Matematički blogovi

Masterclass on mathematical thinking

Terrence Tao - Čet, 2022-01-27 18:16

About a year ago, I was contacted by Masterclass (a subscription-based online education company) on the possibility of producing a series of classes with the premise of explaining mathematical ways of thinking (such as reducing a complex problem to simpler sub-problems, abstracting out inessential aspects of a problem, or applying transforms or analogies to find new ways of thinking about a problem). After a lot of discussion and planning, as well as a film shoot over the summer, the series is now completed. As per their business model, the full lecture series is only available to subscribers of their platform, but the above link does contain a trailer and some sample content.

Kategorije: Matematički blogovi

The inverse theorem for the U^3 Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches

Terrence Tao - Uto, 2021-12-28 19:47

Asgar Jamneshan and myself have just uploaded to the arXiv our preprint “The inverse theorem for the Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches“. This paper, which is a companion to another recent paper of ourselves and Or Shalom, studies the inverse theory for the third Gowers uniformity norm

on an arbitrary finite abelian group , where is the multiplicative derivative. Our main result is as follows:

Theorem 1 (Inverse theorem for ) Let be a finite abelian group, and let be a -bounded function with for some . Then:
  • (i) (Correlation with locally quadratic phase) There exists a regular Bohr set with and , a locally quadratic function , and a function such that

  • (ii) (Correlation with nilsequence) There exists an explicit degree two filtered nilmanifold of dimension , a polynomial map , and a Lipschitz function of constant such that

Such a theorem was proven by Ben Green and myself in the case when was odd, and by Samorodnitsky in the -torsion case . In all cases one uses the “higher order Fourier analysis” techniques introduced by Gowers. After some now-standard manipulations (using for instance what is now known as the Balog-Szemerédi-Gowers lemma), one arrives (for arbitrary ) at an estimate that is roughly of the form

where denotes various -bounded functions whose exact values are not too important, and is a symmetric locally bilinear form. The idea is then to “integrate” this form by expressing it in the form

for some locally quadratic ; this then allows us to write the above correlation as

(after adjusting the functions suitably), and one can now conclude part (i) of the above theorem using some linear Fourier analysis. Part (ii) follows by encoding locally quadratic phase functions as nilsequences; for this we adapt an algebraic construction of Manners.

So the key step is to obtain a representation of the form (1), possibly after shrinking the Bohr set a little if needed. This has been done in the literature in two ways:

  • When is odd, one has the ability to divide by , and on the set one can establish (1) with . (This is similar to how in single variable calculus the function is a function whose second derivative is equal to .)
  • When , then after a change of basis one can take the Bohr set to be for some , and the bilinear form can be written in coordinates as

    for some with . The diagonal terms cause a problem, but by subtracting off the rank one form one can write

    on the orthogonal complement of for some coefficients which now vanish on the diagonal: . One can now obtain (1) on this complement by taking

In our paper we can now treat the case of arbitrary finite abelian groups , by means of the following two new ingredients:

  • (i) Using some geometry of numbers, we can lift the group to a larger (possibly infinite, but still finitely generated) abelian group with a projection map , and find a globally bilinear map on the latter group, such that one has a representation

    of the locally bilinear form by the globally bilinear form when are close enough to the origin.
  • (ii) Using an explicit construction, one can show that every globally bilinear map has a representation of the form (1) for some globally quadratic function .

To illustrate (i), consider the Bohr set in (where denotes the distance to the nearest integer), and consider a locally bilinear form of the form for some real number and all integers (which we identify with elements of . For generic , this form cannot be extended to a globally bilinear form on ; however if one lifts to the finitely generated abelian group

(with projection map ) and introduces the globally bilinear form by the formula

then one has (2) when lie in the interval . A similar construction works for higher rank Bohr sets.

To illustrate (ii), the key case turns out to be when is a cyclic group , in which case will take the form

for some integer . One can then check by direct construction that (1) will be obeyed with

regardless of whether is even or odd. A variant of this construction also works for , and the general case follows from a short calculation verifying that the claim (ii) for any two groups implies the corresponding claim (ii) for the product .

