Katia Consani has kindly written the following post for the NCG blog. Many thanks to Katia!

I like to think that from his grave in St. Petersburg, Euler is probably quite pleased to see how far his studies on ``Analysis Infinitorum'' have been successfully developed in these past 224 years from his death. The book "Introductio in analysis infinitorum"(1748) has been translated into English in 2 books, a few years ago, by John D. Blanton. Stimulated by the nice post of Alain Connes reporting on the "Euler Festival", Euler's arithmetical and geometrical achievements and re-interpretation of the concept of infinitesimal variables in quantized calculus, I rushed to the library to glance again into these books and pleasantly re-discover some parts.

I find these 2 books truly exciting and inspiring for the clarity of the style and of course for the original concepts developed by Euler. These include for example, ideas "on series which arise from products", or "on continued fractions", just to mention the title of some of the chapters. It is also funny to report that in the translator's introduction, Blanton notes that the (final)encouragement for the translation of that book came from a remark he heard from Andre Weil, in 1979, when Weil spoke on the Life and Works of Leonhard Euler at the University of Rochester. Apparently, Weil claimed that he was trying to convince the mathematical community that the students of mathematics would profit much more from a study of Euler's "Introductio in analysis infinitorum", rather than of the available modern textbooks...

Andre Weil wrote also a nice and instructive long chapter commenting on the life and the scientific achievements of Euler in his book "Number Theory, an approach through history" (Birkhauser). Quoting from Weil's book (cfr. par. XVIII, p. 261): "...So much work has been done on the series zeta(n)"--Euler writes in 1735-- "that it seems hardly likely that anything new about them may still turn up..." Nevertheless as Weil remarks, for the next ten years Euler never relaxed his efforts to put his conclusions, for example on zeta(2), on a sound basis. His attention had been drawn also by his earlier work on the gamma-function (the factorial function in Euler's words) to infinite products in their relation to infinite series "...in them [infinite products] the factors proceed accordingto the prime numbers, whose progression is no less irregular [than the terms in Goldbach's series]...". A few years later, Riemann would take the "eulerian products" for zeta(s) as the starting point in his studies in number-theory and he would eventually introduce a"complete" version of zeta(s), inclusive of a gamma factor and satisfying a functional equation.

The definition and the study of what is today called the gamma-function is in fact one of the great achievements of Euler'sstudies in arithmetic. He extended the factorial function n! from the naturals to all real numbers greater than -1 by writing the formula

and observing that the integral on the right of this formulaconverges for non integers values of n, provided that . Later on, Riemann proved that the gamma function (i.e. theEuler's integral on the right side of the formula) which is in fact defined for all complex numbers s in the half-plane , canbe analytically extended to define an analytic function of a complexvariable s, with simple poles at the negative integers and no zeroes. The description of this function has apparently nothing in common with the "eulerian factor" , nonetheless it is only when one completes with a suitable gamma factor that the resulting function behaves very nicely (i.e. satisfies a functional equation). One says that the gamma function is the contribution arising from the "archimedean prime infinity" (i.e. from the archimedean evaluation of Q). Legendre, subsequently introduced the notation in place of the original . This notation prevailed in France and, by the end of the nineteenth century, in the rest of the world as well. With this modification, the set of zeroes of is the set of the natural numbers.

This fact should bring us back on some of the earlier posts on thisBlog, on operators whose spectrum is the set of the natural numbers.In fact, it turns out that the the archimedean factor that enters inthe definition of the complete zeta function, i.e. what is today written as

and also the archimedean factor

that is included, together with , in the description of the complete Hasse-Weil L-function of algebraic varieties defined over number fields (i.e. higher arithmetic generalizations of the Riemann zeta function), both have a nice description as the inverse of a regularized determinant det_\infty(s-T) for the action of a self-adjoint operators T (archimedean frobenius...) on an infinite-dimensional real vector space H (cfr. C. Deninger "On theGamma-factors attached to motives", Invent. Math. 104 (1991) and C.Consani "Double complexes and Euler L-factors", Compositio Math. 111(1998)).

At this point, one naturally wonders whether a precise geometri cconstruction is lurking from such a re-interpretation of the gammafactors and in particular whether NCG, with its sophisticated machinery of spectral triples, can help in proving this expectation. In other words, is there a noncommutative space endowed with a representation in a Hilbert space and a Dirac operator supportingt his classical arithmetic theory? Of course, the first natural candidates for the Hilbert space and the Dirac operator are the aforementioned real vector space H and the unbounded self-adjoint operator T. However, we know that from the mere knowledge of these two objects one cannot expect to reconstruct a noncommutative manifold: the relevant information arises from the definition of an involutive algebra (acting on a Hilbert space).

