Well first I want to thank the organizers for organizing this meeting and in fact running it for number of years as well as for inviting me here. I have to say that I'm myself really new to resurgence at least compared to all the other speakers. In fact 95 percent of what I'm going to tell you I learned from either Marcos or Maxim and I want to thank them for sharing their wisdom so that was very helpful. The only little piece which is new will consist of two concrete theorems that I mentioned toward the end of the talk and these two theorems come from two papers one of which is with Marcos and he's very bright student Pavel Putrov and another is from paper of Pavel and Kumernbaffa. So in this talk I'll try to make connections between two completely different subjects one which already appeared earlier this morning and the other one which has to do with modular forms so I'll try to show and illustrate how we can go back and forth between different subjects first of all by asking questions of one subject in the context of another how the links sometimes can provide the answers to questions sometimes quite surprising answers which are not expected if you're narrowly focused just on one particular topic it's a good illustration that bridging two fields may actually help to understand structure of things that we're trying to study and I'll also raise a lot of new questions by going back and forth and by thinking about this in a broader context will lead me to lots of questions that I'll now use as opportunity to ask during this meeting in fact if any of you happen to know the answers please get back to me and I'll be happy to learn more so like I said I'm new to this and I'll enjoy learning this even further so I'll start with a theory of modular forms and this will be somewhat pedagogical I'll soon get to the place where I want to be but I'll try to frame it in a more canonical context so that if you're interested just in resurgence and have less familiarity with modular forms this won't be too harsh on you so modular objects or modular forms specifically are defined as objects that transform nicely in this case under sl2z modular group so f is our function of variable tau and we require that as if we perform fractional linear transformation on tau f transforms in a nice way here a nice way means that it gets multiplied by c tau plus d to the power k k is usually called the weight of the modular form and this factor is a automorphic factor that again appears in theory of classical modular forms later in the talk I'll show you somewhat more abstract or different versions of modular transformations where you can modify this rule in various ways for example you can add additional terms and so on so depending on how you set up your story you'll get slightly different variants of modular objects called forms and used with other adjectives but this is this is the standard one so let's let's continue from here so our starting point is a function of variable tau and we want to characterize it I also should say that here I'm focusing on standard mobius group which is sl2z you can also look at various finite index subgroups or other modular groups so at least for sl2z the story is generated by two generators conventionally called s and t represented in matrix form as shown on the slide and one of them basically inverts tau is asking how function of tau is related to function of minus one over tau that's interesting that's in physics would be called strong weak coupling duality but t does something a little simpler it basically says that function of tau plus one is the same as function of tau and there is no contribution from this automorphi factor in front so this second fact that f of tau plus one is f of tau immediately tells us that it's a good idea to first of all exponent shape tau introduce q which is exponential of 2 pi i times tau and then this second rule will be implemented automatically also shown here is the fundamental domain and various translates our fundamental domain with respect to sl2z elements so for example shift by plus one takes fundamental domain here by minus one here and s does the inversion so we have this form of the cusp instead of going to infinity and so on so this is how standard theory of modular forms usually begins and just like I mentioned a moment ago the t generator that tau is supposed to be same as tau plus one tells us that we should we better work with variable q which is e to the pi i tau and in the rest of this talk it is this variable q which will be playing the central role or in other words I'll be asking all kinds of questions how q is related to various other types of variables that will appear and we'll be thinking about function of tau actually as a q series as a power series expansion on q with some coefficients a n which are priori are completely unconstrained they can be anything as long as this object satisfies this modularity requirement that under s transform or modular sl2z transforms it behaves in a way we wanted to to behave so that's where it starts so again if you open a textbook and start learning the subject from scratch you may ask what are the simple examples of modular forms let's try to construct them and a standard way to construct something which behaves nicely under the group is roughly speaking to sum over images with respect to the group or perhaps sum over lattice points so then you can pick different entity which you're trying to sum over lattice points in this case say lattice is of the form of complex numbers which are of the form m plus m tau where tau is complex variable m and n are integers and you can try to sum image of the rational function or it's it's a translate set that there's different points which basically gives you this eisenstein series so for each given k again remember k is the weight this defines eisenstein series and you can nicely write it in terms of q q expansion where sigmas or divisor functions these are sums of divisors to suitable powers for example sigma 2k minus 1 of n takes divisors of n positive divisors of n raises them to the power 2k minus 1 and adds them together so this is our first modular form or rather simple class of modular forms which you can massage a little bit further you can do one bear series resumption so on the right hand side we saw that in the q expansion coefficients are given by divisor function this sigma of n coefficient of q of n is sigma of n so standard lambert series resummation says that that also can be written in a much nicer form by introducing denominators which are also functions of q in fact in this you kind of get geometric series progressions which in the end resummon to these divisor functions so the first non-trivial example is eisenstein series e4 first of all we can quickly show that for odd values of the subscript you get zero because you're summing something and there are contributions with pluses and minuses so they cancel so first of all the subscript has to be even and then e2 is a very weird guy you may want from the start even before you start any analysis look at values of the subscript of this index greater than 2 simply for convergence reasons because you're summing over two-dimensional letters and of some of a rational function whose denominator is controlled by this index k so from the very beginning you may want to look at e4 as your first starting point and again just as we discussed on the previous slide this is how it looks like you can use this lambert series resummation to write it in very nice form where in the numerator you have n cubed q to the n and then you divide by this one minus qn factors so after spending some time working with don's a gear i learned my lesson that if you ever have to write a q series or any function it's always a good idea to write first few terms so from now on in my life every time i write any function or any series i always present first couple of terms you can't have negative powers of q uh no no no no uh never at least uh in in this subject so sometimes you may want to require the leading term the constant term to be zero in that case you get cusp modular forms and again you may impose further additional conditions which basically tell you how it behaves at the cusp that's very interesting that constrains the structure or modifies the structure a little bit but you definitely prefer not to go too far in the negative region at least in the traditional theory of modular forms so this first uh non-trivial example of heisenstein series e4 is quite