First introduced by Calabi, and also described by Cheeger in one of his famous papers, these dumbbell-shaped surfaces are a very well studied example in Spectral Geometry. The first eigenvalue of the Laplacian of these surfaces is proportional to the radius of the cylinder joining the two spheres. Thus, considering dumbbells with very thin cylinders, we obtain surfaces with arbitrarily small first eigenvalue.

In view of Lichnerowicz's theorem, this example shows that to find a positive lower bound for the first eigenvalue, one needs some kind of curvature assumption. Together with Kamryn Spinelli and Connor C. Anderson, two undergraduate students at Worcester Polytechnic Institute, we have been exploring this example in the context of integral curvature assumptions. Our exploration lead to a joint publication at PUMP Journal of Undergraduate Research, and the students presented our findings at the 2021 MAA MathFest and in the 2022 Joint Mathematics Meeting.

In view of Lichnerowicz's theorem, this example shows that to find a positive lower bound for the first eigenvalue, one needs some kind of curvature assumption. Together with Kamryn Spinelli and Connor C. Anderson, two undergraduate students at Worcester Polytechnic Institute, we have been exploring this example in the context of integral curvature assumptions. Our exploration lead to a joint publication at PUMP Journal of Undergraduate Research, and the students presented our findings at the 2021 MAA MathFest and in the 2022 Joint Mathematics Meeting.

Lichnerowicz's theorem and Zhong-Yang's theorem (sharpening an estimate of Li and Yau) give positive lower bounds for the first eigenvalue of the Laplacian for closed manifolds, assuming that the Ricci curvature is positive or non-negative. The pink regions in the dumbbell above contain negative curvature and are ultimately responsible for the lack of a lower bound.

In our joint article with Seto, Wei, and Zhang, we proved a Zhong-Yang type eigenvalue estimate for manifolds with a small amount of negative curvature (in an integral sense). Since the example above doesn't have a lower bound, it's clear that it must have a "large" amount of negative curvature (in integral sense). But**is that amount uniformly bounded as we consider thinner and thinner cylinders?** If so, this shows that "smallness" of the integral curvature is a necessary condition and can not be replaced by "bounded" integral curvature.

In our joint article with Seto, Wei, and Zhang, we proved a Zhong-Yang type eigenvalue estimate for manifolds with a small amount of negative curvature (in an integral sense). Since the example above doesn't have a lower bound, it's clear that it must have a "large" amount of negative curvature (in integral sense). But

This is the question that we addressed with Spinelli and Anderson. We constructed explicitly a sequence of smooth dumbbell surfaces and estimated their integral curvature, showing that it remains uniformly bounded as the cylinders become thinner. Although this question had already been answered by Gallot, we approached the problem with the most elementary tools that we could find: an approach that doesn't require any background in the field. The main challenge was to find a way to smooth out a profile curve to generate a surface of revolution in such a way that we would be able to estimate the integral curvature afterwards.

After almost giving up a couple of times, and having tried several approaches, we managed to successfully construct the example explicitly. We wrote the result in an article that has been published by PUMP Journal of Undergraduate Research, and Spinelli and Anderson presented our findings during the 2021 MAA MathFest and in the 2022 Joint Mathematics Meeting. They are currently pursuing their PhD at Brandeis University and at Northeastern University.

However, sometimes it is hard to explain what Algebra does for us (and here by Algebra I mean actual Algebra); and the truth is that Algebra is beneath every single one of these fields! In this post I want to explain one of my favourite theorems, how its roots are essentially algebraic (although it is a theorem in combinatorics), and how it can be applied to the real world: Pólya enumeration theorem.

If I were to ask you in how many ways can you put in line 4 people, assuming that you have some basic notions of combinatorics, you are quickly going to say that you can do so in 4! =4·3·2·1 possible ways (we can choose the first person among 4 people, the second one among 3, the third one among 2, and then we are left with a single choice for the last one).

What happens, though, if I tell you instead that you want these 4 people to sit around a round table and that you want to know in how many*different* ways they can do so? Here the key point is what we mean by *different*: since the table is round, some different ways of arranging the 4 people in line become the same when they sit down around the table. For example, if these people are Alan, Berta, Claude, and Diana (A, B, C, D for short), then ABCD and BCDA are different ways of putting them in line, but when they sit around a round table in this order, the two give rise to the same way of sitting together. The reason is that in both situations, A is always sitting to the left of D, B is sitting to the left of A, C sits to the left of B, and D sits to the left of C (see the picture below).

What happens, though, if I tell you instead that you want these 4 people to sit around a round table and that you want to know in how many

What's so different about the two problems? Well, in the latter, we are considering as the same solution situations that are related by a rotation; in other words, if you can arrange the people in two ways around the circle, but by rotating the first table you can get the second one, then we are saying that the two configurations are the same. In mathematics, when something like this happens, we say that there are symmetries associated to the problem.

And guess how mathematicians study symmetries? Using Algebra! Well, here is the thing: the symmetries associated to this problem are all the possible rotations of the table; however, around the table there are only 4 chairs, so basically we only care about the rotations that move one chair to the other, i.e. rotations of 0º, 90º, 180º, and 270º.

Notice that if we do two of these rotations, we always get another rotation; for example, a rotation of 90º followed by a rotation of 180º gives us a rotation of 270º. Also, every rotation has an inverse, i.e. a partner such that when we do both of them consecutively the result is the table that we had initially, as if we had not done any rotation at all; for example, a rotation of 270º followed by a rotation of 90º gives us a rotation of 360º which is the same as a rotation of 0º (i.e. no rotation at all).

