antivirus scanners I think most people would would have installed an antivirus scanner on your computer so you'd use host-based scanning but many networks also use perimeter-based scanning that is that they at the edge of the network also try and detect and remove malicious software the general approaches or in improvements of antivirus software so either host-based or perimeter-based what does it do well the idea is to identify a malware to detect identify and then remove it once you've identified it so how to how does a piece of software antivirus software detect and identify malware and when we say antivirus software it doesn't just apply to viruses okay so it may apply to all types of malicious software but the common name is antivirus and this goes through the the improvements over time so first generation through the fourth generation so the first type of antivirus software just simple scanners they look for signatures of the viruses and other malicious software so if you know of an existing virus and you know the the structure of the code in that virus then that gives some signature of that virus so what the antivirus software does is when a file comes in to the computer it can it looks at that in that file compares it to the signature of the virus if they match then it makes the assumption that this file contains the virus the signature may be simple look for lines of code in the virus so if the virus contains these 100 lines of code and the file that come into your computer contains those 100 lines of code then assume that that file contains the virus but it can be more complex and that in the signature cannot just be about the exact match on the set of lines of code but can look for patterns in the lines of code for this to work the antivirus software must know about the malware in advance it must know the signature so if there's a new malware that's created then the antivirus will not detect that so the improvements were to follow some more general rules rather than looking for for specific lines of code in which match existing viruses or malware is to follow some rules and do some look for some probable lines of code that would be in the malware so the antivirus would have a set of rules as to what most viruses or most malware would contain and if a file comes in that matches those rules it would flag that file as containing malware so rather than looking at predefined pieces of code based on existing malware try and generalize and say okay most malware has this structure follows these rules in its in its code and try and detect files that come in whether they match those rules the other approach integrity checking is looking at files compared to some reference if a file on your computer system word.exe if we know that word.exe has this exact contents and then the antivirus software runs and find word.exe has a different set of contents then that may be used to try and detect that something's changed the word executable and is infected with a virus so check that the files have not been modified on your computer and you can use hashes and other approaches to do that checking of files. Next approach run or as the malicious software runs monitor what it's doing so malware comes in it's executed on your computer then the antivirus has some program that's monitoring the actions that everything that executes performs and if it's doing some malicious actions then identify that as malware so instead of looking at the structure of the code look at what the code does by letting it run that's the basic approach there so a virus or a file comes into your computer you haven't detected a virus in it but the antivirus software has some software running that when this new file executes and it tries to access particular files it tries to do things on your computer the antivirus monitors that and based upon what files it's trying to access what it's trying to do it can then try and detect if it's a virus or not an example of that some of the things that it may do is here for example the antivirus monitors the behavior of programs for example if some program tries to open files which it shouldn't normally need to try and open then that may be detected as some malicious action so if you know of a set of files on the operating system which normal programs don't need to access but this one does access like the password file then that may be detected as malware or attempts to modify existing executable files so a program runs it tries to change word.exe well in most cases another program shouldn't need to change word.exe to run so if that program takes that action the antivirus software detects that action and then identifies this as a virus or malicious software change system settings for example to change in Windows registry settings to change configuration files in Unix systems if the antivirus detects a program doing this then it again detects this as a virus or malicious software so this is when the malicious software is running it's not detecting it before it runs it doesn't depend upon any signatures about known malicious software the problem is that it run it allows a malicious code to run so therefore if it doesn't detect it then the malicious software still runs and can perform harm. The fourth or the generation here is really just a combination of all the techniques is to improve them and we'll talk about one more technique in a moment but antivirus has progressed from being looking at signatures based upon known malicious software to generating rules and trying to classify files and based upon the general structure and rules to work out with its malicious software to monitoring what software does and if it performs things which will be classified as malicious actions it detects it as malware. One example of that running the malicious software is called generic decryption. Most viruses will be encrypted and to execute they decrypt themselves so that the virus itself is encrypted when the virus runs it decrypts itself and then performs its actions so one approach is to allow that to happen so that's what generic decryption does it allows the virus to decrypt itself and then run but it does it all in a virtual machine not on your real operating system but in a virtual machine set up by the antivirus software so that virtual machine really emulates a CPU so the virus let's say is attached to some executable it's arrived at your computer the antivirus software runs that program in a virtual machine monitoring what it does there in the virtual machine if it does some malicious things like opens files it shouldn't try to open changes executable files then it may detect that as a malware if it's passed as successful was okay it's not malicious then that program can be passed back to the the host computer so basically run the virus in a virtual machine such that if it runs and does damage it only does it in the virtual machine which can be deleted and it's easier when the map the malicious software is running to detect that it's malicious software based upon the actions it takes the general trade-offs here is that okay as a file arrives at your system it needs to be executed in a virtual machine how long do you execute it for so you received a file via email you downloaded a file the antivirus software before it allows you the user to use that file runs it in a virtual machine trying to detect is this file malicious or not but how long does it run it for to detect if it's malicious because that file may only do a malicious action after some period of time so if you run it for too long then it means you have to wait a long time and it degrades the performance of the system if you only run it for a millisecond then you may not see that the malicious events taking place so generic decryption is the process of really allowing the virus to run but run it in a virtual machine to see what it does if it does something bad classify as malware if it doesn't do anything bad then let it be as a normal file yes a sandbox is more general not just for this but yes the same concept as a sandbox you run it in a in a virtual machine that is protected and and kept separate from your normal files on your computer so even if the virus runs it doesn't do any damage to your normal files so that's all the detail we're going to go into with antivirus or countermeasures so just a simple classification that most counter measures look for patterns as they receive files signatures that can be improved to a search based on general rules or to allow files to execute and monitor what they do there are many more aspects of antivirus and malware we've covered some main types of malware viruses and worms and a little bit on social engineering the different payloads is what actions the malware takes there are multiple approaches for antivirus either on a host or on a perimeter of a network for all hosts inside that network and the technique they approach they take of course a lot of this becomes this cat and mouse game of that the the people who develop the malware develop the malware and then the people are trying to block it once they learn about it they develop techniques to stop it but then new techniques are developed so new counter measures need to be developed and there's always continual improvements in both the malware and the counter measures for the malware and a main problem with antivirus techniques malware detection is the performance degradation in that trying to detect to be very accurate in detecting you make it inconvenient for the user either their computer slows down the network access slows down or the ability to access files is reduced so it makes the more effective the