quite nice and interesting again here is the first few terms of its q series expansion and like i mentioned earlier you can obtain modular objects by summing over uh lettuce and and this heisenstein series you can obtain by summing over lettuce in several different ways one is by summing rational function uh with power 2k in the denominator that's how we introduced it another one is actually as a theta function where you sum exponential terms over lettuce point and uh here in this case if you sum over root lettuce of e8 you get a theta function which is in fact the same heisenstein series so this can be obtained in a number of different ways and has many interesting properties so this is very concretely if you haven't seen modular forms in action before that's that's how it looks like here is the next one e6 as i say only even values of the index or argument give you something non-trivial so lembert reclamation gave us that formula again following dawn's advice i write for you first few coefficients explicitly and here below in the plot i show how the real part of this function behaves it's very nice inside the unit disk and it has interesting behavior as we approach the boundary of the unit disk and later in the talk this um a phenomenon of approaching the boundary of the unit disk will play an important role so not exactly the way uh it's shown in a picture but in a closely in a closely related way so that's um our first uh or first two examples uh we could try to go further and look at e8 and so on but one interesting fact about uh modular forms at least traditional modular forms and presenting for you is that the condition that we imposed from the beginning that object f under modular transformation is some factor times f of tau is multiplicative and as a result if you'd have two objects we transform very nicely with respect to modular group you can multiply them so they form a ring and in this case it turns out that the two guys that already showed for you e4 and e6 in fact generate the ring so uh the ring of modular forms is freely generated by these two fellows which means if i now start computing e8 it will be expressed as e4 squared e10 will be basically e4 times e6 and so on i have to get to weight 12 until i start seeing something more than one dimensional um space of of such objects by the way did you notice something else about examples i just showed you integers exactly so their coefficients are integer and this is kind of interesting because the original way we define modular form did not require this integrality all we had is requirement that it was a power series and q that already used the t generator and then this power series and q should have had nice property with respect to s generator which inverts tau but that did not a priority had to give anything with integer coefficients so it's rather peculiar that the answer turns out and in fact in many cases in the theory of all kinds of modular forms the coefficients come out to be integers so natural question to ask is what is the nature of this integer coefficients are they counting something usually in mathematics every time you see an integer number it's counting something so a natural question is what is it counting in in our case the more modern version of the question is if you see a number integer number you can ask does it represent dimension of some vector space or maybe some cohomology theory and in fact already from our discussion we see some hints for it that our integers are not necessarily positive or negative even the two example i showed you indicate that there is no positivity condition so they can be just dimensions of something or counting something our numbers even if they have such interpretation are likely to be of the form of an index where we sum various dimensions with plus minus coefficients so therefore what could possibly be is and that's a big question mark whether our q series expansion can be written as a graded earlier characteristic of some vector spaces which are doubly graded m determines the sign and n determines the power of q so as i mentioned earlier in this talk i'll try to ask lots of questions to which i would love to know answers and maybe on this board i'll summarize my questions these are real questions for you and i don't want to go astray and and use much time of the talk for this but especially after please tell me if you have really good answers or good ideas and my first question is even in the context of modular form is there cohomology theory which is doubly graded which provides which makes this equality true so the fancy name for this finding such spaces is called categorification so anywhere in life where you see a q-series and if it has integer coefficients it's a good habit to ask this kind of question and again modular forms provide lots of examples of q expansion with integer coefficients so therefore i want to ask is there cohomology theory which really makes it true so it is a written index in supersymmetry context in some cases it could be realized as not quite written index but elliptic genus so it is an index but again i don't really know again if you happen to know the answer please tell me i would love to hear i i don't know i honestly i don't know in mathematics the closest i could find is et alcohomology but that's not exactly answering this question for example i cannot just say that h is there are et alcohomology groups unfortunately not so if again you know the real version of of this answer please please let me know i'd be very curious and again i'll try to ask analogous questions as we go so with this frame of mind what are these integer coefficients counting can we realize them as graded dimensions of of cohomology groups of actor spaces let's continue and in one class of examples of modular forms i will give you affirmative answer where the answer will be crystal clear and in fact not just it it's not just exists it's actually useful for people in a completely different area so in that case there will be answer and that makes me hopeful that more generally that can be true so continuing with the crash course on modular forms we should go a little bit further and like i mentioned to you earlier the next interesting thing appears if we start going increasing weight of modular form in weight 12 so first we introduce dedicated eta function which is given by that prototype product of 1 minus q to the n and this function by itself is already a pretty cool function first of all it has a q series expansion as a sum and and it has this theta like expansion with coefficients which i denote chi n these are characters of something of conductor 12 that is not terribly important but the same coefficients will appear later in one of the following slides but already this function itself is counting something cool if you take that infinite product forget about the pre-factor q to the 1 over 24 and just invert it then you see that you have one divided by bunch of geometric series progressions or factors which can be expanded in geometric series and that expansion will have positive coefficients and those positive coefficients in q expansion if you invert this infinite product is actually counting two-dimensional young tableaus or 2d partitions which plays very important role in many subjects from random matrix theory to modern gauge theory now in a course of modular forms that it can either function as interesting but what's even more interesting is its power 24 which gives you function delta called the modular discriminant and this is modular form of way 12 this is not so hard to show by taking logarithmic derivative of delta of tau and like I told you before everything should be generated by our ring of modular forms is generated by e4 and e6 so delta should be expressible in terms of e4 and e6 in fact there are exactly two non trivial well the space of way 12 modular forms is two-dimensional so therefore everything should be combination of e4 cubed and e6 squared and delta is such a combination given by this expression here so you can multiply it out to get the first two terms as Don Segir taught me to do for every function so to understand the meaning of this integer coefficients there is actually an answer in theory of classical form modular forms of course but it's not in the form I wanted it's not in the form of this categorification where you take coefficient and write it as graded holi characteristic it's much more contrived and not satisfying for for our purposes but there is an answer and