And guess how mathematicians study symmetries? Using Algebra! Well, here is the thing: the symmetries associated to this problem are all the possible rotations of the table; however, around the table there are only 4 chairs, so basically we only care about the rotations that move one chair to the other, i.e. rotations of 0º, 90º, 180º, and 270º.

Notice that if we do two of these rotations, we always get another rotation; for example, a rotation of 90º followed by a rotation of 180º gives us a rotation of 270º. Also, every rotation has an inverse, i.e. a partner such that when we do both of them consecutively the result is the table that we had initially, as if we had not done any rotation at all; for example, a rotation of 270º followed by a rotation of 90º gives us a rotation of 360º which is the same as a rotation of 0º (i.e. no rotation at all).

When we have a set with these* properties, we say that the set is a Group. Algebra studies groups, among other mathematical structures; when a problem has symmetries, in most of the cases there is a group hidden somewhere. Groups are defined in a completely abstract way; in our case, the group in consideration is called the Cyclic group of order 4, which is a group generated by a single element (the rotation of 90º) such that when the element is applied 4 times in a row we get back to the initial configuration (and not before we apply it 4 times).

Although groups are very abstract: "sets with an operation that satisfies certain properties", all the finite groups can be thought of as a group of permutations (see Cayley's theorem for a precise statement). This means that we can always think of them as ways of rearranging a certain amount of elements. In the example above, the Cyclic group of order 4 can be thought as a group of permutations of 4 elements, i.e. elements of the Symmectric Group of 4 elements.

The advantage of thinking about the elements of the group as permutations is that this allows us to use the cycle notation. Consider the rotation of 90º in the example above. This rotation sends A to B, B to C, C to D, and D back to A; as a short notation, we can express this information as the permutation **(ABCD)**, where each letter inside the parenthesis is sent to the letter to it's right (and the last letter in the parenthesis is sent to the first one, hence the name "cycle notation"). Similarly, we can represent the rotation of 180º as **(AC)(BD)**, because, after a rotation of 180º, A takes the place of C and C takes the place of A (similarly for B and D). Notice that in this case, there are two independent cycles, both of length 2, while in the 90º case, there was only 1 cycle of length 4. The last two elements of the group are the rotation of 270º, which would be represented by **(ADCB)**, which has again just 1 cycle of length 4, and the rotation of 0º, which would be represented by **(A)(B)(C)(D)**, giving us 4 "cycles" of length 1.

Pólya's enumeration theorem uses precisely this information: take the group of symmetries G of your counting problem, and write its elements in cycle notation. Then count how many elements are there with each possible combination of cycle lengths, and use this information to construct the cycle index polynomial. In our case we have 2 elements with 1 cycle of length 4 (\(2x_4\)), 1 element with 2 cycles of length 2 (\(1x_2^2\) ), and 1 element with 4 cycles of length 1 (\(x_1^4\)). Dividing this by the order of the group, one gets the cycle index:

$$P(x_1,x_2,x_3,x_4) = \frac{1}{4}(2x_4+x_2^2+x_1^4)$$

$$P(x_1,x_2,x_3,x_4) = \frac{1}{4}(2x_4+x_2^2+x_1^4)$$

Now, this polynomial P is an algebraic object: it contains all the information about the symmetries of the problem, but it doesn't know what we want to count yet. At this point, we need to introduce the idea of "colorations". What we want to do is to "assign colors" to the different seats around the table (in our case, "labels" or "names"). We have 4 colors: A,B,C,D. We are not interested in any possible coloring, but only the colorings where each color appears exactly once. What we need to do to find out how many colorings are there of this sort, is to compute the polynomial:

$$F(a,b,c,d):=P(a+b+c+d, a^2+b^2+c^2+d^2, a^3+b^3+c^3+d^3, a^4+b^4+c^4+d^4)$$

where each variable \(a,b,c,d\) corresponds to one of the colors. The number in front of the monomial \(abcd\) will be the answer to our problem: the number of different colorings with each color appearing exactly once, taking into account the symmetries of the problem. In this case, expanding with the help of wolfram alpha (since the expansion has 35 terms) the polynomial:

$$F(a,b,c,d) = \frac{1}{4}((a+b+c+d)^4+(a^2+b^2+c^2+d^2)^2+2(a^4+b^4+c^4+d^4))$$

we see that the coefficient of \(abcd\) is 6. Hence, there are 6 possible ways of sitting around the table.

$$F(a,b,c,d):=P(a+b+c+d, a^2+b^2+c^2+d^2, a^3+b^3+c^3+d^3, a^4+b^4+c^4+d^4)$$

where each variable \(a,b,c,d\) corresponds to one of the colors. The number in front of the monomial \(abcd\) will be the answer to our problem: the number of different colorings with each color appearing exactly once, taking into account the symmetries of the problem. In this case, expanding with the help of wolfram alpha (since the expansion has 35 terms) the polynomial:

$$F(a,b,c,d) = \frac{1}{4}((a+b+c+d)^4+(a^2+b^2+c^2+d^2)^2+2(a^4+b^4+c^4+d^4))$$

we see that the coefficient of \(abcd\) is 6. Hence, there are 6 possible ways of sitting around the table.

