 Professor Lothar Goetje here in our basic notion seminar. He will be speaking today to us about tropical geometry. Lothar, you have the presentation. Okay, thank you very much. So I'm talking about, now it doesn't do it. So I already have technical problems. Okay, I want to talk about tropical geometry. So this is sometimes called the combinatorial shadow of algebraic geometry. So in algebraic geometry, we study algebra varieties, which in the simplest manifestations, manifestation are just zero sets of polynomials, f of x1 to xn in n variables on c to the n. And we want to replace them by a simpler combinator object in tropical geometry. So the hope, so this should be done in such a way that the combinatorial properties of these tropical varieties reflect the geometric properties of the original varieties, algebraic varieties. And then we will find that many results in algebraic geometry have counterpart in tropical geometry. And often we can solve, or try to solve problems in algebraic geometry by translating them into tropical geometry when they become basically problems of combinatorics and then try the corresponding combinatorial problem. So let me briefly introduce why if you want the history. So tropical geometry originally comes from computer science. The kind of is in honor of Imre Simon, which is who's a Brazilian computer scientist who first studied in computer science the max plus algebra, which we will encounter in a moment, which is at the basis of tropical geometry. There are applications of tropical geometry in many fields. So I've been interested in particular in algebraic geometry, for instance, a number of geometry of curves. There's applications in mirror symmetry between algebraic geometry and physics. There's other applications than physics. There are applications in economy. For instance, for the design of auctions and there are also applications in biology, for instance, in computational genetics. So I myself first learned a tropical geometry from the notes of Gartmann, which I have written down here. They are also at the references in the end. And I also consulted them for this talk. There are also this Atom lectures, which I think currently are still going on and the corresponding videos are online at the place I list here and also at the references in the end. So if one is interested in more detail and knowing more about applications and so on, one can look at these lecture series. And finally, there is, for instance, this book by MacLagan and Sturmfelds about tropical geometry, which the references are at the end of the notes. So let me now start with the subject. So as I said in algebraic geometry, we study algebraic varieties, which are zero sets of polynomials, actually common zero sets of several polynomials and several variables. So X is the common zero set of polynomials F1 to Fk, where Fd, Fi are polynomials in n variables on CN, which we also just call AN in algebraic geometry. So explicitly it's a set of all A1 to AN in CN, such that F1 of A1 to AN is equal to Fk and so on, Fk for F1 to AN is equal to zero. So this common zero set. So for instance, if in A2, we take the zero set of just the polynomial X, this will give us a line. At least if we look at the real picture, it looks like a line. With the complex numbers, C2 is actually like R4 and so it is a plane, but as algebraic geometry we think of, still of it as a line. And if we look instead at the compactification of projective space, it will actually be a sphere. I mean, homomorphic to a sphere. In the same way, we can look at the polynomial of degree three. For instance, this one, Y squared minus X times X minus one times X plus one in the plane. So the real points look somewhat like this. So a circle and then this opened up circle going to infinity. If we again take this over the complex numbers, this equation and compactify in the projective plane, we get a torus like this as a zero. So these are some examples. So now we want to do algebraic geometry in a different way to do tropical geometry. So algebraic geometry is based on the algebra of polynomials. So basically polynomials means we add and we multiply. That's what polynomials do. So if you want to change, one way to change algebraic geometry is to change these standard basic operations. So you want to, what happens if we change the rules of algebra? What happens if we change what we mean by addition and modifications? Can we still do algebraic geometry? Okay, so what happens if we change the rules of algebra? So, okay, so I just review what I said. And so we can do the following crazy definition. We look at the tropical semi-field R bar, which is R union minus infinity with a new addition, plus with a circle and a new multiplication, where we have defined the new sum A plus B is the maximum of A and B, and the new product is the sum of A and B. Okay, so for instance, if we take five plus six, this is equal to six. If you take five times six, this is equal to 11. Okay, that seems reasonably crazy. But, strange enough, this gives us almost a field. So there's, so you can see this new addition is easy to see that this is, both addition and multiplication now will be associative and commutative. We can check directly here that the distributive law holds. For instance, if we take A plus B times C, in standard notation is the maximum of A plus B plus C, which obviously is the same as the maximum of A plus C and B plus C, which according to our definition is A times C plus B times C, so the distributive law holds. We see that minus infinity is the neutral element of the new addition, because the maximum of minus infinity and any number is that any number? Zero is the neutral element of the multiplication because the new multiplication is the addition and zero is the neutral element of that. And minus A is the multiplicative inverse of any element in R. So that means any element in R bar without the neutral element of addition. So like in a field. So the only thing which is missing is that we don't have additive inverses. If you have an element in R, it will not have an additive inverse. There will be no B in R bar, such that the maximum of A and B is minus infinity because our element A is already bigger than minus infinity. Okay, so we have a field except for this one action, which doesn't hold. So we can see that it's, it almost seems we have a reasonable structure. And now we want to define tropical varieties as some kind of zero sets of tropical polynomials in Rn. So a tropical polynomial