as an honest guy I'll tell you what the answer is so the answer goes via borrell transform and if I were in Mil Borrell I'll definitely write reference complaint it could go something like this hey there is in a closely related context a very similar transform called borrell transform that you saw earlier this morning which basically does this so given a function f of x you can produce its in this case in number theory usually called melin transform for whatever reason transform which is a function of variable s and I want to point out that it is indeed closely related to borrell transform of earlier this talk and much of what's what you're going to see during this day if you write x in uh exponentiated form if x is exponential of some other variable then basically what this does is precisely the borrell transform which you see in resurgence a lot of time so that's already a very interesting nice connection to the borrell transform and integral also goes over a in this case goes from zero to infinity and so on so we apply this version of the transform melin transform to our modular form to obtain a function which again in the context of resurgence would be some version of borrell transform that encodes some information about integrality or gives partial answer to this integrality that I wanted and what we get is a function and a number theory just like in resurgence it's a good habit to ask about analytic properties of this function it's it's a natural to ask where does it extend in a complex plane where the singularities are what are the poles what are the zeros residues and so on so this transform in this case specifically gives l function and there are several versions of l function one I denote by lambda and the other by l related by this gamma factor lambda is more natural if you want to ask analytic questions about extending this to complex plane and then in fact if you apply this melin transform to our class of modular forms or closely related objects that we introduced before you're going to find that it's an entire function and has two poles one at s equals zero the other is at s equals k where k is the weight it usually has some symmetry where you exchange the two and the coefficient of this the residue of this poles the coefficient vanishes if the leading term in the q-series expansion vanishes for example for cos forms this would really be entire function so for analytic number theory this lambda is more natural for algebraic number theory where we ask about integrality this l object defined by a prefactor gamma is is more natural l is the Dirichlet l function of a modular form a modular object f and you can define it more directly simply by taking coefficients of the q expansion of our original modular form which was sum over n a and q to the power n and dividing them by n to the s so formally this defines some function of s and then again you ask same same type of question what kind of function is it what is it analytic structure that I briefly mentioned on the previous slide and so on so in number theory there is an answer to to to integrality of these coefficients and namely they're counting something they're counting points on varieties over finite fields so first of all you take that l function you write it as product over all primes up to some finite set which may be related to a discriminant of your variety or maybe empty set in fact and you get something like this you have universal factor for for each prime and again weight k appears over there and when k is equal to two in a special case of weight two what happens is that number of points um sorry this coefficients that appear there this ap's these are basically the same coefficients of our original series except that we're taking them at prime value of the index n they're counting number of points on elliptic curves over fields finite fields fp with p elements so ap is often called p defect because naturally you would expect that roughly speaking number of points would be same as number of points in this field plus one which is p plus one and a describes how much your answer is really different so the integrality of coefficients a at least uh for prime values of p and then multiplicatively for others are related to numbers of integer points over elliptic curves so this is a classical result which originally was uh shimura to nyama conjecture and uh then rediscovered by antivay and uh finally made into mathematical theorem by andrew wiles and many of his collaborators followers so this is pretty beautiful subject and it's interesting that the people involved had very interesting character or life so each of these guys is a quite interesting character unfortunately uh teniyama died very young so he committed suicide uh i didn't know this but uh preparing for the talk i checked uh the history of this on wikipedia and discovered that actually his fiancé also committed suicide shortly after and that's actually a very touching story of true love so that's that's quite interesting uh goroshamur is also an interesting character and um at keltec we had a visiting committee earlier this week before i came here and um digross told me about interesting discovery that goroshamura made in princeton so he spent much of his life and how big part of his life in princeton and uh princeton math building is very boring it's like a tower with many floors about 10 or 15 of them and um it's it's uncharacteristic building all floors are identical it's it's a very boring looking structure so princeton students made a very nice prank one day so they basically switched buttons one two three four and so on in the elevator so if you press button seven you would go to fifth floor for example so what else they did they also switched the office labels accordingly so you come out of the elevator and you go to all halls exactly identical so you go to your office and um uh the office in the same place it has your label but then your key doesn't work so this morning uh security on campus got lots of calls from people that that their keys do not work and goroshamur was the one to make a discovery that actually the levels were different simply because he decided to leave his office and take the stairs down and then he suddenly realizes that he has to go many flights many more than he would normally do so he made many interesting discoveries if you so here uh the uh while's theorem formerly known as shimurutin yama conjecture is about way too modular forms which makes a correspondence uh to elliptic curves over uh finite fields but uh in in higher weight and in other variants of modular forms this is part of modularity program uh where you're not necessarily counting integer points of the full variety itself but of its certain piece called motif and uh that's an ongoing interesting research problem so continuing with our crash course on modular forms uh in way 12 as I told you before there are only two non-trivial generators so if you already have our modular discriminant delta and say e4 to the cube you can take the ratio and this object should have no automorphy factor at all so in fact it should be modular function not modular form it should transform without any automorphy factor and that's how it looks like and curious thing since I emphasize counting a lot this object also counts something its coefficients are also integers and it counts dimensions of representations of the monster group the uh maximal sporadic uh the the largest sporadic group so there are lots of connections between this coefficients and counting of things but again they appear as random isolated uh facts and what I would love to see if there is an answer to this question can reconstruct them as index of something so finally for purposes of my talk I want to introduce one last ingredient which also has very nice story or character attached to it namely character of Ramanujan last year there was a very nice movie I encourage those of you who haven't seen it to to go and see it it's also very touching story unfortunately just like Teniyama Ramanujan died very young uh in his early 30s and it's it's very sad story but it's also very touching story and uh another touching fact about it is that he introduced yet another class of modular objects which at that time he didn't even think about modularity he was thinking of q-series expansion and q-series properties so for example a power series like that uh summing over m with q to various powers determined by m is is a uh one of the expressions that that he wrote for from