 Okay, so let me, before even thanking, let me start with some remarks, especially about this young and old age. Our master of ceremony seems to think that his age is young and everything older than him is old. But I have a time-independent suggestion for your consideration. In mathematicians' life, there are only four stages. Undergraduate, graduate, doctoral, and postdoctoral. So after postdoctoral, one never grows. And you feel that in a meeting like this. So I also learn a lot, lot things. Then what little I have to give you, in return I get a lot from young people and we are all postdocs here. So that is it. And secondly about some people I've been kind to mention this Kulkarni limit set and et cetera. And I should thank now that Pepe and as well as Alberto for inviting me here, first of all, for this conference. But more than that, at one point they read that my paper, Groups with Domains of Discontinuity, had a very hard time to get that paper published. But in fact, for two years I think it took and I went to Bonn at one point and it was submitted to Mathematica and Allen. And I came to know the editor, it was harder. He got, tried to get it refereed, no success. But finally he said, okay Ravi, I know it is your paper, I publish it. And so it was published. But very, very few people have read it and I really thank Alberto and Pepe to have read it and have made some use of it. Actually this is a little digression, this is not my theme of my lecture, but that idea is very, very simple. I was thinking in terms of, I knew the Fuchsian groups and Kleinian groups and I was trying to see the n dimensional picture. So groups of conformal transformations of the n sphere. So if gamma is that group then to construct the limit set. So what are the, and classify these groups roughly. So the limit set can be either empty, one point, two points. These two points you really should think of S0. This should be considered like some F minus one. And you consider this as S minus two. But this is a very important case because this means gamma is finite. And we are ignoring this case in this conference. But this is a different world altogether. But I'm going to say something about this in my lecture. And afterwards for every k in n, you consider this, in this case, there is only one definition of limit points really. And the limit set is closure. The accumulation points of orbits. And that limit set should be, it's the least k in which lambda is contained in a round k sphere. And that was the idea. And so that was some classification. And the last case when k equals n, this is the most difficult case and is a generic case. The others are like fiber bundles, et cetera. That was the idea. And I was also reading some papers, a Muirberg at that time, Swedish mathematician or Finnish mathematician, Swedish I guess. And that was about the similar analog in the CPN case and so on. So in CPN case, that is not in my paper. It is only for a very general case. So I started with this L0 and L1 and L2 was just very, very large. I just took the compact set which is not contained in L0 or L1 and take the orbits, orbits of that and take the accumulation points of the orbits. But now in the CPN case, I think this could be filtered more, that you could consider Cp1s, first points, just points correspond to this L1, accumulation points of orbits of points, then Cp1s and then you could continue Cp2, Cp3, CPN. And there could be some very rough classification like this in that case. And this seems to be actually happening in many of your examples. And for the young people who are looking for some problem, I think they should have a look at this. Besides Mexico and France and at other places, also this is an elementary topic people can get into. So this was my digression and now I come to my talk. My talk has not much to do with the, not really to do with the limit sets, but it's a more foundational stuff which I'm attracted to from time to time. And let me just make a couple of remarks about this foundation stuff. I think we all probably agree that mathematical thinking is mainly space and number. In my opinion, the space comes first, but many people think number came first. But besides that, symmetry is a very, very general idea goes beyond mathematics, but mathematicians have been able to pinpoint it in the language of groups that gives some structure. So symmetry and the group theory language. So that may be also considered as a category of mathematical thought in addition to space and number. And so I was thinking about this. Let me tell you, my first topic is dynamical types. So we talk about these dynamical types in all our speaking. Now, for instance, you take the group of isometries of the Euclidean plane. If it is orientation preserving, that's a really very good theorem that is either identity, translation, or a rotation. If it is orientation preserving, then it's reflection or a glide reflection. Then similarly, for the case of the sphere, it is either identity or rotation, that is we usually say. But you know rotation, rotation through which around an axis, but around which angle, around which angle, is the angle is not from zero to two pi because it depends on how you compute the angles. So whether you are looking at that plane orthogonal to the axis, whether you are looking it from the top or from the bottom. So it's better to take that angle between zero to pi if the angle is strictly between zero to pi, then we have this orientation in the plane as well as in the axis. So this axis gets oriented. So that's a different dynamical type than when the angle is pi, when the axis is not oriented. So we say these are the three dynamical types in the spherical geometry. And this hyperbolic geometry, now we know, yeah, we call this elliptic hyperbolic parabolic in the orientation preserving case, orientation reversing case. Again, there are reflections and analogs of glide reflections. Glide reflection can be along the geodesic or about a horocycle. So