Admiral Mark E. Ferguson, Commander U.S. Naval Forces Europe in Africa, visited Naval Station Rota and toured USS Porter and its weapons systems, including the new CRAM, Anti-ship Missile Defense System. He spoke with sailors on the flight deck of the ship and explained how they could educate their shipmates on the new equipment. You are now becoming the experts in the whole Navy on electromagnetic maneuver warfare. Now is your opportunity to change the Navy. You are the experts. Take advantage of it. I've charged the captain to help change the Navy before she leaves. And you can do that through writing tactics, memos, writing new procedures, taking these new systems we've given you and teaching the rest of the Navy how to use them. Petty Officer Jordan Del Gardino, Naval Station Rota.


Shadowrun is an RPG of a different variety for two reasons. Number one, it's not your usual fantasy style fare, taking place in a cyberpunk themed world straight out of Blade Runner. And number two, it got released on a variety of platforms, something that wasn't too common for an RPG. What's weird though is that the Super Nintendo, Sega Genesis, and Sega CD versions were all made by completely separate developers. So all three games are different interpretations of what was originally a pen and paper RPG. Now at full disclosure, I haven't played very much of the Genesis or Sega CD versions, so I'm concentrating on the Super Nintendo version, obviously. But based on what I've read, the Genesis version follows the rules of the original board game much closer, and it's much darker in tone, and that's not surprising considering that's pretty typical for the time, really. Anyway, if you're looking for an RPG, but you're burnt out on the typical Super Nintendo RPG experience, like the Final Fantasy games, the Dragon Quest series, or Lufia, then Shadowrun offers a totally different change of pace in terms of atmosphere and tone. I love the way this game looks, sounds, and feels. The story weighs zero time throwing you right in and getting you killed. Oh, well, at least some Fox thing comes along and casts a spell on your dead body so you can, you know, play the game.  As a result, the game can be a little hard to get into at first, but after a while it's well worth it because the story has a clever way of revealing itself. Well, clever for a SNES game, anyway. Talking to random people gives you clues or keywords where you can go talk to other random people and use those keywords to find out even more information, and so on and so forth. I really like this approach because you can experience the story at your own pace instead of being bludgeoned with unskippable dialogue scenes that are nothing but exposition. The way the story is told forces the player to be independent. As for the gameplay, it's... unique. It's an interesting blend of the usual RPG-leveling-up system combined with a point-and-click mechanics from a PC game. With the isometric viewpoint, the game kind of reminds me of Diablo or the Ultima games, but it's always a little clumsy to operate a cursor with a controller. In fact, this would have made great use of the Super Nintendo mouse. It takes a bit to get used to, you always have to be right next to something and then summon the cursor and then click on it and then choose, examine or open or whatever. It's kind of annoying, but it's not a deal-breaker. Combat allows your cursor to be across here. A novel idea, but the action is kind of lacking at times, especially early on. Die. Come on. Die. Jesus. You level up by collecting karma. Hey, just like in real life, right, Reddit? When you sleep, you choose how you want to level up, whether it's firepower, health, magic skills, or computer hacking skills. You're gonna want to max firepower right away so you can level up quicker. In my experience, magic is just kind of there for the sake of being there. It's really not that useful, but it's nice that the game has it, I guess. It's nice to have a break from always shooting stuff. The computer hacking stuff, or decking, as it's called in the game, gives Shadowrun a bit more variety as the game switches to a top-down view as you move through cyberspace to recover more information. As good as this game is, it's evident why Shadowrun didn't take off back when it first came out because it's just so different than what you'd expect from a Super Nintendo RPG. Also, like I said earlier, the game is a bit difficult to get into at first and it reminds me of a robo-trek in that regard, where it really helps to at least read a walkthrough or watch a Let's Play video just to see what the hell you're supposed to do at first and get the momentum going. I'm not slagging on the game for not holding your hand, just the opposite. Really, I like that it's different, and I like that it makes the player be independent. Like a robo-trek, it's an acquired taste, and it's absolutely worth checking out to see if it's to your taste. Because if it is, you'll frickin' love this game. Of course, I'd be remiss if I didn't mention the Shadowrun revival game called Shadowrun Returns. I found that game pretty frustrating because of the limited save areas, so I didn't even really get that far into it. Finally, I'd rather play the Super Nintendo version.


from Mainz and by the way, we're running a workshop in Mainz next week, so oh wait, you can watch. The deal is. Yeah, so it's not intended here. Oh, okay. Okay, okay. Well, it's a very exclusive. Forget about it. Forget what I said, forget what I said.  These are not, these are not my rules. Yeah, there's an institute, the MITP has all sorts of rules. So the title is 3D BPS indices, Q difference equations and non-perturbatory corrections. Okay, thank you very much. Thank you very much for the opportunity to speak up this nice workshop. By looking at the titles I noticed that maybe I'm working a bit on the exotic topic here. I don't think that there are many talks on 3D. 3D gauge theory, so I decided to try to give a bit of a general overview talk. So this is a topic that I've been working in particular with my collaborator Peter Meyer for a couple of years now. And what I'm going to talk about today is mainly based on a paper which also appeared I think two years ago and some work which hopefully is going to appear. I'm not making it any longer but it should be appearing in the next couple of weeks. Okay, so, so what's the setup for the physicist, it's an n equals three dimensional n equals two gauge theory. And if you have such a gauge theory, then we can, we can have big branches. And for the mathematicians, what this is, there's a manifold involved which we call it target space. So it's a killer target space. It's a base 24x, which in this in the setup is realized as a synthetic portion or if you think more frankly about it, it's some, some sort of a GIT portion. That's a GIT portion that we've also seen in the previous talk. And a gauge theory comes as a, as a compact leak group, a gauge group G, and this GIT quotient arises from a vector space and the vector space is generated by the the Kyle fields. So the matter spectrum in this theory. And it's divided by the complexification of this with the reductive group that we are dividing by the complexification of the gauge group. So if you have such a setup, and this is mainly the quantity I'm going to talk about today, you can calculate a 3D index, or sometimes it's called a 3D path index. And so this is an index of boundary DPS operators. So let's put this theory on a three dimensional manifold, and this three dimensional manifold has a particular structure. So topologically, it's an S1 kind of this too, but it has a bit more structure to it so it comes as a metric in the physical So it's really a disk fibered over the S1, but metric wise the disk fibered over the S1 and when you go around S1, and rotating the disk and this is parameterized by this parameter cube. Okay. So if you have, if you want to count these things, then there's a certain index, and this index is roughly speaking the following, you have to take a trace. And since this is a, this is a space with a with a boundary. And of course, as a boundary, you have to, you have to take a trace where you impose on your field and operate a certain boundary conditions, and the kind of trace you can take. So it's an index. So your way. So for example, if you have a tracing, you have