This concludes the Fourier-analytic proof of Theorem 1. In this paper we also give an ergodic theory proof of (a qualitative version of) Theorem 1(ii), using a correspondence principle argument adapted from this previous paper of Ziegler, and myself. Basically, the idea is to randomly generate a dynamical system on the group , by selecting an infinite number of random shifts , which induces an action of the infinitely generated free abelian group on by the formula

Much as the law of large numbers ensures the almost sure convergence of Monte Carlo integration, one can show that this action is almost surely ergodic (after passing to a suitable Furstenberg-type limit where the size of goes to infinity), and that the dynamical Host-Kra-Gowers seminorms of that system coincide with the combinatorial Gowers norms of the original functions. One is then well placed to apply an inverse theorem for the third Host-Kra-Gowers seminorm for -actions, which was accomplished in the companion paper to this one. After doing so, one almost gets the desired conclusion of Theorem 1(ii), except that after undoing the application of the Furstenberg correspondence principle, the map is merely an almost polynomial rather than a polynomial, which roughly speaking means that instead of certain derivatives of vanishing, they instead are merely very small outside of a small exceptional set. To conclude we need to invoke a “stability of polynomials” result, which at this level of generality was first established by Candela and Szegedy (though we also provide an independent proof here in an appendix), which roughly speaking asserts that every approximate polynomial is close in measure to an actual polynomial. (This general strategy is also employed in the Candela-Szegedy paper, though in the absence of the ergodic inverse theorem input that we rely upon here, the conclusion is weaker in that the filtered nilmanifold is replaced with a general space known as a “CFR nilspace”.)

This transference principle approach seems to work well for the higher step cases (for instance, the stability of polynomials result is known in arbitrary degree); the main difficulty is to establish a suitable higher step inverse theorem in the ergodic theory setting, which we hope to do in future research.

Kategorije: Matematički blogovi

The structure of arbitrary Conze-Lesigne systems

Terrence Tao - Pon, 2021-12-06 16:39

Asgar Jamneshan, Or Shalom, and myself have just uploaded to the arXiv our preprint “The structure of arbitrary Conze–Lesigne systems“. As the title suggests, this paper is devoted to the structural classification of Conze-Lesigne systems, which are a type of measure-preserving system that are “quadratic” or of “complexity two” in a certain technical sense, and are of importance in the theory of multiple recurrence. There are multiple ways to define such systems; here is one. Take a countable abelian group acting in a measure-preserving fashion on a probability space , thus each group element gives rise to a measure-preserving map . Define the third Gowers-Host-Kra seminorm of a function via the formula

where is a Folner sequence for and is the complex conjugation map. One can show that this limit exists and is independent of the choice of Folner sequence, and that the seminorm is indeed a seminorm. A Conze-Lesigne system is an ergodic measure-preserving system in which the seminorm is in fact a norm, thus whenever is non-zero. Informally, this means that when one considers a generic parallelepiped in a Conze–Lesigne system , the location of any vertex of that parallelepiped is more or less determined by the location of the other seven vertices. These are the important systems to understand in order to study “complexity two” patterns, such as arithmetic progressions of length four. While not all systems are Conze-Lesigne systems, it turns out that they always have a maximal factor that is a Conze-Lesigne system, known as the Conze-Lesigne factor or the second Host-Kra-Ziegler factor of the system, and this factor controls all the complexity two recurrence properties of the system.

The analogous theory in complexity one is well understood. Here, one replaces the norm by the norm

and the ergodic systems for which is a norm are called Kronecker systems. These systems are completely classified: a system is Kronecker if and only if it arises from a compact abelian group equipped with Haar probability measure and a translation action for some homomorphism with dense image. Such systems can then be analyzed quite efficiently using the Fourier transform, and this can then be used to satisfactory analyze “complexity one” patterns, such as length three progressions, in arbitrary systems (or, when translated back to combinatorial settings, in arbitrary dense sets of abelian groups).