that can be characterized by what happens at infinity. And so we call these all these dynamical types. We use this language all the time, but here is a moment of truth. What is the dynamical type? That was my question, which bothered me for many years. But before I go to that, I want you, all of you to enjoy these pictures. And I think that these examples, these pictures should be a part of usual training, whether you are learning complex analysis, focusing groups geometry or anything. So this is the picture, this is the picture for the Euclidean group of isometries, just a topological picture. So the orientation preserving isometries of the Euclidean plane are either translation, rotation or the identity. And the whole group is Z goes to AZ plus B in terms of complex numbers, absolute value A equals one. So it is S1 cross D2, S1 cross R2 and R2 is the same topologically as D2. So it's a torus. So how are these dynamical types distributed in this torus? So you can take one arbitrary element as identity and there is a disc through that. So that disc, this meridional disc, except this origin, these are the translations and the rest are rotations. So this is a good feeling about almost all the transformations or rotations or things like that. And similarly, the orientation reversing case is very interesting. I think it's some kind of a foliation. Of the, there is a mobius band of reflection and it is covered by cylinders of glide reflections. I've not drawn that picture here. By the way, this is very elementary. These pictures, I have to take help of my students to draw these pictures. And my students still don't understand how to draw a mobius band. So those of you who are experts in these things. Yeah. So probably you can make these pictures better. And now this is the picture for the isometric orientation presenting isometries of the two sphere, which is the group SO3. You can write it as a three by three matrices with real entries, et cetera, et cetera. But how does it look as a picture? So this is identity at the top. This is like a three dimensional ball where on the equator, you identify antipoder points. So on the equator, you are getting a projective plane. And the rest of the things, they are this parallel family of spheres, rotation through angle strictly between zero and pi. And then the identity. So this is the picture for SO3. And the picture for the PSL2R is the most beautiful. Yeah. Now, you really have to think about this. Maybe this picture is just a schematic picture. You can put some more life into that by putting some metric and other considerations. So here is the identity. Then there are two cones. These are the cones. And the interior of that is elliptics. And the boundary, there are these two cones. This common thing is not in your space. So there are actually, this thing is not in your space. It is the open solid torus. And so this is one cone. This is another cone. And this corresponds to, there are two conjugacy classes of parabolic in PSL2R. And so these two conjugacy classes are these two boundaries of the cones. The inside that are the elliptics. And the outside of that is the hyperbolic. And now you can draw similar pictures for SL2R. This is the double cover of this, or all the way to SL2R, universal cover. So this is infinite family like this. And then you can easily see there is this Lorentzian metric there. But in Lorentzian metric, completeness is not the same as two points can be joined by geodesic, et cetera, et cetera. All these things are visually clear by just meditating on this picture. Okay. Now what I say is the similar thing holds for all classical geometries over real complex numbers, quaternions, octonions, et cetera. In principle, one could draw some pictures and in some of you are drawing these pictures already. So we are not defined yet dynamical types, but we all are confident that there are only finitely many dynamical types. These are these that, that, that, so on, so forth. And so it's really remarkable that the groups are infinite. And so there are infinitely many conjugacy classes. But the dynamical types are only finitely many. After we define them, what dynamical types are. And this is another observation that we also observed that in each transformation, it has a unique spatial invariant and unique numerical invariant. Remember, mathematical thinking means space and number. So we have a spatial invariant and a numerical invariant. So just an example for the rotation in the plane, the center of rotation is a spatial invariant and the angle of rotation is a numerical invariant. So what should we mean by dynamical type? This is the real question of an element and what are its spatial and numerical invariants? And can we give a definition of dynamical type? Now this is really a cognitive question. This is how we are thinking about dynamics. And so usually the space and number part play role in what is dynamics, but this group is also coming. So in this classical geometry is with rich isometry, rich automorphism groups. So the question is, can we define dynamical type? Purely in terms of group alone, purely in terms of group alone. Yeah, and so now you can forget, almost forget geometry and just think in terms of groups. And so this is the notion of G classes. So G stands for centralizers of elements. So let G be a group, two elements X and Y and G are said to be G equivalent if their centralizers are conjugate. And in this three classical geometries for the orientation preserving case, you compute the centralizers and G classes are exactly what we think the dynamical types are. Now in higher dimensions, in classical examples, now you see dynamical type depends only also in our cognitive understanding in terms of space and number and we are thinking in terms of group alone. So that correspondence is not exactly one to one, but it's finite to finite. And so if you want to say finitely many dynamical types, we can as well prove finite, finiteness of G classes in particular groups