will be a polynomial where we replace our usual plus and times by the new plus and the new times. And we will have to change what we mean by zero sets. We actually would not be interested in zero sets, but in something slightly different. So let me first define these polynomials. So I write for variable xi, I write xi to the m just for the i-fold new product of for the m-fold product of xi with itself. So in the standard notation, this is just m times xi. So this is xi to the m. If you have a multi-index, i equal to i1 to n and x is equal to x1 to, this should be written in a standard x1 to xn is the for the variables, you write x to the i to be the product x1 to the i1 times and so on xn to the in, which in standard notation is just i1 x1 plus and so on plus in xn. So this is just the scalar product of the vector i, which is the multi-index times x, which is the vector of coordinates. So x to the i is just i times x, the scalar product. And so now what is the tropical? If you write down something like sum i ai xi, this is supposed to be a function. So the tropical polynomial f times sum i ai xi, where the ai, the coefficients are elements in our bar is the function f. And so the maximum, or so in the standard notation, the maximum over all i ai plus i times xi, so the scalar product from i n to r. So vector b equal to b1 to bn in i n is sent to the maximum over all i ai plus i scalar product with b. So this is a function from i n to r. So in this notation, we will not write the terms where the coefficient ai is equal to minus infinity. So that's the, like one wouldn't write the terms where the coefficient is zero in a standard polynomial, but we certainly write the terms where the coefficient is zero because zero times xi is just xi. Okay. So let's look at an example. We have this polynomial in one variable, minus two times x plus x plus one. So in standard notation, there's the maximum of one x and two x minus one. This minus two times x squared is two x minus one. So I've here given you the graph of this function. You can see it's, it will be equal to one. If x is smaller than minus one, it will be equal to x if x is between one and two. And it will be equal to two x minus two if x is bigger than two. So this is an example of a tropical polynomial in one variable. So I just write this here again. Now you can see, I mean, from the definition, a tropical polynomial is piecewise linear. So one of the, if you have all these monomials, so the tropical monomials, one of the terms will give the maximum and that term is a linear function. So tropical polynomials are piecewise linear. The interesting set will somehow not be the zero set, but the corner locus, which where two kind of regions where the polynomial is linear meet, okay? So for instance, in our, so let me here first, we have this, I write it again. So for the tropical polynomial f equal some i a i x i is equal to the maximum of all i a i plus i times x from i n to r. The tropical hypersurface ZF defined by F is a set of all b and n such that the maximum, the maximum is the value of f of b. So the maximum over all monomials evaluated at b is f of b. So that this maximum f of b is obtained by at least two of the plus monomials. So there would be two, it would be at least two such multi indices, i one and i two. So that is a maximum of a i plus i times b is equal to the maximum of a i a i two plus i two times b at b. No, so that let me say it again. So there would be at least two such monomials where the maximum over all monomials is achieved by these two different monomials. So it's not enough that two monomials are equal, but it should be that two of the monomials give us the maximum of all monomials, okay? So this is, this is the corner locus of the tropical hypersurface defined by F. So in particular in our case, we have that if we look again at our polynomial Z of minus two x times minus two times x squared plus x plus one, we see that the locus where two linear pieces come together are the point one and the point two. So the tropical hypersurface in R defined by this polynomial is the set one two, okay? Obviously the zero sets of tropical polynomials in one variable are maybe not so exciting. So we will now rather look at tropical plane curves. So I'll give a few example of tropical plane curves in R too. So in each, so in each sector, so I write them in the plane in each sector, I will also write which monomial achieves the maximum. For instance, we can look at the tropical line Z of x plus y plus zero. So this is in standard notation, the maximum of x and y and zero. So how does it look like? So of these three monomials, x, y and zero. So when x and y are both negative, the maximum was achieved by zero. So in this region, the maximum was achieved by zero. When y is bigger than zero and bigger than x, the maximum is obtained by y and here it is contained by x. And so the tropical curve defined by this hypersurface is where two of these regions meet for instance, the segment where y is equal to zero and x is smaller than zero. So this part, then this segment and finally also the locals where x is equal to y and both x and y are bigger than zero, or bigger equal to zero. So this is the tropical curve and this is the corresponding function. We can also look at a more complicated case of tropical conic, so degree two curve. So x squared plus y squared plus three times x times y plus two y plus two x plus three. So in standard notation, it's the maximum of two x, two y, x plus y plus three, x plus two, y plus two and three. And in this case, we get this more complicated picture. So for instance, when y is bigger than two and y is bigger than x plus three, the maximum was obtained by two y. Here it's obtained by y plus two. When both x and y are small, the maximum is the constant three, which is this monomial. And so we get the plane is divided into these regions and the locals where two of the regions meet gives us always, this gives us the corresponding tropical curve, which looks like this. Okay, so this would be another such example. And then finally, to just say in general, if we have, if you want to know what a tropical algebraic set or tropical variety is in higher dimensional Rn, we just, so assume we have F1, F2, F3, some tropical polynomials. The tropical algebraic set defined by these tropical polynomials is just the intersection of the corner of the hypersurfaces defined by each of them. Okay, I mean, I will not go into this in general. For the rest of the talk, I will