recreditation with Poissonic and fermion modes, you write this with a fermion number, and then they are some fugacity. So one is q. So q goes together with the Lawrence generator that you have in the theory of this disk together with what is called an asymmetry generator and possibly additional symmetries you have in the theory called labor symmetries. You have to take the kind of index, so you can compute so after is the generator of the flavor symmetry. So it's as you can think of this as a symmetry of this, the quotient. J is the spin generator on this disk and are there is an additional symmetry in the super symmetric series. So it's a generator of the corresponding to one our symmetry. And certainly not an index that that that we have looked for the first time. So this has been introduced for the first time by beam. In 2012, then the group of and put off in papers in 2013, various contexts have studied this index. And what we are mainly using is the formulation. What am I in more. I got your job from 2017. They are many other direction that's on top. That's kept tea. That's kept. It's not the same question. Not as ah, sorry, yes. You can look it up. So they are, they're different, they are different people and yeah it's okay. Okay, good. So for the purpose of this talk, I will actually use a particular simple setup with a simple setup you can think of concerning the gauge group. So if I just take the gauge group. If you want to be one. Then the kind of g it questions we are studying a central very basic examples of growing varieties. And, and moreover, like I said we have to impose boundary conditions on the various fields that we have in the theory. The gauge theory comes with a gauge post on in a particular we also have to impose a boundary condition for the gauge post on and the one we choose or the one which is kind of the riches to think about is the technically boundary conditions conditions for the gauge post on. So if you do this, then you can study. This is for this particular setup, the index takes a very specific form. It now depends on free velocities or it could depend on more but these are the gas that we interested in. So, Q is this parameter which we have universally in this index that will keep the queue is is a global symmetry, which is called in three dimensions. You want topological so that's the dual symmetry to that to that gauge group. But this is dual. G, which in our case is the one. This S and S has to do with this choice of boundary conditions so if you impose directly boundary conditions. It means that we don't have a gauge post on at the boundary in particular because we have set it to zero with this directly boundary conditions. What we have to notice is that since we have this this this gauge symmetry G, which in our case is you won, you still have a remnant of this at the boundary as a global symmetry. And that global symmetry at the boundary is usually in order by you, you want partial, and this is nothing else but G. Okay, so we have a quantity, which depends on these on these three variables. Then this big Q and this S. Okay, so this has a character characteristic expansion. And then there are some quotient functions like and that's you. And let me say a few words about about what you get here. So this is, this is a sum of a boundary. And this is one of the multiple sectors that you have in this theory. This, this whole thing here is a Q. The lateral series. So it is formal.  And this, this, this part here is just something that comes from the matter of fields. And one thing that I haven't talked about, they are in three dimensions, they are particular terms you can add to these to these theories topological terms, but they show up in this index. They have their so-called chance I mean search for their specific choices. So, how does this look like in practice. Let me just give you an example, or let me just be schematic here so how do the terms look like.  So we have a B M where a and B are choices of transignment levels. So a is is you want for us, you want to and Simon's level and be the one times you want our terms items level. And they're suitably quantized. They're both active trends. We'll put the fact that time level, and also how does. How does the matter index look like. Very concrete formulas, matter index for a Kyle. So again, we have two kinds of situations. Either we can impose Neumann boundary conditions say to the scale of field in that multiple and then the boundary conditions of the remaining fields of these multiple are governed by symmetry. And then what you get is something as alpha. Q to the alpha m, Q to the r half, in terms of these Ohama symbols. So this alpha here is the U1, the U1 charge. So the representation with respect to the gauge group, S is this forget that we had and Q is this little Q. And if you have directly boundary conditions. And then you get a similar expression, just, sorry, this is to the minus one, then you get a similar expression, but you get it in the, in the numerator. And one minus r half. So what are these Ohama symbols. Ohama symbols. So for the magnitude of this variable q which we take to be. Here it's still a formal variable, but we can think of this as, as, as, as, as a complex variable, which, which we choose to have the magnitude smaller than one. Another choice would be bigger than one that's actually a girl and at the end of the day, but that's our choice. But if you do this, then the Q half Ohama symbol, just nothing else but this infinite product. And then what do you so depending what, what spectrum you have and what choices of terms I mean terms you have, you can immediately calculate this index, which is, which is taking this formula and writing it as a product of all these contributions that you get from all these matters. Yes. These, these are examples are these. So in this theory where we have the gauge theory it's the one. So if you have, this is the contribution of a single carol field. If it has no one boundary condition. It's, it's one of our popular factor of that work. If you have directly boundary conditions, you have, you have this factor in the numerator, and I should have mentioned.  Yes. Yes, so here appears the index and so you have to sum over. This is a natural formal display lateral series. It's just this infinite some for the integers. I mean what you could do okay so I mean certainly you can talk about more general gauge groups and more general GIT quotients and then you could also calculate that index I will give you a geometric interpretation of that index in a second. Yeah, good. Yeah, so, so this why stands. So one if you want one global. So this stands for labor symmetry and the global symmetry. So one example in this case so you should think of this why in this particular example just of the Q and B s. In general, you're right. I mean there could be more labor symmetries and particularly for your story varieties. You can look at the equity variant setup, and then you would have, you would have for every C star symmetry of your story variety you would have an additional why if you want to keep track of it. Yes. Now a is just in this case. It's just in. It's just integer half integer. So, let's talk about properties 3D index point. So, one, one result that we have, again now in this work in process we make more precise button, it also has been already worked out in collaboration with my and we need that Alexander Pablo is that if you have this X, you have this killer target X that, like I said, if you just think about it, I directly think about it. There is an identification of this you want partial. You can make that you identify as this key for the minus one, which is nothing else but the churn character of age, where H is the hyperplane bundle of X. So in this case where we have a single you want this. We also have a single as a gauge group we have a single you want partial and there is also a single H in this setup, but it certainly has an immediate generalization to more general very great. Or even more general GIT quotients, then we can look at this index. Now given in terms of where we set S to this churn character there or to this inverse churn character q, and then this is proportional up to some numerical factors which I'm not putting a track of the purpose of this talk. You can expand this out into a quantity I of X, which now depends on this. It's H or what