30 years ago, on the 18th of December 1992, the international community laid in place essential standards to protect minorities. It was when UN member states came together and adopted by consensus the declaration on the rights of persons belonging to national or ethnic religions as linguistic minorities. It was a landmark moment, promoting equality, protecting communities, providing a voice, preventing conflict. The declaration aimed to achieve all of that and to provide countries with concrete guidance on how to protect and promote minorities' human rights. 30 years later, we're asking how far we have come. Sadly, the setbacks are startling. Today, minority communities still endure various forms of discrimination. They are subject to hostility, forced assimilation, persecution, and violence. Minorities often still lack equal access to the fundamental human rights of healthcare and decent, non-hazardous employment. As the world reels with the impact of the COVID-19 pandemic, we know that the virus has disproportionately affected minorities. Inter-ethnic tensions continue to surface. Minorities are being stripped of their political and citizenship rights across the world. Hate, prejudice, and incitement to violence against minorities on social media is widespread and occurs with relative impunity. And the combined ravages of poverty and climate change are moving minorities deeper into inequality, marginalization, and oppression. Minorities remain systematically excluded from participating in decision-making around issues that directly affect them. And it is precisely this exclusion that can create fertile ground for violence and conflict. It exacerbates stigma, prejudice, and stereotypes. Minorities' voices batter. Participation benefits not only minority communities but all of society. Participation helps build peace and stability. Only when we involve all voices can we ensure equality and the human rights of all individuals, no matter who they are or where they come from. Minority's rights are not simple statements of intent. They are an obligation. This anniversary serves as a star reminder that governments can and need to do more. They must address common root causes of conflicts and deal with grievances before political forces instrumentalize them. They must recognize that inclusion is a key ingredient to a stable, cohesive, and more vibrant society. Now it is the time to come together to take stock of setbacks and achievements to understand how to best move towards a future where minority rights are a priority. Where a life in dignity and equality is accessible and can be enjoyed by all. My office remains dedicated to implementing the principles of the declaration, where increasing accountability for minority rights violations, where monitoring and responding to crisis affecting minorities and contributing to legislative and policy reforms, while providing training in minority rights advocacy. Under the 2030 Agenda for Sustainable Development, countries have committed to leaving no one behind. As we move towards a post-pandemic future with a new social contract, equality and justice must be positioned at the heart of our action for the people whose rights we strive to protect regardless of their nationality, ethnicity, religion, or language.  As we mark the 30th anniversary of the declaration, let us redouble our effort so that full realization of the human rights of minorities becomes a reality. The minorities declaration is short, but its call to equality is powerful. We are hopeful that more people are starting to listen. Let us unite around the vision of inclusion and dignity anchored in social, economic, political, and environmental justice to shape an equal world for us all and for all generations to come.


Okay, so hi everybody. I'm Damien and I'm supposed to be a security specialist and we are going to show you something called Harry's Security. Yes, my name is Romain. I'm working for Red Hat. I'm supposed to be a Java expert and I know even less stuff about security. I'm together so we are going to talk about security and how to integrate it in your deployment process. The first thing is that security is broad. Where do we start? Okay, so the first thing to start with is what Microsoft did in fact and the OAS project. It's pretty nice. Okay, so it's called OAS on threat modeling. That's very cool to put on resume but what it's about? It's kind of framework or a list of things that you have to consider if you want to put some security in your application. So it's going to allow you to describe your system and reduce it in small modules, something a bit more manageable basically. Yeah, and then perhaps you can fit it into your sprints and manage it and it should support you in finding your quest for the full company.  Interesting. Where do we start with this thing? We start with stride and dread. So first stride. So the goal is to put a category on the thread that you are facing. Okay, so there is like five of them or whatever. We have no time to cover that so just like to know let's pick one and analyze one just to have an idea. What is repetition? What is this arthing for? The repetition is to prove that someone did something for real or at least the user did. Okay, and how this process is helping you mitigate that or fix this issue? In fact, you have to rely on auditors and here you can really prove that at least this login performed the actions. It can be repudiated. Okay, so now let's look at the other acronym because there was two of them remember. Yeah, dread. So here the goal is to put a priority or the mage on your thread. Okay, and by the mage what you mean here is actually the impact on my business, not the mage in general. Exactly. Yeah, let's go quickly over the other part of the acronym. So what is this reproductability? It's how easy to, well, to reproduce the attack. Come on. Yes, and exploitability? Well, how is it to reuse and make something nasty with this attack? Well, I think that the user I guess is transparent but still. Yeah, well, are you impacting 50% of your users? All your users? Only 1% of your users? Only the enemy basically. Discoverability. So here it's important is how it is to, for a complete noob at the other end of the world, to discover again the threat or to exploit it. So globally this is just for having the priority and knowing how damageable it is. Okay, so I kind of like it because what I really like here is that it's a brilliant tool to communicate.  This is a real issue because they have a way to communicate. I like that. It's kind of UML for security? Yeah. Well, if you like UML. Okay, let's make it a bit more concrete now. We are going to think about a simple web app. We're not going to think like a big application, something very simple, something like, you know.  For the example. And to make it even more simple, we can take an internal app. So it's not going to be like, it's going to be an internal app for employees. It's going to be not sexy, nothing fancy, no high-frequency trading, no big data, no ad-doop, no machine learning, not even social app. Yeah, so let's call it a cool app. Yeah, hashtag cool app, exactly. So let's discuss how we're going to build it a bit. So it's a Java application, right? So we need to have frameworks.  And of course we have to build it with Gradle because maybe it's for losers and stuff. The good thing about this tendency is that there is framework for security, like there is for everything else in Java. So let's take one of them and look at what he does. Yeah, cool. I don't know Java. I'm working for Red Hat. This is actually painful to do that.  Please do. Well, actually it's quite appealing. Ouch. It brings an old stack of security pumbering into your application, basically giving you all the stuff you need to have like authorization, authorization, saying that this user is able to access this part of the resources and stuff like that. Yeah, but here you are talking about only your application. So if I think about an attack as admin or security guide, my first idea is demanding the middle attack. Well, the framework supports that. It comes with support for HHS and also ensure that your client is using HTTPS. Yeah, okay. So if we go directly on the front end, so for click checking. Also support for that. You can use this frame option. The same origin thing inside the framework to prevent you from that. Not an expert on this part. And for cross-scripting attacks. It also does that. It provides support for CRSF token, for instance. Yeah, okay. And now if I want my audit logs. Well, actually, wait, we need that. We mentioned earlier we need that. And actually it's also providing everything you need to have to do an audit log, an audit trade. So bottom line is that we listed a long series of threats that are going to be a menace, like a dungeon for your application. And you need, given the amount of stuff you have to face, you need a framework to help you cover all your bases. That being said, we are dealing with an internal app. So we are behind firewalls inside a very compacted VLAN. So we should be kind of safe, no? Yeah, but come on. Be serious, okay? So, you know, like most of the players, they are using laptops and desktop to access the app. So basically there's at least one which has a root access on it. So we can consider that it's unmanaged with a nice malware on it. Well, I have a good news for you. It's not even a laptop, it's tablet on mobile. So it's even better, you know. It's even less managed. Yeah, it's called bring your own device, which is also called bring your own disaster. So to sum up, in fact, so we have a bunch of users with completely unmanaged devices. And those devices all use the internet, which basically means that if somebody hacks your tablets, you're getting directly access to the internal app, the one hidden behind layer of firewall. You just one tiny hop from the internet. Yeah, but you said it. We have firewalls, come on. Well, yeah, so this is a misconception. Firewalls are not about security. Firewalls are about network quality services. That's what they're doing, actually. Yeah, you have to explain this to me. I'm a security, you know. So I love firewalls. Of course you should like, but it's not really helping you in this case because firewalls are designed to block ports, right? True. And generally, you're breaking port that are not used. True. So you need to use the port for the application. Yeah, okay. I can't close all ports, of course. Yeah, okay, so the application has to be reachable at the end. So the attack is going to follow the same path as the regular user, so you can block it. Yeah. That's the issue with firewall. Okay, so what should I do then? Well, the real trick here is to analyze and monitor the traffic itself on the content of it. For instance, if your application is supposed to send an email using SMTP, this is the only thing you should be ordering there, and it should be valid SMTP. So that's the thing. Side note, both of us used to be consultant. We used to be traveling around, like, to different customers. So we've seen a lot of, you know, companies with different security things, with different habits. And we used to grade how crappy the security is, but the number of port being closed. It's just nice when you want to access your mail, you know, to help your customer, and no mail forbidden. Bad. And, yeah, just for security reasons, you know, the VPN you need to access to your support, in fact, to support your customer, well, is also forbidden for good reasons. Yeah, that's very bad, because especially in my case, as a regular employee and consultant, if I go to a customer and contact my VPN, I can't do anything for you anyway. What I love is, in fact, the use case where, in fact, there's only one trusted user by the customer who has access via a stage to the machine, but not new. You are supposed to help him, but you don't have access to anything. So, generally, he goes in, he's logging in, and you piggyback on his access connection, and you connect to the system anyway, useless. Don't laugh, I've done that. We did it. And, well, so at the end, it's nice to block poor and so on, but it's mostly about illusion of security, okay? Exactly, we gave very strong illusion of security because everything is blocked, so everybody's work on delivery is shitty, but there is no real security. What is important here, joke spot, is so you keep an eye on the protocols and you filter the contents. So, generally, use that for this kind of stuff. Where are we now? In the notes? Yeah, yeah, so it's actually quite nice, because if you do filter the content, and if you do implement, you know, invalid request, on the other hand, as a Java developer, the amount of hacking you must be facing are far more reduced.  