counter measures are the more of a performance degradation or denial of service for the normal user of course the one issue that arises is that can you trust the antivirus software what if the antivirus software is malicious or has been compromised then everything becomes a problem because you can't trust the results of that antivirus software so then is what if the operating system has been infected what can you trust at that point well people say once a computer is infected it's you might as well just delete the whole computer format the hard drive and start again because it's very hard to trust any piece of software once one infection has a rise because if a if a malware infects your computer and it has found a vulnerability in the operating system as infected the operating system then the operating system then can control the antivirus software and when new files arrive it can the infected the operating system can tell the antivirus software to to accept those new files to pretend that everything is okay so future attacks can be go undetected and the ways around that lead to things that we're not going to cover but interesting trust of computing making sure the whole chain of the bios when your computer boots the bios that loads the operating system the operating system all the applications the on the operating system are trusted cryptographically trusted that is using signatures and so on and some other topics that you may explore but we will not questions on malicious software yes so that the antivirus software that you install or is commonly available at least for home computers today again they modify key files on your system and they effectively integrate into the operating system and they have their own set of rules for detecting they don't just look based upon signatures of known malware they look up based upon actions of other software rules being programmed into them and it may be possible that they think other antivirus software is doing malicious activity as it meets its criteria is doing something that's unexpected it's commonly recommended that just one antivirus software be installed but which one's best I don't know and it may change over time any final questions on malware so just understand the concepts what's a worm what's the difference between a virus and a worm what what antivirus software does what is generic decryption what else with respect to the the upcoming midterm exam I would not ask you about any details of this particular virus or so on just the concepts be able to compare and maybe give an explanation of what is fishing what is a key logger what what are some actions of a logic bomb the details about a specific virus will not be in the exam although I know we had examples last lecture but we'll not cover them in the exam what types of viruses there may be what's the difference maybe simple things about a simple questions about this simple virus if it was given again you don't need to remember this but if if I gave you this you should be able to answer some questions about okay which line of code is considered a logic bomb or or or take some conditions as to perform some action okay you note that trigger pulled if return true if some conditions hold so here we could implement our logic bomb of say the conditions are if the date equals the 2nd of January 2014 then take this action okay enough


nada, casi nada y bueno me ha gustado especialmente el chocolate, me ha parecido estupendo. ¿Qué chocolate? Había chocolate, lo siento, estaba muy bien. Bueno, sí, vamos a hablar de algo que yo personalmente sé que me hace muchísima falta, o sea de cómo organizarnos. Después de lo que nos ha contado Wendy antes, me hace aún más falta y creo que la charla que vamos a escuchar ahora es perfecta después de y ya lo veráis, ya veráis por qué, porque yo tengo el privilegio de ya haber conocido a Paula y a sus compañeras y socias, una que tengo aquí delante, el año pasado en Work and Verona, donde escuché la charla de Alessandra que está aquí que es socia de Paula, que era también sobre organización y es que flipé, o sea las cosas, el tiempo que todos perdimos, yo no sé si os pasa a vosotros también, a mí por cierto, yo llego a especialmente los que trabajamos solos sin tener un equipo, sin tener un jefe ni nada, el tiempo que me pierdo yo con y ni siquiera veces redes sociales, o sea puede ser de todo, mis gatos, lo que sea. Entonces, todos los consejos, bueno, consejos, sistemas, Wendy antes nos ha dado sistemas, lo que Wendy nos ha contado de cómo sacar provecho de clientes con poco presupuesto ha sido sistemas. Paula y sus compañeras, pero hoy Paula, van a hacer lo mismo y Paula merece muchísimo la pena escucharle porque ella es súper organizada, porque es nómada digital, o sea ella vive, lleva su maleta y vive sacando las cosas de la maleta, entonces si tienes que ser organizada, yo no sé si vosotros viajáis, yo cuando viaje viajo desastres, que viajo bastante a menudo, he mejorado un poco pero todavía no lo tengo bien claro. Entonces no veo la hora de que Paula nos cuente cómo, no, cuál es el ingrediente secreto de una organización de tu tiempo, bueno perfecta, no, nadie, no queremos ser perfectos, pero lo mejor posible para sacarle más provecho al tiempo libre, porque creo que es eso no, para ser más felices y tener más tiempo y viajar bien sin dejarse las cosas, los cables del iphone en todos los hoteles y tal y o se dado la presentación en español porque Paula entiende muy bien español pero la charla para que quede claro es va a ser en inglés y ahora muchas gracias a Paula para haber venido, por haber venido aquí. So thank you for being here, today we talk about time, so let's get it started but first of all let me introduce myself, I'm Paula, the hidden one, I'm Italian and I'm a professional organizer, have you ever heard of this job, someone did, someone no, not at all, I'm a consultant and I help my clients to be organized during their days to set their priorities, to get their things done, so their activities and to find an important work-life balance which is something that interests the majority of us, so I am co-founder of Organizatessen, we are a group of four professional organizers and we work with companies, with freelancers, we give classes and we make one-to-one sessions, so today we will talk about time in general, so I'm sure you've heard a lot about time management, you've used these expressions a lot of times but I'm sorry to inform you that we cannot manage time, so you will be disappointed but the best thing that we can do is manage ourselves in our day, why is there a need of a professional organizer or a consultant like me for example because life got really complicated in the last 10 or 20 years, there are so many things that we have to do, so many things that we want to do and so little time and energy to accomplish them, so our schedule already started, we start our schedule already in a hurry in the morning and most of the times we arrive at the end of the day that we are exhausted and we still have a lot of things to do that we have to postpone to the next day, so we have to fit all the pieces together and this is why it is important, like Wendy said before, to focus on the important things in your day, meaning in your work but also in your personal life, so this is very important to have a good time management, so we talk about time but time is limited, we don't have that much time we would like to have, we have just 24 hours in a day and this is for everyone, I know we would like to have an ocean of minutes all for ourselves but we have just 24 hours and let's say we have a pool of minutes and in a pool you cannot add more water than it fits, so although we would love it but we cannot save minutes that we didn't use properly and we cannot get the ones that we use back, so we have to cope with what we have, sorry but my English sometimes it's like okay so we have to party with a pool, I could teach you a lot of ways to be productive and Rocio was looking for some more productive productivity tools but I'm sorry I wouldn't give you any today, no we will just focus on this, that productivity is of course a solution, in our day we can learn techniques and strategies and we can add tools in our day to get more things done but the point is we cannot get 36 hour days, although we are very efficient with many tools but still we cannot stretch our time, so we have to deal with this and we have to be really aware of this, so what, what am I doing here, I'm not talking about time management, I'm not giving tools, what am I doing here, now I want I would like to focus on something very important to me that I think could be very important to you as well, how long does a minute last, you would say 60 seconds, okay do we agree on that of course, okay fine this is what we know but in fact how long is a minute aware of how long does a minute last, when we are aware of what we are doing compared to a minute in which we are not aware of what happens, for example when we are scrolling our feeds like Wendy told us or if we are focusing on something that matters to us, so I would like to make an experiment with you, I will set a time watch on one minute and we will spend this minute just being aware of the time passing by, so you just stay there and we will see what happens, alright, okay let me take out my phone, are you ready, okay let's start, okay that felt awkward right, you weren't at ease probably, we started laughing no no no no no no no you weren't at ease and probably you had the impulse to take out your phone probably to scroll down the feed or maybe someone did it, I don't know, but this is what happens all