I mean by this the churn class of this hyperplane one. Plus, and then, since, since, since this is, well, if you think of this as this divisor, then this automatically truncates as the dimension the complex dimension. I mean the highest form you can have this is complex dimension, this manifold X, so all the terms that are higher in degree of age, they vanish upon this identification to do this expansion. And this I acts of age comma. You, this has a very geometric meaning. This is sometimes called a period. It's called the key theoretic. Given I function of this, of this target space of this GT quotient X. So what is this I function. This I function. I mean we have kind of heard it was in the previous talk the homological version of this, this I function is a generating function of certain invariance you can calculate on these target acts and what are these. These, this is a generating function. This is a generating function of polymorphic Euler characteristics on the modular space of stable maps of genius zero. And I write an example that say this one map point of X, which is the following form. So Lee has constructed a student of him tell has constructed a structure sheaf. I should say that the professor in the field has established a virtual structure sheaf on this modular space of stable maps and this allows you to actually calculate Euler characteristics, which are the following form. So if you have on this modular space of table maps one mark points, you can take for instance the evaluation map. And it's like this, ours, he was on power, take this bundle is the line bundle key, some power K. And divide by possibly in this is where this Q shows up, where we have this technological bundle on this particular space of stable maps, but this is the code tension line that we get from these mark points. These are Euler characteristics, which are encoded this generating function. So, these numbers are integers. And just as a little side remark, you recover from. Yes. This is the modular space of stable maps from Tina zero is one mark point into X, the virtual dimension of that guy depends on. Well, I don't have the formula or the top of my head it's a usual formula that you know and lock. I mean that's, I don't know. That is right I mean this formula depends on the first, I mean the virtual dimension of this formula depends on the first chunk of the dimension of the first turn class and of the genius and this generating function that we are discussing other genius zero. Okay, just as a little side remark to what we've, what we've seen in the previous talk. So if you send q to one, then you recover in this limit. You recover quantum homology. And as such, what you also recover.  The rational numbers rational genius zero from a fit in invariance. And, well, a particular instance that we physicists love to study is if you have a color be our threefold, then we know that there's a re resumption of these from a fit in invariance to go for come a buffer invariance which are integers. And this resumption is kind of based of also relating these grimo fit in invariance from a targets base perspective to to one dimension higher by adding an additional as one. And this, this Euler characteristic, yet another resumption of the grimo fit in invariance, which are integral, but they are not specific to any club house anything that works in general. And this is a kind of a form of physics perspective a very similar idea, because there is no reason for the grimo fit in invariance to be integral because they don't stem from an index. However, if you, if you calculate, if you calculate these, these generating function of this counting function of the operators, then it's very natural that they are integral numbers. And so what we're saying is that that these integers from that perspective. They kind of a world she world, world she world she theory left from two dimensions to three dimensions by also including additional one, which make them integral. No, no, no, you have to be careful so the the Euler characteristic is always integer. We are not doing intersection theory. So, actually, you're raising a very good question since this is a stack so you have possibly some overfold action on this. It's a bit tricky and this makes it actually also complicated to calculate these numbers to define these numbers properly. And there is what is called an inertia manifold that which has several components that the attribute to the stack. And then you calculate contributions from each component in this inertia manifold and that is always guaranteed. Yes, so you're right. So yes, indeed. Yes, yeah. Thank you. Okay, so these are actually rational. I mean, integral coefficients with rational functions in Q but just to be in the safe time. Okay. Yes. Yeah, so this is in general so you're when you take the Q to one limit, you always recover the quantum pomology invariance the chromo fit and invariance. I'm just saying that in the case that this is calabi I mean you can do this also for fun or in principle if you're powerful enough to the results that we are previously there should also be a limit. So if you look at the picture constants of these rings that we talked about in the previous talk, they should also rise in that they should also rise in that. Is there a relation to higher genus so what is known. Okay, so this is an interesting question to higher genus for color via three volts. Okay, we can also discuss that afterwards a bit, but for color via three volts in principle these, these things I have been defined. It's difficult to compute them. Let me put it this way. I mean, so far. How you calculated. Okay, so you one way is you calculated by localization or the other way that I've been sketching here. You just calculated from this index. So we know, so you look at this spectrum, you can write down this. So either you do a direct counting of these BPS operators or you get it from localization. Okay, but for you one series are for. Well, I mean more general for general group G. I mean these these formulas are known so in principle you can as a starting point you can write down this index. When we set q to minus one. I think sending q to roots of unity is an interesting question and I'm afraid I don't have a good answer for this, and I've been asked this many talks, but I don't know what, what, how to interpret geometrically Okay, today I don't want to really talk too much about these, these invariance, but I want to look at it at more the analytic properties of these of these indices. Sometimes sporadically, and we also don't understand the system ethics of this so there's an interesting connection to this this block group, and these CFTs that show up in this nouns list of CFTs. And we get these modular functions that we can construct from the local considerations sporadic or specific choices of age groups and matter of the other in terms of levels, but we don't understand the systematic levels. Maybe you can also discuss what's going on. They are, I think they are relations to that. Yes, but I'm afraid I don't really have to. Okay. So, what's interesting to know is that these, these, these eyes one. So these I axis, they fulfill to what is called a Q difference equation equation. So in other words, so we have a Q difference operator L, which is a finite some. So this difference operator has some order and, and it has some coefficients. So Q, Q we, we think about having a particular value, but you can also think of this as a polynomial both and Q and a big Q and little Q. But if you set little Q to a particular complex value which is absolutely small value than one then these are polynomials in in Q. So if you have this, this difference operator, Sigma K, where Sigma acts on Q, or if you commute it has Q, it just multiplies it was a little cube. Okay. And now the statement is that. And I guess I want to know, I'm going to zero and I'm going to zero. So you can find such an operator that these, these, these I functions, I function on the lead. Okay, so this also has a physical interpretation. So this, this, this difference operator is actually award identity. Well, Wilson lines, lines in this maybe gauge theory. Okay, but let me, let me maybe focus a bit more on this difference equations. So actually, there are nice analytic properties of solutions. There are a few difference equations. And one thing is actually, and maybe as