You're not going to get some kind of crazy URL with, like, buffer overflow or whatever because the reverse proxy is going to drop that. Yeah, okay, so let's go on another area of the application. Yeah, the application better, of course. Also, you have the data. In this case, we have no real business. It's just an old app, so there's no back, no real user, and it's not very interesting. So it's not about wire transfer or whatever. No value to hack and sell somewhere else. Yes, you won't be studying our money to put it on a dark web. This is an internal app. However, we do have, like, typical, you know, employee productivity data. So stuff such as employee address, phone number. Yeah, so what is called, in fact, personally identifier information. And by law, you have to take care of those. So also, you may have salaries, financial packages passing on the wire. So, of course, here you have to restrict access based on the user now. And there is always a bit of business in this kind of stuff. So you may have information about deal, opportunities. Yeah, this is confidential.  And by secure it means encrypting everything from one end to the other end. Yeah, so encryption. Okay, on the front-end side. So if you want to encrypt something, we have to encrypt first the communication. So it's SL and HTTPS. Exactly. So that's pretty easy. I think anybody here has ever, like, you know, deployed HTTPS or SSL on his own little engine hacks or Java web. It's like 10, 10 minutes to do this. Keeping it safe, keeping it, like, keep running a bit of that sort of work. It's heavy. It's massive, in fact, because it's an operation that you have to remember the most of the time. And you can do it only once a year.  And I'm not even talking about a leak certificate, which is then still valid, but, well, you shouldn't use it. Yeah. So let's imagine right now that the infrastructure is correct and we have done that. Well, we're good, right? We have SSL in place. Yeah, nobody moved, and I, in fact, but SSL has been now. Yeah, you still have to keep up your knowledge. That's exactly the point, yeah. So we've deal with the front end. Let's look at the back end now. Well, before that. So now we also have to encrypt everything between the app and the databases because this could also be spied on by hacker. And that's actually complicated because databases are not very good with... No, yeah. So we have it by experience. With also MongoDB, for example, it's really hard to have and to maintain encryption of data. And in the case of MongoDB, well, the feature was missing for years. Yes, I remember that, actually, the first version of this talk, my co-worker Francois, my former co-worker Francois, had to basically implement that in MongoDB. So, or back port it in MongoDB. Anyway, let's say that we have secured this part. There is still one little detail. Database is running on a physical system on virtual as a one and it's throwing on the disk. Hacker can also access that. So we need to think about, do I need to encrypt my data on it? Well, of course, yes, come on. But by seeing that, I'm not solving the problem because encrypted everything is pretty heavy. It's expensive in terms of cost, well, money or performances. And encrypted everything, in fact, doesn't really help, especially when you select what you want to encrypt and it's not the good data. Yeah, so you may remember this story about Ashtam Edison last year. It's a perfect example of that. Basically, they didn't make anything by the book. They did encrypt the password, for instance. There was a hash of password, but they did not encrypt the name of the user, neither the email, neither the physical addresses. So you add everything you needed to blackmail people. People committed suicide because of this hack. Yeah, this can escalate pretty fast. That went dark, but yeah. So, one being that now we have encrypted everything, so we need still to access it. We need to have legitimate user access the application. How do we implement that? Yeah, well, we have to prove who they are and ensure who they are. So it's called strong authentication nowadays. Exactly. So the thing is that you cannot just use the same password. You need to have more, because Ashtam has password R. Yeah, LDAP, KBOS, it's not enough. So this is why you can't have the same password? And also because old passwords can be stolen. So anyway, you talk about something I call 2FA, I think. What is this thing? Yeah, so to factor out for authentication, so to identify someone, you have three factor of authentication. Who the user is. So it's a biological, well, a fingerprint, for example, what the user knows, like a password, or what the user has, which can be a mobile phone or a security token, for example. And 2FA is, well, you pick two out of three. Well, 2FA is definitely the best way right now to implement strong authentication, but it has a big issue with it. It's that people don't like to always put out their phone to take their token or have a token. Users generally fight it and company quite rightly. This is arming their productivity. Yeah, so if you want to have 2FA, you need definitely to have SSO, a single sign-on process. Otherwise, well, the user will be your enemy at the end. Yeah, so for instance, in Red Hat, I'm pretty lucky because when I'm logging in my email and I get access to the Wiki also, I don't need to re-log all the time. Not much, at least. Yeah, so what is really important is that the security is not an excuse for the user's experience. The good example here is, in fact, Slack, for example. Nowadays, we don't even need a password and it's still safe to log in. The user experience is just brilliant here. Okay, so let's go back to our app. We have it secure. We put every encryption everywhere. We are good, except that recently, if you pay attention, the scope of the battlefield for security has kind of extended because there is also a way to arm your app apart from actually expecting a defect. For instance, you can hack it at the source and hack your CI. Yeah, so the CI is also a very sensible system. In fact, it's a system which has a lot of power. For example, it has the right to download your dependencies. It has the right also to access production because when it is triggering a deployment, this is what it is doing and it is doing on something that it has built. So if CI is accessing production, CI's have your secrets, which means that you need to handle and manage your secret properly. This slide is difficult to read, but this guy committed password into his GitHub. So guess what happened? So that's complex. You need a tool for that. People are generally using something like Ansible Vault to do that. This is what we try to do, in fact. But the secret are not the only weakness in the end. Yes, because if you're doing things right, hopefully I wish for you, when you're building your app, your app is 95% not your code. It's dependencies fetching from everywhere. Especially if you're doing Java, you're going to fetch Apache command lines, Apache whatever, Wildfire Blast, Spring Boot Singly. And when you're fetching stuff from everywhere, it's from the internet. So you are basically implicitly trusting the internet there. So it's definitely not funny here, but you have to download carefully your dependency, you have to check what you're downloading, and just check the checksums. Okay, it helps. Exactly. So by the way, Red Hat has been pushing this model for years with Satellite, already with RPM. Way before we bought J-Bus, we were already pushing people to say like, have a repo, sign everything, know what you're handling. Yeah, you already focus on Red Hat, don't you? But at the end, manage your artifacts. They do pay my bill, you know. Okay, so application is built, CI is somehow safe, so now we have a nice little war. Let's run it, let's deploy it. Yeah, just tell me where. Yeah, I don't want to deal with this shit, I'm too old for that. I'm going to do what the cookies are doing, I'm going to go cloud native. So in fact, you don't trust the internet, but you trust someone else's computer. Well, I actually believe that cloud is safer than in-house server. Whoa, really?  Yeah, okay, okay, that's true that we have some servers at home, and well, they're still not patched. Okay, so, okay, point again. Yeah, it's been 10 years, that is. Also, like, if you look at Netflix, for instance, Netflix has a very cool model, and if you access a movie this weekend, the VM you're going to run on is at worst two days old. At worst. Well, it's just nice, but even the problem is, even with a fresh system, sometimes, well, with zero days, the system is still sensible to it to attack. For example, Meldon hasn't been fixed for days, weeks. Yes, and this is bringing us to the very next point, and the most important point is that security is not about being, you know, patched and, you know, fresh. It's also new guys to be ready to fight an attack. Actively. So, let's take an example. Do you remember a few years ago when GitHub was attacked? GitHub faced a very strong distributed denial of services attack, like passing two years ago now, and they were very communicative about that. They were saying on Twitter, hey, we are under attack, so the site is off, we're doing that, we're doing that, we have to be able to blah, and they were very efficient about that, and that was very interesting to see that they were both, you know, able to patch the system, but also actively raised an attack, and they were very transparent also. Yeah, what is really important is that your business is not an excuse for not being transparent to your customers. So, hey, this is my turn. So, I'm working for a bank, which is a startup, and we have, even for a bank, we have a status page where we show the state of our system. We are transparent for our customers, so in case of storm, well, we hope they trust us a bit more. Yes, if you remember Equifax last year, this is a mess by hiding these leaks that we're having. For weeks, they didn't say anything, people were in danger, and they should have really much come forward and said, hey, we've been hacked, like, be safe about that. Yeah, they tried to run away from them responsibilities, and if they're in danger, way more invite them customers. They even hide details for fixing the breach. Yeah, so the point here is actually that it's not a question of are we going to be hacked, it's a question of when are you going to be hacked, and what you're going to do at this point. So, let's take the metaphor of fighter fighters, you know. Firefighters, for example, are always trained to fight, well, fire. So, in fact, you have to be trained to face an attack. Exactly, so basically your app is born to be hacked at some point, so you have to prepare for that. So, to go along with the firefighter analogy, if you want to protect the building against fire, you're going to first install, for instance, a smoke detector. Which, in IT sense, is more ideas, so intrusion detection or prevention system.  Yeah, and for your radar approach, it would be as a Linux. Yes, exactly, everybody's running with a Linux here. Yeah, and of course, we're doing Java, so you're all running with a security measure on, no? Wait, everybody's doing containers nowadays, and I'm sure they care about that second profile. Yeah, point taken. Anyway, it's time to wrap up and sum up what we've been hobbling for 20 minutes. So, last slide. So, yeah, I survived. You first. Yeah, so what is really important for our security is to keep in mind the big picture, to not lie to yourself, in fact.  And don't ever think you're safe. You can always be asked from somewhere. Yeah, VLANs and firewalls are just an excuse. Come on, no, there is no real defense. And your data isn't a feature. You have to encrypt what needs to be encrypted. This is a hard part. Yeah, and encryption means managing secrets and using strong notification. Which doesn't mean that, in fact, you have to forget usability to face security. There's no trade-offs here. Yes, and also don't forget that Battlefield has extended up to CI and cloud and other environment. So, you have to also take care, as part of your application, the continuous integration system, how you deliver code and how you deploy even in the cloud. So, basically, feel free to be ready to fight fire, in this case, and act, because you are going to be act. And that's it.