specialize to plane tropical curves, which are kind of easier to visualize and are also sufficiently complicated to study. And so these will be tropical hypersurfaces in R2, which are given as corner locals of tropical polynomials in two variables, X and Y, of polynomials in two variables, X and Y. So okay, so this is the case of plane tropical curves. So first give you a slightly wrong definition and then I will correct it. So I mean, in, you know, the degree of polynomial, some Aij X to the I, Y to the J is the maximum over all I plus J for monomials, which actually occur. So where the coefficient is not zero. So for a tropical polynomial, you would say the same. The degree of a tropical polynomial F is the sum. So F equal to sum Ij Aij X to the I, Y to the J is the maximum of the I plus J, which occur. That means where the coefficient is not minus infinity. And the coefficient is allowed to be zero. So therefore, plane tropical curve degrees should be the hypersurface defined by a tropical polynomial of degree D. Now, this doesn't quite work because we have to slightly change it. So I say that the tropical polynomial, F equal to sum Ij Aij X to the I, Y to the J has degree D if, as I said before, D is the maximum of the I plus J which occur in the polynomial. But I also require that the monomial X to the D occurs with some coefficient. Y to the D occurs with some coefficient which is not minus infinity and the X to the zero, Y to the zero also occurs. Somehow the extreme values and the extreme powers should also occur. Okay. So, and then the plane tropical curve degree D is the hypersurface defined by such a tropical polynomial of degree D. So for instance, we had already the case of a line which is a polynomial of degree one, X, Y and zero occur. X, the coefficient of X to the zero, Y to the zero is just this. So this looks like this. We can look at the, we have this example of this conic because monotropical curve had this shape these are always unbounded edges. And then we had here we have written down without writing down the equation, the example of a curve of degree three. So what you can see here, if you look at it and is that, I mean, and if you ignore that I didn't draw as well that such a curve of degree D has always D unbounded edges, one in this direction minus one zero, one down zero minus one and one diagonally up in the direction one one and D of those where D is the degree. So here it's one in each of these directions, here it's two and here it's three in each of these directions. Okay, so we observe that. We'll see in a moment why this is true and actually why it's almost true. It's not quite true in general, but almost. And so actually now I want to describe the shape. So let me, so this was, so we see something about how they look like. And now we want to somehow in some sense prove that the tropical curves actually do look like that and understand in general something about their shape. So this is in some sense the most difficult part of this talk where it requires the biggest concentration because I'm actually trying to prove something and you have to use your imagination and also follow closely and we'll see whether it works. So our aim is to find out what a tropical curve of degree D looks like. So assume we have gamma is equal to the hypersurface defined by F, the plane-topic curve degree D where F is this tropical polynomials which I write more in the standard notation, the maximum over AI times XBI times Y plus CI where AI and DI will be some integers. These are in the standard notation, this is X to the AI time. So in the tropical notation, this is X to the AI times Y to the BI. And so AI and BI are non-negative integers whose sum is at most I plus J and if I plus G must occur. So this AI BI occurring here are distinct integer points. So points with integer coordinates in the triangle delta D. So the set of all AB and R2 where A is non-negative, B is non-negative and the sum of A and B is smaller equal to D. Here I have drawn down this triangle for D equal to two together with its integer points. And we have required that the AI BI among those, we have the point D zero. So the extreme, so the vertices of this triangle should also be among them. Okay, so now how do we get an edge of the corresponding tropical curve? This occurs if the, you know, this points occurs at the point X, Y and R2. If F of X, Y, which is the maximum over all these monomials is obtained by at least two of the monomials. So if there exists an X in R2 such that AI BI, AIX plus BIY plus CI is equal to EJX plus BJY or CJ for some I different from J, then there will be a line segment in the corresponding tropical curve. And if you think about it for a moment, you see that this line segment will be orthogonal to the line connecting these two points AI BI and AJBJ. Because the slope of these things is that and then where they meet is orthogonal to that. So then gamma will have an edge in the orthogonal to this line. So now we take these lines between these. So we only now write down these EI BI where, which actually occur, so we, which actually occur here and we connect them by a line and if we are in this situation that there is an X, Y, such that AI BI, AIX plus BIY plus CI is equal to AJX plus BJY plus CJ is equal to the maximum of all of them. So if one thinks about this for a moment, this will give us a subdivision of our original triangle delta D. And we will find that among these vertices which occur here are also the extreme points. Because somehow as these are the extreme points, at some point they will become, the corresponding monomial will become bigger than all the others. And so they will always occur in this thing, will always contribute to the maximum at some point and they will therefore always occur here. So these lines give a subdivision of delta D into polygons with integer vertices. So here in this case we assume we have just these monomials which actually contribute, so which occur in such an equation. And then we have the subdivision into these two integer triangles. So okay, I'll just repeat what I had. And now let's see. Now we will choose. So as I said, the condition that X to the DY to the D occur implies that among these vertices that occur here are actually the ones at the edges, at the extremes of the triangle. And that among the edges that are used in the subdivisions are also, so that this subdivision of this triangle will also give us a subdivision of the outer edges of the big triangle. So these all occur. And