the time, we are not aware of the time passing by, so if you knew a minute would last so long, how many things would you do in a minute, so today I would like to introduce a new point of view, instead of struggling to get hundreds of tasks done the whole day and running and switching from one task very rapidly, so why couldn't you just focus on a task, why couldn't you just dive deep in your time and feel it and leave it, would that be so difficult, maybe yes because we are not used to that, but still this is this is the thing that we should start to learn because in fact switching very fast from one task to the other rarely gives us the feeling that we are doing well, that we are doing something with real quality, so this is something that we have to learn, so the keyword of today is awareness, awareness is the real way to get more time in your day, could you imagine to dive deep in the pool instead of being just on the surface and moving from one task to the other, so how would that feel if you just dive deep in your time in the tasks you are doing, at the beginning you would see everything blurred probably and dark, you wouldn't recognize shapes, but then all of a sudden you would get a customer to it and you would start seeing all the details that earlier weren't clear, so you would see all the things that are important to you and that are the real treasure in your day, what has organization to do with this, organization is one of the ways, I know, to raise awareness in your day because all the time we are very, we have a crowded mind, we have so many things in mind, we have so many things to remember, things to do, worries or dreams as well and they all fill our head, so organization is a way to clear your mind, to free it, to free the energy, to focus on the things that matter to you, to the things that are important in your work or in your life, so this is something that we have to learn because we are not used to be aware like we saw in the experiment earlier and I know what you're thinking, there are so many things in your day that are a burden to you that bother you, that feel like a pain in the neck, something that you don't like, probably some tasks or some activities like, I don't know, accounting maybe, or writing that email to the client who's not paying you, do you have any other example, yes, he thought about the client who's not paying, okay, do you have any other example of something that bothers you but still is on your to-do list, I guess you have, so could you imagine, no, yes, tons, okay, could you imagine how would it be to get through these things faster, to focus on them and to accomplish them, can you feel how satisfied you would be once they are gone, once you cancel them from your to-do list, it could be very, very important to you, it would free a lot of time and of energy because all the time that you think about this task you're not doing, you're still wasting time and energy, so if you start doing so you will start to develop your strength and to be satisfied of what you're doing because this is very important to us, if you focus on the things that bother you and you just complete them, you will feel like superpowers, you will feel very satisfied, so this is something that we should learn, we all should learn, but the problem is someone told us that multitasking is a great skill, that we will be faster if we switch from one task to the other and we do at the same time, two or more things at the same time, unfortunately multitasking is a big lie because you are forcing your brain to do a lot more effort to switch from one task to the other instead of focusing on one task per time, so we are not computers, we have to admit it to ourselves first of all and start doing one thing at a time, I know this sounds very, very difficult but if you still don't believe me, let's play, if you have a sheet of paper and a pen just take them out and we will do some exercise together, ok, are you ready? Perfect, now you will have to write the English alphabet in a row in capital letters, just like this, some doubts about the English alphabet probably, ok, that was an easy task, now on the next row you have to write the sequence of numbers from 1 to 26, ok, easy right, ok, now on the next row you will have to alternate letters and numbers, just like this, so A, 1, B, 2, C, 3, until Z and 26, ok, how is it going? Good, ok, it's tiring actually, ok, done, ok, nice, ok, this is an experiment of what happens when you multitask, so switching from one task to the other meaning from the letter to the number is a big, big effort to your brain because it has to concentrate on which was the last thing you did, which is the next step you are going to do for every sequence, so every time you switch from one to the other, your brain is making a big, big effort and this can bring to burnout and the quality of the job you are doing of course is poor and this was a stupid example and say, but can you believe it what happens when you are working and at the same time you are just chatting on WhatsApp or when you are on the phone speaking to a client and at the same time you are writing an email, so which is the result that you are not listening to the phone and that you are writing the email like an eight year old kid probably and that you will ask the person to repeat what he or she was saying and then if you read the email again you have to rewrite it, so multitasking is not helping you to be faster it makes you make more mistakes and then it will ask you more time to fix these mistakes and you will feel more exhausted in your day because of all of this switching, so it's important to learn to focus on one thing at a time, this is my suggestion, do one thing at a time if you are on the phone, be on the phone, if you are writing the email be present on your email because the quality of your work will increase and the satisfaction that comes through this will increase as well and we were talking about satisfaction earlier so it's very important to feel that we are doing well in our day and doing one thing at a time helps us to do this and can you believe how it feels by doing the things that you don't like? It's just the same, being big satisfaction and I know that in reality it could be very very difficult to do one thing at a time because we are not used to this this is why I suggest to use a timer a timer can save you, it means it can help you to focus on the activities you are doing this means you can set the timer on 10 or 15 or 30 minutes depending on how long you think your task will last and then focus on that activity and only that one and by focusing I mean turning off the programs that could distract you and switching your phone to flight mode and telling the people around you that you are very busy for that 10 or 15 minutes and then dive deep in this task it will feel very very different and I swear that you will do all the things also in less time than you thought and even before the timer rings so we've been talking about time management what is time management? it's spending our time well and wisely and not wasting it is about getting our things done and get them done well the ones we like as well as the ones we don't like it means focusing on every activity and putting your heart, your mind and soul in everything you do and this means raising your awareness, focusing means paying attention to what you're doing to your job, to your personal life, to the persons you are with and this increases your satisfaction of course and so it's only by doing all of this that you will start realizing that you have a treasure in your day which is your time and you will realize as well that you have even more time than you thought so this is my take away for you please do one thing at a time and try to focus and be aware of what you're doing your time will explode and expand so thank you so much bueno le pasamos hola hello Paola, thank you I use the pomodoro for my work and I want to know what do you think about that if you think that it's useful for me it's useful because I work I have a pomodoro of 15 minutes and I don't know, I only want to know what do you think about that pomodoro for who doesn't know it it's a technique that says to set a timer on 25 minutes and then have breaks and to focus on that 25 minutes I think it's a good solution if it works for you of course because techniques must fit to everyone's work or day and if you feel like it's helping I'm sure it's a good solution some people say 25 minutes it's too short time to focus and then taking a break is not helping them to get things done but still my suggestion is the same one use the timer to focus to dive deep in the things you're doing because this helps you to get them done faster so if it works for you, yes, absolutely hola Paola si, si, thank you for the presentation I want to, you said a couple of times getting things done what do you think about the GTD method ok, I'm not really... because what I heard is deep work and GTD so... I'm not really a big big fan of the whole GTD thing but I think there are some... sorry, what is? GTD is getting things done, a method by David Allen who's really like the productivity boss or whatever, he's American and it's a method, it's a very specific one I'm a little bit afraid of that method of course but it has a big problem for me because you need to feed it you need to feed this method yes you waste a lot of time just keeping all the things organized it's what I think, it was asking you yes, just like for the Pomodoro technique every technique is good if you feel it if you are comfortable using it because otherwise it feels like it's another job y