All right, so we were talking about automorphic forms on PGLN. Let me briefly recall what we did last time. So our group G is PGLN, and we have three important subgroups, N, A, and K. And we talked about the set D of differential operators, G invariant on the upper half plane. And we had the Harris-Chandra C function, which measures the spectral density. And we talked about Whitaker functions. Whitaker functions are eigenfunctions of all the differential operators. And we had some formulas for Whitaker functions, and so on. So the first thing I would like to start with today is the Fourier expansion. And well, there are several possible Fourier expansions, depending on which group you want to Fourier expand on. And well, one of the most common versions is to expand with respect to N. And Fourier expansion is always easy if the group is a billion. And if N equals 2, then N is a billion. But if N is not 2, then N is not a billion. And it has only a smaller billion part. So the billion part of N is sitting here on the first off diagonal. And everything else is a bit harder to treat. And that's why the Fourier expansion looks a bit complicated. If you have a mass form phi, then it has an additional sum over a lower dimensional group, gamma in SL N minus 1Z, modulo N N minus 1Z. So N is the unipotent upper triangular group of dimension N minus 1. And then a sum over the coefficients, M1 up to MN minus 1, different from 0. I'm being a bit sloppy with signs here. There are plus and minus signs that I'm going to ignore. A M1 up to MN minus 1 normalized in such a way that these are of absolute value 1 on average. Again, there should be some absolute value signs, perhaps. But I ignore them. So the MJ occurs with a certain power. And the power is J times N minus J over 2. It's not so important. It's just a normalization. And then a Whitaker function. The Whitaker function has index mu, which is the spectral parameter of the cusp form phi. And then a diagonal matrix M that I'm going to write down in a second. Then a matrix that in the upper left block, N minus N block contains the matrix gamma and a 1. And then the point z. And M is the matrix 1M1 up to MN minus 1M1. So you mean the product value? So here, this is a matrix product. This is a product. This is just a single entry. This is a single entry, yes. So this is a diagonal matrix. Yeah, so diagonal matrix. Yes. OK, these are the Fourier coefficients. AM1 up to MN minus 1 are Fourier coefficients. This is what we call Fourier coefficients. And they are related to the Satake parameters and to Hecker eigenvalues. In fact, we'll see later that they are Schuh polynomials in the Satake parameters. OK, and you can recover the Fourier coefficient by integrating the form phi over the group N. So AM1 up to MN minus 1 divided by this product of the Mj's to the power j times N minus j over 2 times the Whitaker function. This is just the product of phi of uz times E of minus, let's see, M1 U1 minus, and so on, minus MN minus 1 UN minus. So you take a character of this group here and integrate it against the mass form. And this produces the Fourier coefficient. OK, so now I would like to speak a bit about Hecker operators and develop a bit of Hecker theory on PGLN. So let P be a prime. We fix a prime, once and for all, and consider Hecker operators at P. So Hecker operators are parametrized by double cosets. And so the typical representative is a diagonal matrix with P power entries. And we fix an n tuple of exponents. And for notational simplicity, we write capital A. It's not this group. It's just a name for a diagonal matrix. It's the diagonal P to power A1 up to P to power AN. And then, so gamma is always SLNZ. And we decompose this double coset into single cosets. It's a disjoint union of single cosets. OK, and to each double coset, we can associate a Hecker operator. I call it ta of P or simply ta. And it maps a function F to the function that I get by summing or by applying all these coset representatives. So this is for the argument. And this is obviously well-defined. And again, gamma invariant, because this was a double coset. OK, and it's clear that taAAAA is just the identity, because then you have a matrix with all entries the same. And this is also permutation invariant. ta is the same as t sigma of A for all sigma in the vial group SN. And the adjoint of ta is t minus A. OK, in particular, without loss of generality, I can scale this in a way that, well, first of all, I can order it so that the smallest of the ANs is this. So I can order it in descending order, say. And then I can make sure that the last one is 0 simply by scaling it. So without loss of generality, I can assume that A1 is bigger than A2 is bigger than, and so on, AN, I can assume that AN is 0. I write absolute value of A for the sum of the AJs. And then I define something that I think of as the volume of such an n tuple. So again, if they are ordered in descending order, then I define v of A to be sum j AJ. So here, the ordering is relevant, of course. So the first one gets weighted just a little, and the last one gets weighted a lot. All right, and then I define the vial orbit. W of A is just the orbit of A under the vial group. So it's all sigma A, where A, sorry, where sigma is in W. OK, one of the main problems in every HECA algebra is the multiplication of two HECA operators. If you want to multiply two HECA operators, you want to write it as a linear combination of other HECA operators. And that's combinatorially possible, but a very complicated thing to do. Multiplication is complicated. But of course, we like to do complicated things. So if you multiply two HECA operators associated with two double cosets, then you can write it as a linear combination of double cosets with respect to certain matrices D. That's clear. But the question is, for which D is this coefficient alpha D non-zero, and what is it? I mean, can you compute given? I mean, of course, it's determined.  Well, you can write down an abstract formula, but this helps very little. So only those D appear where the double coset gamma D, gamma is contained in gamma A, gamma B, gamma. That's something you don't want to compute. And then alpha D is given as the number of pairs, J, K, such that gamma D is gamma alpha J, beta K, where alpha J and beta K are the given coset representatives. OK, so in principle, you can compute it, but it's very complicated. Algebraic combinatorists have worked on this a lot. And there is something called Young Tableau. So if I give you two tuples, A and B, you can write them in a certain tableau, and then you can find out which of the D's occur. And this is called the Littlewood Richardson condition. OK, so this is combinatorially challenging. OK, HECO operators, of course, commute with the differential operators in D, because these operators are gene-variant and with themselves and are normal with respect to the usual inner product. And I write, denote the eigenvalue of ta for a mass form f by lambda a of f, or perhaps lambda a of p comma f, if you want to emphasize the prime. OK, I said earlier that Schuer polynomials play a role. And for those of you who haven't seen Schuer polynomials, let me define what they are. Schuer polynomials. So if you have an n-tuple, A, natural numbers, including 0, in descending order, we define the Schuer polynomial S A of x to be a ratio of two determinants. So x has n entries. It's a polynomial in n variables. x1 to the power a1 plus n minus 1 up to xn power an plus n minus 1. And then you decrease the powers one by one. And finally, x1 to the power a1 up to x1 to the power an. This is one determinant. xn to the power b. xn, yes. And you divide this by a Fundament determinant. x1 to the power n minus 1 up to xn, n minus 1, x1. So it's just one. OK, so this thing is the Schuer polynomial. It's associated with the tuple a. OK, it's obviously a symmetric polynomial. It's a symmetric polynomial of degree absolute value of a. And it turns out that the Fourier coefficients that I defined up there are just Schuer polynomials in the Sartake parameters. So a of p to the power k1 up to p to the power kn minus 1 equals, OK, I need a bit of space for this, the Schuer polynomial attached to the following n tuple, which is k1 plus up to kn minus 1. And then you remove step by step a term. And so the last entry is just, well, the second last entry is k1. And the last entry is 1. So this is obviously descending order. And the alpha is the n tuple of Sartake parameters. So in particular, a1111p. So then only the first entry appears and everything else is 0. By the way, this is not 1, this is 0. S1 0 0 0 alpha, which is just, as you can see from the definition, the sum of the alpha j. So this is what is usually called the Peth Hecker operator. OK, one way of trying to understand the Hecker algebra is to use a kind of spherical map, namely the Sartake isomorphism, the Sartake map. The Sartake map attaches to each Hecker operator a polynomial. Sorry, yeah, a polynomial. And it's a ring isomorphism. So multiplication becomes very easy because you just multiply two polynomials. And everybody knows how to multiply two polynomials. The disadvantage is that it's hard to compute the image. It is useful to renormalize the Hecker operators. I renormalize them as follows. T tilde, for me, is the same thing, but multiplied by P to the power V of A. OK, and now I can write down the map omega. The map omega maps TA of P tilde to a polynomial with entries rational with only P denominator. So I need a bit more space. So in z1 over P and then x1 up to xn. And it's symmetric. It's a symmetric polynomial with coefficients having only P denominator. And as I said, the image is very hard to compute. It depends on the explicit coset representatives. So you have to know explicitly the coset representatives and then do something with them, depending on explicit coset representatives. But it turns out if you don't want to know exactly what the image is, but just roughly what the image is, then it's very easy. This equals up to a small error term, just the sure polynomial. And the error term is of size 1 over P. So by this I mean a polynomial whose coefficients are bounded by 1 over P. So if P gets large, then you can very well approximate the image under this map by the sure polynomial. And in this notation, you can write down the eigenvalue. So the eigenvalue of f for the HECO operator ta is you take the unnormalized ta of P, apply the map omega, then you get a polynomial. And in this polynomial, you substitute P to the n plus 1 over 2 times alpha, where alpha are the satake parameters of f. Let's say alpha f. So in particular, we can guess what we expect how big. So how big should lambda AB in this normalization? Well, the sure polynomial has bounded coefficients. So the expectation is, so this is what you would expect, that the size of lambda APF in this normalization is roughly, I have renormalized this by a factor P to power V of A. So P to power minus V of A. And this is a polynomial of degree n, so of degree absolute value of A. And I plug in P to the n plus 1 over 2, so plus n plus 1 over 2 times absolute value of A. This is roughly the size that we expect. OK. And to give you an explicit example how we can actually work with these HECO operators, I would like to show you how to construct an amplifier on GLN. How to construct an amplifier on GLN. And my aim is to produce a very explicit amplifier. The fact that abstractly an amplifier exists can be proved abstractly. And there is a beautiful treatment in a paper of Akshay together with Leo Silberman. They give an abstract argument why an amplifier should exist and how you construct it. You can learn a lot from reading this. I would like to construct a very explicit amplifier. So see also Silberman-Benkertisch. So the whole idea of amplification is based on HECO relations. So we need a HECO relation like lambda p squared equals 1 plus lambda p squared. This is the prototype of a HECO relation, and we want to have something similar in general. OK. Obviously, there must be some relations in this algebra, but I want to have something explicit. So let me first show you in the language of the Satake map what this looks like. OK, I have a bit of space here left, and then I need to clean the blackboard. OK, under the Satake map, we have. So let's look at the case n equals 2. So what are the HECO operators of interest for us? There is the HECO operator T11. This is just the identity, and the image is x1, sorry, this is supposed to be T10. So that's really the p-th HECO operator, and the image is x1 plus x2 over p.  