so that means the edges of delta B are unions of some of these lines connecting in these vertices. So now let's choose an inner point in each of the polygons into which I have divided my triangle. And we connect these inner points in adjacent such polygons, so adjacent with respect to this subdivision by lines which are orthogonal to the line connecting them. So note that this direction orthogonal to this line connecting this edge is precisely the direction into which one of the edges of the tropical curve will go. Because if there is such an edge connecting them, there will be a piece of the tropical curve orthogonal to that direction. So if we do this, so we take a point in each of them, in each of these and we connect them by lines orthogonal to the edges of the subdivision, we will get a graph. And now we have that the edges of our original tropical curve which is zero set of F have precisely the same directions as the edges of this curve that we got here by the subdivision. And with a bit more thought, we also see that the incidence of the edges is the same. So here in this, these three edges here meet in this point B2. And these three edges with the same directions meet in one point and so on. So we find that this curve gamma zero that we get from the subdivision is equal to our original tropical curve gamma except for the lengths of the edges and the position of the curve in R2. So therefore this gamma zero describes perfectly the shape of gamma. So if we do this thing and we make the corresponding subdivision we get our tropical curve gamma except that it could be stretched in some directions and it could be put anywhere in the plane. Okay, so now we want to squeeze out a little bit more out of this. So we have found this. We want to squeeze out a little bit more information about the shape out of this observation. Namely, we want to now get to the balancing condition. So let's concentrate on one of the polygons of the subdivision. Here we take this lower one. So if we just take the difference, so assume we have A1 B1 until EK BK are the vertices of this polygon, say clockwise. So if we take the difference AI BI minus AI plus one BI plus one, if I sum them all up, it just means I go once around the triangle. So the sum of these vectors must be zero because it's just going once around the triangle. Now we can also define this BI which is BI minus BI plus one AI plus one minus AI. This will be the number of vectors minus AI. This will be an outer normal vector to the triangle. So at each, so this one, this BI will be orthogonal to the edge connecting these two points. And in fact, it is just the vector connecting the two vertices turned by 90 degrees. So therefore, if we sum up, if we take this VI's and sum them all up, we also get zero because we just go once around the original triangle turned by 90 degrees. Okay, so we get this condition that the sum of these vectors will always be zero. And so we can try to put this in a slightly nicer form. So in this particular case, we see that for instance, this vector B2 is minus one one B3 is one one and B1 is two times zero minus one and the sum of these is zero. At least I hope, yeah. And so we will write this now, this VI write as WI times UI, where UI is a primitive integer vector in the same direction. And WI call them multiplicity of the corresponding edge of the topical curve. And this is equal to what would be called the lattice length of the edge of this polygon. That is, if you go along the edge of the polygon, if you leave out the first one, how many lattice points do you encounter until you get to the end? So it's the same as the number of lattice points on the edge minus one, namely the point you started. So this is the lattice length. And so then we can transform this to translate this fact that the sum of the VI is equal to zero into so-called balancing condition. So for every vertex of our gamma, we look at the edges going in the outgoing directions. We can write the outgoing from the vertex. We look at all of them, outgoing from the vertex. You write them as VI equal to WI times UI, where UI is a primitive integer vector and WI we call the multiplicity of the edge, or the weight of the edge actually normally. Then we have the sum over all the outgoing vectors, WI times UI is equal to zero. That's the balancing condition of topical geometry. So in our example, we have these three vectors, V1, V2 and V3. V1 is equal to two times zero minus one. And the other one are these minus one one and one one. And the sum of these is zero. So this is in this picture. And we will assign the weight of the edge will be equal to the weight that we have here, the lattice length. So for instance, for our curve gamma zero here, corresponding to the subdivision, the weight of this edge is two because this edge of the triangle, the vertical edge of this triangle has the lattice length two. And it's the same here. So our corresponding tropical curve looks like this. So with one edge going horizontally to minus infinity, one vertical going to minus infinity, but each of them have weight two and then two going diagonally up. Okay. So, and there's, okay. And so let me just, and so to just finish this off. So the edges of delta D have, we have said that the edges of this triangle are union of the segments of some of the segments. So thus it follows that gamma zero has an edge which is orthogonal to this edge here. It has as many edges obviously as there are segments dividing it. But the lattice length of each of the edges of delta D is D. So therefore we get D unbounded edges of gamma or gamma zero in each of the directions minus one zero, which is orthogonal to the vertical thing, zero minus one and one one if you count them with weights. So in our particular case, the, you know, the curve has degree two. So counting with weights, there are two edges in each of the three directions. Okay. So this is, and so this I sum up here what we have found. A plane-topic curve to degree D is a balanced weighted graph. So balanced means we have this balancing condition and weighted means that edges have weights where we, which with edges of rational slopes with D counted with weight unbounded edges in each of the direction minus one zero, zero minus one, one one. And at each vertex V, they satisfy the balancing condition that the sum I equals one to K, WI UI is equal to zero, where the VI are the outgoing edges at V with weight, WI and UI is the primitive integer vector in the outgoing direction. Okay. So