it's something that is too much and if you have this feeling then maybe GTD is not the system for you but you can take some of its suggestions actually I like the brain dump part for example so brain dump is like taking out all the things that you have in mind and writing them this is very helpful of course but the rest of it's... yeah, yeah I know, it can be... a different cause yes, exactly so maybe you can take some parts of it the ones that are useful for you, sure and this is what we do actually in our job to split it and to use what is okay for you and this is what we do actually we just try to make the best solution for the person actually and not just taking a method and applying it completely because if you don't feel it, it doesn't work thank you Hello, I was wondering if you have... because the point of the talk was to get very focused very aware and mindful of what you are doing in the moment you are doing it and I was wondering if you have ever worked with people who were like physically unable to focus like people with ADHD and how do you... and if you recommended them some exercises or how did you deal with them, it has happened okay, no, I never worked with ADHD persons but I think some of the techniques might work for them as well for example, I am thinking of children, for example that cannot focus on their task y maybe help them to be focused one or two minutes and then three until five, etc. this helps them to get used to some methods but specifically with these problems, I don't know actually I have a question if I may so I know exactly how long a minute is it's between 14 and 16 breaths depending on how deep I breathe because I do meditation, I do yoga I hold poses for very long and so on but still I have a huge, huge problem focusing when I'm working because I am addicted to my phone to Slack, to Facebook, to Twitter, to Instagram it's terrible, I have recently I mean I'm a very grown woman I mean I'm nearly, you know but I still I feel like a child we age with an attention deficit problem and it's induced by phones and it's become recently like a real problem for me if I have to focus also because I work for myself I don't have a boss, I don't have a team I am not accountable to anyone but myself and it's a real issue these days that I get to the end of the day I'm like what have I done maybe I have got a few things finished but really it's become a problem so this talk is really important for me so I guess I was thinking maybe the pomodoro technique is something that I should be doing because maybe yes, start with small chunks and then as you were saying like with a child but really I am cursing my phone because I wasn't addicted and I have to admit I am addicted to my phone and then if I turn the phone off I still have Facebook and Twitter and Slack on my computer and the thing is that I think that I should just switch off the internet just turning it off but I can't do that because a lot of the tasks that I need to accomplish actually need the internet so it's a problem so I think that what you do is really it's incredibly relevant these days I guess it's not a question as in, help but so I would say pomodoro technique yes, but there are also some tools programs on the computer for example, I can imagine of self control you just set the pages that you don't want to see while you're working so Facebook out of it and then you make the list and then I know but help yourself I know it's hard it's hard because we are all addicted actually I have an app like that because I know the problem and it's called freedom and once it's in working you can't turn it off you can't turn it off and it's horrible it is actually horrible because I have my freedom app on for social media between 10 at night and 10 in the morning because I want to get because my loss of time was before I went to bed I was endlessly scrolling through the Instagram and Facebook and whatever and when I got up it was the first thing I did so I eliminated those and it is like so refreshing so it's amazing but I cursed the app millions of times already so it is but it's working because you can't turn it off I think also I hope freedom God, I don't what I was thinking as well is that my excuse is that some of the people I work with communicate with me via messenger it's just because we're used to it we're friends on Facebook and they write to me in messenger so I guess that the story is that you just have to tell everybody it's like if you want to speak to me you need to email me and then we make so I don't open messenger answer immediately because you don't have to I know, but I feel I know, but I get anxious because I think maybe someone's written to me and I'm going to die I'm going to die because I can't see it you're right I'm sorry, no, you don't need it no, I don't need it okay, after my rant is there any more questions? what do you do it with not work related tasks because I am pretty good at doing focused work yes when it's work but then I go cooking and cleaning and ironing and walking the dog and I like that okay so how do you do how do you do you also use the timer technique? you can use it if you're doing the cleaning and you don't like cleaning you just set the timer 15 minutes, very concentrated and then you just can map you give it a break and then you do something else hello, hello there, sorry so thank you for the presentation very interesting my question is when we are a lot here entrepreneurs and when you're entrepreneur sometimes you have to be multitasking because in my case at least I receive calls and then that call brings to one thing that I have to answer and do right away or I was doing something else and then something else happened and it's very difficult to just say no, or for example I cannot put my phone I can't not put my phone into airplane mode it's impossible because in my case if I miss a call that means one sale that it's gone so in this case I cannot say I have developed multitasking because that's not true but I have learned more or less to do several tasks kind of at the same time I won't say I'm very successful at it but it has to be done like this at least for the moment because I am alone doing my own stuff maybe when I have something else or somebody else working with me that could be different but for the moment sometimes you have at least in my case I have to do it like this and I was talking to my colleague here we're friends and we're less where I have the same issue but I guess we are not the only one probably doing that so how do you deal with these cases? what is your recommendation to that? Well, I start of understanding if you're working in a hospital or an emergency room so this is not the case of course so we are like we can take a breath, okay? so it's okay to we have a lot of things done but it's not that you have to save lives and you have to do right everything at the same time so this means probably we can just focus on one thing maybe you are on the phone just be on the phone and then you take a note on what you have to do next but just it's a sequence of what you have to do it's not you're not doing a good job actually if you're not focused on what you're doing because you're thinking of too many things altogether so this is the point is just being aware that you have to do it in sequence it's not possible to do a good job and to be focused and to be satisfied of what you're doing if you just run from one thing to the other it's just like being on the surface all the time and you're not really getting what you're doing I understand there are so many people that tell us that no, I cannot switch my emails off that we have to be on the phone all the time and that I communicate with my clients like this it's all true, we all do it all the time but we're not saving lives in my case, I'm not saving lives in that moment so I have to make choices and stay focused on the one thing that I'm doing at that moment and then I go on in setting priorities as well and then making a list actually of what you have to do in sequence because okay, you are able now to multitask or to switch very fast from one task to the other but you feel exhausted from this you're probably not happy either so I think that they were not saving lives I mean, I'm saying this to myself as well but we're not saving lives I think that's the most important thing now it's in five minutes there's the round table in track one and I cannot thank Paula enough for the wonderful, wonderful talk that she's given us