a reference. I mean, there's also many recent literature but many of the things I learned from some old work by the way from 2003. There's also by so long and collaborators and this was a part of the PhD thesis PhD thesis. This, this has been discussed. So one, one important aspect of these two different equations is, so if you have a meromorphic solution. Say in some open ring around zero. You smaller epsilon. Then what automatically automatically follows that you have a meromorphic you can extend this to a meromorphic uniquely extend this to a meromorphic solution over C star. And this is this peculiar feature. And this is one of a few difference equations. So, if you think about this what does this, this difference operator do. I mean it is, it essentially just replaces the argument of any function. By multiplying by little q, but since the magnitude of little q is unequal to zero. In our case it's smaller than zero by acting with the sigma or the inverse sigma, you can gradually always analytically extend using this difference operator. This to the next epsilon step. The only thing you're going to do is, since these polynomials in q, the worst that can happen is that you introduce by dividing by these coefficients, you introduce another poll, but that doesn't violate the property that's going to be a meromorphic function so. So if you have a meromorphic function in this in some small ring around zero, it automatically analytically extends to meromorphic solution. So the other thing similar what you use for differential equations you can also write down a first order matrix equation. First order matrix equation. Sigma F some connection matrix a which also depends on you but like I said q is think of a fixed constant for now. Where in the usual way, where we write, where we write this vector F, where we try to find the solution over it F is a solution then the vector after the solution this type of minus one to the F. Okay, so if you have now a set, the complete set of local solutions. Say X zero F one through and so if you have an end order difference equation and they're also in solutions at say zero smaller q small epsilon. And let's say we have another set of solutions. G one. So G and at zero smaller to inverse epsilon. Then we already know that these solutions by what we've said before, they all. So they should be meromorphic it should say morphic also meromorphic. We can automatically extend to solutions of whole of C star, and one consequence of this is that we must be able to rewrite this, this matrix of solutions by some matrix P, and this P is called a book of matrix. So this P has a specific property FB act with Sigma on P. Then it must be invariant because if it were not invariant, then you can easily convince yourself. If this is a solution and that is a solution of a consequence this P has to be invariant. So here, consequence of this. So we have now a matrix, translating these two solutions which is invariant and the Sigma. So a consequence of this is that that P is actually a matrix of elliptic functions. On the elliptic curve, it's how, which is C star divided by q to the C, q, q, and this is one reason why it's nice to choose q is more than one, you can identify so that the complex structure parameter of that elliptic curve is the function of the exponent of this of the cube. Yes. Yeah, so in small, so in small q. This is actually also an interesting thing which has was related to this previous question. So at least for these in the context of these given functions, it has also a very specific poll structure in small q, namely it has only polls at risk of unity. So, yeah, but yeah, that's, yeah. Okay. The infinite products I wrote down. Yes, so hold on. And that that that is true, but it has. So the poll structures you get. So okay so this depends a bit what you plug in for these gases but if you said all these accurate in parameters to zero at the end. The polls you get, and this is very specific to this case theoretic give and try functions. You only have polls and little q along risk of unity but yeah. Okay, so I'm actually this. So, so this is very nice to know that this, this matrix P is a matrix of elliptic functions but well I guess you're not powerful enough but for many examples, even though we can determine these local solutions. And because these local solutions are often given in terms of an infinite series. So we know that they are meromorphic functions over C star. We have a hard time finding analytic expressions. Actually, for these, for these specific for these book of matrices. I mean working out these elliptic function in general is in general not, not so straightforward. So some simple examples and one example I would like to present here because it's particular simple for the Connie fold. And I tell you in a second what geometry I mean by this, you can actually calculate that P explicitly and then there are a few other examples that we can do this. So, this consists of four Carl fields in this case. And again, the gauge group was you want, we said, so it has a single you one charge. So, we have Kyle fields plus one plus one, and we have Kyle fields minus one minus one. So for this simple geometry. We have actually this theory has two Hicks branches. And they both, they're related by geometrically they're related by a flop. And they look. The two GIT questions you can construct out of this, that the geometric are these are these two places, which are these results. And now we can go ahead and go through this program. And actually, so I mean write down the corresponding operator L solve the solutions so let me, let me just give you a flavor of this. So the solution at q equals zero. How does this look like. So, so there's going to be if you do this. So this is expansion in in age, if you do it here. So you have a zero and a two form class. So this is morally speaking, age to the zero, age to the one. We also would like to have some classes for these for these non compact direction and this goes back to I think questions, we actually have to go to a covariant setting now and calculate them a very formalogy classes. But more or less speaking, if you do that, what we can extract is also four form and six form class with compact support. And then to restically, to think of this as for bonding to something like it's two squared, it's three squared but really what we're calculating you know some equity variant pomology classes but let me not go into this discussion here. And then, if you if you do this whole thing at your end and at the end you can again send the equivalent parameters to zero if you want, then we get a period solution around zero. And this takes the following form. So one, in this case is a solution. Minus LQ is a solution and tell you in a second what this is LQ over two plus a huge generalization of lead to, and then, let me not write the third one, because it gets a bit more involved so it involves a generalization of the acute generalization of the three. So what are these things. Well, log Q is essentially the logarithmic derivative of a theta function of a theta function on this elliptic curve. And the important thing of taking this logarithmic derivative. The theta function is this is meromorphic, the way we've written it meromorphic in C star. And then if you take the logarithmic derivative we still get something more amorphic in C star, but the interesting property is if you if you act with this difference operator on LQ, then we get back LQ plus one. And that saves a bit like, like the monogamy of a logarithmic when we act with the Q difference operator. Okay, and like I said, this lead to Q is a certain generalization of lead to in the sense that when we send Q, again to one to the chronology limit, you get the ordinary lead to, and it takes the following form. The series one one infinity. Okay. And then there is also a choice of transiments the choice of transiments level enters here that I mentioned before. And then there is one. Okay, spread. And like I said there appears something a generalization of the three and the third line. And this has the characteristic features it has denominators which go with the cube. And again if you take the Q equal to one limit, you get lead three of that, that function. What do you mean this ordinary. So you mean the father of quantum dialogue. Now this will appear later on. So this is not just this is not just a pohama symbol. This relates to. So if you start differentiating this, you get similarly, similarly like you are from the ordinary.  