And then we have T11, which is simply the identity, and that's x1 times x2 over p cubed. OK, and now we want to cook up an identity of these three operators, and we know what we have to look for. The sum of these two is the Tp squared HECO operator, and this is the Tp HECO operator, so we want to square this one and sum these two, and then there should be an identity. And indeed, so you can check that T10p squared minus T20p plus T11p equals x1 x2 divided by p squared, and that's just p times the identity. And recalling that our HECO operators, these Tp's that I defined here, are normalized a bit differently, we would expect that the Tp HECO operators roughly square root of p, so if you square it then you get p and you get precisely the HECO relation that you expect. OK, and now we want to do something similar in general. Any questions? Are the Fourier coefficients multiplicative? Yes. Yes, they are. Of course, not completely, where's the hook there? OK, of course, not completely multiplicative, but they are multiplicative. Well, they are multiplicative if you have an eigenform of all HECO operators. And that's simply the Chinese remainder theorem. If you multiply two HECO operators for different primes, then you can just multiply the diagonal matrices. So you can really treat one prime at a time. OK, so let's do a bit of algebraic combinatorics. I define capital pi n to be the set of all partitions of n by which I mean the following. All n-tuples, a1 up to an, in descending order, such that their sum equals n. So 0, of course, must be allowed. So some of them can be 0, and then some 1s and so on. And the sum has to be n. That's what I mean by pi n. And I consider the vector space of polynomials with, say, complex coefficients, although this completely irrelevant, in n variables that are symmetric and of degree n. So all symmetric polynomials of degree n in n variables, and it's obvious that the dimension is just the cardinality of pi of n. And I will construct a HECO relation by looking at various spaces of this vector space. So let's consider various spaces. First basis is the obvious basis. I take all monomials where c is in the vile orbit of sum a, and a runs through pi n. So this generates v. These are obviously symmetric polynomials, and they have the correct degree and every polynomials of that form. OK. So this is sort of the canonical basis for this space. Then I would like to look at the second basis, which looks as far. Basically, I take the sure polynomials as basis elements. So I take all a and pi n. And I look at the t-eighth HECO operator, and I normalize it. Then I apply the Satake map. This gives me a polynomial. And I take the collection of these polynomials. And this is, well, essentially the sure polynomials because the image is up to a small error, the sure polynomial, plus small error.  So I can really write here an equality. OK. So to give you an example, in the case n equals 4, we have the HECO operators t 4 0 0, then 3 1 0 0, 2 2 0 0, 2 1 1 0, and t 1 1 1. So pi of 4 has five elements. These are the partitions that we get. And the corresponding five sure polynomials form a basis of the vector space. OK, so the most interesting basis is the third one that I'm now going to define. And I define by bracket J the n-tuple J 0 0 0 0 0. And I look at the HECO operator J 0 0 0 0 0 0. And its image under this attack map is essentially just the sum over all monomials of absolute value of degree J plus a small error term. OK, and I claim that the following is a basis. And I will prove this. And having proved that the third basis, which I'm going to define, is a basis will automatically produce the HECO relation. So this is probably the only proof that I'm going to include in this lecture series. So I claim that the following is a basis. I claim that V is generated by the following set. I take this attack a map of the following product of HECO operator. So I take an n-tuple a in pi n. And then it has entries a1 up to an. I take the entry a J 0 0 0 0 0. And I multiply this together, J from 1 to n. And I claim that the collection of these polynomials generate the space. So again, let's look at an example just to see what is happening here. It's very simple. It's just notationally a bit complicated. If n equals 4, then these are the following HECO operators. And there is a tilde here. t4 0 0 0 0. Then I have 3 0 0 and 1 0 0. This is the second. This is the product. Then I have t2 squared, t2 0 0 0 squared. This corresponds to the tuple 2 2 0 0. Then I have t2 0 0 0 and t1 0 0 squared. And I have t1 0 0 to the fourth power. These are the five HECO operators whose image I'm going to take. And so you see the important thing is that this collection does not contain the identity. And so if this is a basis, then I can write the identity as a linear combination of these, and then I win. So what I have to show is that this is really a basis. OK, here's the proof. Proof. So we take a partition a. And so as I said somewhere up there, the image of t tilde aj is this thing there. So I take the product, j from 1 to n, and then sum x to the power b, where b has absolute value aj, plus a small arrow of size o of 1 over p. But I ignore this for a moment. And I just compute what this is. So this thing is this polynomial. And I want to compute what is this polynomial. And then the collection of all these polynomials should be a basis. Well, it turns out that this is you sum over all a prime in the set of partitions. And you sum over all c in the vile orbit of a prime, x to the power c. And you have some coefficient here that depends on a and a prime. And what is this coefficient, where c a prime equals the number of matrices cij with non-negative entries, such that the row sum or column sum, I don't know. So the sum over the i's is aj. And the sum over the j's is ai prime. So if you prescribe a and a prime, then you have two n tuples. And you construct all matrices with given row sums and given column sums. So the vector of row sums should be the a vector. And the vector of column sums should be the a prime vector the other way around. It's irrelevant because it's obviously symmetric. OK, so this is the coefficient. And proving that this is a basis is equivalent to showing that the matrix cAA prime, if I go over all possible pairs of partitions, is invertible. So I need to show that this matrix is invertible. If you state it like this, it's a perfect competition problem. So you construct the matrix as follows. Here's the matrix. The matrix has dimension pi of n. And each entry is the number of matrices with given row and column sums. OK, show that the matrix is invertible. OK, if you are a good competition mathematician, then you can perhaps solve it on the spot. I'm not good at these problems, so I just did the first few cases by hand or by Mathematica to see if I find any pattern here. Show that the matrix cAA prime, where a and a prime are in pi n, is non-singular. OK, so I computed the determinant for n equals 1, n equals 2, n equals 3, n equals 4, n equals 5. Turned out the determinant is 1. So this cannot be a coincidence. So show more strongly, show that the determinant equals 1. So here's, in the case n equals 4, the matrix looks like this. It's 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 1, 2, 3, 4, 6, 1, 3, 4, 7, 12, 1, 4, 6, 12, 24. And has the term. Obviously, it's symmetric. It has to be symmetric. Show that the determinant is 1. Well, in this case, you can just compute that the determinant is 1. Or you let Mathematica compute it. But how do you show this in general? And I played for a while. I computed eigenvalues. Doesn't help anything. But then I computed the Cholesky decomposition. Turns out it has a very beautiful decomposition as A transpose A, where A is 1, 1, 1, 1, 1, and then something funny. So in this case, it's just 1, 1, 1, 1, 2, 3, 1, 2, 3. And so obviously, the determinant of this thing is 1. And this cannot be a coincidence. I mean, typically, the Cholesky decomposition has square roots. I mean, this is an integer matrix. And this cannot be a coincidence. So it must happen in general that this matrix has a Cholesky decomposition where these numbers count something. And OK, well, then I looked it up and I found what it is. So algebraic combinatorics, people have known this for a long time. This is the so-called Kostka matrix. And these are the Kostka numbers that are well-known in the theory of Schur polynomials. And so this is well-known that this matrix has this decomposition with an integral matrix here. And once on the diagonal and then everything is clear. OK, so as a corollary, there exists a linear combination with bounded coefficients. Oh, and by the way, so all the time I ignored this error 1 over p. I mean, if I really work with the images of the Schur poly, of the Hecker operators themselves, it's a total mess. But it clears up if you work with Schur polynomials, if you let p go to infinity. So if you just look at sufficiently large primes, it turns out that this matrix, if you define it properly, so really with the images of the Hecker operator, is still invertible, even for p equals 2.  So there exists a linear combination with bounded coefficients such that, so the coefficients I call y a such that sum over all partitions, y a product j from 1 to n t tilde a j equals, well, equals, yeah, t 1, 1, 1, 1, 1, which is the identity. Well, it's a scaled version of the identity. It's n times n plus 1 over 2 times the identity. And as a corollary, and this is what you typically need for an amplifier, corollary, there exists 1 less than j less than n. So your amplifier has depth n. You have to go at most to the nth power, just as in the case n equals 2, you only need to go up to p squared. Such that lambda j of nf is bounded from below by what you expect. This is the expectation.  And you can compute what this is. It is p to the power j times n minus j over 2. Simply because here you have a sum of HECA operators and their eigenvalues cannot all be small if they equal the identity. At least one of them must have a big eigenvalue. Otherwise, their sum cannot be the identity. OK, any questions? All right. OK, so this was a little bit of a crash course on HECA theory on PGLN. And I want to spend the remaining 10 minutes on just briefly mentioning Eisenstein series. So far, we have been talking about cusp forms most of the time, although this discussion is completely independent of whether your form is a cusp form or not. HECA theory applies in the same way to Eisenstein series. I would like to briefly mention Eisenstein series. Which one? OK. OK, Eisenstein series. So abstractly, they are attached to parabolic subgroups. So, OK, so here the amplifier is at humanity 1 over n. Yes. Is it a general feature of, suppose you have a split group of rank n that an amplifier would be 1 over the rank? So you have your amplifier, so you need to take the operator up to P to the power n. I think you can't get lower than that. Yeah, so I'm wondering if for any other group, if you can achieve it. So what would be the invariant which is the rank? Yeah, I think it's the rank. No, it's no more than the rank. No lower than 1 over the rank. 1 over the rank, you know, is the rank of the affinity for the question. In general, for general split groups. Maybe we'll talk about it afterwards. OK, so let me move to Eisenstein series. Just a very brief discussion. So abstractly, they are attached to parabolic subgroups. And I just show you how it works for n equals 3. So for n equals 3, let me start by the case n equals 2, because somehow this is an inductive process. So let me start by the decomposition of the standard upper half-plane modular SL2Z and the L2 space. So this is the case n equals 2. And this decomposes into the Eisenstein space plus the hospital space plus the constant function. And the constant function should be seen as a residue of Eisenstein series. OK, and from this, we can go one level up. There are three types of Eisenstein series corresponding to this decomposition. There is the minimal parabolic subgroup, minimal parabolic subgroup, which is the group that Akshay mentioned at the beginning of his talk. This measures the frequency of attendance in a lecture series. And the corresponding Eisenstein series has a spectral parameter mu just as a cusp form and a complex variable z. And it can be defined by summing, well, basically, I mean, this is gamma infinity in usual notation. And you take the power function that I used also to define Whitaker functions. So this is a very close analog to the usual Eisenstein series on GL2. It has Fourier coefficients that are essentially divisor functions. So the Fourier coefficients are of the following form. d1, d2, d3 equals m. And then you take mu1, mu2, mu3. And this is the m1, m1st Fourier coefficient. And then by the Hecker relations, you get all others. So these are the three langlines parameters. They add up to 0. And so this is a generalized divisor function. OK. So what's i mu of gamma z again? So i mu is the power function. It only depends on the y-coordinate. So of this matrix, you decompose it in Iwazawa decomposition x times y. And you'd only take the y-coordinate. And i mu of y is a sum over yj to some linear polynomial So it's some powers of yj.  