Okay, so we were finishing the chapter about dimension. So we were talking about dimension and transcendence degree. So I don't really call the definition of transcendence degree, which is relatively clear. So we had, there was this alternative definition definition of the transcendence degree that, so we wanted to prove the following theorem. So if X is a variety, then the dimension of X is equal to the transcendence degree, I don't even know how I denoted it, of the function field, the field of rational functions over K. So the number of algebraically independent elements here. And we had known that these algebraically independent elements can be taken as a subset of the classes of the coordinates, for instance, if you are an AN. And we, so this we are about, we proved this model of the statement. So we, so prove this, assuming the following. The theorem that every variety is vibrational to a hypersurface, okay? And this is somewhat surprising result, because somehow one would think that varieties are much more general than just hypersurface and they also are. But there's, there are open subsets, which are isomorphic to open subsets of hypersurfaces always, okay? And now we want to prove this theorem. So first, so for simplicity, I will prove the theorem under the assumption that the characteristic of K is equal to zero. The proof that I will give can be modified to work for any characteristic, but it's a bit more complicated. And so I will just, so for simplicity assume. So I mean, this I use because I want to use the theorem of the primitive element. Now, one can prove that in the situation where I am, the theorem of the primitive element can be applied. You need a whatever, a separable field extension and the field extension we're looking at will be separable, but one would have to prove that and I don't want to. And in characteristic zero, we know that all finite extensions are separable. So that would be fine. So, so we have the theorem of the primitive element. So let K be a field of characteristic zero and let L over K be a finite field extension. Then there exists a primitive element. So then there exists an element B in L such that L can be obtained by adding, joining this single element B to K. And I expect you had this in the ultra course. So it's not, so now we want to use this to prove our theorem, this one, whatever. So we have to somehow make this hypersurface, no? We have to find the equation of a hypersurface. And we do this by using what we know about this transcendence basis. So we say, so we have Kx, the function field of X.  There is some transcendence basis to choose one. So then we know by the theorem that I cited the other time that if I take Kx over Kx1 to X are is a finite algebraic extension. Now we had this theorem. If I have a finite degenerated field extension of K and we divide by K a joint transcendence basis then the quotient, then this is a finite algebraic extension. So now we apply the theorem of the primitive element. So there exists one element which I have to join to this field so that it becomes Kx. So there exists an element Y in Kx such that Kx, large Kx is equal to the small field Kx1 to X are and I join also Y. And Y is algebraic, Kx1 to X are. So if Y is algebraic over Kx1 to X are that, I mean there's a polynomial of which it is a zero. I can basically take the minimal polynomial of this. So thus, so there exists an irreducible polynomial say F which with coefficients in this smaller field Kx1 to X are with variables in a polynomial in X with coefficients in this field such that if I take F of my element Y, this will be equal to zero. It's irreducible so it's not zero anyway. Okay, now we want to basically use this F to define the hyper surface. So this is supposed to be finally the equation of a variety which is by rational to X. But in order to do this, you know, we cannot have it like this with coefficients in the field. You want to clear the denominators. No, this is, so we know that Kx1 to X are is isomorphic to the field of rational functions. Kx1 to X are is isomorphic. Or I could also say is the field of rational functions in variables X1 to X are. So I can write F, see some whatever was some finite sum of some else GL from X1 to X1 to X are divided by HL times X to the L where the GI or the GL and the HL are some polynomials with coefficients in the small k in X1 to X are. No, because we know the elements of this quotient field of this field of rational functions are just quotients of polynomials. Okay, so now we can clear the denominators. So multiply F by the product of the HL and divide afterwards by the greatest common divisor of the new coefficients. So after multiplying, I get some new coefficients of X to the L, I take the greatest common divisor. So then I, in this way, I obtain a new polynomial. So in this way, you get F, which is some polynomial in X1 to X are times the original F, which is equal to whatever G we'll call it H tilde times F, where H tilde is a polynomial in K X1 to X are and also F is a polynomial K X1 to X are and X. We just have cleared the denominators and I have, and in fact, F is a primitive polynomial. So let's see. I write it like this, I can, this is the same thing by definition of the polynomial ring, but I know the polynomial ring and R plus one variables is the same as the polynomial ring and R variables as polynomials in X with coefficients, polynomials in one variable less. But just, if I viewed now as a polynomial in X, it's a primitive polynomial because I've just said that I wanted the coefficients to be relatively prime or divided by the greatest common divisor. And we know that if I take the original F, so the original F is irreducible in this field therefore also F itself, the small F is irreducible in K as element in K X1 to X are comma X because it's after all just the big F multiplied by a unit in this, in this ring. No, it's just the constant from the viewpoint of this. And so by the Gauss lemma, F is irreducible as a polynomial in this ring. Okay, so we have this irreducible polynomial, I don't know whether. So, and now I want to say, as I said, I want to take this polynomial which is just a polynomial in R plus one variables. I want to take this as the polynomial which determines a hypersurface in AR plus one. So let Y be the zero set of F in AR plus one. This is a hypersurface, it's an irreducible hypersurface that's the zero set of an irreducible polynomial. And if I look at the function field of Y, what is it? It's the quotient field of this ring. The field of rational functions. So this is, so we know if we take the, how is it called? The coordinate ring of Y, this will be K X one to XR X divided by the ideal generated by F. And so the function field is the quotient field of that. Yeah, indeed. And now, yeah, here I'm maybe a little bit fast, but so you could, I claim that this is the same as the function field of K X one to XR and then like this. So you just have to see that.  Anyway, you can try to think about it's quite easy to see. And this is just equal to what we had here. I have unfortunately wiped it out, but we had seen that this was precisely, if this is nothing else as Q S K X one to XR, a joint X divided by F. And this is precisely what we had gotten as K X. K X. Well, if you remember, we had that K X was isomorphic to K X one to XR, a joint some element Y, which was the zero of a polynomial. So if I were here, we have the F, but I can replace over this thing, the difference between the small F and the big F is just a unit. So I can also write F here. And then it's precisely how it was given. And so we see that K X is equal to K Y, so they are birational. So it's a somewhat unexplicit proof because we don't actually see any open set, which is isomorphic to any other open set. We just prove, abstractly, that the function fields are isomorphic. And we also have not really explicitly done anything at all. We take this transcendence basis, which exists by some general existence result. So somehow how you find this hypersurface, which is birational to our given variety is completely mysterious from the proof. But anyway, we have proven that is one. Okay. It's, yeah, yeah, X is birational to Y. That's, K X is equal to K Y. K Y is, and therefore X is birational to Y, sorry. That's, you know that two varieties are birational, if and only if the function fields are isomorphic. Okay, so this proves this result. We will also use it again later in this particular theorem. So either at the end of today's lecture, but that's slightly unlikely, more likely in the next lecture. So with this, we actually finish the story of dimension. So I have this other definition of dimension. So one was with this change of irreducible closed subsets. The other one was with a transcendence degree. And we find that they are both the same. And so as I said, the transcendence degree, you really can think of it as somehow in some sense, at least if you view this XI somehow as coordinates. So the transcendence basis, you can see how many coordinates you have in some sense locally on your variety, so that it has something to do with that. And the other definition is this other thing that each time you cut down by some creation, it gets, the dimension gets smaller by one, and then this way you see the dimension. So there's two quite different views on dimension, which are, which turn out to give the same result. Okay. And so now we want to talk about tangent space, singular and non-singular points. So, you know, if you have a manifold, then the manifold has a tangent space at every point. An algebraic, and this tangent space has always the same dimension as the manifold. If you have an algebraic variety, it will have a tangent space, and this tangent space will not always have the same dimension. It depends on the point where you are, which dimension it has. But there will be some open subset where the dimension will always be the same. And where the dimension is actually, of the tangent space is actually equal to the dimension of the manifold or of the variety. And these are the non-singular points, so where the thing looks smooth. And where the dimension is different, of the tangent space is different from the dimension of the variety, these will be the singular points. And it somehow also coincides, you know, when you see, if when you make a picture and you see something which looks singular, you will see that this actually is a singular point according to this definition. Okay, so let's see. I will have to do this a bit slowly. First, I do it for hypersurfaces in AN, and then I do it for a fine varieties, and then I do it in general. So we start with the case of hypersurfaces in fine space. So singular and so case of hypersurfaces in AN. So first I talk about the differential of a polynomial, and I will use