peace point of view and now this class of objects goes by the name of mock modular forms and again there are various interesting subcategories and uh these are modular objects of 21st century I probably should say even though they're quite old their properties and structure we're still about to discover so that's that's the last class I want to mention and then I want to mention interesting intriguing property of this class which actually has a very old history Don Zagir calls it strange identity and it involves uh an object which is also uh close causing of uh this mock modular forms in fact a special case of it that I showed in the previous slide and what's strange about this identity is that the two sides of this identity are never well defined at the same time the left hand side written as a sum of one minus q times one minus q to square and so on is only defined if q is a root of unity and in fact uh was uh Don was inspired by talk that Maxime gave exactly 20 years ago it was 1997 so now we should celebrate 20th birthday of of this expression and in fact Maxime actually was looking for analytic continuation of written Rishi Tichin tribe and parents so I'm sure that somewhere in the back of his mind uh it was closely related to theme of this entire discussion we have today that uh in particular the rest of my talk so this side of the equation is only defined as roots of unity so if you picture the unit disk so this the left hand side only makes sense where this sum terminates and these are precisely roots of unity points on the unit the right hand side on the other hand is nice q series expansion in fact it's roughly speaking half derivative of the datacind eta function so coefficients chi are exactly the same as on the previous slide and if you roughly differentiate uh datacind eta function which was also theta function like this it didn't have n in front and the sum so if you differentiate datacind eta function with respect to tau you'll pull down n squared and that's what half derivative refers to if you pull down just n and not n squared by by differentiating with respect to tau you get this sort of expression so it's very much like datacind eta sum where I stick n times the rest of the q series and this is perfectly defined inside the unit disk so right hand side is defined inside the unit disk when absolute value of q is strictly less than one and left hand side is defined on the unit disk but of course what relates this guy is is the property that if you take q to be so this is q plane if you take q from inside the unit disk and ask about the limit as you go to root of unity you recover the left hand side so that's what this identity means and that's formal expressions so what it can make kind of I think also formal expansion near any root of one the left hand side and the right hand side as well um probably yes exactly yes yeah exactly yes yes yes but that's right yeah so the the limits and their expansions coincide in uh in the vicinity of root of unity so that's what the strange identity means and my goal in the second half of the talk is present to a context where first of all the strange identity is not strange at all where it's natural and it happens a lot and also answer the other question about a role of integer coefficients the function when it's very discontinuous on the on the disk or on the on the subject because it's only for a fraction that you can make some sense of it so the left hand side is only for fraction meaning on it's yeah it's very discontinuous it's only defined at at roots of unity and yeah but this this is defined inside so what happens if you reach any other point well again it's it's pretty badly divergent I would say I don't know what happens because if if it's not the root of unity maybe I don't know that's a good question at least I don't know that okay so the second half of the talk is going to be completely different subject so and that's where like I say making connections and bridges is natural in fact the question about modular forms that I ask about interpreting the coefficients as index of something actually comes from this other field of mathematics that has to do with three manifolds and low dimensional topology so there my question and I'll also pose this as a question is how to associate a q-series to a three manifold so that's that's my second question and I don't really know the answer so if you know anyway how you associate q-series to a three manifold of course for this you probably would be low-dimensional topologists please tell me because the answer is either going to be related to what I'm going to tell you next or it's going to be something new and something cool okay and I would be really interested I'm not joking I'm I'm really interested then in this field we can also ask analogous question if you have a q-series just like we had in the previous the first half of the talk it will be expansion with respect to q if it happens to have integer coefficients it's also natural to ask and people in this field ask a lot this kind of question what are the vector spaces that have the graded vector spaces whose graded dimension this integer coefficients are encoding so for from a viewpoint of low-dimensional topology this is a starting point for asking more interesting question about vector spaces homology but first you have to have a q-series with integer coefficients and again I actually don't know how to associate q-series to a three manifold or other objects in low-dimensional topology so I'll be extremely interested to hear so one standard variant of three manifolds which is very powerful is Wittner-Russey-Seeking-Pryven variant it's computed by Sharon Simon's path integral exactly as Marcus explained in his morning talk and it's a path integral over gauge connection A whose exponent is k times Sharon Simon's action S is the Sharon Simon's functional that Marcus wrote on the blackboard and k is the level integer that has to be integer because the integral needs to be well defined so in this story as well as in many other quantum field theories you can start Feynman diagram expansion which is basically saddle point expansion of the integral this is something that physicists do and love and that's precisely a good starting point for resurgence because you obtain perturbative expansion or saddle point approximation to the integral you have vastly growing number of diagrams in fact this is stolen from paper of Marcus where this table and examples of diagrams show you the number with different number of loops r is the number of loops maybe minus one and d is the space of such diagrams so like I told you in the beginning both Marcus and Maxim influenced a lot of what's going on here so again I want to thank them for sharing that wisdom and hopefully not blaming me for stealing the pictures from the papers so like Marcus said we can start doing perturbative expansion to the of this path integral and try to approach it with resurgence theory so in this context we'll use two versions of resurgence theory one is roughly 200 years old goes to Picard left shits talks and so on and unfortunately it will not be enough to treat some of the questions that arise in your assignments theory and its analytic continuation so we'll have we'll have to resort to resurgence of Picard left shits theory which no pun intended is called resurgence so that's only 20 plus something years old and again there are many key players here in France so this will be version 2.0 of Picard left shits theory that will be able to track certain technical difficulties that that we're going to see in in this analysis if you are naively applying Picard left shits theory to your assignments path integral you expect roughly the following you expect that if you complexify your variables or fields in this case you naturally will get expansion where you have subtle points that are various flat connections and again if you complexify this will be complex flat connections on a three manifold m3 and for each one of them you have this kind of trans serious expansion where you have exponential factor that I made very explicit is basically trans simons functional of a given flat connection times perturbative serious expansion in 1 over k so for purposes of this talk I use several variables so q is exponential of h bar and this is same as 2 pi i divided by level k so the small parameter is roughly 1 over k okay so so that that perturbation over entry is real dimension yes this is three-dimensional manifold critical points