this is the description of plane-topical curves. Now I want to give some kind of elementary applications and analog of a standard result. I want to talk about the genus of topical curves. So an important invariant of a small objective algebra a curve over C is the genus. So it equals the number of handles of C. So it's also equal to one half the first petty number of C. So for instance, genus zero would be a sphere, genus one would be a torus and genus N would be a torus with N handles. And in addition, we want to look at the so-called geometric genus. So this is the genus of the normalization. So if the curve is singular, so like here it has a singular point, then we look at the genus of the genus of C is the genus of the normalization. So we are allowed to pull the singularity apart. So if you pull this part here, you get something homomorphic to a sphere. So the genus of this is zero, whereas this curve has genus one. Okay. So now I want to do the same for plane-topical curve. So the genus of a plane-topical curve is, in some sense, similar. G of gamma is just, now the dimension is just the first petty number of the topical curve. I mean, this is gamma. So not one-half, but gamma. So the number of independent cycles in gamma. So for instance, if I have a line like this, this has genus zero. If I have this curve, this cubic curve, there's genus one because we can see one cycle. And on the other hand, we also want to have something like the normalization. So if a cycle gets closed by intersecting two edges like here, we think of having kind of pulled them apart a little bit so that it has opened. So this thing would have genus zero. So a cycle doesn't count if it's done by intersecting edges. So this would be the genus of a plane-topical curve. So it's a standard theorem that if C is a smoothly G complex curve in P2, then it's genus of degree D, then it's a genus is D minus one times D minus two by it by two. And if C is singular, the genus is smaller. So this is a standard elementary result. Now we want to see whether similar things hold for a topical curves. So first of all, we should have some idea what it means for a topical curve to be smooth because obviously it's not smooth in the obvious sense. It's a locally linear graph. So if we, so we make the following maybe slightly ad hoc definition, if V is a trivalent vertex of a plane-topical curve and we have the corresponding unit outgoing vectors along the edges of V at V with weights W1, W2, W3, we pick any two of them of these outgoing vectors because of the balancing condition, what we define is independent of which two we pick. Then the multi-mikardian multiplicity at V is M of V, which is the part of the weights times the absolute value of the determinant of the matrix we get by putting next to each other these two vectors in R2. So this will be, this is a matrix with integer entry. So this will be an integer and it takes absolute value. So it's a positive integer, because they are linear independent, it's positive and then we multiply by this. So we get a positive integer. So in other words, we take the parallelogram spent by the vector W1 times U1 and W2 times U2 and we take its area. This is the Mikardian multiplicity. Incidentally, if we look at our decomposition of our triangle delta D into polygons, we see that this parallelogram we see here is kind of twice the triangle. In the subdivision corresponding to the spec vertex. So, you know, turned by 90 degrees. So that means the Mikardian multiplicity of this vertex V is twice the area of the corresponding polygon in the decomposition of delta D into triangles. We will use this in a moment. Okay. And now if we do, I have said that the Mikardian multiplicity is this. So a plane topical curve, gamma of degree D is called non-singular if it is trivalent. So all vertices are trivalent. And for every vertex D of gamma, we have that the Mikardian multiplicity is one. So the area of this parallelogram is one. So a line, it's for instance, non-singular. You can see that this would be a non-singular cubic because if you just look at the angles and there's no, you will find that the Mikardian multiplicity is one. So now we want to show that if gamma is a smooth plane degree D topical curve, then its genus is D minus one times D minus two divided by two, the same for a smooth degree D curve in P two, you know, over the complex numbers. And if the curve is singular, its genus is smaller. Actually, we will only prove the first part but if you look carefully at the proof, you can easily show the second part. I mean, you can easily do it as an exercise. So we write gamma zero to be the number of vertices of our curves, gamma one, to be the number of bounded edges of our curve. Then I claim the genus of gamma is after the number of cycles is one plus gamma one minus gamma zero. I assume the curve is, the topical curve is connected. One plus gamma one minus gamma zero. How does one see this? Maybe you can maybe just see if it, if there's just one edge or something is obvious, but you know, and then if we add a cycle to our topical curve, it means, you know, we will add a certain number of edges and we add one vertex less. For instance, here, in order to add this cycle, we had to add one vertex and two edges in order to add this cycle, we have to add three edges and two vertices. You can easily see this always true. And so we get this form. Furthermore, we see that gamma has 3D unbounded edges. Because the mechanical multiplicity at every vertex is one that can be no edges with weights with a different format. Every vertex is trivalent, we have required that. And every bounded edge will connect two vertices. So that means the total number of bounded, the total number of edges. Let me see. So we have all together gamma zero vertices. Each of them has three outgoing edges. So from this point of view, there are three times gamma zero edges of which 3D are the unbounded edges. And then there are the bounded edges, which are gamma one, but each of them connects two vertices, so it counts twice. And so we get this formula. And finally, the area of the triangle delta D is D squared halves. We have just seen that the triangle corresponding to each vertex of V has a mechanical multiplicity one. So the area of the corresponding triangle is one half of that. So it's one half. So we find that the number of vertices is D squared. And so if we just put these three formulas together, you will immediately see that G of gamma is this number. Okay, so this