that to define what I want. So, or rather partial derivatives. So let, so if f is a polynomial in x1, xn, we can consider, so we have, we can consider the partial derivatives, df by dxi, this would be also some polynomial. And you know, these are just, you compute them formally, you know, it's not an analysis thing, you can do this over any field. So if x is a zero set of f in AN, it's a hypersurface, and we assume that the ideal of x is equal to a deal generated by f. So here I'm not assuming that f is irreducible. So if f is irreducible, for instance, the ideal of x is equal to that, but if it could be that x is reducible, but the fact that the ideal of x equal to ideal generated by f means that if you write it as a product of factors, no, none of the factors has a power in it. So, and so anyway, I assume, so x is allowed to be reducible, but it must be that the ideal of x is just f. So a point P and x is called a singular point, where if all partial derivatives of f vanish at this point, so if and on f, the f by dx i at P is equal to zero for all i equals one to n. Okay, so if I have a point which lies in x, so f is zero there, and all partial derivatives are also zero, it's called a singular point. Otherwise, it's called a non-singular point. And maybe one says that the set of non-singular points at x-rag is defined to be the set of all P in x. P is non-singular. So I want to stress one point. A singular point of x must be point of x, okay? It's not, so it's not enough, so by definition, a singular point of x is a point of x. So the condition is not just that all partial derivatives are zero, but also f must be zero. So when I, in the homework and final exam over the years, there was always somebody who got this wrong. So therefore, I stated it again explicitly. So a singular point of a variety is in particular a point of the variety. Okay, so example, so I just want to make some examples. So first, so this condition that the idea of x is f is some more important for this. Yeah, you can also say regular point, yeah. So that's why it's written x-rag here, yeah. But I, you sometimes also say smooth point. One can maybe say that x is called smooth if, and only if x is equal to the set of regular points. So all the points of x are regular. So first, we really want that we really need that the ideal of x is equal to f, not just that x is the zero set of f. I mean, for instance, if I take the zero set of y squared in a two, so the variables x and y, this is just zero set of y, it's just a line, and this is non-singular. It's smooth, so it's called non-singular or smooth. So it's non-singular because you could take the derivative d by dy of this is one, and this will not be zero anywhere. But on the other hand, if you do it for y squared, at every point in the zero set of y, so d by the x of y squared is always zero, and d by dy at every point where y is equal to zero will also be zero. So you would, from this definition, would get that every point of this is a singular point, but that's not true. Actually, every point is non-singular, okay? We can look at some simple examples. If we take x equal to c of x minus y, I do it this way, y minus x squared, then you can easily see that this is a conic, so a parabola. If you make the real picture, it somehow looks like that, so it looks kind of smooth, and indeed you find that this is non-singular. So all the points of it, if you just take d by, d by dy, this will always be one, and well, this is two x, so unless x and y are both zero, anyway, you will see that this will work. Okay, this is a non-singular right, what? So let's just look at, for every point you have to check. So let's check at the point, so if I take d by dy, yeah, and okay, and so it's non-zero. No, but we don't need to show for x. We have to show that not all are zero. So it's a single point if all the partial derivatives are zero. And so here the first one is always non-zero, so it's completely different. Okay, so we can also look at x equal to the zero set of y squared minus x squared times x plus one. So this is, we know is a nodal cubic, so you somehow have a picture, so I always have the same examples, no? Because there are not so many examples which one can actually deal with explicitly. So it looks somehow like that, and you can see there is one point which looks a little bit different, which is the point zero zero. Now when x and y are both zero, something happens. And this, if you look at the real picture, you see really that you have two branches which intersect like that. So you would expect that this is a singular point, and you can easily check that zero zero is a singular point. Now if you just take d by dy, this will be two y, and so it's zero one, y is equal to zero, and d by dx, y you can work this out, it's also obviously zero if y is equal, if x is equal to zero. And in the same way for a cuspular cubic, so you can check that all the other points are non-singular if you want. Yeah, it's the only singular point. And this would be the cuspular cubic, you know that the picture is somehow like this, and we have again a special point here, zero zero, and we find again that zero zero is the only singular point. In fact you can obviously see that this is a singular point, you have the one derivatives two y, and the other one is three. What do you mean? No, this has one singular point. So there's only one, so if you take away the point, it's non-singular, but it is singular. We call it something singular if it has at least a singular point. What, no, maybe please ask a question. Yeah, no, it's called non-singular if all the points are non-singular. No, in fact, we will see in a moment that for any hypersurface, most points are non-singular. It's always an open dense subset where it's non-singular. So therefore, you don't call something non-singular because it's almost everywhere non-singular. You call it non-singular if it is everywhere non-singular. No, it's singular if it is somewhere singular. So otherwise, so it's called smooth if and only if x is equal to x-ray. So if all the points are non-singular, it's called singular if there's at least one point where it's singular. Yeah, yeah, but anyway, I didn't want to actually introduce this, but if you ask whether this is non-singular, no, it's not non-singular. It's non-singular only if it's everywhere non-singular, otherwise it would be. And that, yeah, this is non-singular. It's everywhere non-singular. But these two are singular. Okay, so let's, what? I don't know, that was maybe calf. It sounded like something. Okay, so now let me see. Now I want to prove that if I have an irreducible hypersurface, then it has an open dense subset which is non-singular. So then the set of non-singular points is open and dense. So proposition, so let x in an irreducible hypersurface, so zero set of an irreducible polynomial, then x-ray is open and dense, which means the same as non-empty, but I write it to intense in x. So you could say, okay, so we have the, want to see why this is true. So again, I'm a little bit cheating. I will assume for simplicity that the characteristic of our field is zeroed. Okay, so one can again look a little bit more carefully and see that the proof works with minor modifications if k is, if the characteristic is not zero, but I don't want to take the trouble. So I just will assume the characteristic of k is zero. So we take some polynomial f in k, x1, xn, of which is, so the polynomial of which it is zero set. So irreducible such that x is the zero set of f. So then we know that the ideal of x is the ideal generated by f, no? It's a prime ideal. So now we can say, so maybe I call xing to be the set of singular points of x. And now if you remember the definition, I claim that the singular points of x, what are the singular points of x? These are the points of x, where all the partial derivatives of f vanish. The points of f of x are the zero set of f. And then we want also all the partial derivatives to vanish, so we take also, so it's this zero set. These are the singular points. And so obviously this is a closed subset of x. So this is contained in x by definition and it's closed in x, because it's a zero set of some more polynomials. So what we have to show is that the compliment is dense, which is the same as saying that the compliment is non-empty because the compliment is an open subset of an irreducible of a variety. If I have an open subset of an irreducible variety, it is dense, as long as it's not empty. Obviously when it's empty, it's not dense. Okay, you send this a bit too long. Yeah, anyway, so we have to show. So the only thing we are left to show is that x is not equal to the single locals of x. Well, assume it is. So then we want to come to a contradiction. So you know that then I claim, so the ideal of all polynomials that vanish on x is the ideal generated by f. So therefore I claim that, you know, if x is equal to xing, it follows that df by dx i, if I take the zero set of that, this contains x, no? Because if I take, the zero set should not become smaller by intersecting with it for all i, all i. So in other words, we have that df by dx i is an element in the ideal generated by f, for all i. But now if you think of it, this is more or less impossible because df by dx i has certainly a smaller degree than f. So how is it possible that something of a smaller degree is contained in ideal generated by something? There's only one possibility, namely that this thing is zero. It follows that df by dx i equal to zero for all i. But so now we use the characteristic zero. If the characteristic is non-zero, then there are some ways how this could actually be