or subtle points of this action functional are precisely flat connections and m is the set of flat connections but do you say complex how do you come because in resurgent analysis as well as in per car left shits theory natural thing is to complexify the original variables so over the three manifold you have some right connections but connections are real well originally we had g bundle over three manifold m3 and then naturally we passed to gc bundle basically meaning that if you had connection a you now which had some reality condition for example it was hermitian anti hermitian now you make it a complex valued and declare that all complex values of its components are allowed so just formally complexify because doing this may help you find subtle points which are not necessarily on a real axis or or and then they still may have important contributions so you formally complexify everything you had before and then it decompose right so i should probably not see okay so there are two versions or two ways to look at such integrals using subtle point approximations one is where given each critical point which i denote by alpha here you associate with all trajectories of steepest descent that marcos mentioned earlier and they usually sweep out some surface called left shift symbol so that's a picture of a left shift symbol and then you can project this left shift symbol onto what will be called borrel plane this is basically projection by your action functional so s of a projects you to complex plane so values of s is precisely the what i'm going to call borrel plane later so the thing is so then what you get at least naively you would expect that your with nourishing thrive invariant in this case has expansion of this form where this is perturbative series around each subtle point some over settled points with some coefficients and alpha which are related to what marcos called c and determine the choice of the control the problem is and this is one interesting thing where picard left is not going to be enough is that in turn simon's theory each flat connection has infinitely many lifts and that's a delicate thing so the space of all gauge connections modular gauge transformations is not simply connected you may go on a closed loop kind of like this and each alpha has infinitely many copies that's related to the fact that in four dimensions we have instant on number and that's also related to the fact that turn simon's of alpha is the same as turn simon's of alpha plus any integer namely that turn simon's functional itself is only defined mod integers so what happens then is that instead of one subtle point you have infinitely many towers in infinite towers of subtle points so that's already kind of tricky and that's why I introduced here this notation bold alpha which denotes the lift of your subtle point into into this tower this is interesting thing and I should I think I should have interesting applications to other problems of trigonometric or hyperbolic nature so if you have two difference equations if you have trigonometric or hyperbolic integrable system you'll have similar phenomenon you'll have sometimes this infinitely many lifts of a single subtle point if your equations are written in trigonometric form so that's that's because that means that now we are working on the universal cover so instead of space of all gauge connections I have to introduce the universal cover and z tells me on which sheet of the universal cover I am because right so theta will appear in a second and this bold phase alpha is again the lift so this basically says that if you had if you naively thought that your manifold had say three flat connections now what I'm telling you is that it has to have infinitely many copies but ends should be such contributions of all of them of all this tower should be such that each individual and alpha and bold phase alpha sum up into what we used to call an alpha so an alpha is a contribution of this connection alpha and bold phase alpha means means its integral lift so that's that's first delicate point that you encounter again on gauge theory trigonometric hyperbolic integrable systems things like that Q difference equation your gauge group is here sl2c and you're descending down to su2 oh here here I was saying that okay sorry that's that's a good question indeed so here this last part of the equation says that originally this written Rishi Hinton drive invariant is defined by su2 gauge theory so our contour is such that all real points should contribute with coefficient one at least that's what we well that that's that's basically part of the choice of the constant c that Marcus was telling us in the first part of the talk so it's part of its additional data it basically tells you how how to set up your problem and the problem is set up in such way that you want the coefficients to be one if alpha is as you do flat connection and zero if it's not so that that's the definition if you wish it's part of the definition of how we want to define trans-heimans path integral we want to define it by integrating over real slice in this complexified space of fields so that that's basically the second exactly and yeah I'm that's that's what I believe I say on the next slide indeed it's not well defined and this left shift symbols exhibit the car left shifts monodrome is so if you start there in parameters such as h bar or q or other parameters you will have stocks phenomena and stocks phenomena are basically saying that if you go from one configuration to the other where I just draw for your projection on the borrel plane what happens is that a given contour gamma receives contributions from other contours and their intersection numbers will determine for you how what's what's the coefficient so that's another interesting phenomenon I want to emphasize in a finite dimensional problem the stocks coefficients or at least in picard left shift theory these coefficients are integers because they're intersection numbers of some cycles left its symbol and left its anti-symbol so what happened there may be zero cycles and you can you are integrating over the zero circle for the cycles which gives you zero so you must specify the cycles in usual picard left shift theory this cycles are mid-dimensional in the space of complex fields so their intersection number is again if you're in a nice situation it's just the number it's integer because it counts the intersection points of two mid-dimensional cycles so it's just an integer but I want to emphasize that here because of the first issue that we had infinitely many subtle points what's going to happen is that effectively this numbers will become fractional and that's a pretty cool phenomenon because we'll have some of integers some of say plus minus ones and they will be they'll need to be regularized resummed and as a result you get some interesting fractions so this is another tricky point which usual picard left shift theory cannot handle but resurgence does extremely well if it's very clever so that's I think where I'm heading next yeah so you want to treat this kind of problem using resurgent analysis which projects everything to the borrel plane you don't have to think about this infinitely many copies in the space of fields it will do it for you by itself even if you are very naive and but if you care about what happens in the space of all the variables you can do that too but it will take care of it and very naively what you do is the following you take your perturbative series such as this h bar expansion you divide by our factorial which are this our loop coefficients exactly as marcus explained in his talk and produce borrel some bz of psi and then just like a number theory you ask about analytic structure of this function you don't just do the summation you ask does this have analytic continuation to complex plane and if so what are the singularities what are the residues very much like what we did for l functions now let's see so this picture is illustration of borrel resumption where you only sum not up to infinity but just take first couple of terms and this is what you can easily do with mathematics if you're like me and don't want to spend too much time on either thinking about abstract argument or even in mathematics you don't have to have high power marcus is extremely good at this i know only how to do say five terms or three terms it's quite amazing that this actually is very nicely behaved and pretty fast convergent at least in good examples so this is done