of automorphisms. And that indeed is true, yeah. And what are the special and numerical invariance in terms, so this is a general result on G classes and finiteness of G classes in classical geometries. And the interesting thing is that we do not keep that in mind that these classical geometries are actually defined over reals, complex numbers and quaternions. So think independently of this usual question. Now Euclidean geometry, there is a theoretical issue to show some uniqueness of this Euclidean geometry under some hypothesis, under some making statement, making some precise definitions. The role of real numbers is implicit in Euclidean's construction itself. And then how it was explained by Dedekind and other people. Either real numbers are really, really real close fields or in that category, in that category, the Euclidean geometry is unique. If you think of Euclidean geometry over the rationals, say take Q cross Q, the two dimensional vector space over Q and take a positive definite quadratic form. And that defines the metric, although the metric is not, distance is not rational, but the square is rational. But consider that structure. Then of the five postulates of Euclide, fifth postulate is also valid. Given a line and given a point not on it, there is a unique line parallel to that, et cetera, et cetera. But where are the real numbers coming? They come in the fourth postulate. And so this is something worth keeping in mind. So over the rationals, the fourth postulate fails. Fourth postulate fails that not all lines are equivalent in this, in this geometry. And so Euclidean metric geometry. And so the Euclidean metric two dimensional geometries are really classified. This is number theoretic invariant. Quadratic form gives you some quadratic extension of the rationals. And then there is a norm group of that extension. So norm group modulo Q star square. So this is a huge infinite copies of the two, the twos. And those, though they classify this Euclidean geometry over the rationals. So keep this example in mind in understanding the further development. And now this special and numerical invariants can be explained by general theorem. And this finiteness of the classes is due to this use of real complex quadrennians. And if you take, define these geometries over more general fields, if the field has more arithmetic, then there will be arithmetical invariants. And that will destroy the finiteness of the classes, but you will get a different understanding. Now let me just say that this Weyl's theory for compact league rules, the structure of compact league rules is one of the most beautiful chapters of mathematics. If you really look at what he's proving, there's a very large just purely group theoretic part of that. So that is true. It uses neither compactness, nor the fact that it's a league rule. So this, he doesn't have a G class there, but he has effectively the torii, which is the Z class of a regular element. And the Weyl group, et cetera, these are all dynamical notions and can be defined purely in terms of groups. And so this is the, I shall explain that little later, but here is a typical result in general. If you consider GLN of a finite field, a GLN of a field. Now, if you want a finiteness result, so what are the finiteness assumptions coming? So suppose there exists only finitely many field extensions of a fixed degree of F. So that covers real numbers. Of course, algebraically close field complex numbers. Then that also covers periodic fields, complete periodic fields. There are only finitely, there are many extensions, but of a fixed degree, there are only finitely many finite extensions. So there also this finiteness result holds. Okay, so at the analogs of this theorem, you could prove in the symplectic case and orthogonal case and so on. And now this is a really general theorem on any group action. Any group acting on any set X. Now, part of this has appeared in this Wiles analysis and some other analysis for algebraic groups, et cetera. But this is a purely group theoretic result. And so let G be a group acting on a set X. And we have orbits of elements and then there is the orbit class. So on an orbit, if two elements are in the same orbit, then that orbit is always of the form G mod, the stabilizer subgroup at that point. Now the stabilizer group depends on X. But X and Y are in the same orbit. Then G sub Y, the stabilizer there is conjugate to G sub X. So this happens for the orbit. The orbit class is just the conjugacy class of the subgroup itself. So they are the same orbit. They are of course in the same orbit class, but the orbit classes are more general. So to fix the idea, take the action of rotations around an axis of the sphere. So here the north pole and the south pole, the whole group fixes it. So that is an orbit class by itself. And all the other elements, the group acts, this group is just SO2, the stabilizer subgroup is identity. So stabilizer subgroup is identity everywhere. So the subgroup is identity everywhere. So stabilizer subgroup is identity everywhere. So there are only two classes, two orbit classes. And here such finiteness results have been proved by topologists in much more general context. So a compactly group, I guess even a compactly group acting on any reasonable space, it has only finitely many orbits, etc. These are the theorems that are proved in by the topologists. Okay, now this is a rather detailed analysis. It may be difficult to keep all these things in your mind right away, but let me just go through that. So G sub x is a stabilizer at x, Gx is the orbit of x, and Rx the orbit class of x. f sub x is the fixed points of the stabilizer at x. So that is a subset of x. Then f prime x are the generic elements in this fx, and by generic I mean the fixed, if an element is in the fixed point, then the stabilizer subgroup at that element contains G sub x. But generic elements are those for which the stabilizer subgroup is exactly dy equals Gx. Then take the