It turns out though, that the definition of the real vector space H is in fact quite geometrical! H is an infinite direct sum, indexed over the natural numbers, of real de-Rham cohomology groups associated to smooth, projective algebraic varieties (a point for the case of which are naturally related to the definition of these zeta-functions. Moreover, with a bit more of technical effort (cfr.C. Consani and M. Marcolli "Noncommutative geometry, dynamics and infinitely-adic Arakelov geometry" Selecta Math. 10 (2004)) one can show that the archimedean part of the zeta-function of an arithmetic surface of genus at least 2, indeed coincides with a precise zeta-function in a family of zetas associated to a noncommutative manifold. This manifold supplies the arithmetical data in two possible ways, either with a construction performed in its interior or with a second construction carried out on its boundary.

I like to conclude this post by saying that these results give evidence to the statement that NCG successfully carries on Euler's geometrical legacy!

## Monday, June 25, 2007

## Friday, June 22, 2007

### Infinitesimal variables

I was recently at the "Euler Festival" in St. Petersburg, for the 300-th anniversary of the birth of Euler.

The wonderful hospitality of the great scientists there, such as Ludwig Faddeev, created a perfect atmosphere for the event. With the long evenings, added to the charm of the city, there was plenty of time to try and get in the right mood to get more familiar with some of Euler's thoughts.

The wonderful hospitality of the great scientists there, such as Ludwig Faddeev, created a perfect atmosphere for the event. With the long evenings, added to the charm of the city, there was plenty of time to try and get in the right mood to get more familiar with some of Euler's thoughts.

I spent time reading the little booklet that was distributed to participants. It contained the talk given by A. N. Krylov in October 1933 on the occasion of the 150-th of Euler's death, at a special session of the USSR academy of sciences. One of my down to earth motivation was to find some connection between the topic of the talk I had planned to give and the works of Euler, but that was easy since as we all know he is "everywhere dense" in mathematics.

In volume 1 of his book "Introductio in Analysis Infinitorum", he gives his computation of zeta values at even integers plus a lot of other things, like partition number generating functions etc. He also gives the numerical zeta values (which he found first) in 23 decimal digits. He also gives the value of pi in 127 decimal digits!

The second volume is devoted to analytic geometry. According to the article of A. N. Krylov, "it is marked by astonishing simplicity and clarity, and Euler uses only tools from elementary algebra and trigonometry"; "the aim of the second volume, as Euler understands it, consists not in analysing properties of curves given geometrically, but, on the opposite, in using curves and their properties for visual presentation of functions given by equations of the first, second, third, fourth, and greater degrees".

In fact I got most intrigued by the discussion which follows in Krylov's talk, of the relation of Euler with the calculus of infinitesimals. In the preface of the above book, Euler wrote:

*"Many times I saw the difficulties of those who start to study analysis of infinitesimals, stem from the fact that they want to acquire the knowledge in this higher branch of analysis having insufficient prerequisites in elementary algebra. This not only causes obstacles they encounter from the beginning, but also gives a false idea of the infinity, whereas a true treatment of this notion must lead in studies"*

The article of A. N. Krylov then gives a detailed comparison of the points of view followed by Newton and Leibniz on the calculus of infinitesimals. One striking point of the discussion is the role that "variables" play in Newton's approach, while Leibniz introduced the term "infinitesimal" but did not use variables. According to Newton:

*"In a certain problem, a variable is the quantity that takes an infinite number of values which are quite determined by this problem and are arranged in a definite order"*

*"A variable is called infinitesimal if among its particular values one can be found such that this value itself and all following it are smaller in absolute value than an arbitrary given number".*

In the classical formulation of variables as maps from a set X to the real numbers R, the set X has to be uncountable if some variable has continuous range. But then for any other variable with countable range some of the multiplicities are infinite. This means that discrete and continuous variables cannot coexist in this modern formalism.

Fortunately everything is fine and this problem of treating continuous and discrete variables on the same footing is completely solved using the formalism of quantum mechanics.

The first basic change of paradigm has indeed to do with the classical notion of a "real variable" which one would classically describe as a real valued function on a set X, ie as a map from this set X to real numbers. In fact quantum mechanics provides a very convenient substitute. It is given by a self-adjoint operator in Hilbert space. Note that the choice of Hilbert space is irrelevant here since all separable infinite dimensional Hilbert spaces are isomorphic.

All the usual attributes of real variables such as their range, the number of times a real number is reached as a value of the variable etc... have a perfect analogue in the quantum mechanical setting.