with very poor accuracy but you already can start seeing that something is probably happening on imaginary axis that you may have some singularities this is supposed to be profile plot for for this borrel sum indeed what happens you have poles on the imaginary axis and again i want to emphasize with the slide that what you have to care about to keep in mind is what are the singularities on the borrel plane and what are the residues that basically at least for me it tells what resurgence is supposed to produce and encodes the most essential data about the problem so it's like a snapshot like a photograph of a borrel plane that that says a lot so in this case again coming back to our example you perform lateral borrel summation and because everything was on a all the singularities on an imaginary axis you can deform contour and basically sum of their residues but the only thing that's going to happen is you're summing over infinitely many sedals so in the carolafshad theory it's very hard to develop the homology theory or any sort of homological machinery which deals with this infinite number of intersections and so on but in the borrel plane that's very easy we just have infinite sum of residues and that's not a big deal and it produces for you some final answer in the form again that Marcus explained and here I think this quote by Hadamard is very appropriate especially in this occasion and right so the two key ingredients are the coefficients and alpha beta which tell us how corrections to saddle point alpha will come from other saddle points these are called trans serious coefficients and then position of singularities on a borrel plane determines for you the exponentials this is precisely what Marcus explained in his introductory talk so since he already spent quite a bit of time on exponentials I'll emphasize for you the structure of these trans serious coefficients and alpha beta as I already mentioned they come at least in this kind of problem or in any other trigonometric or hyperbolic problem from infinite sums and you get sums of plus minus ones which are basically the intersection numbers the only thing you have to sum infinitely many times plus minus ones you can do it using zeta function irregularization you get this kind of expression which essentially then boils down to some fractions okay and that's a very interesting phenomenon which in this context leads to to a miracle that I'm going to summarize for you in the form of a theorem so this theorem comes from paper of myself Marcus and Pavel and it says something interesting so if you have in the trans-timeless theory you have different flat connections which have different stabilizers so part of the gauge group may still fix your flat connection and if that happens the flat connection is called reducible so it turns out if you think about it from the viewpoint of Picard leftist theory or resurgence theory that reducible flat connections are such that they can receive contribution from non-Abelian ones as trans series but not the other way around so there is directionality so irreducible and reducible are not the same at all and again part of this has to do with the fact that they have stabilizers so that's an interesting theorem and a consequence of this theorem is something cool it says that you can decompose your Wittenrushy-Tiching-Triven variant into some of blocks called them basic classes for instance by analogy with Donaldson theory how it decomposes into cyberquid and variants basic blocks or basic classes labeled just by Abelian flat connections and it turns out that these guys do have nice analytic continuation inside the unit disk even though original fellow doesn't again I have no idea how to analytically continue Wittenrushy-Tiching-Triven variant from roots of unity so that means reducible you yes in sorry I'm talking about SU2 groups so for me reducible Abelian are the same but in general you're absolutely right reducible so Wittenrushy-Tiching-Triven variant has only defined at roots of unity and the big question on this field is how to give a Q series again I don't know the answer to this question but what I do know is that to a given three manifold you can associate not just one but collection of Q series expansions labeled by Abelian or reducible connections each one is well defined in the unit disk and they do have exactly this property that if you approach roots of unity from inside the unit disk you get the corresponding value all the complex that went away right these are all unitary connections these are unitary connections no no no this this is complex this is complex with a point that if K if Q approaches root of unity they have to reproduce Wittenrushy-Tiching-Triven which is the real slice so I meant the connections are flat connections are unitary connections of complex in this discussion in principle uh if you're if you're inside the unit disk if Q is complex any any Q with less than one norm then all contribute all all flat connections real and complex contribute to this expression so all of them so but from the point of view partition function we are calculating partition function of Church-Simons on M3 of unitary and Simons or unitary but then it decomposes into some of all possible contributions correct and it's a billionized in a very funny way so again it's a billionized in a way where you take um so maybe that that's another question I wanted to raise where you have I can't think about z a z with a subscript a m3 as some kind of a billion partition function but you're not sure I can't think of it you can but I don't know how to do it in terms of Simons theory in using some string theory language you can so I feel that that this may be getting a little abstract let me see a show of hands I'm going to finish in five minutes don't worry uh how many of you are happy with Picard-Lafcher's theory just to make sure that I'm not losing everybody okay just Picard-Lafcher's theory okay how many of you are happy with Picard-Lafcher's theory with infinitely many Picard-Lafcher's monodrome is happening at the same time okay not so bad so for the rest of you uh that's the luxury of the physics talk okay I have a demonstration so I just want to explain what's going on here okay so we have a theory where we have infinitely many critical points subtle points and let me represent each subtle point so by by bin so this is going to be um reducible flat connection this is going to be irreducible flat connection and that's another reducible guy so to make it easier I'll denote irreducible ones non-abilian flat connections by red uh crystal balls there are infinitely many copies of each flat connection each subtle point because of this lift to universal cover so I represent them by infinitely many stones red stones sitting here so that's going to be reducible so middle one will be irreducible you have infinitely many copies because of this infinite cover issue or hyperbolic nature of the problem of a subtle point and this is going to be another this is going to be another uh non-abilian sorry abelian flat connection so reds are abelian white is non-abilian so what happens in this process of Borel resumption or resurgence is that this guy gets distributed among the abelian ones so the contribution is still there just like it was in the written Rishi-Tihin-Triven brain written Rishi-Tihin-Triven brain sums all of them with a coefficient one but what happens here is that first of all you have infinitely many copies that's the first phenomenon and second is that because of the Picorilevschitz monodrome is that a plus minus one summed infinitely many times this guy non-abilian fellow gets distributed with coefficients say one third and two thirds so what I should do to produce to produce this object z a on the right hand side I should redistribute so these are now zas and they do have everything we had before where total now is divided just into reducible fellows okay so hopefully this will clarify for you what Picorilevschitz theory with infinitely many leftist symbols means so and maybe as a question and I think I should probably conclude here I want to ask for other phenomena maybe a number theory where the following happens you have some entity which is total and for us this was total written Rishi-Tihin-Triven brain and on my example it was contribution of reducible irreducible plus another reducible and the middle guy gets redistributed into these two with some coefficients say one third two thirds such that the total sum is still there and I want