We return now to the complexity two setting. The most famous examples of Conze-Lesigne systems are (order two) nilsystems, in which the space is a quotient of a two-step nilpotent Lie group by a lattice (equipped with Haar probability measure), and the action is given by a translation for some group homomorphism . For instance, the Heisenberg -nilsystem

with a shift of the form

for two real numbers with linearly independent over , is a Conze-Lesigne system. As the base case of a well known result of Host and Kra, it is shown in fact that all Conze-Lesigne -systems are inverse limits of nilsystems (previous results in this direction were obtained by Conze-Lesigne, Furstenberg-Weiss, and others). Similar results are known for -systems when is finitely generated, thanks to the thesis work of Griesmer (with further proofs by Gutman-Lian and Candela-Szegedy). However, this is not the case once is not finitely generated; as a recent example of Shalom shows, Conze-Lesigne systems need not be the inverse limit of nilsystems in this case.

Our main result is that even in the infinitely generated case, Conze-Lesigne systems are still inverse limits of a slight generalisation of the nilsystem concept, in which is a locally compact Polish group rather than a Lie group:

Theorem 1 (Classification of Conze-Lesigne systems) Let be a countable abelian group, and an ergodic measure-preserving -system. Then is a Conze-Lesigne system if and only if it is the inverse limit of translational systems , where is a nilpotent locally compact Polish group of nilpotency class two, and is a lattice in (and also a lattice in the commutator group ), with equipped with the Haar probability measure and a translation action for some homomorphism .

In a forthcoming companion paper to this one, Asgar Jamneshan and I will use this theorem to derive an inverse theorem for the Gowers norm for an arbitrary finite abelian group (with no restrictions on the order of , in particular our result handles the case of even and odd in a unified fashion). In principle, having a higher order version of this theorem will similarly allow us to derive inverse theorems for norms for arbitrary and finite abelian ; we hope to investigate this further in future work.

We sketch some of the main ideas used to prove the theorem. The existing machinery developed by Conze-Lesigne, Furstenberg-Weiss, Host-Kra, and others allows one to describe an arbitrary Conze-Lesigne system as a group extension , where is a Kronecker system (a rotational system on a compact abelian group and translation action ), is another compact abelian group, and the cocycle is a collection of measurable maps obeying the cocycle equation

for almost all . Furthermore, is of “type two”, which means in this concrete setting that it obeys an additional equation

for all and almost all , and some measurable function ; roughly speaking this asserts that is “linear up to coboundaries”. For technical reasons it is also convenient to reduce to the case where is separable. The problem is that the equation (2) is unwieldy to work with. In the model case when the target group is a circle , one can use some Fourier analysis to convert (2) into the more tractable Conze-Lesigne equation

for all , all , and almost all , where for each , is a measurable function, and is a homomorphism. (For technical reasons it is often also convenient to enforce that depend in a measurable fashion on ; this can always be achieved, at least when the Conze-Lesigne system is separable, but actually verifying that this is possible actually requires a certain amount of effort, which we devote an appendix to in our paper.) It is not difficult to see that (3) implies (2) for any group (as long as one has the measurability in mentioned previously), but the converse turns out to fail for some groups , such as solenoid groups (e.g., inverse limits of as ), as was essentially shown by Rudolph. However, in our paper we were able to find a separate argument that also derived the Conze-Lesigne equation in the case of a cyclic group . Putting together the and cases, one can then derive the Conze-Lesigne equation for arbitrary compact abelian Lie groups (as such groups are isomorphic to direct products of finitely many tori and cyclic groups). As has been known for some time (see e.g., this paper of Host and Kra), once one has a Conze-Lesigne equation, one can more or less describe the system as a translational system , where the Host-Kra group is the set of all pairs that solve an equation of the form (3) (with these pairs acting on by the law ), and is the stabiliser of a point in this system. This then establishes the theorem in the case when is a Lie group, and the general case basically comes from the fact (from Fourier analysis or the Peter-Weyl theorem) that an arbitrary compact abelian group is an inverse limit of Lie groups. (There is a technical issue here in that one has to check that the space of translational system factors of form a directed set in order to have a genuine inverse limit, but this can be dealt with by modifications of the tools mentioned here.)