In Spring 2017, I conducted an undergraduate research project on this topic (see the group pic that we took above). After understanding the theorem, we investigated several examples, showing how the cyclic index depends on the group action, and not only on the group of symmetries.

Because of its interdisciplinary flavour, between combinatorics and algebra, with applications to chemistry, graph theory, and statistical mechanics, it's a very attractive project for undergraduate students. I believe it could even be studied at an elementary level by high school students, at least in its simpler versions (like Burnside's lemma). In fact, in countries where students must write a research-like dissertation to finish high school (like in Catalonia), I think that this topic would be an exciting way of discovering modern mathematics, beyond the school level, helping to make an informed decision about whether to pursue a math degree in college.

Last summer I participated in a summer school on Geometric Analysis in the beautiful town of Como, Italy. A part from having a great time and meeting a lot of mathematicians, I was introduced, by professor Christina Sormani (CUNY), to a new notion of convergence of metric spaces: the Sormani-Wenger Intrinsic Flat Convergence.

What is this all about, you might be asking? Well, before jumping on this rather involved concept, let's try to understand what is a metric space and what do I mean by convergence of metric spaces.

What is this all about, you might be asking? Well, before jumping on this rather involved concept, let's try to understand what is a metric space and what do I mean by convergence of metric spaces.

Geometry, as its name suggests, is the science of measuring things on Earth, i.e. quantifying the distance between objects. In modern mathematics, there are many different subfields under this name, and not all of them deal with the notion of distance. However, one of them, called Metric Geometry, remains having *distance* as its main object of study.

A set with a distance is called a*Metric Space*. There are many different examples of metric spaces: the most basic ones are the real line, the plane, or the three dimensional space with the usual (Euclidean) distances. However, these notions of distance are not always useful. For instance, if you are a taxi driver in Barcelona, and you want to know how far is Sagrada Família from Plaça de Catalunya, the usual distance between two points in the plane is not going to be the actual distance that you are going to drive, since cars, like humans, can not go through buildings. Instead, you will probably need to use the Taxicab distance, which computes the distance between two points on a grid, like the streets in l'Eixample of Barcelona.

A set with a distance is called a

To be able to make sense of this new notion of distance in the plane, mathematicians needed to generalize the definition of distance; they did so by keeping only the key ingredients in the intuitive notion of distance:

Given a set S, a distance function in S is a map that assigns to each pair of points (elements in S) a number, and that satisfies:

Given a set S, a distance function in S is a map that assigns to each pair of points (elements in S) a number, and that satisfies:

- The distance between two points is always non-negative, and it is zero only when the two points are actually the same point.
- The distance between A and B is the same as the distance from B to A.
- The distance from A to C is, at most, the distance between A and a third point B plus the distance from B to C.

My favorite Metric Spaces are the so called Riemannian Manifolds. A manifold is a generalization of the notions of curve and surface: they are n-dimensional smooth spaces. If this idea scares you, just think of your favourite surface; for example, a sphere! We live in a spherical planet, and we talk about distances between cities. However, the distance that we are interested in is not the distance between the two cities in the 3-dimensional space where we live in (because to go from one city to the other we can not make a tunnel through the Earth!). Instead, we want to know the shortest distance between them when travelling on the surface of the Earth. This notion of distance is called Riemannian distance, or geodesic distance.

Notice that the definition of distance above is extremely general: it allows us to define distances in any set! For instance, one can define distance in a graph, or in a set of functions (see for example the distance induced by the uniform norm in the set of continuous functions).

Once a distance is specified in a set S, we can define the notion of convergence. You might be familiar with the notion of convergence of numbers: consider the sequence 1, 1/2, 1/3, 1/4, 1/5, ... ; we can write it in general as x_n = 1/n. It is a decreasing sequence of positive numbers which approaches x = 0 when n goes to infinity. What we mean by that is that the distance between x_n and x becomes as small as we want by picking n large enough.

Similarly, given a sequence of elements x_n in the metric space S (i.e. an enumerated collection of elements x_1, x_2, x_3, etc.), we say that the sequence converges to an element x of S if the distances from x_n to x are as small as we want, for n large enough.

Different notions of distance give us different interpretations of convergence. In general, having a sequence x_n converging to an element x tells us that the elements x_n are becoming more and more similar to the element x (in some sense that changes depending on the chosen notion of distance!). For example, what does it mean that a sequence of functions f_n converges to f? In the uniform norm, this means that the functions f_n are approaching f at every point with "the same speed" (see the image below), i.e. uniformly in all the domain.

Once a distance is specified in a set S, we can define the notion of convergence. You might be familiar with the notion of convergence of numbers: consider the sequence 1, 1/2, 1/3, 1/4, 1/5, ... ; we can write it in general as x_n = 1/n. It is a decreasing sequence of positive numbers which approaches x = 0 when n goes to infinity. What we mean by that is that the distance between x_n and x becomes as small as we want by picking n large enough.

Similarly, given a sequence of elements x_n in the metric space S (i.e. an enumerated collection of elements x_1, x_2, x_3, etc.), we say that the sequence converges to an element x of S if the distances from x_n to x are as small as we want, for n large enough.

Different notions of distance give us different interpretations of convergence. In general, having a sequence x_n converging to an element x tells us that the elements x_n are becoming more and more similar to the element x (in some sense that changes depending on the chosen notion of distance!). For example, what does it mean that a sequence of functions f_n converges to f? In the uniform norm, this means that the functions f_n are approaching f at every point with "the same speed" (see the image below), i.e. uniformly in all the domain.