Okay, so now you can do the same story for the, for the other for the other solution as well. Let me not go into this but let me maybe. I'm running out of time. Let me just make an observation so if you, if you do this we can now work out. I mean, we work out this solution the solution at the other end we can work out the. And again what I should maybe stress, it's this function, which contains the the chronological that the case the erratic grimoire put in your answer the K invariant these are these are the characteristics that I've been talking about so this is the function which contains the information. And similarly you can extract it from the lab here as well. Okay. So, one observation is that the spirit of matrix of matrix. Cancels holds a solution holds in the solutions infinity. So, when we do this, for instance, for the poor form part. Namely for the part that I've written down here rather explicitly, you get the following formula that you can write this particular period period in the following fashion. So here you get an elliptic function, which is essentially given by the by a constant plus the wire stress p function. I've written in this multiplicative variables, times the period one, which we also have on the other side for you inverse equal to zero, plus, plus a minus one so minus one is also clearly an elliptic function a constant is an elliptic function. Where q tilde is essentially the same quantity that we had here just written in the inverse variable that w, that w is the inverse. So now what we see here is the following property. So we first of all we see that we have two elliptic functions that these are the ones which come from the book of matrix. And then the other thing we see this that this actually when you analyze these functions more carefully. This is holomorphic at q equals zero. This is holomorphic at w equals zero. And these, and, and these they are meromorphic meromorphic functions on C star. So what you see is that the poles of of this of this function, they get by these meromorphic functions, they get canceled because the poles that we have here if you go further out in q. They are not up here so this thing has different poles. And the, the, the, the job of, of this, this factor but with this essentially comes from a John Simon's term, but a particular of this factor is to cancel the poles that we have in one expansion with the poles of the, of the other expansion. Okay. And unfortunately I'm a bit running out of time, but let me just say, how we now interpret this or how we use this. I'm going to use a bit to Murat's, well maybe Murat's question. Yes. What the little. Now this, this important this does not have a logarithmic French cut. It just has an essential single. So the price you pay is that it has an essential singularity at q equals zero, but it behaves. So if you use this as a logarithm if you send little q one or, or as a matter of fact if you send little q to one. Then this is an interesting story in its own right this is described already in this thesis by so raw. If you, I mean, it's a bit. It depends how you precisely take this limit for these functions q goes to one, but then what you get is is actually a logarithm. Just maybe conclude by, by saying the following. So if you have now this equation. Um, so this is maybe part three. Non derivative corrections. We can now in this works in general, but for the conifolded bricks because particularly nicely and you can carry it out more. Yeah, essentially analytically. So if you introduce also as dual variables variables, which come from an S transformation maybe q tilde is if you take this elliptic curve and do an S transformation of the modulus of this elliptic curve. There's a natural associated q which we, let's call this q tilde which is q divided by one over q. But this is essentially e to the two pi i u divided by tau q is e to the two pi i u. And similarly, there's q, which is e to the two pi i divided by the minus one. And, and this q has the property if these act by construction if you act with this difference operator on this q, it actually commutes or it gives me to the two pi i u tilde sigma. So, you can also use functions in this q as as as coefficients in your series expansion and if you do that, then you, you can write down new solutions. Which have nice of the analytic properties. Let's say the, the, the solution of this form around zero. It now is a function of q and q tilde. It, this is now an ordinary logarithm of q, which we can attribute to a transience term. And that I can tell you afterwards more about this then we have this q that we had before. But then there's another piece which let's call this l plus, which collects the polls. Which collects the, the polls which enter in order to generate, I mean to communicate non polls here to pull there we collect all these polls bring polls to this side and the other to the other side.  This is also a solution to this two difference equation, but it has nicer analytic behavior because it has no polls. But it still has, and this can, like I said, can be pretty attributed to transience terms it had, has branch cuts with respect to q tilde and w tilde which is the inverse of this q to the minus one. Okay, so let me just say what what this interpretation is so they are a couple of comments if you do this. It actually does diagonalize the book of matrix this is another way how you can describe it. It also has an interpretation so these, these variables also have an interpretation of physics interpretation. So this q was related to inserting. I mean adding with the q difference operator corresponds to inserting a Wilson line acting so there's a corresponding dual difference operator. It also corresponds to inserting a dual war text line, and then, in a certain dual symmetry. So what you're saying is that the non perturbed completion contains these, these, these dual war text lines as a background. And the other thing that there's a melon Barnes type of interpretation and if you describe a certain contour, which comes from a deepest this descent method. You also automatically collect these terms, unfortunately have time to talk about this so there's another way to derive this. Examples like just a single carl multiply which is the simplest, what you get is the, the Fadeff quantum dialogue and as for instance described in this paper by gaurib. And, and so maybe we actually hear more about this in your talk, because it turns out, if you look at the conifold the conifold this generating function they actually coincide for the conifold geometry. And it's not totally clear to us why this is the case with the topological string partition function, we have this little q corresponds to the genius parameter, but just for this case. And then it corresponds to the to the. I mean, so, so the completion that we got here is similar to the completion that more intense in his, his works as well so maybe let me stop. Well, we've had a lot of questions, but there's time for exactly one more before coffee. Sorry so you mean when we do so you come back to this counting problem. Yes. Okay, so, so yeah so this is one way how you can interpret it you can actually also. So this is kind of a land of Ginsburg kind of formulation in our first week, we had speculated a bit about this. What's the generalization of the holy water mirror. And they are there structure wise which things which look quite similar. So this is related to this melon Barnes type of story that I haven't really time to cover. So there's also similarly as as as in to the there are certain integral representations of these periods. And this is. So to say by some contour integral to rescue kind of prescription, another way how you can describe and this is maybe analog to the model, how you also can describe period. So if they arise really from this index, or from this localization perspective so you don't need to talk about these periods if you're just interested in this invariance but these, these generating functions, they will build this to different situation than you, then you can go from there and try to get the mirror. Okay, let's thank Hans once again. So, for 15, we have a talk by David's of arrow, it's on remote. So we have to be here on time.