There is an additional technical issue worth pointing out here (which unfortunately was glossed over in some previous work in the area). Because the cocycle equation (1) and the Conze-Lesigne equation (3) are only valid almost everywhere instead of everywhere, the action of on is technically only a near-action rather than a genuine action, and as such one cannot directly define to be the stabiliser of a point without running into multiple problems. To fix this, one has to pass to a topological model of in which the action becomes continuous, and the stabilizer becomes well defined, although one then has to work a little more to check that the action is still transitive. This can be done via Gelfand duality; we proceed using a mixture of a construction from this book of Host and Kra, and the machinery in this recent paper of Asgar and myself.

Now we discuss how to establish the Conze-Lesigne equation (3) in the cyclic group case . As this group embeds into the torus , it is easy to use existing methods obtain (3) but with the homomorphism and the function taking values in rather than in . The main task is then to fix up the homomorphism so that it takes values in , that is to say that vanishes. This only needs to be done locally near the origin, because the claim is easy when lies in the dense subgroup of , and also because the claim can be shown to be additive in . Near the origin one can leverage the Steinhaus lemma to make depend linearly (or more precisely, homomorphically) on , and because the cocycle already takes values in , vanishes and must be an eigenvalue of the system . But as was assumed to be separable, there are only countably many eigenvalues, and by another application of Steinhaus and linearity one can then make vanish on an open neighborhood of the identity, giving the claim.

Kategorije: Matematički blogovi

A math rave

Terrence Tao - Pet, 2021-11-19 00:04

As math educators, we often wish out loud that our students were more excited about mathematics. I finally came across a video that indicates what such a world might be like:

Kategorije: Matematički blogovi

Venn and Euler type diagrams for vector spaces and abelian groups

Terrence Tao - Pon, 2021-11-08 04:53

A popular way to visualise relationships between some finite number of sets is via Venn diagrams, or more generally Euler diagrams. In these diagrams, a set is depicted as a two-dimensional shape such as a disk or a rectangle, and the various Boolean relationships between these sets (e.g., that one set is contained in another, or that the intersection of two of the sets is equal to a third) is represented by the Boolean algebra of these shapes; Venn diagrams correspond to the case where the sets are in “general position” in the sense that all non-trivial Boolean combinations of the sets are non-empty. For instance to depict the general situation of two sets together with their intersection and one might use a Venn diagram such as

(where we have given each region depicted a different color, and moved the edges of each region a little away from each other in order to make them all visible separately), but if one wanted to instead depict a situation in which the intersection was empty, one could use an Euler diagram such as

One can use the area of various regions in a Venn or Euler diagram as a heuristic proxy for the cardinality (or measure ) of the set corresponding to such a region. For instance, the above Venn diagram can be used to intuitively justify the inclusion-exclusion formula

for finite sets , while the above Euler diagram similarly justifies the special case

for finite disjoint sets .

While Venn and Euler diagrams are traditionally two-dimensional in nature, there is nothing preventing one from using one-dimensional diagrams such as

or even three-dimensional diagrams such as this one from Wikipedia:

Of course, in such cases one would use length or volume as a heuristic proxy for cardinality or measure, rather than area.

With the addition of arrows, Venn and Euler diagrams can also accommodate (to some extent) functions between sets. Here for instance is a depiction of a function , the image of that function, and the image of some subset of :

Here one can illustrate surjectivity of by having fill out all of ; one can similarly illustrate injectivity of by giving exactly the same shape (or at least the same area) as . So here for instance might be how one would illustrate an injective function :

Cartesian product operations can be incorporated into these diagrams by appropriate combinations of one-dimensional and two-dimensional diagrams. Here for instance is a diagram that illustrates the identity :

In this blog post I would like to propose a similar family of diagrams to illustrate relationships between vector spaces (over a fixed base field , such as the reals) or abelian groups, rather than sets. The categories of (-)vector spaces and abelian groups are quite similar in many ways; the former consists of modules over a base field , while the latter consists of modules over the integers ; also, both categories are basic examples of abelian categories. The notion of a dimension in a vector space is analogous in many ways to that of cardinality of a set; see this previous post for an instance of this analogy (in the context of Shannon entropy). (UPDATE: I have learned that an essentially identical notation has also been proposed in an unpublished manuscript of Ravi Vakil.)