In metric geometry, one of the main concerns that we have is to know when two spaces X and Y are isometric. This means that one can get from one space to the other by a one to one and onto map that preserves distances (called an isometry). An example is illustrated in the picture below: consider X to be the square of vertices ABCD and Y to be the square with vertices EFGH. We can consider them as metric spaces (with the distance function inherited from the usual distance function in the plane). As sets, X and Y are completely different things (they contain different points). However, we can get from X to Y by using a rotation and a translation; rotations and translations preserve distances, hence X and Y are isometric as metric spaces.

In other words, two metric spaces are isometric if they are "the same" from the point of view of metric geometry (the two spaces in the example below are both squares of side 2, so in that sense, they are both "the same" object).

In other words, two metric spaces are isometric if they are "the same" from the point of view of metric geometry (the two spaces in the example below are both squares of side 2, so in that sense, they are both "the same" object).

Now, here is the **key idea**: wouldn't it be great if we could define a notion of distance between metric spaces in such a way that convergence in that distance means that the metric spaces are getting closer and closer to being isometric (i.e. being "the same" metric space)?

Yes, you heard me right: I am saying that it would be awesome to consider the space of all metric spaces, and to try to define a distance function in that space (so we would be measuring the distance between two metric spaces, not between the points in the metric space). The idea is that we could be asking ourselves what is the distance between any two sets where we have defined a distance: like the distance between the sphere and the plane.

Not only that, but we would like to define a notion of distance in such a way that two metric spaces are at distance zero exactly when they are "the same metric space" (i.e. isometric). So, for instance, the distance between a sphere and a plane would have to be always strictly positive, since these two spaces are not isometric.

This is exactly what the notion of Gromov-Hausdorff distance (first defined by M. Gromov, in the picture below, based on the Hausdorff distance) does: it defines a distance between any two (compact) metric spaces in such a way that the distance between two spaces is zero only when they are isometric.

Yes, you heard me right: I am saying that it would be awesome to consider the space of all metric spaces, and to try to define a distance function in that space (so we would be measuring the distance between two metric spaces, not between the points in the metric space). The idea is that we could be asking ourselves what is the distance between any two sets where we have defined a distance: like the distance between the sphere and the plane.

Not only that, but we would like to define a notion of distance in such a way that two metric spaces are at distance zero exactly when they are "the same metric space" (i.e. isometric). So, for instance, the distance between a sphere and a plane would have to be always strictly positive, since these two spaces are not isometric.

This is exactly what the notion of Gromov-Hausdorff distance (first defined by M. Gromov, in the picture below, based on the Hausdorff distance) does: it defines a distance between any two (compact) metric spaces in such a way that the distance between two spaces is zero only when they are isometric.

This notion of distance has been used a lot in the last 40 years, among other things, to study sequences of smooth manifolds and their limit spaces (this is what I do in my research!). For example, a sequence of spheres of radius 1/n converges in the Gromov-Hausdorff sense to a point. If we fix a point p in the sphere, and we study the points nearby (say the points at distance 1 from p), and then we let the radius of the sphere increase to infinity, it turns out that the sequence of neighborhoods of p converges to a neighborhood of a point in the flat plane in the Gromov-Hausdorff sense, i.e. a very large sphere is almost the same, as a metric space and near any fixed point, as a plane (that's why plane geometry works fine for our everyday lives, even if we live in a sphere!).

However, there are other sequences of metric spaces that we would like to converge in the GH sense (short for Gromov-Hausdorff) and that do not converge! Christina Sormani explains them very well in her article: consider a sphere with a smooth cusp. At every step, add more cusps, but make them thinner and thinner, always of the same length. After several steps, we get something that would look like a sea urchin. Tom Ilmanen proved that this sequence of "hairy spheres" (or sea urchins) does not converge to a sphere in the GH sense; however, he conjectured that the sequence should converge in some weak sense.

However, there are other sequences of metric spaces that we would like to converge in the GH sense (short for Gromov-Hausdorff) and that do not converge! Christina Sormani explains them very well in her article: consider a sphere with a smooth cusp. At every step, add more cusps, but make them thinner and thinner, always of the same length. After several steps, we get something that would look like a sea urchin. Tom Ilmanen proved that this sequence of "hairy spheres" (or sea urchins) does not converge to a sphere in the GH sense; however, he conjectured that the sequence should converge in some weak sense.

In 2010, Christina Sormani and Stefan Wenger came up with a new notion of convergence between (a special kind of) metric spaces, called the Intrinsic Flat Convergence, in which this sequence of hairy spheres converges to the usual sphere.

So to the question: is a sea urchin a ball? The answer depends on your metric, tell me how you measure distances, and I will tell you whether they are the same or not!

]]>So to the question: is a sea urchin a ball? The answer depends on your metric, tell me how you measure distances, and I will tell you whether they are the same or not!

When a mathematician reads something like this, after realizing that it is a nice result, the first question that comes to his mind is: is it possible to construct a finite set of n points in the plane that do not lie in a straight line, and whose distances are all integers?

Well, it is easy to see that one can do that for n=3 (just consider any Pythagorean triple, such as the triangle with sides 3, 4, and 5). But is it possible to do that for an arbitrary n? The answer is not obvious at first, unless you have thought about the problem before (although that this is probably a very well established result). Actually, the proof of the statement above, called the Erdös-Anning theorem, was published in 1945, and the proof already contains a solution to the case with finitely many points. However, although beautiful, the example that the paper provides is overcomplicated, from my point of view.