Here on ThinkTek Hawaii, Dan Osterman for another, I usually say exceedingly exciting. This one probably not exciting, more on the profound side than the exciting side. I'm going to kind of pick up where we left off last week and talk about Hydrogen 101, except I'm going to start there, but by the end of the show today, I want to talk about Hydrogen today and give you actual proof that Hydrogen is no longer a dream, it's a reality, and it's going to become part of your life just so you might as well get ready for it, or if you're into investing, probably a good time to start thinking about stocks you want to invest in to deal with Hydrogen. To start off with I'd like to show a video that I didn't get to show last week on the show. I wanted to, because last week was about Hydrogen 101, and a little bit of basic training on Hydrogen, and it's a video I've shown a couple times on the show, and I really think it's important because it lays, especially for transportation, it lays some of the fundamental facts of Hydrogen that we'll be talking about today. So let's go ahead and roll the video. Hydrogen, the simplest element, and also the most abundant. Hydrogen makes up roughly 75% of all mass in the universe. Hydrogen also powers most of the stars in our universe, so it's only fitting that it has come to be recognized as a viable alternative energy source, and we need alternatives because fossil fuels are problematic. They're messy, dirty, expensive to obtain and not secure, and they're limited. Hydrogen, on the other hand, is everywhere. Hydrogen can be produced from a wide variety of sources, including water itself, using other renewable energies. That means it's clean, really clean. As a zero emission fuel source, the only byproducts are water, heat, and electricity. Easily transported, Hydrogen can be stored and distributed on a large scale as either gas or liquid. As a fuel, Hydrogen itself is very light. In fact, Hydrogen is 472 times more efficient by weight than lead acid batteries. And it isn't just for transportation. Hydrogen can also effectively produce and store energy for power grids. Hydrogen gas is transformed into energy within a fuel cell. As Hydrogen passes through a fuel cell, electrons are released and an electrical current is produced and captured for use. Electric vehicle motors powered by Hydrogen fuel cells are twice as efficient as gas or diesel engines. They can travel farther distances than lithium batteries, especially in heavy vehicles, and can last for decades. Hydrogen-powered fuel cells are scalable to buses and commercial fleets such as trucks, trains, ships, and aircraft. Fuel cells allow for fast, easy refueling, and Hydrogen can be easily adapted to current refueling stations, making it a convenient fuel source for everyone. It is a proven, safe, clean, and efficient energy source currently in use worldwide. Hydrogen is everywhere, including our clean energy future. So that's a really good review of Hydrogen in general. What I wanted to really talk about today is a summary of all of the things that make Hydrogen so really applicable. And I want to really focus on these as basically the facts and fundamentals that make Hydrogen or storing energy a superior product for the future. Now, what I've found over the last few years is that sometimes I feel like I'm screaming into a hurricane. I'm talking to politicians. I'm talking to business people. I'm talking to all kind of folks about the benefits of Hydrogen and they just kind of shake their hand and go, yeah, yeah, okay. But they can never get past that and they go back to what they're comfortable with. So I wanted to talk a little bit about what makes something a fad versus what makes it a real solid trend going forward. And how I know for a fact that Hydrogen is really starting to move into the reality where it used to be always kind of the hypothetical. Yeah, it seems a good idea. Yeah, it has some cool stuff, but it's just never going to make it. And I know it's making it now. I see it happening. I see it because I look for it. If you're not looking for the signs, you're probably never ever going to pick up on them. And just like other things where people go, gee, I wish I had thought of that first or gee, I wish I had bought stock in that company a long time ago. I'd be a millionaire by now. I'm telling you right now is the time to really start looking at Hydrogen for the signs that'll say it's here, it's going to make a difference, and it's here for good. So first of all, let's talk about some of the characteristics of Hydrogen that I've been pushing for a long time that are the solid truths, the things that don't change, the things that no matter what, these are always going to be there and give Hydrogen advantage. The first thing to recognize is, and I start off in hydrogen transportation, the first thing to recognize is that transportation, if we're going to go fossil free in the next 15, 20 years, you're going to have to go electric. I mean, makes sense, right? We're all looking at electric cars and hybrids and things like that. Well, a hydrogen fuel cell vehicle is an electric vehicle. So what you have to understand is that when you start moving towards electric transportation, whether it's a hydrogen fuel cell car or truck, or whether it's a battery plug-in vehicle, it's electric, and you got to charge it from somewhere. And that somewhere is the grid. And the grid is going to need to literally double in size to keep up with all the transportation requirements that we have. So let me just point out that when we start talking about ground transportation, cars, trucks, buses, trains, it's all going to be electric. And it's all going to run off either the grid directly or batteries or hydrogen, one of the three. And right now, I think we need to be looking at all three. But eventually, I can tell you hydrogen is going to win. And here's why. When it comes to transportation, it's all about weight. Everything is about weight, whether you're designing a car, a truck, or the space shuttle, or an airplane. When you have to deal with weight in airplanes, you end up dealing with drag, and you end up having to deal with all kinds of issues with carrying that weight around, costume or fuel. So number one, you've got a weight thing that hydrogen can't be beat. Hydrogen is 14 times lighter than air. So you can pack an awful lot of energy into a small space and a light compartment, if you're using hydrogen, hydrogen is most energy energy dense element that you can imagine. It's, it's like the video showed all the batteries for one tank of hydrogen. That's literally true. The other thing that's been interesting in the transportation sector is that a lot of people that are supporting electric vehicles, they look at hydrogen as competition for batteries. And so they actually start fighting the laws and the and things that are meant to help hydrogen, because they see it as competition. Most of what I've seen in the in the world is that if you're a hydrogen person, you actually like batteries there, you need them when you have a hydrogen system.  So there's always attention. And nowhere does that tension show up more than with Elon Musk, and his fuel cell test his cars versus fuel cells. The second area I want to talk about, oh, and also the cost of infrastructure and transportation, there's people don't think about this, but you think about the infrastructure for charging batteries. And everybody goes well, yeah, battery charging system costs X amount of dollars. Remember what I said at the beginning, when you start to go electric, it all runs off the grid. Right now, almost all of it. So if you're going to charge your Tesla, and the Tesla's being charged off a grid that runs on coal or oil, you're not really being clean. But also, your grid is going to have to double in size to start charging all these vehicles. What happens when the grid doubles in size? It's not just the charging station that you got to pay for. It's the substation. It's the transformers. It's the power line upgrades. It's all that infrastructure that also has to be paid for by somebody. And the somebody is the rate payer for the utility, or the person who owns the building that's got to be upgraded for the power. The second thing, because this leads right into it, is the grid. If we're all going gung-ho for clean energy, hydroelectric is very clean energy, and we have that, but we don't have enough of it. Nuclear is very clean energy, but most people don't have a tolerance for nuclear based on the Three Mile Island and Chernobyl and things like that. Nobody really wants to have nuclear now. Japan is basically getting rid of all their nuclear. But it's clean power. But once we're getting comfortable with it, and the ones that are getting really cheap and really ubiquitous are wind and solar, they're just great. But the one thing about electric energy that you have to understand is, you either have to, when you generate it, you