As with Venn and Euler diagrams, the diagrams I propose for vector spaces (or abelian groups) can be set up in any dimension. For simplicity, let’s begin with one dimension, and restrict attention to vector spaces (the situation for abelian groups is basically identical). In this one-dimensional model we will be able to depict the following relations and operations between vector spaces:
  • The inclusion of one vector space in another (here I prefer to use the group notation for inclusion rather than the set notation ).
  • The quotient of a vector space by a subspace .
  • A linear transformation between vector spaces, as well as the kernel , image , cokernel , and the coimage .
  • A single short or long exact sequence between vector spaces.
  • (A heuristic proxy for) the dimension of a vector space.
  • Direct sum of two spaces.

The idea is to use half-open intervals in the real line for any to model vector spaces. In fact we can make an explicit correspondence: let us identify each half-open interval with the (infinite-dimensional) vector space

that is is identified with the space of continuous functions on the interval that vanish at the right-endpoint . Such functions can be continuously extended by zero to the half-line .

Note that if then the vector space is a subspace of , if we extend the functions in both spaces by zero to the half-line ; furthermore, the quotient of by is naturally identifiable with . Thus, an inclusion , as well as the quotient space , can be modeled here as follows:

In contrast, if , it is significantly less “natural” to view as a subspace of ; one could do it by extending functions in to the right by zero and to the left by constants, but in this notational convention one should view such an identification as “artificial” and to be avoided.

All of the spaces are infinite dimensional, but morally speaking the dimension of the vector space is “proportional” to the length of the corresponding interval. Intuitively, if we try to discretise this vector space by sampling at some mesh of spacing , one gets a finite-dimensional vector space of dimension roughly . Already the above diagram now depicts the basic identity

between a finite-dimensional vector space , a subspace of that space, and a quotient of that space.

Note that if , then there is a linear transformation from the vector space to the vector space which takes a function in , restricts it to , then extends it by zero to . The kernel of this transformation is , the image is (isomorphic to) , the cokernel is (isomorphic to) , and the coimage is (isomorphic to) . With this in mind, we can now depict a general linear transformation and its associated spaces by the following diagram:

Note how the first isomorphism theorem and the rank-nullity theorem are heuristically illustrated by this diagram. One can specialise to the case of injective, surjective, or bijective transformations by degenerating one or more of the half-open intervals in the above diagram to the empty interval. A left shift on gives rise to a nilpotent linear transformation from to itself:

In a similar spirit, a short exact sequence of vector spaces (or abelian groups) can now be depicted by the diagram

and a long exact sequence can similarly be depicted by the diagram

UPDATE: As I have learned from an unpublished manuscript of Ravi Vakil, this notation can also easily depict the cohomology groups of a cochain complex by the diagram

and similarly depict the homology groups of a chain complex by the diagram

One can associate the disjoint union of half-open intervals to the direct sum of the associated vector spaces, giving a way to depict direct sums via this notation:

To increase the expressiveness of this notation we now move to two dimensions, where we can also depict the following additional relations and operations:

  • The intersection and sum of two subspaces of an ambient space ;
  • Multiple short or long exact sequences;
  • The tensor product of two vector spaces .

Here, we replace half-open intervals by half-open sets: geometric shapes , such as polygons or disks, which contain some portion of the boundary (drawn using solid lines) but omit some other portion of the boundary (drawn with dashed lines). Each such shape can be associated with a vector space, namely the continuous functions on that vanish on the omitted portion of the boundary. All of the relations that were previously depicted using one-dimensional diagrams can now be similarly depicted using two-dimensional diagrams. For instance, here is a two-dimensional depiction of a vector space , a subspace , and its quotient :

(In this post I will try to consistently make the lower and left boundaries of these regions closed, and the upper and right boundaries open, although this is not essential for this notation to be applicable.)