Plus, I didn't know the answer to this question, and I realized that many of my fellows didn't know it either, so I started to try to construct examples of sets of non-aligned points with integer distances between them.

From the example of the triangle, it was easy to construct a set of n=4 points with this property, just by considering the rectangle of sides 3 and 4 (whose diagonal has length 5). But again, being able to do it for n=4 points doesn't mean that it can be done for any number of points, and I didn't see how to generalize that construction.

Plus, I didn't know the answer to this question, and I realized that many of my fellows didn't know it either, so I started to try to construct examples of sets of non-aligned points with integer distances between them.

From the example of the triangle, it was easy to construct a set of n=4 points with this property, just by considering the rectangle of sides 3 and 4 (whose diagonal has length 5). But again, being able to do it for n=4 points doesn't mean that it can be done for any number of points, and I didn't see how to generalize that construction.

After n=4, my goal was to construct a set with many other points. One idea that came to my mind was to consider a regular hexagon inscribed in a circumference, together with its center, a set with n=7 points. I considered this example, because in order to have control on the distances, it is better to have as much symmetry as possible in our set of points. Unfortunately, I realized that it is impossible to construct a regular hexagon whose distances are all integers, no matter what the distance between two adjacent points is (the distance between two non-consecutive points in the hexagon is always an irrational number if the side of the hexagon is an integer).

Hence I decided to come back to the idea of using Pythagorean triples, constructing some kind of symmetric set of points. I realized that an easy way to construct a set of n=5 points was by using four triangles with sides 3, 4, and 5 to form a rhombus with sides of length 5. The vertices of the rhombus together with the intersection of the two diagonals are a set of 5 points with the desired property.

Hence I decided to come back to the idea of using Pythagorean triples, constructing some kind of symmetric set of points. I realized that an easy way to construct a set of n=5 points was by using four triangles with sides 3, 4, and 5 to form a rhombus with sides of length 5. The vertices of the rhombus together with the intersection of the two diagonals are a set of 5 points with the desired property.

With this construction in mind, I realized that if we add points in the big diagonal of the rhombus at some integer distance l from the center (i.e., from the intersection of the two diagonals), in such a way that its distance x to the points in the small diagonal is also an integer, then we get a construction with more than n=5 points that do not lie in a line and whose distances are all integers. Unfortunately, it is not possible to do that with the Pythagorean triple that I chose, because the smallest side of the triangle, 3, doesn't have enough divisors (it is prime).

That's why I decided to try this construction with a rhombus build from four triangles with sides 8, 15, and 17. Since we want the distances x and l to be integer, when applying the Pythagorean theorem to the triangle with sides 8, l, and x, we get x²=l²+8², hence 8²=x²-l²=(x+l)(x-l), so we have to consider all the possible solutions where x+l and x-l are factors of 8²=64. It turns out that this system has one non-trivial solution for l=6 (hence x=10), so we can add two points in the big diagonal at distance 6 from the center, creating a set of n=7 points with the desired properties.

That's why I decided to try this construction with a rhombus build from four triangles with sides 8, 15, and 17. Since we want the distances x and l to be integer, when applying the Pythagorean theorem to the triangle with sides 8, l, and x, we get x²=l²+8², hence 8²=x²-l²=(x+l)(x-l), so we have to consider all the possible solutions where x+l and x-l are factors of 8²=64. It turns out that this system has one non-trivial solution for l=6 (hence x=10), so we can add two points in the big diagonal at distance 6 from the center, creating a set of n=7 points with the desired properties.

Similarly, if the rhombus is build out of four triangles with sides 12, 35, and 37, with sides of length 37, we can add six points in the big diagonal at distances 5, 9, and 16 from the center respectively, creating a set of n=11 points with the desired property.

First of all, notice that any power of 2, say 2^k, is the length of the cathetus of a Pythagorean triangle, the Pythagorean triple being (2^(p+q+1), 2^(2p)-2^(2q), 2^(2p)+2^(2q)), where n=p+q+1 and p>q are integers. The way I came up with this triple is by using Euclid's formula to generate Pythagorean triples. Now, we can use the same method as before to add points in the cathetus of length 2^(2p)-2^(2q).

If a point is added at distance l from the center, and x represents de distance to the top (or the bottom) vertex, by the Pythagorean theorem we have 2^(2k)+l² = x², so we have the equation 2^(2k) = (x+l)(x-l). Since we want x and l to be integers, we have to solve the systems of equations x+l = 2^(2k-i), x-l=2^i, for i any integer between 0 and k (since we can assume x+l>x-l, restricting to l>0). This system always has a solution; however, some of the solutions are not valid for our purposes, namely when i=k (in which case l=0), when i=2q+1 (in which case we get a point that we already had), and when i=0 (in which case we get a non-integer). Hence, at the end of the day, we can add 2(k-2) points to the rhombus, which already has 5 points, so we get an example with n=2k+1 points. Of course, this is the case as long as all the solutions that we get are different; but this is something not too difficult to prove (you can see the details in the notes for the talk that I gave in the Math Club at UCR).

Hence, this proves that it is possible, using this construction, to get a set with any odd number of non-aligned points in the plain whose distances are integer numbers. By removing one of the points, we get the result for any even number of points, thus for any number.