normalizer of G sub x in G, that is a part of the group, and the while group at x is just the quotient group. And now there is an action of the normal, this Nx on G mod Gx. This gets a little technical, you just have to bear with me on that. And it also acts on fx, and the action on f prime x is free. And this action is also free, when you pass on to the while group. And so there is a free diagonal action of while group on to G mod G sub x cross f prime x. Now there is a mapping from G mod G sub x cross f prime x into this orbit class of x. And that is the action, that is very fine. And for each z in the image, this map is surjective. This map for every element in the image, the inverse image consists of an orbit of Wx. And so the induced map is a bijection. So yeah, I take some time to convince yourself that all these things are true. It's a theorem like you just once verified, once in your life, or if you don't want to verify it, you just assume it and just go ahead. But this is the complete generality about actions. And now the most important case is when the group is acting on itself by conjugation. Then the stabilizer at a point is the same as centralizer of that point. And the six points of the centralizer is exactly the center of the centralizer. And this f prime x is the generic elements in that f of x. And yeah, this is the center. So this is an abelian group. And so this f prime x will be a subset of the abelian group. Now you see from the group point of view, if there were only abelian groups, then we would not even develop the group theory. That will be some extension of number theory. So this numerical invariant is coming from this f prime x. And this, yeah. So we have these two types of vibrations, g mod g sub x, cross f prime x, and that model of the wild group. And so this is the bijection. And here you take the quotient, you take the partial quotient on one factor. And this g mod g, g x is a homogeneous space of g. So that is your special invariant. And this f prime x, when g is acting on itself by conjugation, this is a subset of the abelian group. That is the numerical invariant. And this is the way it's a complete generality for any group actions. Yeah, okay. Now it is even more algebraic now. But I would suggest that you take any group that you like. You consider any action on any space that you like. Consider the homeomorphism group of the real line, or homeomorphism group of the circle, etc. and carry out these constructions. They can be described explicitly, fairly explicitly. And you will get some understanding of the internal structure of the group. So that is my general observation. And now the next observation about, so we know now the g classes in a group. These are the conjugacy classes of centralizers. And when I first wrote this paper, I completely missed this point. This first paper I published in 2007. But then while working out some g classes in finite groups, and especially p groups, we came across this beautiful notion by Philip Hall. Philip Hall was a great mathematician. Was a really great mathematician. And he discusses the notion of isoclinism. And that is very relevant here. And this, probably the people do not know this here. So you consider the map g cross g into the derived group. Namely x, y goes to x inverse, y inverse, x, y, the commutator. You see the element is in the center, that is in the kernel. So this map factors through g mod, the center to g prime. And now you consider two groups, this g1 and g2 are isoclinic. There is an isomorphism of g1 prime to g2 prime, and an isomorphism phi of the centralizer quotient, such that it commutes with this diagram. So isoclinism is a, this phi psi, this is an isoclinism. There is a certain notion, there may not be a map of one group into the other, but there will be a map of these quotients and these sub-objects. And that is an isoclinism. And the one thing is that the two isoclinic groups, there is a canonical bijection between g classes. So even an infinite group can be isoclinic to a finite group. For instance, infinite abelian group is isoclinic to identity group and so on. So the g classes from a group theoretic point of view is also capturing how non-commutative the group is. So this is the general feeling. And now I can tell you that in general a g class in a group will be unions of some cosets of the center. That is one restriction. And so on and so forth. So I think if you want to know more about g classes, you can see my this first paper in 2007 and the other paper is yet to appear. But it has been accepted in journal of algebra and applications. And this is my joint work with my student Rahul Kitture and Vikas Zador. So I want to tell you something about the sociology of mathematics here. So we wrote a long nice paper explaining everything, taking how the notion has come from geometry and so on. Sent to a good journal in algebra. Referee, the editor returns it after four months. Are there any errors in the proofs or something? He doesn't want to say. Nothing. But it's rejected. Then we sent to another journal but chopping off the paper. And so at Harvard I was once a teaching assistant to Brauer for his course in algebra. And Martin Isaacs, who is in Wisconsin now, he was two years ahead of me. And so he was an editor of one journal so I sent to him. And Martin Isaacs remembered me. So he asked me, you are writing this. But then he says this geometric motivation, chop it off. Simple thing about Z-class is a union of concepts of ZG. Chop it off. This is too easy. And so the final paper now will appear. But neither the geometers will learn from it. What the group theorists, pure group theorists, especially finite group theorists will learn from it. I just cannot decide. But it's a nice 15-page paper and long things. So in my remaining time, let me just wish to say about this, applying this philosophy of Z-classes to finite groups. I'm backward. Sorry. Right. Now you should, as geometric topologists, we should have developed some