But now we can depict some additional relations. Here for instance is one way to depict the intersection and sum of two subspaces :

Note how this illustrates the identity

between finite-dimensional vector spaces , as well as some standard isomorphisms such as .

Two finite subgroups of an abelian group are said to be commensurable if is a finite index subgroup of . One can depict this by making the area of the region between and small and/or colored with some specific color:

Here the commensurability of is equivalent to the finiteness of the groups and , which correspond to the gray triangles in the above diagram. Now for instance it becomes intuitively clear why commensurability should be an equivalence relation.

To illustrate how this notation can support multiple short exact sequences, I gave myself the exercise of using this notation to depict the snake lemma, as labeled by this following diagram taken from the just linked Wikipedia page:

This turned out to be remarkably tricky to accomplish without introducing degeneracies (e.g., one of the kernels or cokernels vanishing). Here is one solution I came up with; perhaps there are more elegant ones. In particular, there should be a depiction that more explicitly captures the duality symmetry of the snake diagram.

Here, the maps between the six spaces are the obvious restriction maps (and one can visually verify that the two squares in the snake diagram actually commute). Each of the kernel and cokernel spaces of the three vertical restriction maps are then associated to the union of two of the subregions as indicated in the diagram. Note how the overlaps between these kernels and cokernels generate the long exact “snake”.

UPDATE: by modifying a similar diagram in an unpublished manuscript of Ravi Vakil, I can now produce a more symmetric version of the above diagram, again with a very visible “snake”:

With our notation, the (algebraic) tensor product of an interval and another interval is not quite , but this becomes true if one uses the -algebra version of the tensor product, thanks to the Stone-Weierstrass theorem. So one can plausibly use Cartesian products as a proxy for the vector space tensor product. For instance, here is a depiction of the relation when is a subspace of :

There are unfortunately some limitations to this notation: for instance, no matter how many dimensions one uses for one’s diagrams, these diagrams would suggest the incorrect identity

(which incidentally is, at this time of writing, the highest-voted answer to the MathOverflow question “Examples of common false beliefs in mathematics“). (See also this previous blog post for a similar phenomenon when using sets or vector spaces to model entropy of information variables.) Nevertheless it seems accurate enough to be of use in illustrating many common relations between vector spaces and abelian groups. With appropriate grains of salt it might also be usable for further categories beyond these two, though for non-abelian categories one should proceed with caution, as the diagram may suggest relations that are not actually true in this category. For instance, in the category of topological groups one might use the diagram

to describe the fact that an arbitrary topological group splits into a connected subgroup and a totally disconnected quotient, or in the category of finite-dimensional Lie algebras over the reals one might use the diagram

to describe the fact that such algebras split into the solvable radical and a semisimple quotient.

Kategorije: Matematički blogovi

Spectral rigidity of almost circular ellipses (after Hezari–Zelditch)

Disquisitiones Mathematicae - Pon, 2021-05-03 09:56

In 1966, M. Kac wrote a famous article asking whether Can one hear the shape of drum?: mathematically speaking, one wants to reconstruct (up to isometries) a domain from the knowledge of the spectrum of its Laplacian.

In his article, M. Kac showed that one can hear the shape of a disk because of the following two facts:

  • the area and the perimeter of a smooth domain are determined by the eigenvalues of its Laplacian via the asymptotics of the trace of the heat operator:
  • the isoperimetric inequality says that we can recognise the disk from its perimeter and area: indeed, any smooth domain satisfies , and the equality holds if and only if is isometric to a disk of radius .

(In particular, a smooth domain with the same Laplace eigenvalues of has area and perimeter , so that the isoperimetric inequality ensures that and are isometric.)

In a preprint from 2019, Hezari and Zelditch showed that one can also hear the shape of an ellipse of small eccentricity . As it is explained by Zelditch in this video here, a first-order approximation to their basic strategy to hear ellipses of small eccentricities is to replace “trace of the heat operator” and “isoperimetric inequality” in Kac’s argument by “trace of the wave operator” and “dynamical rigidity of ellipses”.