is a very elementary proof of this fact by a simple combinatorial. So now in the, I want to briefly talk about tropical normative geometry, which is what kind of interests me about this subject. So let's talk about normative geometry of curves. You want to count curves satisfying suitable conditions. So we want to count curves in some space which satisfies some conditions so that there are only finding many. For instance, there are applications of such questions in, for instance, in string theory in physics, but also in other things, also in symplectic geometry. So let's look at it. So we look just maybe at the curves in P2. So the space of single curves in P2 of degree D and genus G, so we know if the curve is smooth then the genus is determined by the degree. But if the curve is singular, genus can be smaller. So one can check that in a suitable sense there is a space of such curves in dimension minus one. If we require such a curve to pass through a fixed, to a given point in the plane P2, this will cut down the dimension by one. And so if one does this carefully, one finds that if you take 3D plus G minus one general points in P2, there will be finitely many curves of degree D and genus G passing through all of them. And then you could ask yourself, how many are they? It seems interesting question and it is also somewhat difficult. So what is this now? And so this number is called the severity degree because it's the degree of the so-called severity and DG, so it's the number of degree D, genus G curves in P2 through 3D plus G minus one general points in P2. And one can prove that this number is independent of the choice of the points in P2 as long as the points are sufficiently general so that they're finitely. So for an instance, everybody knows that the N10 is equal to one. There's one line passing through two points. No, you can, as long as the two points are not the same, there will be one line connecting. And now, so these severity degrees are kind of difficult to determine also if one looks at other spaces than P2. So to determine the severity degrees, one uses advanced tools of algebraic geometry. This particular problem of computing the severity degrees was studied by, or solved by Kapoor and Harris. Some, I mean, I don't know, maybe 15 years ago. And but it uses some advanced tools of algebraic geometry to study these spaces parameterizing this curve. So you want to instead look at the same problem from the viewpoint of tropical geometry. So instead you want to count tropical curves. For instance, we see, at least that through general points in R2, there's a unique tropical line. So if you fix these two points, a tropical line always looks like this. One line going like this, one line going down, one going diagonally, depending on the relative position of the points, the points will either lie like this, or will lie like this, or they will lie like this. But there's always one tropical line passing through them. So like in algebraic geometry. So in general, if we want to count curves using tropical geometry, we have to count them with some multiplicity. And the multiplicity is the one we already defined, the Michalkin multiplicity. So we recall the definition of the Michalkin multiplicity. So if V is the trivalent vertex of a plane tropical curve gamma, and we have these three outgoing vectors, primitive integer vectors outgoing along the edges of V, at V, and with the corresponding weights W1, W1, W2, the Michalkin multiplicity is W1 times W2 times the absolute value of determinant of the matrix given by U1 and U2. And then if gamma is a trivalent tropical curve, so all vertices are trivalent, then the Michalkin multiplicity of gamma is M of gamma, which is the product over all vertices of gamma of the Michalkin multiplicity of the vertex. This will always be a positive integer because Michalkin multiplicity of each vertex is a positive integer. Can only be one if all Michalkin multiplicities of each vertex are one, otherwise it's bigger. Okay. And so now we can define the tropical severe degree as by counting the tropical curves through these. So given 3D plus G minus one general points in R2, there are finally many, so one can prove, there are finally many degree D genus G tropical curves gamma through all these points, PI, and all these will be trivalent if the curves are sufficiently general. And the tropical severe degree is then NDG chop, which is the sum over all these curves gamma of the Michalkin multiplicity of gamma. So gamma runs over all degree D genus G tropical curves through all the points PI. So again, the Michalkin multiplicity of the curve is the product over all vertices of the Michalkin multiplicity of the vertex. And now there's a theorem proven by Michalkin, which says that the severe degrees and the tropical severe degrees agree. So the severe degree is equal to the tropical severe degree. So this is proven by some kind of degeneration argument. I think there's another proof also by Siebert and Nishinu in a more algebraic setting. But this means that we can compute the severe degrees by tropical geometry. So by computing the tropical severe degrees, so by just counting these tropical curves and this is much easier than to do the algebraic geometry and studying these complicated modernized spaces of curves. Okay, so this allows us to do a number of geometry of curves via combinatorics. Now, still, maybe I can see a few words more. So I wanted to introduce one combinatorial tool to make this task of counting these tropical curves a little bit easier so that one can get some kind of feeling. Of course, I mean, if you imagine you have general points in the plane and you want to count all the tropical curves through them, that might still be not so easy. You have to kind of think how these tropical curves can look like. So counting tropical curves is easier than counting complex curves. Still, the combinatorics is complicated. There's a combinatorial tool, which are the flow diagrams, which make the task a bit easier. And I want to really introduce this. So we want to count degree D in C to the curves to 3D plus G minus one general points in R2. So now we can choose the points are supposed to be in general position, but they have to be in general position from the tropical point of view. And from the tropical point of view, they are in general position even if they all lie on a line with a very small irrational slope and they are stretched out very widely along this line. So here you can see this