to ask if you know of any other phenomena for example a number theory or elsewhere where some objects of slightly different nature there should be some analog of reducible irreducible maybe a billion non-abilian get redistributed in very similar way for example I could find L packets in the length lines correspondence being somewhat analogous where again you have certain representations and you package them together you group them in classes with certain coefficients which are kind of analogous to this trans series coefficients but I would love to hear of more more examples so finally I don't want to take more of your time I want to point out that once you start with this perturbative series expansion and one over k in trans simons theory you do this clever borrower summation resurgence is very smart it takes care of all this pole summations and so on you produce this object z a and these guys now have first of all by definition because we were summing over steepest descent contours they're well defined inside the unit disc so this is actually q series and second of all they turn out to have integer coefficients so some magic happens where for any three manifold in this trans simons theory not only you continue your written recitation drive invariant away from the roots of unity inside the unit disc for each of these basic blocks z a they start having integer coefficients as functions of q so that's to me magic I have no explanation of this from mathematical point of view physics suggests why it should be the case but that's that's at least the result so it turns out that these are precisely mock modular forms that we saw on the previous slide at least on the context of cypher manifolds probably not in general but at least there okay I think my time is up so I should so they are basically yeah so there is a question of notation in our paper there are z a's and z a hats and there are simple linear combinations of each other so I promised you two theorems and I managed in my time to get only to the first second theorem would tell you which linear combinations have integrality but that's that's I would say slightly simpler issue regarding the categorization of q series the first question so I as far as I recall there is a paper by specter he actually used this written index and derived zeta function as some measure of some physical system something so I think this kind of configuration exists in the physics literature then I think allen cones and barnel barnel number theorists so he also has similar things but derived from the specters specter paper things I think specter of two papers one is halfly broken on half the broken super symmetry and it is one is and what is the q series network I think he constructed something out of that means I don't require so for something information as well I would love to have the reference and again for some isolated things of course this is easy I hope my question is how general would that be because so here for example this is for class of mock modular forms I would love to have it for some big classes not just one single example but anyway I'll take over this for the zeta function you say somehow related to some green silver foliar construction so okay thank you very very useful so explain why the theorem one is true okay so theorem one is true for the following reason again that should be also a good lesson for resurgence because so what happens is you have subtle points one and and irreducible guys are non-degenerate subtle points reducible ones are very degenerate because you have further action by stabilizer of the group so it's they're actually sticky so what we're dealing with we're dealing with the car left shits monodrome is or crossing types of singularities were so you're saying it's not a one it's not a one no not at all right so from singularity theory point of view and in terms of this is unavoidable because trivial flat connection a equals zero is always a flat connection and that's highly reducible it's all of the group G stabilizes that guy so in terms of this theory you cannot have a situation where all your critical points are nice more so even more spot you're in a situation where there is this big degeneracy and what's happening is basically the denominator the quotient by additional stabilizer shifts the more syntax in such a way that leftist symbols if you think about it from picard leftist monodrome point of view don't even see each other so they there are in a different grading or different dimension so from more theory point of view there are in their own position not mid-dimensional anymore so that's the difference between reducible or reducible so as a smoking gun for a surgeons or quantum field theory or anything else that you may be interested in if your singularities are not just poles or not just simple singularities and picard leftist theory there is some degeneracy you should expect phenomena like this where there will be directionality so in this case again reducible can pick up irreducible as pieces but not the other way around it's all about singularity is independent of m3 right whatever yeah yeah yeah it's it's independent of m3 and in fact it's independent of this infinitely many copies it's just about behavior of picard leftist symbols for degenerate subtle points so and unfortunately they're completely unavoidable insurance items theory so we had to deal with that so the being of the talk you ask question one about some commercial origin of as a stensiles they know there's very some as a stensiles that gives you as a stensiles of the form that you wrote the leftist sum typically it comes with a character in the numerator and it's associated with cycles so for example you can get a version of the e4 series with some character from three cycles on prego varieties it's it's called venison construction and from that you I mean you can lift up by computing summation or something that is induced by the I mean the homology or the homology of the of the the main one I see so there is some hmn some doubly graded homology groups that that give you eisenstein's here so there's there's something that is attached to to the sun curves that is called the as a stensiles that leads to some as a stensiles there is an orenomorphic one with characters and and they are just you can put in some time well excellent I would I would love to learn more about it


Shopify has over 2,000 apps on their app store. With that many apps at your disposal, it's kind of hard to choose the right one. Choosing the right one will take some time, effort, a lot of testing, and overall, it's a fairly tedious process. Choosing the right app can in fact influence how our customers of your store. Some of them help us give better customer service while others help us upsell. But don't you worry because that's exactly what we're gonna be covering in this video. We're gonna be covering the top 10 best Shopify apps for your dropshipping store. What's going on everyone, Mar here with AutoDS and if you like videos on the dropshipping business or how to start and scale yours, make sure to hit that subscribe button as well as leave a like because that's part of what we cover on this channel. So if you don't want to miss out on any future content to be able to scale your business, make sure to subscribe. Having the right Shopify apps in our store can really make the entire business process a lot more manageable. Shopify apps can really help us with conversions, higher profits, better customer service, and the list goes on and on. A few examples of what apps can actually do for our store is for one, they can help us fulfill entire orders. They can also help us chat directly with customers and give the best possible customer service. Now apps really do have the potential to help us scale our dropshipping stores to a whole nother level. So choosing the right ones is crucial. So without further ado, let's kick off the countdown for our top 10 picks for the best dropshipping apps for your Shopify store. Kicking off the countdown in at number one, we have none other than AutoDS. AutoDS is a platform for dropshippers made by dropshippers. They have an impressive 4.8 out of five on the app store and have a lot of very powerful tools to be able to help start automating and scaling your dropshipping business. They have an extremely handy product research and product import tool. This tool helps you find and import products to your store with just a few clicks. You also have complete and total order fulfillment through your buyer accounts or through the AutoDS platform so you can still use your current suppliers. Now along with the entire order fulfillment, they also have automatic tracking number updates. So as soon as your order ships, you can rest assured that your customer is gonna be getting an email with a tracking number. Now one of my personal favorite features from AutoDS is the fact that we have price and stock monitoring. With a price and stock monitoring, you never have to worry about running out of inventory or price fluctuations. AutoDS will automatically update your stock or adjust prices for price fluctuations. Now another very handy thing that we offer over AutoDS is resources. If you need help, we're there for you. AutoDS has tons of resources like this YouTube channel, blogs, eBooks, mentorship programs and more. So if you wanna check us out over at AutoDS then make sure to check out the description down below. The link will be there. And that doesn't just go for AutoDS. Any of the other dropshipping apps that I'm gonna mention in this video will be there as well. AutoDS starts at just $9.90 a month on the annual plan. Now coming in our second spot, we have Printful.  