After proving that, the Erdös-Anning theorem becomes even more interesting and mysterious!

Notes_Talk_DiscoveringATheoremByErdos_MathClubUCR.pdf |

General Relativity has always fascinated me; it is probably one of the main reasons why I decided to study mathematics and physics, and it is definitely an important reason for studying Differential Geometry and Geometric Analysis, even if my research is not focused in Einstein's equations. Why is it so fascinating? Well, that's what I am going to explain on Friday, in the Graduate Student Seminar (1pm-2pm, Surge 284, UCR). But in case you can not make it, let me explain you a few things about relativity.

By the end of the XIXth century, if you were a physics major you would be studying:

- Newtonian Mechanics (Galilean transformations, inertial observers...)
- Electromagnetism (Maxwell's equations)
- Thermodynamics / Classical statistical mechanics

Hence, in the beginning of the XXth century, physicists were trying to reformulate either one of the two theories, in order to make them compatible. Einstein decided to modify Newtonian Mechanics, as shown in his first famous paper of 1905; to do so, he had to forget about the idea absolute time and space (common for all the inertial observers), which are very natural ideas for the human brain (it seems reasonable that two inertial observers should measure the same amount of time between two events, and the same length between two points). But how can one do physics if two observers do not measure the same length intervals or times between two events? Well, using mental experiments, Einstein build his theory of Special Relativity from the following postulates:

- The laws of physics are the same for any inertial observer.
- The velocity of light, c, is a constant of nature.
- Space & time are not independent, they are together in a 4D space called spacetime.

Notice that the bilinear form, usually denoted by η, is symmetric but it is not positive definite! Hence, it is not defining a dot product, nor a metric in the space; however, it is non-degenerate, so we say that it defines a pseudo-metric. In this space, there are vectors with zero norm which are not the zero vector (those relating two events that can only be connected by an observer travelling at the speed of light), or even vectors with negative length (those between two events that can not be reached by any observer travelling at a velocity below or equal to the speed of light).

However, Special Relativity has several weaknesses:

However, Special Relativity has several weaknesses:

- It does not explain gravity: Newtonian gravity involves a vector field connecting instantaneously spacelike events (events separated by negative distances).
- How do we distinguish, a priori, between an inertial frame and a non-inertial frame?

Inspired by the equality between the inertial mass (the one in the formula F=ma) and the gravitational mass (the one that gives us the gravitational force mg), Einstein realized that there can not be a theory of gravity without a theory that allows observers that are being accelerated. Hence, he decided to build the Theory of General Relativity starting from the Equivalence Principle, that states that the laws of physics must hold for any observer (not only for a specific class of observers, namely the inertial ones); in other words, gravity is equivalent to accelerated frames, it is a geometric property of spacetime, and any motion described by a freely falling object (that might be accelerating with respect to us) is due to a gravitational field. This removes the philosophical problem that all the previous theories had, about distinguishing a priori between inertial and non-inertial observers. It also allows to build a much more geometric physical theory, were "arbitrary" (or smooth enough) transformations between observers are allowed; the requirement of the transformations between observers being isometries (transformations preserving η) vanishes.

Allowing arbitrary transformations implies that the velocity of light is not constant anymore, nor are the coefficients of the bilinear form η. However, locally we expect the General Theory of Relativity to reproduce the Special Theory of Relativity, so there is some sort of bilinear form at each point, that changes from point to point, but that in the right coordinates looks, locally, like η. That is to say, our spacetime is a manifold (we no longer require a flat spacetime), together with a pseudo-metric: an inner product (pseudo-dot product) that changes smoothly from point to point. This object is called a pseudo-Riemannian manifold, and in General Relativity spacetime is modelled by a 4D pseudo-Riemannian manifold.

So we have a pseudo-Riemannian manifold, whose geometry determines all the gravitational effects. We know that in Newtonian gravity there is a relationship between the mass distribution and the gravitational field in the space, so since we expect to reproduce Newtonian gravity at small scales, that means that there should be an equation relating the mass (or energy, equivalently) distribution in spacetime to the geometry. What kind of equations should we look for? Since they have to agree with Newton's theory of gravity, they should be second order differential equations involving, for example, vector fields. Moreover, according to the Equivalence Principle, they must not depend on coordinates (as physicists say, they must be covariant); this is very easy to understand from the point of view of Differential Geometry: the equation should be an equality between sections of some vector bundle, thus independent of the chosen coordinates (for example, an equality between vector fields, as defined on a manifold). More generally than just vector fields, we look for an equation between tensor fields.

On the left hand side we will have a tensor coming from geometry (the Einstein tensor), and on the right hand side we will have a tensor with the physical information (the Energy-Momentum tensor, already known in fluid mechanics). Einstein's tensor is constructed from Ricci's curvature tensor, the scalar curvature, and the metric. An additional term, proportional to the metric, can be added, but it is not considered as part as Einstein's tensor, since he refused to add this term, that would have as a consequence a non-static universe (if he had considered it, he would have been able to predict the expansion of the universe). Summarizing, Einstein's field equation looks as follows:

On the left hand side we will have a tensor coming from geometry (the Einstein tensor), and on the right hand side we will have a tensor with the physical information (the Energy-Momentum tensor, already known in fluid mechanics). Einstein's tensor is constructed from Ricci's curvature tensor, the scalar curvature, and the metric. An additional term, proportional to the metric, can be added, but it is not considered as part as Einstein's tensor, since he refused to add this term, that would have as a consequence a non-static universe (if he had considered it, he would have been able to predict the expansion of the universe). Summarizing, Einstein's field equation looks as follows:

Here the unknowns are the metric coefficients, while the Energy-Momentum tensor T is some given data (in the same way that the force F in F=ma is given by the problem that we are studying); all the other tensors are constructed in terms of the pseudo-metric g. This system of equations is a very non-linear system of 10 PDEs; even when studying empty spacetime (T=0), and considering only the first two terms in the left hand side, solving the system seems really difficult! However, surprisingly, only a month after the publication of the equations by Einstein, Schwarzschild published one of the most useful solutions to these equations: the Schwarzschild solution for rotationally symmetric spaces in vacuum. This solution can be applied to understand the motion of the Earth around the Sun, by neglecting the mass of the Earth and considering that all the curvature comes from the Sun; the Earth is a freely falling massless body, hence it follows one of the geodesics. Unfortunately, the two-body problem is still an open problem in General Relativity, and it can only be solved numerically. But so far, this solution was enough to locate General Relativity in the center of modern large scale physics.

]]>Let me be honest with you. When I applied to UCR for my PhD I didn't think I would end up studying here, I didn't know too much about this school, nor about its faculty members. However, the Department of Mathematics in Riverside has a lot of good and worldwide famous mathematicians. Among them, John Baez caught my attention when I applied to UCR, and he is, together with my PhD advisor Qi S. Zhang, the main reason I decided to come here.

John Baez is a very active professor, not only in the university, but also online: in his webpage (one of the first "blogs" in the world wide web) you can find, literally, all sorts of things; I love to get lost surfing through it! He is an awesome instructor, I had the opportunity to attend one of his classes, MATH209A, an introduction to measure theory, and after that I decided to do my best to attend as many of his classes as I could. That's the reason that I have been attending his seminars on Network Theory and on Category Theory!

John Baez is a very active professor, not only in the university, but also online: in his webpage (one of the first "blogs" in the world wide web) you can find, literally, all sorts of things; I love to get lost surfing through it! He is an awesome instructor, I had the opportunity to attend one of his classes, MATH209A, an introduction to measure theory, and after that I decided to do my best to attend as many of his classes as I could. That's the reason that I have been attending his seminars on Network Theory and on Category Theory!

Last year, his seminar was on Network Theory, one of the topics in which he and his students are working. The idea behind it is to apply Category Theory to everyday physics, such as electrical circuits or control theory, as well as to any other subject that uses diagrams or networks (such as biology, chemistry, or theoretical physics). The goal is to unify the language of all of this fields through Category Theory, in the same way that it has happened in different areas of Mathematics. The first four sessions can be found on youtube:

This year, however, the focus of his Seminar is on the applications of Category Theory within mathematics, so that the graduate students in the Math Department learn how can Category Theory help them in their own fields. So far, the seminar has been an absolute success, it is hard to find an empty seat, and all the sessions have been really exciting (and I must say that I am not particularly interested in Category Theory). He is talking about different topics that were suggested by the students over the summer, and relating them to Category Theory.

Among the topics he has covered so far, I particularly liked Lecture 2, where he explained the duality between commutative C*-algebras and Compact Hausdorff topological spaces, which are opposite of each other in the categorical sense, even if that sounds crazy. I also loved his categorical approach to Galois Theory; I really think that any mathematician that has ever studied Galois Theory using the classical approach (studying fields through groups) should have a look at the general setting were Galois Theory is defined! You can find some notes on the Seminar in his website, taken by some of the students.

Next quarter, professor Baez is offering a class on Category Theory, with a more formal approach to the field. I am looking forward to it, attending John Baez classes is always worth it!

]]>Among the topics he has covered so far, I particularly liked Lecture 2, where he explained the duality between commutative C*-algebras and Compact Hausdorff topological spaces, which are opposite of each other in the categorical sense, even if that sounds crazy. I also loved his categorical approach to Galois Theory; I really think that any mathematician that has ever studied Galois Theory using the classical approach (studying fields through groups) should have a look at the general setting were Galois Theory is defined! You can find some notes on the Seminar in his website, taken by some of the students.

Next quarter, professor Baez is offering a class on Category Theory, with a more formal approach to the field. I am looking forward to it, attending John Baez classes is always worth it!

The buildings consist of a Main Hall, containing the dining room and most of the bedrooms (really nice bedrooms, by the way! With incredible views to the mountains), and the library (one of the best libraries on Mathematics in the world!), which also contains some classrooms. In addition, there are some Bungalows, used by the visitors that are enrolled in certain programs, such as the Oberwolfach Leibniz Fellows.

When I got here, I suddenly realised that Oberwolfach was going to be a complete experience: everything in this research center has been designed to improve the mathematical productivity and the interactions between mathematicians. For example, there is no WiFi in the rooms from 9am to 9pm, to encourage you to go to the library or to the dining room, and meet other visitors. In addition to that, during the meals, you can not sit anywhere you like. The seats are assigned randomly so that you get to meet other mathematicians every day! Since most of the people stay here just for one week, this is the best way to get to know everybody before leaving the center!

If you need to do some exercise, there are nice hikes to the mountain, and there is a small gym next to the library, where you can play table tennis (or table football!); there is also a music room, with a Steynway & Sons grand piano, and other instruments (that was a really big surprise!).