The range is the spectrum of the operator, and the spectral multiplicity gives the number of times a real number is reached. In the early times of quantum mechanics, physicists had a clear intuition of this analogy between operators in Hilbert space (which they called q-numbers) and variables.

What is surprising is that the new set-up immediately provides a natural home for the "infinitesimal variables" and here the distinction between "variables" and numbers (in many ways this is where the point of view of Newton is more efficient than that of Leibniz) is essential.

Indeed it is perfectly possible for an operator to be "smaller than epsilon for any epsilon" without being zero. This happens when the norm of the restriction of the operator to subspaces of finite codimension tends to zero when these subspaces decrease (under the natural filtration by inclusion). The corresponding operators are called "compact" and they share with naive infinitesimals all the expected algebraic properties. Indeed they form a two-sided ideal of the algebra of bounded operators in Hilbert space and the only property of the naive infinitesimal calculus that needs to be dropped is the commutativity. It is only because one drops commutativity that variables with continuous range can coexist with variables with countable range.

Thus it is the uniqueness of the separable infinite dimensional Hilbert space that cures the above problem, L^2[0,1] is the same as l^2(N), and variables with continuous range coexist happily with variables with countable range, such as the infinitesimal ones. The only new fact is that they do not commute, and the real subtlety is in their algebraic relations. For instance it is the lack of commutation of the line element

**ds**with the coordinates that allows one to measure distances in a noncommutative space given as a spectral triple.## Thursday, June 21, 2007

### Journal of Noncommutative Geometry, Issue 3

The third issue of Journal of Noncommutative Geometry has just appeared. This is a newly established Journal devoted to the subject and its application. In the description of the Journal we read:``The Journal of Noncommutative Geometry covers the noncommutative world in all its aspects. It is devoted to publication of research articles which represent major advances in the area of noncommutative geometry and its applications to other fields of mathematics and theoretical physics".

Four articles appear in this issue. `The first Hochschild cohomology group of quantum matrices and the quantum special linear group' by S. Launois and T. H. Lenagan; `Base change and K-theory for GL(n) ' by Sergio Mendes and Roger Plymen; `Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: cell operads' by Ralph M. Kaufmann; and `Conformal structures in noncommutative geometry' by Christian Bär. Enjoy reading!

Four articles appear in this issue. `The first Hochschild cohomology group of quantum matrices and the quantum special linear group' by S. Launois and T. H. Lenagan; `Base change and K-theory for GL(n) ' by Sergio Mendes and Roger Plymen; `Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: cell operads' by Ralph M. Kaufmann; and `Conformal structures in noncommutative geometry' by Christian Bär. Enjoy reading!

## Wednesday, June 13, 2007

### Determinant, Trace, and Noncommutative Geometry

Recently there was some discussion in the n-category cafe about linear algebra and specially about determinants and their place in a linear algebra undergraduate course. The whole discussion was in fact triggered by a rather polemic paper entitled `Down with Determinants' which seemed to suggest that determinants better be introduced in the graduate school first! This `extremist' point of view was, understandably, challenged by several people who pointed out various algebraic and geometric aspects of determinants and their importance. You can also find a rebuttal in Lieven Le Bruyn's blog here. I would like to echo those sentiments and say: long live determinants!

I don't want to say much about determinants or how linear algebra should be taught to undergraduates. This is not what this blog is all about and in fact I am not sure I am qualified enough in that regard. Others are of course most welcome to comment on all aspects of these issues as they see fit. I just wanted to mention one aspect of determinants and its relation with traces that have some implications for NCG. But first a bit of early history of determinants is perhaps in order.

Determinants have a long history going back to some work of Leibniz and the Japanese mathematician Seki Kowa, also know as Takakazu, in the 17 th century. Cramer's rule of mid 18th century seems to be the first general result on determinants. In the 19th century Sylvester suggested determinants be called `Bezoutians' in honor of Etienne Bezout (see this newly published English translation of Bezout's old text, General Theory of Algebraic Equations ). For more history and in particular to get a glimpse of what happened in the 19th century see this Wikepedia article which also cites a 3rd century BC Chinese text!

Now the point of introducing determinants was not to show that matrices (over algebraically closed fields) have eigenvalues-this came later of course. There are equally interesting applications of determinants, however (see below for just a few), and postponing a proper introduction of determinants to graduate years I don't think would be wise.