They have a 4.6 out of five on the App Store and they have some very high quality prints. They offer custom branding as well as packaging so you can have your entire order customized to your liking or to your brand colors with your brand logo, whatever it may be. They also have a very powerful built in mockup generator. So you're able to make mockups directly on the app. There's no need to go on Photoshop and start doing all the work through there. Everything is right on Printful to make it easier for you. The Printful app starts at just $9 a month. And number three we have CJ Dropshipping. CJ Dropshipping helps with product sourcing, product listing and complete order fulfillment. Now one of the cool things about CJ Dropshipping is the fact that they have the ability to source products that are not yet on their platform. So if there's something that you like that you don't see on the website or the platform then you can go ahead and request it and they'll get it for you. That could give you a competitive advantage against other stores that don't have that product. CJ Dropshipping has an impressive 4.7 out of five on the app store and they have amazing customer service. Now the app is free to install but some service fees can apply. In our number four spot we have Printify. Printify is another print on demand supplier with affordable print on demand products and high quality custom designs. They have a wide variety of products such as t-shirts, phone cases, mouse pads and a lot more. They offer fast delivery with local warehouses and they also have a mockup generator. They have a 4.4 out of five on the app store and their paid plan starts at just $29 a month. Landing into our number five spot we have PageFly Landing Page Builder. Try saying that five times fast. Landing pages typically are the first thing that a customer sees whenever they land on your website. Hence the term landing page. PageFly gives you the tools to help you create high converting landing pages to be able to help start increasing profits and scaling your business. They have over 90 easy to use and easy to customize professional templates to really make your store stand out and they also offer a very convenient and handy PageSpeed Optimizer. What the PageSpeed Optimizer does is it optimizes your page so that way it loads a lot faster without any hiccups. It's been proven time and time again that the longer a customer has to wait for a website to load, the more likely they are to just click away and forget about the site. PageFly also has some very powerful marketing and sales tools, such as a countdown timer. Countdown timers can be great because they create a false sense of urgency. It gets the customer thinking, I need this deal now before they run out. They also have things like badges, variation swatches and a lot more. Their paid plans start at just $25 a month. So really quick before we continue this countdown, if you're finding this information useful, make sure to subscribe and leave a like. Now, let's get back to it. Optimizing our number six spot, we have SEO, speed and image optimizer. Without getting too deep into it, SEO or search engine optimization is crucial to getting our pages found. Without the proper SEO, it can be a lot harder for a customer to find our webpage or for us to rank higher in search engines, such as Google. This app is built to help improve your customer's experience on your website. With its user interface optimizer, this app can do just that and it also helps with higher conversion rates because your customer's getting a smoother experience on your website. It also has SEO enhancement tools which help you start ranking higher in search engines and an image optimizer as well.  The SEO, speed and image optimizer app starts at just $199 a month and let me tell you that it's worth it. It has a perfect five out of five on the app store. In our number seven spot, we have Frequently Bought Together. Frequently Bought Together has been around since 2017 and for good reason. It's one of the best apps currently available that help you increase profits by upselling and recommending products. This app uses a proven AI recommendation system to be able to help you upsell those products. You're also able to customize product bundles with a bunch of different relevant products that work together and tell them at a discounted rate, increasing your average order value. This app also helps by recommending products that are frequently bought or used together. It has a 4.9 out of five on the app store and starts at just $9.99 a month. Being pinned to our number eight spot, we have Pinterest. Most people, when they have a question or they need an answer, they're gonna go to Google but when they have an idea or they need to get inspired and get some ideas, they're gonna go to Pinterest. People absolutely love going to Pinterest for ideas and inspiration to do almost anything, especially when it comes to events.  Now this can really come in handy if you have a relevant product for what people are searching for. This app helps you reach more customers by automating posts to the ever-growing and very popular social media platform, Pinterest. A really cool feature about the Pinterest app is the fact that you can track your content's performance using the Pinterest tags. You're also able to sync all of your Shopify listings to Pinterest itself and that helps you bring in a lot of organic traffic. The Pinterest app has a 4.8 out of five and is totally free to use. In our number nine spot, we have Shopify inbox. I absolutely love Shopify inbox. It lets customers send you direct messages and it lets you reply back to them. Those messages you can receive in the form of a chat or through email. A cool thing about this is the fact that you're able to integrate it with social media platforms such as Facebook Messenger, Instagram Chat and a few others. If a customer has any question regarding their order, this app helps you to correctly answer them and give them the right information by being able to see the items that they've given in the past, their past orders and your message history. The Shopify inbox app helps you automate responses as well as frequently asked questions. Last but certainly not least, in our number 10 spot, we have Judge Me Product Reviews. Judge Me Product Reviews is a fantastic app because it helps collect reviews after an order has been fulfilled. It also has a very handy review importer.  Now on top of that, you're also able to manage those reviews and pick and choose which ones you wanna include and which ones you don't. This app has a perfect five out of five in the app store and the paid plan starts at just $15 a month. Now I'm really curious to know, did your favorite app make it to this list? If it didn't, make sure to drop it down in the comments below so that way I can check it out and maybe even include it in a future video. Also remember that if you wanna check out any of these apps, I'm gonna have a link to the article down in the description below. Once again, my name is Mario with Out Of The S and if you like videos on the drop shipping business and how to be able to grow and scale yours, make sure to hit that subscribe button as well as leave a like so that way you don't miss out on any future videos. Huge thank you to everyone for watching.