Also, if you want to read the news, there is a good selection of newspapers, and if you want to play some table games, the main hall has several of them (not to mention the giant chess board in the first floor!). Of course, there are coffee machines everywhere, and you can also buy some postcards, drinks (or even chocolate if you know where to find it!) that are located in different places in the buildings.

Also, if you want to read the news, there is a good selection of newspapers, and if you want to play some table games, the main hall has several of them (not to mention the giant chess board in the first floor!). Of course, there are coffee machines everywhere, and you can also buy some postcards, drinks (or even chocolate if you know where to find it!) that are located in different places in the buildings.

There are also some special events during the week. For instance, we did a little hike on Wednesday, we visited the museum of Minerals and Mathematics (MiMa) in Oberwolfach, and after trying the famous cake named after the Black Forest (Schwarzwalde kirschetorte), we had a barbecue outside! I must say that the food is abundant, and it is pretty good; you can dangerously get used to this lifestyle!

]]>The Mathematisches Forschungsinstitut Oberwolfach is an international research center situated in the German Black Forest. It has one of the best mathematical libraries in the world, and is famous for their seminars, conferences, and mathematical gatherings.

The first time I heard about MFO was when I collaborated with the Exhibition Imaginary/BCN, in Barcelona; this exhibition started in MFO, after Herwig Hauser created a series of beautiful pictures of algebraic surfaces with interesting mathematical structures, with the goal to catch the attention of prospective PhD students; later, the pictures became part of a very nice exhibition, that has already travelled to many cities from around the world!

It was a big surprise when I saw that the director of the MFO, Professor Gerhard Huisken, together with Professor Simon Brendle (from Stanford University), organized one of the Oberwolfach Seminars on Singularity Analysis for Geometric Flows, a topic in which I am very interested (it is essentially the field of my PhD thesis). So I didn't doubt about it, and I showed my interest in attending the seminar.

The first time I heard about MFO was when I collaborated with the Exhibition Imaginary/BCN, in Barcelona; this exhibition started in MFO, after Herwig Hauser created a series of beautiful pictures of algebraic surfaces with interesting mathematical structures, with the goal to catch the attention of prospective PhD students; later, the pictures became part of a very nice exhibition, that has already travelled to many cities from around the world!

It was a big surprise when I saw that the director of the MFO, Professor Gerhard Huisken, together with Professor Simon Brendle (from Stanford University), organized one of the Oberwolfach Seminars on Singularity Analysis for Geometric Flows, a topic in which I am very interested (it is essentially the field of my PhD thesis). So I didn't doubt about it, and I showed my interest in attending the seminar.

MFO offered me funding to attend the seminar, and I am very excited about it! I am sure that it will help me a lot to develop my understanding on this field! So, it looks like I am visiting Oberwolfach in September.

Thank you MFO!

**Singularity Analysis for Geometric Flows**

What does that even mean? Well, let me try to explain it, without going into too much detail. Geometric Flows are differential equations that appear in a geometric setting, usually involving geometric concepts such as the curvature of a surface or a curve (or a geometric object of higher dimension). One example is the Curve Shortening Flow (a particular case of the Mean Curvature Flow), were the shape of a curve evolves with respect to a parameter (say time) depending on its curvature: in the regions where the curve has a high curvature, it evolves fast, moving in the normal direction to the curve, while in the flat regions it almost doesn't move. You can see an example in the following video:

Thank you MFO!

What does that even mean? Well, let me try to explain it, without going into too much detail. Geometric Flows are differential equations that appear in a geometric setting, usually involving geometric concepts such as the curvature of a surface or a curve (or a geometric object of higher dimension). One example is the Curve Shortening Flow (a particular case of the Mean Curvature Flow), were the shape of a curve evolves with respect to a parameter (say time) depending on its curvature: in the regions where the curve has a high curvature, it evolves fast, moving in the normal direction to the curve, while in the flat regions it almost doesn't move. You can see an example in the following video:

As it is shown in the video above, sometimes the curves might shrink to a point; similarly, when dealing with a surface, the whole surface, or just a curve in it, might shrink to a point. When this happens, we say that the flow has encountered a singularity. It turns out that studying how the flow evolves if we "forget" about these singularities, by keeping them under control, might bring us to interesting information about the initial curve that we were studying. This was shown, for instance, by Grigori Perelman, who used the analysis of singularities in the Ricci flow to prove the Geometrization Conjecture and the Poincaré Conjecture, a major problem in mathematics that was worth 1,000,000$. Here is a video of the evolution of the surface of a peanut under the Ricci flow:

Well, I think that this is enough for today, although that I will definitely talk more about geometric flows in the future (with more detail!), and I will share with you my experience in Oberwolfach. If you would like to know a little bit more about geometric flows and the Poincaré conjecture, you might find the following video very interesting! If you want to learn it seriously, you can look for Professor Huisken's lectures on the Mean Curvature Flow on Youtube, I found them very enlightening!

This is the first post in my professional blog! The aim of this blog is to share ideas, discoveries, exciting news, academic experiences, and much more with all of you!

Not only the blog is new, but also, I have build a brand new professional website! You can find there my CV, my contact information, a summary of my research, and my teaching experience. Enjoy!!

]]>Not only the blog is new, but also, I have build a brand new professional website! You can find there my CV, my contact information, a summary of my research, and my teaching experience. Enjoy!!