line. Here we have these points. And now we look at the tropical curves. See in this case, it's degree two passing through them. And if in this case, that the point conditions are like this, gamma will have a very special shape. It has a so-called floor decomposition. I will briefly explain what this is. So it is decomposed in some horizontal edges. So here we have, and if you throw away the horizontal edges, then we have certain components. The horizontal edges we call escalators because traditionally one would think of the whole thing 20 by 90 degrees. And so the escalators go up and the connected components of the rest are called the floors. And the following properties hold. Every floor and every escalator contains precisely one mark point, as we can see here. So only the escalator, so the horizontal lines can have weights different from one and any vertex has multiplicity. Mikaik multiplicity one, unless it's adjacent to an escalator, whose multiplicity is, whose weight is not one, in which case the multiplicity of the vertex is the weight of the edge. So one can show that the shape of the curves will be like this. So let's look at this. We look at a tropical curve through a horizontal stretch configuration of points and we have the associated. And you want to associate to that a so-called floor diagram. So the escalators are the horizontal segments and the floors are the connected components of the complement. So in blue I have surrounded the floors. So there's one mark point on every floor and one on every escalator. So we put one black vertex for every escalator, more or less at the position of the mark point. I mean at the horizontal along the line of the black, of the corresponding mark point. And the white vertex for every floor at the position of the mark point along the line of that. So for instance, to this tropical curve we get this diagram. So we connect the floor to an escalator if the corresponding, we connect a black dot to a white dot if the corresponding floor escalator is connected to the floor. So in this case, the diagram corresponding to this curve is this. And we keep the weights of the escalators. So we have here, we place this tropical curve by this simpler graph. And so, and then to count the tropical curves we can instead count the floor diagrams. There are some rules which follow from our definition. Every bounded edge contains a black and a white vertex. Every unbounded edge connects to a black vertex. I mean, every edge that ends with a black word, I mean, which ends on the left will end with a black vertex. And every black vertex is connected to two edges, one incoming from the left and one outgoing to the right of the same way. And white vertices can have several incoming outgoing edges. So the number of incoming edges is always one bigger than the number of outgoing edges if you count them with weight. So here we have an example of a floor diagram of degree three. So the floor diagram of degree D and genus G, so corresponding to a degree D, genus D tropical curve has D incoming edges of weight, one no outgoing edges and G cycles. So this is a degree three weight floor diagram of genus one. And so, and then to count the tropical curves, you instead counted the floor diagrams. And so we get the tropical count is just the sum of all floor diagrams of degree D of genus G of the multiplicity of the floor diagram. And this is just the product of all edges of the weight of the edge. And so we apply this to count use floor diagrams to see how we can count degree three of genus zero in the plane. So what is the number of degree three genus zero curves in the plane through eight general points. And so we have to look at all possible floor diagrams. If you think of it the moment, there are three different ways how the floor diagram can look like. Either we can have first two edges come together in a white vertex, and then this will afterwards join with another white one. And then it goes like this. And now how many different possibilities of this? So the different floor diagrams correspond to the different ways how these points can be distributed along the line where the points were supposed to go. So there are five positions for P. So this curve counts as five. The kai-ken plus multiplicity is one because all edges have multiplicity one. So these are five curves. Then we can look at this case. First three edges come together and then it split up into two branches like this. Now we're going to look at the kai-ken just like this. Now one should remember that here I've written one of them up and one of them down, but they are undistinguishable. So the possible cases is how are these two points divided between these two things? So I have to divide these four positions into two parts with two. And so there are three possibilities. Either the points five and six are on the first and seven and eight are on the second and so on there, these three. So this counts for three. And the final possibility is that first all three come together, but then we have an edge of way two going out and then going on. And then it finishes like this with an edge of weight one. And so in this case, we see we have two edges of weight two. And so the kai-ken multiplicity is two times two is equal to four. But here in this case, you have no choice how the points can be, they are all kind of lined up. So therefore we have one possible curve that counts with multiplicity four. So this counts for four. So the total number is five plus three plus four is equal to 12. And so this is also a classical result by going back to the beginning of the 19th century that this number is indeed 12. So that we found here. Okay, that was all I wanted to say. Maybe here's the list of references which you can then consult if you want to know more. Okay, thank you. That was all I wanted to say. Very nice, Lothar. Very nice lecture. Should we open a little bit for questions, Klaudia, or should we finish the video first? I cannot resist, Lothar, but ask you two questions before we finish. First one is that I noticed in the first slides that this was motivated by a science scientist, a computer scientist, right? So do you know what was his original motivation for studying this variation of the algebra where the sum is by the max and the multiplication is changed by the sum? And how do people came up with the... Yeah, no, unfortunately, I don't know. I didn't study it, so I cannot tell. I mean, the thing is he studied this algebra and it was some question in maybe theoretical