feelings for finite groups. Yeah, it's a different area altogether for our usual thinking. But it's a very rich field. Now if you say finite group, it's after all a compactly group. So the Weier's theory applies. Yes and no. The crucial statements in compactly group about Weier's theory use the hypothesis that the group is connected. That is not quite appreciated by other people. So the statements that depend on this connectedness, they just go away. Now for example, if you take a compact soluble league group and compact connected soluble league group. Now compact connected soluble league group is Abelian. And Abelian compact connected league groups are just the products of S1s. So the theory is finished. So the crux of the matter there is disconnected groups. And finite soluble groups is really an area by itself as I'm finding out. And their analogy is with non-compact soluble groups. Non-compact soluble league groups. There are analogies but not with compact league groups. And among finite soluble groups, finite nilportant groups is one very important area. And many, many people are working in this area. And it's really a jungle. So this Philip Hall's idea of isophenism gives you some real insight in this. Working out, say, the natural question, work out the isomorphism types of finite p-groups. So again with this computer help, I guess by hand you can do the groups of order p-q, groups of order p to the 4. With much more work you can do groups of order p to the 5. And with computer help, the Australians have a big group of mathematicians here. So they have pushed this thing up to groups of order p to the 8. So this is up to isomorphism. Up to isophenism the things are little, little, little better. So I'm really giving you some, making some statements of which I'm not an expert. But you may like to know these things. And the other extreme is the classification, the finite simple groups. So that classification was harder than the classification of simply connected groups. And so, but that is a major achievement of the mathematics in the 20th century. And really a major worldwide mathematics, mathematical activities to understand and simplify the proof and develop a general theory of finite groups. So in this lecture I report only a few beginning results on Z-classes in finite groups. The results of interest are, the finiteness result is not of interest for finite groups, but the actual bounds on the number of Z-classes. Then connection with linear representations. Now this is another thing. The people working in representation theory of finite groups or even league groups, they quickly go to representations over complex numbers. But here, what is relevant is representations over the rationals. There is some connection with representation over the rationals and Z-classes. And I think our, this is a good work in whether it is acceptable to journals or not. I think it is a good work. Actual characterization of P groups with maximum number of, with minimum number of Z-classes. So that's a, that's a, that's a neat result. Now there are some analogies. You know, in a normal subgroup is a union of conjugacy classes that we know. How, how good for Z-classes? So there is, in any group, a maximal among the abelian normal subgroups is again a union of Z-classes. Now a non-abelian finite group contains at least three Z-classes. This is a nontrivial theorem and it's connected with Wetterburn's famous theorem that a finite division ring is a field. Again, one of my other students, Ronny Gurage, he, he observed it. This is the connection with the representation theory. The conjugacy classes of cyclic subgroups is the same as irreducible representations over the rationals. This is stated as one of the theorems in Seier's book on rationality questions in representations. So one easy upper bound, for a finite group, the number of Z-classes is at most the number of irreducible representations of G mod the center over the rationals. And then there are nice things to work out exactly what these groups are, et cetera. So for a finite P group, if the center has index P to the K, there's at most that 1 plus P plus P square plus P to the K minus 1 plus 2 Z-classes. Now, a necessary condition to attain this bound is that this is another family of groups that finite group theories have studied. A P group is said to be special if its center and commutators of group are same. And if there are elementary abelian P groups, and it's said to be extra special if that elementary abelian P group is actually Cp. So this is how these things develop. And so he forgot to write down, I forgot to write down this Aij Rn Zp, Rn filled with P elements. So this group has the exactly that upper bound and those many exactly Z-classes. Now a lower bound, a non-abelian finite P group has at least P plus 2 Z-classes. And we can characterize exactly which these groups are where the lower bound is attained. So this lower bound is attained exactly when G mod the center is either Cp cross Cp, that is cyclic group of order P, cross cyclic group order P, or the following holds that G has a unique abelian subgroup of index P and center of G mod ZG has order P. So in a more fancy language in finite P group theory division into classes, so it is the student sufficient condition for P plus 2 Z-classes if and only if it is isoclinic to a P group of maximal class with exactly one abelian subgroup of index P. And this is the kind of thing that comes about. Now this initial cases, this groups of order P cube you can do it by hand and by some cleverness. But groups of P to the fourth also you work hard and you do it. But without having the help from computers I do not see how this work can proceed. So fortunately both of my students have very good in this gap program. They developed expertise in the gap program and historically we worked out the all Z-classes in groups of order P to the fifth and then we saw the pattern and then we proved these results. Okay, that is about it. Thank you.