have to either use it, or you store it. Because you can't just take, you know, sunshine, when the sun's shining, you have to take that energy while you have it. And if you can't use it all, you need to be storing it. So how do you store it? Well, there's, you can do pump hydro, pump water uphill, and then run it through turbines downhill. We talked about this stuff last week, so I'm not going to go over it again. But there's a lot of ways to store energy, flywheels, hydrogen, pump hydro, could stored air under pressure, all kinds of ways to do it. But you have to store it. And just think we're doubling the size of the grid or more just to accommodate transportation. And we're going to be getting all of our energy, or a lot of our energy from intermittent renewables like solar and wind. And you're going to have to store all that energy that right now we store in oil or gasoline, or coal or something else to keep it on the side until we need it. What happens if you don't have oil or coal or natural gas to store that energy? You're going to store it all in the battery? And you start looking at the cost of batteries as you scale up, you get to about 100 megawatts, and batteries do not pencil out. They're too expensive. They're totally being used inefficiently. And you have to replace the batteries when they start to wear out. And the batteries have hazmat issues, the batteries have material issues when you build them, that the materials come from foreign countries, and we don't have a lot of control over things like that. The batteries also have normal batteries. The technology we use now mostly is lithium cobalt technology. The cobalt, number one, there's only about 20 years of it left in the earth at the present rate for using it. And again, we're going to be all transportation and all. It's not going to last. So what happens when you run out of materials for your batteries, all of a sudden they're not getting cheaper, they're getting more expensive. They're getting mined by kids that are working in mines for 50 cents a day and things like that. You just can't do it. So when it comes to transportation and grid, they're interconnected. And the way to connect them the best is hydrogen, because that hydrogen can be used to store energy, to put back in stationary fuel cells, where you need power instead of running power lines, to charge cars and things like that. And it can be made in the same location off of, what it will take off of houses and those kinds of things. So transportation and the grid are going to be tied in the future. And the best storage medium for a number of reasons is going to end up being hydrogen. I end up talking to a lot of folks about the military, because we did all of our research work for the Air Force. And one of the contractors that we had, when I first met him and talked to him and he told me, you know, some of the great things about hydrogen from his perspective, I said, well, what does it mean to the military? He goes, it's clean and green. And I go, the military drops bombs on people and blows things up. They don't care about clean and green. They go in to make a mess. They'll do clean and green because the federal government tells them they have to spend money on that. But they really don't care. It does nothing for them in the field. What does hydrogen do for the military? Even the military didn't know that. And the things they thought they knew, they really didn't know well. So I had to explain it to them. Number one, the military needs stuff that's silent. Hydrogen fuel cells and that technology is nearly silent. Not like an internal combustion engine that's got mufflers and making all kinds of noise. I could tell you stories about sneaking up on buildings. So fuel cell vehicles versus trucks that are run off internal combustion engines. There's no comparison. Number two, fuel cells put off very little heat. They do put off heat, but not much and it's easy to conceal. When the enemy is looking for you at night, they look for hot spots. They take a thermal imager and they're looking for you with a heat source. Fuel cell vehicles don't make much heat, whether it's in a stationary site or whether it's in vehicles in a convoy. Hydrogen fuel cells don't put off any carbon monoxide or any noxious gases. So you can run 24 hours a day in a hangar and you don't hurt anybody at all. They can breathe it. It's just making water and heat. So it's not making any carbon monoxide or anything. You've also got the point that if you were going to have all of your support equipment running off of batteries in a combat cycle like we do in the military, 24 hours a day, seven days a week for week after week after week, you have to have three batteries for every piece of equipment. One that's running the equipment, one that's being charged, and one that's being cooled down after charging so it can be put back in the equipment. So you have to ship three times as much equipment in the form of batteries to run your equipment and that all translates into logistics costs and costs in an aircraft fuel and weight to ship all those things overseas. The next thing is safety. And if I say the word hydrogen to most people, the first thing they come back is Hindenburg or, and I apologize for the rooster, you just woke up. Hindenburg or hydrogen bomb. In reality, the space program has been using hydrogen not only for rocket fuel to get air rockets into space, but in making oxygen aboard spacecraft just like we do on submarine using an electrolyser that takes water and turns it into oxygen and hydrogen. We've been doing that for years, like half a century. And we've been shipping liquid hydrogen from where it's made in Canada to keep Canaveral on tanker cars. And have you ever heard a big story about a major hydrogen spill or hydrogen accident in the last 50 years? Quite frankly, I look for hydrogen accident news and it's hard to find. You can do searches and everything. You just can't find it. So I wanted to say one more thing. Safety is, is just, I could do a whole show about safety, but this is another one of the uses for hydrogen. This is actually a hydrogen cooker and that that silver puck in the middle of the thing with a nice flame coming out is an experimental burner that was built by Paul Pontio at Blue Planet Research and it does two things. It shows you, number one, how great a flame you can get for cooking out of hydrogen. But also, if you look up on Wikipedia, it'll tell you that hydrogen flames are invisible. Well, they're not invisible. They're invisible in daylight. But if you're indoors or if it's nighttime, guess what? It looks just like this. We're going to take a quick break now and when we come back, we're going to talk some more about some of the advantages that let me know or let us know that hydrogen is here for the long term. So this week, I got two emails. One from J. Fido, who's there to think tech, with a comment on one of my earlier shows. In fact, the show is over a year old that somebody sent in and he was bad mouthing hydrogen and said Elon Musk is a genius. Therefore, battery plug-in cars are the way to go and Elon Musk sends rockets into space and has underground tubes for transportation and all kind of cool stuff.  And I didn't even answer the email. I mean, usually if somebody sends in a question or something, I'll answer it.  Elon Musk is a brilliant man. He designed a beautiful car. He built an incredibly efficient factory with great workers. But I've got a couple things to tell you about Elon Musk. Number one, even the most brilliant people on the planet in history have screwed up and have made mistakes. Some of them have made huge mistakes.  But if you watch, you'll see them. In fact, one of the most recent ones, a guy was driving his Tesla. He went off the road and hit a tree and the impact made him unconscious.  