In fact, it's perhaps not even a sum, right? Right, it's a product. Right, it's a product. Yes. OK, and the exponents are chosen in such a way. So the important thing about this power function is that it has the correct eigenvalues for all the differential operators. So you just choose the powers appropriately to make sure that it's an eigenfunction of all the operators. OK, so this is the minimal parabolic subgroup, then maximal parabolic subgroup. And of course, in general, there are many groups in between. But for n equals 3, there is only minimal and maximal. So this is the group that looks like this. And here you have a bit of room. You have a 2 by 2 block. And here you can insert a gl2 cusp form. EU of z, s, u is now a cusp form on gl2. And that's some gamma in gamma mod p. p is this parabolic determinant of gamma z to the power s times u of, well, you would like to write gamma z, but this doesn't make sense, because gamma z is in gl3. And you want gl2 entry. So you first project this to the upper left corner with the map pi. So pi sends 1x2x3x111 times y2y1y11 to 1x21 and y21. So this makes sense. And the Fourier coefficients are twists of the Hecker eigenvalues of u. So they are basically convolutions with the identity. So the Fourier coefficients are d1, d2 lambda u of d1 times d2 to the power 2s times d1 to the power minus s. And the third type belongs to the constant function up there. So you can view it in two ways. Either you can view it by just putting the constant function here, or you can view it by taking a residue of a minimal parabolic Eisenstein series. So that's a degenerate case. u is the constant function. What do you think? What are the Fourier coefficients in this degenerate case?  There are no Fourier coefficients. At least no non-degenerate Fourier coefficients. The only Fourier coefficients that exist for this degenerate case are the Fourier coefficients where at least one of the variables is 0. So a m1, m2 with m1 and 2 both being non-zero doesn't exist. OK, any questions? OK, well then I think that's a good point to stop. And next time, we'll do something more analytic and look at moments of L functions. Questions? Yeah. Can you give a reference to the combinatorial part of the NP-tire? Yeah, well, I hate to reference my own papers. But so you find this in various preprints on the archive by myself and Peter Marga. And the whole Hecker business, like sure polynomials and things like that, you can find in a book by McDonald. Is this correctly spelled? I hope. No? I'm not sure. So OK. And it's called, I think it's called, whole polynomials. I think it's Cambridge University Press. I think it's Oxford. It's Oxford? Yeah. Right, right, right, it's Oxford, yeah. Maybe it's called symmetric functions and whole polynomials or something like that. This is where I learned all these things with sure polynomials and so on. Actually, third chapter of Seymour's arithmetic automorphons for that day. Very nice. OK, so let's quote this too. What is the arithmeticity of automorphine? OK.


We are joined by the number two draft pick in the first round Saquon Barkley and the Saquon. We thank you for joining us How you doing doing good? Thank you guys for having me. All right. Everybody wants to know so we'll ask how's the leg the leg It's feeling pretty good. You know taking a day by day. You know the strength I know the training staff and the coaches and strength staff Have a great game playing with me and you know, I've been trying to attack today and try to get better Everything but do you feel as if you're losing anything by not being able to practice because of the leg?  You know still still day-to-day You know, I obviously I would hope to play I would love to play. I would love to play every opportunity I get But like I said, you know, I'm just kind of just falling along Instructions and seeing seeing what they got playing for me.  I mean you spoiled all of us. We expect that every single time What was that like to get out into the open and you're running free and this the first time you run in the NFL game, you know It's just so so many emotions were just going on just not because of the run Just, you know, every met life stadium growing up I was a Jets fan, but I always wanted to play in that stadium and to be able to finally play in that stadium and be able To play with my brothers and my teammates was a special moment and that the first run, you know It was a decent run. The O-line did a tremendous job up front blocking Evan Ingram did a great job blocking to and the second level blocks. I think it was red rep me to also unbelievable block also I got find a way to get an zone and eliminate plays there That's the mentality that I had in college and that's the mentality. I'm gonna continue to have in NFL Patience was what I saw from it. How much patience does it take to be able to kick it outside like that? Yeah, you definitely got to be a patient back in this league, you know The thing where I feel like I could I can continue to grow is Understanding the down and distance understand the sticks understanding when it take to two and when it take to three and when it's tried Do a little too much just being aware of the whole the whole game and whole situation But yeah, definitely patience if you look of all the great backs in the league where I'm trying to get to that one day You look at Levy on look at Zeke hunt like Alvin the list goes on all those guys have great vision and great Balance and great patience and I got continue to you know with repetition and I've seen more more reps in the game We'll continue to get better there Now you obviously play at a very high level in the big 10, but everybody always tells us it's such a quantum leap Is it a quantum leap in terms of speed out on the field and that you're playing for paying now rather than you know a scholarship? You know, I wouldn't say it's such a huge leap in my opinion everyone's different for me speed is there's not the The biggest different in my opinion. Um, what is it?  So it's not really the speed It's just where you have to be and getting there before the defender. How about the schedule, you know 16 games only having the one by How do you anticipate being able to handle the extra workload? Yeah, that's that's what i'm trying to figure out You know trying to continue to learn there. Um, you know, that's what the the trainers Been telling me every time I try to get frustrated saying, you know This is a 16 16 game season and hopefully got one and even more The last game of the year is I think like December 3rd or something like that for me I would just be getting ready to play a bowl game and already have two or three weeks off right during that bowl game So, you know, just obviously the wearing and tear on your body is it's going to be a lot different this year But also with more time in the season and more time with games There's also more time to recover and take care of your body and uh, I thought I was doing a good job I would take care of my body. Um, obviously with this, but you know, just a minor step back and just kind of, you know Was a slap in the face. Let me know that even though you thought you were doing enough you got to do more Who's what veterans so far have been the most?  Uh, I would say that you know, there's there's so many of them elide snacks tree I would say definitely Odell and Shep Just, you know, obviously just from the friendship that we had before prior to coming here and camp and uh, just training with each other building that relationship there You know, definitely, I think uh, oh and Shep are two great guys to Mild yourself after and the way that they come to work and the way that they carry themselves And I think they've been doing a great job That's as leaders and as taking the young gun under the wing Talks with john to do it. Just the list goes on and on. Yeah Talk with sequan barkley here on the michael casio A lot of questions because the offensive line has been reconfigured. What's your take on them so far? You know, yeah, um, I don't know. I was here last year. So I don't I can't really compare them to what happened last year But in in my opinion, I think that they're doing a great job. Um, I think that they're doing an unbelievable job Uh, not only just in the game, but you know, just a little extra stuff that we do on the side Even in meetings standing after in meetings, uh, you know, Nate nature. It's a great leader Will wills been very impressive. He's very physical and uh, they get to the next level They get to the next level really fast and makes my job a lot easier They mentioned you grew up a jet fan. You may not get a chance to play on friday, but how Cool would it be to play against the team you grew up following? You know, just name just playing against him just being in that that stadium with them will be will kind of be a little cool moment for me, but very short moment for me because Obviously, I grew up as a jet fan, but that that's all in the past now It's all i'm all blue now in all new york giants.  This is going to be a special moment Is there any fear whatsoever that because of the leg you won't be ready for the first game? Uh, no, I mean, there's there's not really any fear. Um, I think that I've been moving really well The only thing I think that can happen if god forbid if some something not to be something again While getting back, but I think the the the schedule i'm on and the the process i've been going to going through and The strength to have and the training staff. I think that that they have me Oh, I think they're they've been doing a great job and like I said, I feel really great But i'm just falling falling really now. It's weird and I'm sure you haven't been asked this when I graduated from college And then I started working I can't it felt weird that I didn't have to go to class. Is it weird that you don't have to go to class that this is now your job? Uh, it's not weird. I would say it's um more of a life-saving thing because Classes, uh, definitely, especially at Penn State and you get class checked and you got to be there and uh, You know just the whole standard that you have at Penn State. Um, I want to say it sucks But it's definitely a lot better now That uh, I can just focus on football now my life is uh, you know, come in here Take care of your body come in here and practice work your butt off and spend time with your family and repeat repeat repeat So uh football is a lot more easier definitely without classes and it makes sense too. I mean, okay This is professional football. I'm making air quotes, but that's big-time football that you're playing So you're preparing to play big-time football. Oh, by the way, you have to go to class and pass Yeah, exactly. So that's tough man. Exactly. Yeah, so michael didn't tell you as he kept going back to for them Even after he graduated, but I didn't have to go to class. I just You're probably way too young to remember the movie same apples fire, but it was like that We just do the rob low care. Just keep going back Just to hang around. Ask him how it turned out. Saquon had it turned out. It worked out all right I'm just saying don't be that Penn State's not that far away. Don't spend your off days going back there. Move on Michael never moved on. See you soon. Thank you That is Saquon Barkley the number two pick in all of the NFL and the Giants expecting great things from him And what could be an explosive offense