computer science that he... But I... I mean, the reference I've put here, so... And this was around what time? In the 80s? Yeah, that was in the 80s, yeah. So tropical geometry is relatively new, yeah. Very nice. Yeah, and yeah. And so there is... If you look at the internet, you can also find a little movie which unfortunately is almost free of mathematical content of some kind of self-advertisement of the economy department of Oxford where some economics professor tells how he uses tropical geometry to design options. But he doesn't actually explain anything. He just very proudly says that he uses such modern tools as tropical geometry. So there might be another movie somewhere where more details are given, but anyway. Very nice. The other curiosity that I had in listening to your talk was how fast, I mean, the last computation that you showed for giving a number of points in genetic position and computing the curves with the given genes and degree, how does this grow computationally? If I give you a big number, is this computationally feasible or do things grow really fast to calculate these floor diagrams? I mean, if you organize carefully what you're doing, it leads to recursive recursion formula. And I kind of programmed this recursion and maybe you can easily with the computer, if you are willing to wait maybe a couple of hours, compute maybe until everything until 40 or 50 or something. So, you know, these are huge numbers, but you can, it's quite suitable for computation. I mean, also on the laptop. I mean, I've, I mean, there is this, Caporazo-Harris gives some kind of recursion formula and you can prove the same recursion formula using tropical geometry. But this was completely say inaccessible before this analogy between the tropical count. No, no, no. The original proof here by Caporazo-Harris is using algebraic geometry. So you use some deformation theory and so on. And they also have these, they also get some strange multiplicity which come from some deformation theory. But in the tropical geometry, they actually come from this macaulking multiplicity. So there's much easier to understand. It's a quite difficult paper, but you can do it with the tropical geometry. But tropical geometry has also other applications. So if it was just to improve this, this would maybe be so bad. One thing is for instance, I didn't have time to mention that you can also count real curves. So if you want to know how many of these curves are real, there's a so-called version change in variance which count the real curves with certain signs and gives some therefore lower bounds for the actual number of real curves. And you can show that these can also be computed using tropical geometry. And this gives very similar recursion formula for the real curve. And so that's something which was, I think not known before like that. So that comes from the tropical geometry. And anyway. Nice. Claudio, anything you want to add or comment as of now? Yes, no. I had a curiosity more, very, very, I mean, very basic, but you are kind of almost answered now, but I mean, passing from the complex now, I mean the theorems that you are connecting between the classical algebra geometry and tropical geometry seems to connect between complex algebraic geometry to this very strange structure of the real numbers. Which is for a foreign mathematician is very strange. I mean, is it possible to say something about, I mean, why does it work? I mean, so at least I can give some kind of, I was, if I had more time, I would have explained, tried to say more in my lecture, because so you can, the connection, I mean, at least in the Nicarician point of view comes from what they call amoebas. So if you have, say a complex, so assume we are in the plane, so you have a complex curve. So then you take the logarithm of the, so you have F as a polynomial, you take the logarithm of the absolute value of F. If you look at the, if you look what you get here, it will be so-called amoeba. So it's a sort of form with some kind of tentacles going to infinity in the same directions. Now, if you kind of let the base of the logarithm go to infinity, this thing comes thinner and thinner and the final limit is a tropical curve. And so you can somehow see that going from the, I mean, that's at least some motivation that going from the algebraic geometry to the complex geometry is a, can be viewed as a limit process. That's one way of seeing it. There's also an algebraic way, which is quite different, which is via working, if instead of working over the complex number, you work over the field of posterior series. So power series with possibly rational coefficients, then you can define what is called a tropicalization. So if you have a zero set of some polynomial in this thing, you can somehow, and I don't remember precisely how it goes, but you just remember the valuation of the corresponding terms with, the variation is meet which power of the variable. So field of posterior series means you have K of X to the one over N for all possible values of N. So all of these added. And so the valuation of X to the one over N is maybe one over N. And so if you now have a usual polynomial, you can take the tropicalization of H, which means you just keep kind of the, in a suitable sense, the leading term of everything, just the valuations of the leading term. And this will give you, will associate to, so this gives you some number. And now you can somehow see that if you take all the points on the corresponding complex curve and take their valuation, and not the field over the curve over the field of fugitive series, you take the variation in the two variables, say, then so each of these points will give you a point in the real plane. So actually with rational coordinates. And so if you just look at those, you'll get some kind of image. If you now take the closure of that in the plane, this will be the tropical curve associated to the thing. So it's just the set of all valuations of the points on the curve, however closed up so that it's actually something. Again, you have this limit procedure. So that's another way how you can get the tropical curve from the curve over in algebraic geometry. And I mean, yeah, anyway, and this is very close to another description in terms of what you call Berkowitz spaces, which maybe I don't want to. Very nice. Let's thank Lothar again. We get this very brilliant, very nice lecture and I think where we wrap up our presentation today. Thank you very much, Lothar. Okay, bye.