He wanted to get the guy out of the car, except that the doors wouldn't open because the doors were locked and the only way he can lock him is to do it from the guy who runs the car who's unconscious in the car on fire. So the bystanders got to watch a guy get burned to death and to get pre-mated in his own vehicle. Now battery plug-in cars are great, but everything has its limitations. And I'm not saying that there would never be a hydrogen fire in a hydrogen car, but he is not perfect. He does amazing work. He does great things, but just because he's pooping hydrogen doesn't make him the rocket scientist that he has proven himself to be, which is he does amazing stuff, but he can be wrong. It happens that he's wrong. Hydrogen is going to outlast battery, outperform battery, until a newer, maybe digital, maybe some kind of cosmic thing we haven't even thought of yet comes along. And most battery people that I work with say that's at least 15 years away and hasn't even been proven yet. So we can't just keep looking at people like Elon Musk and saying, oh, he's so smart. He knows everything. We're just going to go with him. That's the mentality of the society we live in now, where we're looking for the quick solution, quick answer, and we go to the smarter person we can figure out, call a scientific, make a decision, and go. That's not doing your homework, and that's not doing it right. So I got a second email from a good friend of mine that you have on ThinkTech that does say to clean energy, and he sent me this thing called an article that was really an infomercial by some guy pushing stock, that he was a stock wizard. The first couple seconds on the infomercial I went, you know, this guy's talking about hydrogen, but he didn't even call it hydrogen. He called it blue gas, and there's no reason for that. When you don't want to have that, remember I said that when I say hydrogen people say Hindenburg or they say Hbomb. Well, if you say I've got a new name for hydrogen and you just start selling this new product, this guy called it blue gas. He calls hydrogen blue gas. Okay, you suddenly don't have the negative perception of hydrogen. So he's talking for 20 minutes about blue gas, and I'm going, he's talking about hydrogen. Every example he gives, they go, that's hydrogen. Even the pictures he's showing, the car is a hydrogen fuel cell car. The pump at the station he's fueling his car. It's a hydrogen fuel cell pump. It's a J2601 standard pump. Sure enough, after he gets finished selling his website or his blog or whatever, he says it's hydrogen, and he's talking about hydrogen. He's talking about how great it is. Well, let me tell you something. The time of hydrogen has come. In 2017, in January of 2017, it was a turning point in hydrogen. That's when a group got together. They decided to call themselves a hydrogen council, and they brought together the biggest companies that deal with energy and hydrogen.  And I've been streaming at legislatures and congressional staff and mayors and governors for 10 years about how hydrogen is so great. We could have put in a hydrogen station in Honolulu for between three and five million dollars, which is not chicken feet, but I couldn't get them to do anything. I got a legislature to float a bond for $1.25 million, and then my department head wouldn't let me put out an RFP for the bond because he couldn't back it up if it defaulted. So I get nowhere with politicians. But at the same time, the city of Honolulu had a three billion, that's with a B, not an M, three billion dollar rail project that is currently grown to a 10 billion dollar rail project that during this COVID virus has bankrupted because we no longer have the tax revenues coming in to pay our bills has literally bankrupted this county and the state in short order.  Now for a bunch of politicians who are screaming about global climate change and the existential threat to climate change and they're spending 10, 13 billion dollars on a rail project when five, six, eight, 10 years ago even farther back they could have started putting in the infrastructure for hydrogen and changed our whole economy to a energy self-sufficient economy. Don't talk to me about climate change. I mean that's as simple and critical as you can get. But the industry wasn't talking about climate change. They were solving climate. The industry has turned the corner. The industry has said we don't need subsidies, we don't need big companies to back us up, we need to make economic sense and we need to take off and do stuff. So I want to put up a chart. And this chart is kind of important. It's actually a stock chart. It's a one-year chart that shows from November of 2019 to November of 2020 the growth of a company. I won't tell you what company it is but that's one I really like. I've had the CEO on my show several times. But the thing to look at at this chart it doesn't matter what the company is. The thing to look at at this chart is how it's grown. On the left-hand vertical scale is dollars, zero at the bottom and I think it's 40 dollars at the top maybe 30 dollars at the top. If you look at any hydrogen company that's established you'll see it looks almost the same as this where for the last three to five years it's been flat like on the left side of that chart for the whole time. And you look at starting the middle of this year it just starts moving up and up and up and up. So on that chart if you bought that stock back in December you would have paid 350 a share, three dollars a share of that stock. On the right side that's like 27 dollars a share on the right-hand side. The stock went from three dollars a share to 27 dollars a share in six months. That's happening in almost every established hydrogen company that I know of and I could pull up five more charts that look just like this. The exception would be and it's the one that the guy commented on is Nikola Motors which by the way Nikola is run by another very sharp man like Elon Musk. He doesn't do rockets he just does big trucks and unfortunately when he was trying to market his equipment he did a little bit of buffoonery in a video and got caught and so there's a big scandal and his stock went from 20 dollars a share to almost 100 dollars a share then back down to 20 dollars a share and as of today it's back up over 30 dollars a share it's recovering. But the interesting thing about that company is even before he did the shenanigans with the video Toyota had actually run the same test and showed that that truck could outbeat a diesel truck in real time with no shenanigans. So all he was trying to do was reproduce a test done by Toyota with his own equipment and it wasn't really ready for market yet. But the real story is that that company has the backing of General Motors has the backing of a whole bunch of large large industrial investors and folks that make the equipment and that company is going to take off. The company is Nicola Motor just keep watching it. It has kind of a bumpy graph not like the one I showed but it's definitely moving the same direction all up. So don't let me be telling you I told you so a year from now or two years from now or 10 years from now. If you're serious about clean energy if you're serious about clean transportation if you're serious about clean grid start looking at hydrogen and start looking at the companies that do hydrogen and start watching their stock prices go up. And if you want to invest I'm not an inside trader I'm just telling you.  Great Thanksgiving. Hello.


The second term of the 2020-2021 academic year is set to resume following closure on October 16, 2020, due to COVID-19. Teachers and auxiliary staff of Schools Island-wide reported to their respective schools on Monday, January 4, 2021, in preparation for the return of students on January 11, 2021. Due to the community's spread of COVID-19, schools had remained closed with special allowances being made for the education sector to facilitate students of tertiary-level schools in medical and natural science programs that require laboratory and practical evaluations, students of secondary schools in Forms 4 and 5, who are preparing for regional and international academic evaluations, and primary school students in Grade 6, preparing for national and regional academic evaluations, particularly the common entrance exam. Cognizant that the health and safety of students and staff is paramount, the Department of Education, Innovation and Generations has been and continues to consult with stakeholders to ensure the mode of operation provides for the safest possible environment for all. The Department will further update the public on finalized plans for January 2021 in a subsequent communique. As primary and secondary schools prepare to reopen, the Sa'afa Lewis Community College says it will continue instruction on the virtual platform when it reopens on the 4th of January. College Communications Officer Natalie Jolie-Fanis explains, We have a population of close to 2,000 students. So the logistics of having students physically on the campus in a classroom space is not as easily manageable as in other schools where there is a set class room for a group of students. There are constant changes in movement with our classes, with our labs, etc. So we remain online. Our students, as you know, are more mature students. So we come back as of January 4th into our online mode. The Sa'afa Lewis Community College looks forward to prompt resumption of classes in January to catch up on course study in the previous semester. We had a later than normal start for the new semester. We started, I think, around the middle of September, whereas traditionally we would start either the last week in August, the first week in September. So our semester, that's another reason we really can't delay the break, our semester has been pushed into January. So when the students return, they will be completing semester one. Following exams to wrap up semester one, Jolie-Fanis says semester two will officially commence in February 2021 and run for its full 15 weeks, pushing later into May than usual.