When you're a phone salesperson, visually three things you need to know, okay? And I'll put them in priority order. Number one is preconceived notion of what they need. So every single person you talk to has a preconceived, very few people, even if they say they don't know what they want, almost everybody has an idea of what they need. Number two is risk tolerance level. Are they a low risk taker, medium risk taker, or high risk taker? And number three is their cash in the bank. Okay, you gotta know those three things. I've been on some calls with people who work for me and I'm like, how much cash do you think? And they're like, oh, they got a hundred grand of cash. Then I get on the phone, I extract always. I can always extract how much cash someone has. The guy I just talked to has 700,000 of cash. I know, because I extracted it very carefully. I'll record a separate video on how to extract, like if you're a CIA agent, okay? I can extract from anybody.


Hey, it's Bridget. Welcome to Sunday Morning Coffee with Bridget. Today, we're going to get inspired by a daily meditation reading, a passage from Melody Beatty's book, Journey to the Heart, Daily Meditations on the Path to Freeing Your Soul. Her work, Melody Beatty's, is often used in recovery. It's often used when you talk about codependency. She actually is from Minnesota. She doesn't live that far from where I roam. Let's say that. Okay. So, and how this book is set up is like, each day, you can go through it by the year, each day, and just read something for that day and then use your journal to contemplate whatever passage or piece of information comes through for that, or it can help to give you something else to focus on or think about besides something else that might be more distracting or destructive to you. So it kind of helps set your mind, your thought. Okay, some of it's deep too, so I'm just going to give you a heads up. So we'll find out what our topic is after we read this. Okay. So it says, go a little further. I arrived at Oregon's Willamette National Forest after dark. Suddenly I found myself at a fork in the road. To the right there was a chained gate marked foot travelers welcome. And to the left there was an open road marked nature sanctuary authorized visitors only. There were both signs. Then I headed to the left. I didn't see anything that looked like lodging and I began to feel uncomfortable. Like one of the unauthorized visitors the sign warned about. I, I backed the car out to the fork turned around and left. There I still hadn't found the retreat. I was tired worried about, about running out of gas. I tried to remember what I'd been learning that, despite that despite. I tried to remember what I'd been learning that desperation attracts more desperation. That's true for anything right. I realized and visualized, I relaxed and visualized myself finding the retreat being given a key to a room and going to sleep in a bed. I visualized it until I could see the scene clearly in my mind. Before long I found myself back at the fork. I mean no harm, I thought. So I'll just drive down the nature sanctuary road again, the one for authorized visitors only. I drove so far. I drove as far as I had been before then decided to push ahead a bit more. I rounded the bend, and there it was the parking lot, the night office and a man who could give me a key to my room within 20 minutes I was in bed for the evening. Sometimes we need to go farther than we thought we could. We're past our fear past our uncertainty past the bend, we can't see beyond. If we stay on the course, give it that extra push and go around the band. We may find what we are looking for. Let's just let that settle for a minute. I can't really see the image of the corner, don't you? And I know that we all feel like this. The level of uncertainty, the unknown is challenging. It's always going to be a challenge. Uncertainty has always been here. We just never were that it's psychically in tune to it. The basics of uncertainty or of unknown have become more obvious to everyone, not just the people who consider themselves empaths or not just people who are intuitive or psychic. Everyone feels that. It's this feeling of something bad versus something good. What we're trying to do in our mind is to go toward the bad to prepare for that inevitable other shoe to drop or to help to prevent us from some pain or from deeper pain, because if we know about it, and we are preparing our reaction or response to that bad situation or scenario outcome, we will feel better when it actually happens. When the truth is, as you have well heard and many different other experts speaking about this topic, your body reacts and responds just like as if it is happening. So when you project or think or feel and then make in your mind's eye the storyline of something not good happening, then your body is experiencing over and over and over again. You're the worry, the trauma of the not event, the not event. As in this story in this passage by melody baby. She visualize she was able to calm down, not feel so desperate, calm or soft down, which doesn't mean you turn yourself off.  Your body is giving you senses sensory responses to give you information so that then you can make choice. It's not giving you information for direct action it's giving you information for awareness that then creates a action that is appropriate and aligned with you. It's reaching to a desperation or a fear. This is always a challenge. This will always be a challenge this is always going to be the challenge we're always going to air on the side of being afraid of what we don't know, instead of bracing the opportunity for the universe to surprise and delight us. Well, I know I feel this too. Me too, me too that way, but these times that we are living in have brought us to a point of challenge where we have been so forced into a constant state of unknown of uncertainty that we have begun to revolt from it or become rebellious against it. And it's kind of like well if you can't beat him join him. That's not at all going to help us. I was I was speaking with a client yesterday and this came up to me.   And it works. If you want to let go and not have any kind of choice in your life or responsibility or accountability to how you are actually creating your life. And I get it, people sometimes like early on in my psychic career people will come to me to get answers.  I give them the connection of the energetic information that I see and sense feel and hear that gives them insight information it doesn't give them answers it doesn't tell them what to do I do not tell them what to do. But people are, and have been when you're pushed into a state of unknown and uncertainty and you're afraid which is a natural human things natural emotions fear, and you're anxious and you're desperate. You got to know you got to know something and people are in that place, willing to give their ability to choose to somebody else so that they can make the choice so that if it goes wrong or it's bad then they don't have to accept that they didn't do anything about it. You just trusted this other person there for this other person, you whoever coach some motivational speaker a doctor whoever gave you information and they must know better about life, your life than you and so therefore because they have some special skill, whether it be a lawyer or a doctor psychic or whatever that they obviously no more they're an expert therefore I'm just going to blindly, or just laid on my own autonomy, and follow them.  And yet, when we're in times of unknown and uncertainty that this instinctual just just decide for me just do this for me just tell me what to do. And, excuse me. In times like that. In times like this. It's not about what to do. It's not about uncertainty is not cured by doing something. You cannot cure the unknown by doing something. An action is certain it's a tangible thing, yet the action itself in alone by itself isn't going to just magically change everything. It's a start an action step is a start, and you can try different action steps to see what works just like melody did in the story. I tried action steps I tried this I tried that I tried this and then I did this. She tried different things that's what action is about trying different things decisions are plural. You make a choice you make a decision, then you make an action and then, if that doesn't work you're back at the choice again. It's a circle right it's your back at the choice again. So then you make another decision. Right, and then that results in an action. So, it's a process and it's continuous. So you don't just make one big decision and your life is great.  We're lost in us. I think there's this perception of we make a change and magically something happens like somebody's rise to stardom or rise to fame or somebody on YouTube that seems like all of a sudden there an overnight success it's like the overnight success took 15 years. We lose sight of that in this fast paced instant connect like world of constant information. And yet, really. This is something that sure feels like it's something that's amplified by the excess of our desire to connect. Oh, no, that's not right. The uncertainty is being amplified, because what's being shared like on social media is all the energy of desperation. You share your feelings and your opinions and that kind of thing. However, the flip side of that is, don't just come on then when you feel bad, come on when you're feeling good, come on and show examples of being okay. Just, okay, because most of life is about the center point and being aligned and when you're aligned things are flowing. They don't feel spectacular and they don't feel crappy. They feel just okay, like, you're open. And there's still uncertainty that exists all around you and it's always here and it's always been here. It's just now we're using it as this poster child for all of our fears, our feelings of mental health, our challenges with mental health like depression and anxiety, and we're using with feelings of shame and guilt and not enoughness. That's why uncertainty feels so big. It's not big, and it's a land of mysterious opportunity for sure, for sure. Because if everything was planned out, would you really be all that excited? Like if you knew how everything was going to go, would you really want to be here? You probably spent most of your time trying to change the stuff you didn't necessarily like about it anyway. And therefore then create a totally different experience and so the unknown and the uncertainty, although it totally is a pain in the butt sometimes, is a good part of our process. It reassures us. It gives us opportunities. It reassures us that our spirit is here, that our intuition is here living in our body and that we have to rely on something else just besides our brains. And we have to pull in our resources into our center core and when we do that we connect with ourselves and that's the purpose of uncertainty, connecting with yourself, pulling yourself into alignment. It's not about dis, it's not about disengagement from others or the outside world. It's about being with yourself and then with the world, moving with the world in the natural rhythm of things. And that feels about right. All right. This is Bridget. Thank you so much for being here for Sunday Morning Coffee with Bridget. This is my weekly podcast here on Above Life Channel. Sometimes it's audio, sometimes it is a video. And you can expect it on Sundays. I'll see you on Mondays for the weekly channeling video with afterlife celebrity guests, which is designed to inspire your spirit and fill you with some hope. And you can also join me on Fairy Grasshopper, my other YouTube channel as well, to learn more about intuition, to learn more about me. I share a lot of blogs there. And I talk about intuitive talk topics and also different kinds of divination tools and things of that nature that will help you as a spirited being living a very real human life. Thanks for being here.