true, which one then has to exclude. But if the characteristic is zero, then this implies that f is constant. And this contradicts, I mean, f is supposed to be hyper-surface, so it defines a hyper-surface, so it must be non-constant polynomial. So this was this? Yeah, I expect it can also be generated that, but I now just, I mean, certainly we, yeah, I think with a bit more care, one can also do it if it's irreusable, but okay, maybe that you can think of as an exercise, but I think it shouldn't be very difficult to generalize. I just was a bit lazy. Okay, so this was, for the moment, everything I wanted to say about hyper-surface is, I, now I want to talk about tangent space for defined varieties. So the tangent space and singularity and non-singularity. Yeah, I haven't talked about tangent space, we'll see in a moment that what the tangent space was in this case. So one actually finds that according to the definition we are going to give, for a singular point, the tangent space will be the whole of A n, okay, for hyper-surface like that. But let's do it, okay, so let me see. So, I mean, we are not going to get so far, but it is so that there's a big difference in difficulty between treating varieties which are non-singular and varieties which are singular, okay? So many, so one of the reasons is that over the complex numbers, a non-singular variety will be a complex manifold. So you can also, and in fact, so one can then also use differential geometry to study them for once. But at any rate, many other methods of also algebraic geometric methods depend, I mean, or become at least much simpler if the variety is non-singular. So therefore one wants very much to, you know, often wants to restrict the tension to non-singular varieties. So I, in my work, I very seldom encounter singular varieties because I don't, because I like to also use topological and differential geometric tools. And so therefore it is important to, you know, to have to be able to talk about this. So also, for instance, if one studies homology and homology of these varieties, it's much easier to describe if they are non-singular. So it's related to, and so we, if one goes on in the study, one would most of the time want to study non-singular varieties, develop special tools for that. And we will actually, at the end of this lecture, we'll have maybe one or two lectures or one and a half about how it looks like for non-singular curves. That you somehow have a different way of describing the local ring of a non-singular curve. And this can be used to study, you know, many things about non-singular curves, which would be the last chapter of these notes, which we are not going to talk about, I think. But if we did one finds that, for instance, morphisms of non-singular curves have a much nicer description because of the fact that they are non-singular, and therefore the local rings of the non-singular curves have very special properties. So there's really some, I mean, it is really quite useful. And obviously if one is fearless, later when one will also deal with singular things, there is any way one cannot always avoid singularities, one also has to study them, and then it's maybe important in order to understand things, then to understand what the singularities precisely are. And so it's really, I mean, it's really quite important for many things. I mean, algebraic geometry is not only about classification. I mean, to say when two things are isomorphic, that's a very coarse way of looking at it. Obviously it's also true if you have two varieties, and you have an isomorphism between them, then it must map non-singular points to non-singular points and singular points to singular points. So that also you can find something about whether things can be isomorphic from looking at singularities and tangents based. And this is, as you can see until now, as compared to what we did about dimension, somehow talking about tangent space and singular and non-singular is in some sense more elementary. You know, you have actually concrete computations. The only thing why it comes afterwards is that the general definition of non-singular means that the dimension of the tangent space is equal to the dimension of the variety. So obviously you can only make sense of that if you know what dimension is. But by itself, dimension is in some sense more complicated to handle concept than this. Okay, so now we want to study the tangent space and often a fine algebraic set. So I also do it for a fine algebraic set. I don't necessarily assume that things are irreducible because I don't need it. So let x in an in a fine algebraic set and I want to define the tangent space. So first I define the, so definition. First I define the differential of a polynomial. So let f in k x1 xn be some polynomial and I take p a point, do I need to call it? Yeah, maybe, no. p a point in an, then I can look at the differential at p of f. So the differential of f at p is equal to the sum i equals one to n. I take the partial derivative at p and then I multiply this. So this is just a number, no? I take the partial derivative, which is a polynomial I evaluate at p, this is a number and I multiply this by xi. So this is a polynomial of degree one. A linear polynomial. And therefore it's, I can also view it as a linear map. And now the tangent space to x at p. So if p is a point in x, again the tangent space only defined for points in x. You don't define the tangent space to x at some point which does not lie in x. And is a tpx, which is defined to be the zero set of all the differentials for f in ex. So you just take, so you take all the polynomials in the ideal of x and you take the differential at p. So you see therefore the tpx is actually a vector space. So it's a zero set of linear polynomials in an. So it's a defined subspace of an, which goes through the origin. So it's just a vector space. And now, so p in x is called a non-singular point. If the dimension of this tangent space, you can either take the dimension as a vector space or also the dimension of sub-variety. This is the same because it's an defined space. The dimension of tpx is equal to the dimension of x at p. So what do I mean by that? I think I haven't introduced this. So the dimension of x of p is equal to the maximum of the dimensions of components, of irreducible components of x passing through p. And otherwise p is called a singular point. So in particular if x is irreducible, then x is called a non-singular point, if and only if the dimension of the tangent space at x is equal to the dimension of x. And I should also say this actually is a bit misleading, namely if I have a non-singular point, then this will always lie only on one component. So one can't show, but we cannot in the moment, and I mean it's slightly more, it takes a bit of algebra which we haven't developed. So can show that if p in x is a non-singular point, then p lies only on one component of x. So if the point p lies on the intersection of two components, it's always a singular point. But it's not so... For hypersurfaces you... it would be an exercise to prove that, but in general it takes a bit of algebra. Okay. And again so we have x, equal to the set of p non-singular points of x, and by x-ing is x, set of singular points. And x is called smooth if x is equal to x as before. So there's... Here it's a little bit annoying that we have... So we have to find tpx to be zero set of the differences of all elements in the ideal of x. So that's very many polynomials. So that might not be very easy to check. But so therefore we want to maybe have a way to use less. Just take a set of generators of the ideal. And this is a simple remark. So if the ideal of x is equal to the ideal generated by some polynomial f1 to fr, then the tangent space of x is equal to zero set of dpf1. Okay. So we need to only take the generators. And this is a rather straightforward exercise. So maybe as we have the definition here, I can leave it there for a moment. So by definition obviously the tangent space of x is contained in this common zero set. Because by the definition here, the tangent set is the common zero set of all elements in the ideal of x. And these are certain elements in the ideal of x. So you have to prove the converse inclusion. So if h is an element in the ideal of x, we can write it h equal to some hi fi. We can write it as a linear combination of the fi. Well, then if you look at the definition here of the differential, you can see from the definition that it will fulfill the product rule. I mean the Leibniz rule. Just as a derivative always does, and this is done by derivatives, you find that you have a Leibniz rule, dp of f times g will be equal to dpf times g of t plus f of p times dpg. I mean it's an exercise for a very simple exercise to do this computation. So we apply this here. So if we have dph, this will be equal to some i equals 1 to r, dphi times fi of p plus hi of p times dpfi. Now the fi were supposed to be in the ideal of x and p is a point in x. So p is in x and fi is in the ideal of x. So fi of p is equal to 0. So this term is just not there. And then so you see that this is a combination of the dpfi. So that means that if we instead of taking just the dpfi, we take the dp of all these, this generates the same ideal and so the 0 set is also the same. OK, do I need this? Yeah. OK, so I want to also, OK, so this is the definition. I also want to talk about the, ah, so maybe I should, as an example, we have to see that our previous definition of non-singularity somehow coincides with this one, no? So if x is the 0 set of f in a n plus 1 and the ideal of f is equal to, no, the ideal of x