More concretely, Hezari and Zelditch considered a smooth domain which is isospectral to an ellipse (i.e., for all ) of small eccentricity , and they took the following steps:

  • in Section 2 of their paper, it is shown that is necessarily close to ; in fact, after parametrising with an arc-length parameter , we can control the -norms of the derivatives of the curvature of from the asymptotics of the trace of the heat operator: indeed,whereand, in general, for a certain constant and an adequate “universal” polynomial (here, is the -th derivative of with respect to ); in particular, since and are isospectral, for all , and Melrose explored this fact to get a pre-compactness bound  for all (via a bootstrap argument where the Poincaré inequality and the Sobolev embedding theorem are employed to convert bounds on , , into bounds on , , ); in other terms, after Melrose, the shape of any isospectral to is bounded; by reworking Melrose’s argument, Hezari and Zelditch actually show that is almost circular in the sense that for all ;
  • in view of a theorem of Avila, de Simoi and Kaloshin, an almost circular domain is isometric to an ellipse provided  is rationally integrable, i.e., for each , the periodic trajectories with rotation number  in the billiard table determined by are tangent to a smooth convex curve (usually called caustic);
  • in Sections 3 to 6 of their paper, it is shown that the portion between and of the singular support of the trace of the wave operator  coincides with the set of lengths of periodic trajectories with rotation number , , in an almost circular billiard table ; in particular, if is isospectral to , one concludes (in Section 7 of their paper) that, for each , all periodic trajectories in with rotation number have the same length and they form a caustic, so that is rationally integrable (and, a fortiori, isometric to after Avila–de Simoi–Kaloshin).

A natural question raised by Hezari–Zelditch work is to determine the magnitude of the upper bound on the eccentricities of the ellipses which can be heard from their methods.

In this direction, I would like to conclude this short post by noticing that I asked a group of 6 undergraduate students (in their 2nd year) at \’Ecole Polytechnique to follow closely the articles by Avila–de Simoi–Kaloshin and Hezari–Zelditch (while trying to explicitly compute as many implied constants as possible), and, after 6 months of work, they produced this report here (in French) concluding that is not bigger than . (Of course, there is plenty of room for tiny improvements here, but one will probably need some new ideas before reaching a “normal size” [e.g., ].)

Kategorije: Matematički blogovi

New Theorems

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has 'new acquisitions': Theorems no. 247-248
Kategorije: Matematički blogovi

New Theorems

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has 'new acquisitions': Theorems no. 249-251
Kategorije: Matematički blogovi

New Theorem

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has 'new acquisitions': Theorem no. 228, 229, 230, by R.A. Fisher, Poncelet and Ore
Kategorije: Matematički blogovi

New Theorems

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has 'new acquisitions': Theorem no. 231, 232, 233, 234
Kategorije: Matematički blogovi

New Theorems

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has 'new acquisitions': Theorem no. 235, 236, 237
Kategorije: Matematički blogovi

New Theorems

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has 'new acquisitions': Theorem no. 238 and 239
Kategorije: Matematički blogovi

New Theorems

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has 'new acquisitions': Theorem no. 240 and 241
Kategorije: Matematički blogovi

New Theorem

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has a 'new acquisition': Theorem no. 242, the Polya-Redfield Enumeration Theorem
Kategorije: Matematički blogovi

New Theorems

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has 'new acquisitions': Theorems no. 243-246
Kategorije: Matematički blogovi

New Theorem

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has a 'new acquisitions': Theorem no. 227, Cauchy's Theorem in Group Theory
Kategorije: Matematički blogovi

New Theorem

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has a 'new acquisitions': Theorem no. 226, Wolstenholme's Theorem
Kategorije: Matematički blogovi

New Theorem

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has 'new acquisitions': Theorem no. 223, 224, 225, by Tutte, Green and Regiomontanus
Kategorije: Matematički blogovi

New Theorem

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has a 'new acquisitions': Theorem no. 222, Faulhaber's Formula
Kategorije: Matematički blogovi

New Theorem

Theorem of the Day - Sri, 2019-11-20 16:40
Theorem of the Day has a 'new acquisitions': Theorem no. 221, the Inclusion-Exclusion Principle
Kategorije: Matematički blogovi