If you've been following this channel for a while, you've heard me talk about chocolate a lot. It's one of my absolute favorite foods. And in my opinion, it's one of the most delicious foods you can eat on the plant paradox.  But even after giving you the go-ahead to eat plenty of bittersweet chocolate, I'm still getting lots of questions about chocolate. So today, I want to put together a quick chocolate 101 for you. But first, I want to ask you a question. Are there any other health topics you'd like to know more about? If so, reply in the comments and let me know what you're curious about. After all, I created this channel for you and I want to make sure you're benefiting from it. So if there's anything you want to know, go ahead and leave a comment below. Now, I know you're here for the chocolate. So let's get right to it. Chocolate 101 with Dr. Gundry. First of all, what kind of chocolate can you eat? The best kind, bittersweet chocolate. It's like I always say about leafy greens and other vegetables. More bitter, more better. And when it comes to chocolate, it couldn't be more true. Your chocolate should always be at least 72% cacao. Personally, I'd prefer it to be above 80% cacao, though I understand that's too bitter for a lot of people. But wait, chocolate has sugar, right? Well, if possible, look for chocolate that's sweetened with stevia or monk fruit. Lilies is a great brand. But if you're not able to find stevia sweetened chocolate, that's okay. Just buy the most bitter, regular chocolate you can handle. There'll be a little sugar in it, but the amount of sugar is pretty negligible. And if you're only eating a little square of chocolate a night, it's not a huge deal. The benefits will outweigh the drawbacks. Is there any milk chocolate I can eat?  For one, milk chocolate is significantly higher in sugar than dark chocolate. But even if it's stevia sweetened, milk chocolate is often made with milk that has casein A1, making it terrible for your health. Plus the milk will bind to the polyphenols in chocolate, ruining those health benefits. Now, if you're running around Southern Europe and you find a stevia sweetened chocolate made and sold there, chances are you'll be okay. But remember, you won't get the polyphenol benefits. And if you do find that chocolate, write in and tell me about it. What about white chocolate? There's exactly one place that white chocolate belongs, and that's the trash can. Not only is it loaded in sugar, but it's got none of the health-boosting polyphenols that dark or even milk chocolate has. Plus, white chocolate bars have about as much milk as milk chocolate, sometimes even more. And if it's A1 milk, that just spells trouble for your health. So that harmless-looking white chocolate bar is simply a lose-lose for your body, especially your gut. And in my opinion, it tastes pretty lousy, too. Can I eat cocoa butter? If you can find pure, raw cocoa butter, absolutely. Like coconut oil, it's just fine to eat in moderation and adds a really interesting flavor to foods like adding just a hint of chocolate, but no heat. But if you're into baking, it actually makes a pretty good butter substitute.  But like I said, it adds a pretty interesting flavor. One you may or may not enjoy, but it's rich in some pretty interesting phytochemicals and fatty acids. So if you're into the flavor, go ahead and add it to your pantry. Just remember, when it comes to healthy fats, nothing.  So you shouldn't use cocoa butter instead of good old extra virgin olive oil. But you don't have to avoid it either. What's the story with cocoa powder? For some reason, people are really confused about cocoa powder, probably because there are so many different varieties on the market these days. And when it comes to supporting your health, one type really is a cut above the rest. And one should absolutely be avoided at all costs. What should you avoid? Sweetened cocoa powder.  For one, it's loaded in sugar. One popular brand has over 30 grams of sugar in just a quarter cup of powder, making it almost all sugar. Plus you find cocoa mix or powdered cocoa rather than actual cocoa powder. You're getting more than just chocolate and sugar. That's because a lot of these shelf stable mixes have a one milk in them, as well as stabilizers and other additives that are often made from ingredients like soy. And if that weren't enough, most of these mixes are made with the wrong kind of cocoa powder, dutched cocoa. You may also see it labeled as European style cocoa, alkalize, black cocoa, or dutch process. And dutched cocoa is cocoa that's treated with an alkaline solution to neutralize its pH and mellow out its flavor. It also makes for a richer color. Now, a lot of people find dutched cocoa to be delicious, but there's a problem. That alkaline treatment does more than just change the pH of chocolate. It zaps all the stuff that makes it healthy too. So you won't enjoy the incredible benefits that come from eating dark chocolate. So what's the solution? Natural cocoa powder, also known as non-dutch cocoa. Good news, not only is this stuff even more delicious than dutch cocoa, in my opinion, but here in the US it's more widely available. Brands like Hershey's and Ghirardelli both use non-dutch natural cocoa powder.  The flavor of natural cocoa powder is complex, robust, and fruity. It's fantastic. And because it hasn't been scientifically tampered with, it's incredibly nutritious too. Unsweetened cocoa powder is one of the highest polyphenol foods on the planet. Just remember, chances are you're going to want to add a little bit of stevia, monk fruit, or xylitol to anything you're cooking with cocoa powder. On its own, it's pretty bitter, like an unsweetened chocolate bar, but it's great in a chocolate almond cake, believe me. Now, before I go, let me answer one last question. How much chocolate can I eat? The answer, have a little piece about this size every single night, if you want. Just remember, more bitter, more better. And if it's less than 72% cacao, it's a sugar bomb and best avoid it. Because I'm Dr. Gundry and I'm always looking out for you and my extra dark chocolate.


In this module we will talk about the medium sterilization, but in detail we will try to discuss the calculation of the del factor during heating and the cooling. As we know that the overall sterilization process is depend upon the viable number of cells, but actually when we talk about the stabilization of the medium, we have to take this process by dealing with the heat stabilization process. So, as we see in our previous module that the overall del factor include the del factor of heating, cooling and the holding time. So, let us try to see in detail that how we can calculate the del factor. So, as we have seen in our previous modules that the relationship of the temperature we can see in this equation that the del factor is equal to A which is the Arrhenius constant and T and E and then minus E over R T. So, this equation actually shows the relationship of the del factor with the temperature and the time. So, this T is the time. So, however during the heating and the cooling period the temperature is not constant because when the process of sterilization start temperature gradually increase. So, when there is a required temperature of the sterilization requirement then there is a holding time. So, the increment in the temperature with respect to the time then actually what happen that also effect on the del factor. So, that is we have to need to calculate the del factor on each integration of the above equation. So, as you can see in this slide that reported by the Indover and Humphrey in 1959 produced the integrated form of this equation for a variety of temperature to time profile including the linear exponential and hyperbolic. So, however the regime observed in this practice is frequently difficult to classify and then making the application of these complex equation problematical. Richard in 1968 demonstrate the use of graphical method to illustrate this relationship. You can see this relationship in the next slide. So, in this relationship we can see that how the time axis is divided into different increment according to T 1, T 2 and T 3 and so on. So, Richard basically suggest that 30 different reasonable number to increase the temperature. So, as concern you can see in this slide that on y axis is a temperature on x axis is a time. So, in first step so, this one and this two T 2, T 3, T 4, T 5, T 6 and so on. So, this is basically the graphical integration method that applied to the increase of the temperature. So, with the passes of the time temperature is gradually increase. So, if all these intervals which Richard said that that maximum should be the 30 which is the suitable number. So, then we can say that for each increment the temperature corresponding to the midpoint time is recorded. It may now be approximated that the total del factor of the heating up period is equivalent to the sum of the del factor of the midpoint temperature for each time of the increment. So, the value of the specific death rate in case of bacillus stereothermophilus spores as we have seen in our previous module. We are taking the example of the del factor of bacillus stereothermophilus spores. So, in this case then bacillus stereothermophilus spores at each midpoint temperature may be deduced from the Arrhenius equation using the thermal death characteristics as published by D. Inrofer and Humphrey in 1959. So, the value of the del factor corresponding to the each time increment may then be calculated from this equations. So, as concerned in the first increment del factor 1 that is equal to k 1 and t del 2 k 2 over and t and same on in each increment we can calculate the del factor of each step. So, the sum of the del factors for all the increments will then equal the del factor for heating up the period. So, with the same pattern when we are dealing with the del factor in the cooling pattern. So, the pattern will be the same. So, if we are talking about the heat up system and then the cooling system and then in between the two is the holding time. So, the del factor is basically the mixture and the addition of all the del factors include in the heating up system and the cooling down the system. So, in later module we will talk about in detail that how we can further elaborate this del factor.