is equal to the ideal generated by f, then so we have, there's only one f. So the tangent space of x will be equal to the 0 set of dpf, no? And dpf is equal to the sum i equals 1 to n df by dxi at p times xi. So basically, so if, so this is a linear polynomial, so the 0 set has co-dimension 1, except of course, if this polynomial is 0. So we have either, we have that df by dxi at p is equal to 0 for all i equals 1 to n, then it follows that the tangent space of x at p is equal to the whole of a n. And this means that the dimension of it is not equal, the dimension of the hypersurface is by definition n minus, is by our theorem n minus 1. So, and thus p is a singular point. Or otherwise, we have that at least one of these derivative is non-zero, then the dimension of the, then we have indeed the hypersurface, and so the, I mean the tangent space is a hypersurface, and so it also has dimension n minus 1. And so we see that the definitions do indeed coincide, and also what tangent space is in this case. So now I want to say just very briefly, if you have a morphism of a fine varieties, then it also induces a map between the tangent spaces at the corresponding points, and this is given in terms of somehow the Jacobian of this map, which also I can write in terms of differentials. So, definition, so this is a bit like, you know, the second tier of analysis, what? Oh, yeah, yeah, I don't know why I wrote, I mean, it could have been n plus 1, but at least afterwards I had decided it was n. So the Jacobian of f1 to fr in kx1 is r, is the matrix n times n matrix, what is it n times r times n matrix, so j of f1, fr, which is the matrix, df1 by dx1 until dfr by dx if 1 by dxn, and here dfr by dx1 dfr by dxn. So this is a polynomial, so it's a matrix in this. So matrix is coefficient in this polynomial, and maybe I should note just that you, if I, I mean, I can view the differential at p as linear map, so as a matrix. And so the matrix, so if I take the differential of, if I take the differential at p of f1 as a matrix, this is df1 by dx1, and so on, df1 by dxn. So therefore I see that I could also say that dp f1, so for instance if I take the 0 set of dp f1 to dp fr, this will therefore be equal to the kernel of the Jacobian evaluated at p, because it's just a different way of writing df1. I can also write a corresponding column here, it's just, okay, so now I want to define the tangent map definition, so let x in n, y in m, close sub-varieties, or you don't need, maybe I don't actually need them to be a reducible node, because I have defined what the morphism or polynomial map between, anyway, so let p be a point in x, and q a point in y, and let p be a morphism from x to y, so I can write it as f1 to fm to y, where the fi are some polynomial in the variables on the source, and we assume that p of p is equal to q, so then I want to just say what the map on, induced map on tangent factors is, so the differential of phi at p, I can write it in two ways, this is equal, is dp phi, which is equal to dp f1 to dp fr, and I claim this is a map from tpx to tqy, so by definition dp phi is just, one can see that the coordinates on a n are in some way mapped to the coordinates on a m, so it is a map from a n to a m, and it's an easy exercise to show that it maps the tangent space of x to the tangent space of y, I mean, just if you go through the definitions, it's basically obvious, so but check, it maps tpx to tqy, and as you can see here, as I just said in words before, you can also write this, if you want instead of writing it in components like this, you can also write it as a matrix, and the matrix is just a Jacobian, so you can also write dp phi, okay, this is this map, and you can see that, I mean, just the obvious things that if I take the, do this for the identity, then this is the identity, if you take the identity from x to x, it's easy to check that if you take the dp psi composed with phi, this will be d phi of p psi composed with dp phi, like, you know, you learn also in analysis, and so thus, if phi is an isomorphism, then dp phi from tpx to p phi of p, y is an isomorphism of vector spaces or whatever, for all p in x, so this actually can be quite useful if you want to prove that something is not an isomorphism, it's enough to find one point where this tangent map, which is very easy to compute, is not an isomorphism. So this can be useful. So for instance, if we take our standard example, so if, again, we take c to be the cascaded cubic, so then we knew we had this nice morphism from a1 to c, which was given by sending a point t in a1 to t, t squared, t to the 3, no? Now, then it follows that d, so if I take a point a in a1, dA phi will be 2A 3A squared, and so we see that this tangent map is actually the 0 map at the point 0, 0. And it happens very often that when you have a morphism, which is not an isomorphism, then either the morphism is not injective or the tangent map is not injective. I mean, I'm not saying it's injective, but if you want to check it in examples, it's something you should check first, instead of looking for more complicated ways. Yeah, yeah, no, I mean, well, then... No, no, it is, we are also just starting, no? I've got my notes, so I have to... No, no, I mean, the beginning is quite easy, maybe because I decide to make it easy in the beginning. I mean, to start slowly. Okay, I don't know why I didn't bring it, but so let's... Now, this does it for the moment for the case of a fine variety. Now I want to, in general, define the tangent space, and I will give a completely different definition, which is much more abstract, but it is intrinsic. So, here we always have defined the tangent space in terms of the equation by t. The thing is somehow embedded in n, and then given by some equation and we get the tangent space somehow as a sub-space of the defined space. Now, we want to define the tangent space in such a way that it is automatically you can automatically see it only is invariant and isomorphism, and in fact you want to define it in terms of the local ring at the point. We have defined the local ring at the point and in terms of that we want to define the tangent space. This is not so great in many cases if you want to do concrete computations, but it is for theoretical things it's much more powerful to have some intrinsic definition. If you want to prove any kind of theorem it's not so good with such things. So now let's make tangent space and it is what generalizes best for general varieties. So let's just let the definition let X be a variety I want to say what the tangent space is and it's kind of strange. So we have the we'll be in terms of the local ring so let OXP so let P in X a point so the tangent space TPX is TPX it's the dual vector space MP we take MP the maximal ideal so P a point and we should have written it OXP the local ring and MP the local ring the maximal ideal so then the tangent space is you take MP model MP squared now MP is a no it's an idea MP model MP squared is a vector space no so it's a certainly a K vector space and I take its dual so if you I don't know whether you have encountered this in differential geometry so one definition in differential geometry of the tangent space is precisely that we look at the functions I mean the differentiable functions in a neighborhood of P modulo the differential functions in a neighborhood of P so which vanish at P modulo those which vanish modulo the square of those and this is one possible definition of the tangent space so it is the same definition as in differential geometry but no here in this algebraic setting but you most like you had another definition I don't know but there is a definition which looks precisely like that so more explicitly I could say this is all the the linear maps so K linear maps from well this by definition if it's a dual vector space it's just from mp modulo mp squared to K but it might be more useful to say it as linear maps from mp to K such that maybe I call them V from mp to K such that V restricted to mp squared is equal to 0 obviously that's the same thing I cannot really identify and then P in X is called non-singular if Tpx the dimension of Tpx now this is a vector space so if the dimension of this vector space is equal to the dimension of X and then we have the usual thing remember it's called singular so we have singular if it's not non-singular and we have X, we have the singular locus and we have regular locus and if X is equal to X we have X is equal to if X so we have set this three times today so if X is equal to Xreg then X is called non-singular or smooth okay this is just the same kind of sermon each time okay so this is this definition now this looks a bit in the moment it is maybe not even completely evident that this dimension has to be finite we have this mp which is a some some subspace of the of the local ring the local ring is certainly infinite dimensional as a vector space divided by mp squared and now we say that this would be regular if the dimension is equal to the dimension of X we don't even know whether it has finite dimensions so that doesn't sound so great but we will have to see that this would be okay and if for instance X is in a fine variety a close sub-variety of a fine space we have now two different definitions of the tangent space so they had better be equal otherwise this is not okay and so this is the the first thing that we will prove next time that these are equal which is a slightly tricky thing not very but you just have to set up the definition and then then we will go on from there and I think the next time also start talking about non-singular curves in the last bit okay thank you