What's good y'all Chris from jassescasebreaks.com doing the last break that's sold out for our all spots sold giveaway For a break cut which we did first thing tomorrow morning or tomorrow afternoon 3 p.m. Pacific with Jason This is 2022 Bowman draft baseball first edition two box break random team number five So there's a 26 spots sold with four combo teams the Padres giants the A's Astros the Rockies Moreland and the Cardinals Brewer's are all combined spots. So we have 26 spots here. There are teams and Our names here as well from Kevin all the way down to Nick We've got the last seven spots to fill out this break. Thank you Nick. Appreciate ya We will do our dice roll for both randomizers. Thanks again. Y'all good luck two and six eight man, I deleted my Player list to grab this one more time. Sorry y'all takes a second. There we go So once again, two and six eight times. Good luck one two three four five six Seven last one eight They need pure concentration There is no there is no splash zone in the the World Knife throwing League three four five six Seven and last one eight. So eight times both for here are teams from the Texas Rangers down to the Brewer's and Cardinals and eight times on our player list From Adam down to Robert. I I'm looking this very close. I mean these guys are professionals. So you would assume that There's not gonna be a lot of errant throws, but you never know man. You could you know turn something or pull something and All of a sudden you got a knife going pretty fast in your general direction with no netting no chain mail That's kind of tough. All right Adam with the Rangers Kevin T with the twins and Dodgers Robert with the white socks Kevin T with the podgers and Giants combined spot Nick S with the O's Guardians Angels and Royals Adam with the D-backs Devon with the Tigers Nick with that last spot mojo truly gets the Phillies Robert will get the Nationals a second half here Bartholomew gets the Mets Adam K gets the blue is Kevin and with the red socks Robert with the rays Chad D with the Pirates Robert with the Reds Bartholomew with the A's Astros combo Nick with the Mariners Robert with the Rockies and Marlins combined spot Nick with the Yankees Adam with the Braves Dennis with the Cubs and Robert with the Brewer's Cardinals combined spot We got some offers in the chat already. We'll pause real quick for any trade chat when we come back We'll rip this two box break of Bowman draft first edition. Thanks again. Be right back All right, we were back a few offers up, but nothing too serious nothing in concrete. So we are going to Get this trade window close get this printed out and we're gonna rock and roll This is an all-cards ship break and it's just two boxes But uh, you know that first edition right above on the right Actually kind of has a nice value on these One of the first breaks we did of this for the That Jackson holiday auto refractor, which is the thumbnail on the break site That's a very nice card that first edition only kind of amplifies it, but here we go poor steering bar didn't deserve that man So these are these come in sealed boxes like this So I'm just gonna And I'm actually not gonna figure this away because it's someone who does the shipping as well You would be surprised to see how you use for these ten six two boxes actually are They are perfect for shipping sealed boxes Like this like perfect size for a hobby box My manager has requested that the breakers keep these if we do them cool my Refuse it's my job All right, let's do it 2022 Bowman Draft first edition all 26 of these spots on the All in the Random all spots randomizer Barmy got a bad shake man Any other fan in the world will have done the exact same thing and it only got worse because was it moist as a loo? Like pointed and shouted at them. I don't know. It's a rough shape Like I remember the only thing that's kind of close to that When it comes to just the occurrence of it. It was like a regular season game It was a Reds game. They were in Cincinnati some the batter pops it up and Joey Vada was like running and trying to track it along the foul base. They're the first baseline going into the crowd and No, a fan in fears not air fears interferes, but gets in the ball and catching the ball and Joey Vada like looks at him looks at his red his registers and it's like Well, he pulls out his jersey and I think like immediately like next inning apologized gave him his jersey and everything like That's kind of how you handle it. It happens man. They're a fair fan You pay for seats that good like you have a chance at a foul ball go for it and that ball was foul It was not in tears. It was not in play Like what's the real was this little was not getting to that ball That'd be a very interesting foreman signs the Moises la Lou card He probably would never do that when that dude that dude's life got ruined because of that He was getting like death threats and harassed like he got like the news found out where he lives and everything It was horrible sports fans Pretty nasty sometimes Alright, let's do it These are kind of non-numbered baby wounds. I'll put to the side so we could get a sleeve on them Like these they're not numbered, but I'll put them to the side for sleeping Dylan beaver's yellow second rounder for the O's 20 out of 75 Baltimore and Nick Stover People wanted the O's Held on to him and it's a nice one to start Henry Bolts Second rounder for the A's another nice one here first rounder Brooks Lee Out of 150 for the Minnesota Twins. Let me go to Kevin Trying to trade couldn't trade mojo. He was willing to give up some spots For the O's so nice little connection there People are getting very brave at sports games So tomorrow Johnson and that non-numbered baby blue All the tomorrow Johnson's going to Chad doll in the Pittsburgh Pirates I've Melendez 105 out of 150 Arizona Dumbabacks and Adam cupperman Set the collegiate single season record for most home runs Nice one there nothing here or tomorrow Johnson Danny DeAndrade Ellie Dela Cruz top prospect. He's 48 out of 150 for the Reds and Robert Runkle Okay, we all in cones There's a Jackson holiday Henry Williams Out of 150 lots of blues which is a nice one Padres In that dual combo spot going to Kevin T. Padres Giants Marking last second box one shouldn't have her auto in here Unless I'm completely blind as an orange 14 out of 25 Jacob Maurice Marcy One more for the Padres Giants combo Kevin top sixth rounder. What's a fun fact here Earned two three high schoolers in football all the performer and two in hoops. Wow He's an athlete Alexis Hernandez non-numbered blue Jordan Walker 48 out of 50 nice gold 48 of 50 Jordan Walker Our autos not guaranteed in here I've never done one of these actually No, I guess not chrome models are one in every 207 packs Huh, well, thank you Thank you. Tuck I've uh, I've as you can see I've never done one of these But that is interesting, but what your parallels definitely helps He walks the damn Eesh Let me grab a tub actually So that Jackson holiday was like really nice then jeez louise. All right. Damn Joe's cooking Let's see if we can snag one. Huh second box here second of two boxes. Let's do it y'all. Thanks again. Good luck It's a good name by the way. It's like your cookie. It's a good name I'll grab another first rounder parallel here. Shall we? You are welcome They usually report around uh my birthday and my birthday isn't a month. I'm turning 28 oh my god 20 Yes, all right. Yes. Good box two of two good luck tomorrow Johnson one more on those non numbered baby blues Chandler Simpson Yellow drew Gilbert. That's a nice first rounder 74 of 75 for the Houston Astros. She's in that combo spots with the eight 28th pick in the draft That's going to go to Bartholomew Oh I feel a lot older than 30 Ryan Reckley blue 30 out of 150 I like to make uh, maybe it's the the jerk in me, but I do like making people feel older than they actually are Or making them realize how old they are Adele out of the autumn or 75 out of 75 for the Rockies That's the combines up with the marlin that's going to roberts Like uh, my best friend is a is like 36 27 He's a few he's a good number older than me And we were talking about Star Wars one day because that's just what we do They're both fans And he's like, yeah, I remember camping out and waiting in line I got out of school early that day Because he wasn't like, yeah, I got like freshman year of high school. I got out see it like I don't remember seeing it. I was like five and he just stared at me like blank face like what do you mean? Like Ryan I was five on the movie came out and he couldn't process it Joe Ampe out of 150 for the Guardians Nick Stober And he was like so mad at me Because it was such a big moment in his life It was the first Star Wars movie that either of us had seen like in person in theaters when it came out new And he's just like, I hate you right now I'm like, sorry bud It's funny to me though Oh, how's your green and the baby blue nice one for the Nets and Robert Jackson holiday Cooper hereby Tomorrow Johnson No shops Kumar rocker 101 out of 150 for the Texas Ranger the third pick in the draft. That's going to Adam Cuperman Nice one right there Little blue color match there And a gold Brandon Barrera another first rounder for the Toronto Blue Jays and Adam Cuperman Gold 26 out of 50 Love the color in this box. See if we can grab one more nice one for someone before we call it Umbreak 20 Bernabelle Right here last stack. Thanks again, y'all Got a blue Of Jacob Misurusky 111 out of 150 for the Brewer's In the combined spot with the Cardinals. That's going to Robert Runkel. I can rent the car now. It's pretty cool Last card a Parker Messick All right Thanks again, y'all nice little fun break there two boxes I'm going to add random team number six on the site right now check back soon for more baseball on jashescasebreak.com Thanks again. Bye. Bye