As for the existence of eigenvalues for matrices or more generally the non-emptiness of the spectrum of an element of an algebra I know at least two very general situations where one can show that the spectrum of any element is non-empty. They both work in infinite dimensional situations and neither use a determinant function (which may not exist after all). When the algebra is a complex unital Banach algebra this result is due to Gelfand. In fact the whole notion of a Banach algebra is due to Gelfand from the late 1930's who called it normed rings. This was then used in a crucial way, by Gelfand and Naimark, in the proof of their celebrated theorem on the structure of commutative C*-algebras. A second case is an algebra over an algebraically closed field where the dimension of the algebra is less than the cardinality of the field. The argument in this case is similar to the proof of the existence of eigenvalues used in the `dwd' paper mentined above. Notice that the dimension of the algebra need not be finite now. This added generality is not a luxury and comes quite handy in proving things like Hilbert's Nullstellensatz (for fields with uncountable number of elements-see the first chapter of this book). Notice that Nullstellensatz is an algebraic analogue of the Gelfand-Naimark theorem and both results are pivotal for the general philosophy of noncommutative geometry.

On the educational side, see here for a nice story on `how to compute determinants', or perhaps how not to compute a determinant! Halmos' old (1940's?) book on finite dimensional vector spaces has gone through many editions but is still highly readable and one of my favorites. It is written with a view towards functional analysis and operator theory on Hilbert space. So one learns early enough about spectral theorem, polar decomposition, and determinants which are defined using the exterior algebra and volume forms. It does not cover much in multilinear algebra though. Another favorite of mine is Manin and Kostrikin's more modern and wonderful book Linear Algebra and Geometry . It covers a lot of topics well beyond the standard stuff on canonical forms. Things such as multilinear algebra, determinants and Pfaffians, reverse triangle inequality in Minkowski space and the twin paradox, foundations of quantum mechanics, and fast multiplication algorithms. It has even some small item on Feynman rules in QFT! Another favourite of mine is Prasolov's problems and theorems in linear algebra. You should also definitely check P. Cartier's A course on determinants (in: Conformal Invariance and String Theory, 1987) for a nice and modern survey of determinants. It covers much including things like infinite dimensional Fredholm determinants, and superdeterminants.

In the 19th century when people wanted to prove that the sum and product of algebraic numbers is again an algebraic number they would use determinants. Nowadays of course this is done using vector spaces and the formula dim_E K=(dim_E F)( dim_F K) for field extensions E C F C K. But imagine you really want to find a polynomial with rational coefficients P such that P (a+b)=0, or P(ab)=0, assuming a and b are algebraic. What would you do? Here is a modern adaptation of the classical method to find such a polynomial P quickly. Notice that a complex number is algebraic iff it is the eigenvalue of a matrix with rational coefficients. If a and b are eigenvalues of A and B, then a+b is an eigenvalue of I \otimes B + A\otimes I (remember the Hamiltonian of a combined system of two particles in QM?) and ab is of course an eigenvalue of A \otimes B. We should then just compute the characteristic polynomials of these matrices which is pretty straightforward. A similar proof applies to show that if a and b are algebraic integers then so are ab and a+b.....

Determinants and Physics: Pauli's exclusion principle in quantum mechanics can be formulated mathematically as saying that if the Hilbert space of states of a fermion is H then the Hilbert space of states of a pair of fermions should be H /\H, the exterior product of H with itself. By the same principle the Hilbert space of n fermions should be the n-th exterior power of H, /\^n H. For bosons, on the other hand, the appropriate n particle Hilbert space is the n-th symmetric power S^n H. Now, mandated by the special theory of relativity, quantum field theory and the second quantization of fields tell us that the number of particles can not remain constant and so, in the case of fermions, we go over to the so called fermionic Fock space

/\ H = C+ H + H/\H + H/\H/\H +..........

An operator A: H -> H induces an operator

/\ A : /\ H -> /\ H, (/\ A) (v_1 /\v2 ....../\v_n)= Av_1 /\Av_2 ..../\Av_n

Similarly for bosons the apprpriate Hilbert space is the bosonic Fock space

SH =C+ H + S^2 H + S^3 H +......

with the associated operator

SA: SH -> SH

Now if H is n-dimensional, /\^n V is one dimnsional and we see that the exterior algebra gives us a formula/definition for the determinant in terms of trace:

Det (A) = Tr (/\^n A)

We see a direct link here between the exterior algebra, fermions, and determinants. The above formula is the beginning of a series of formulas relating the determinant and trace. For example the beautiful MacMahon Master Theorem states that

(1) Det (1+tA) =\sum t^k Tr (/\^k A)

and if we put t=-1 we obtain

(2) Det (1-A) =\sum (-1)^k Tr (/\^k A)

To prove this one can first assume A is diagonal(izable) in which case the proof is really easy and then use the fact that diagonalizable matrices are dense among all matrices, plus continuity and invariance under conjugation of both sides. There are of course more algebraic proofs that work over any commutative ring.

We would like to write the RHS of (2) as trace of /\A acting on /\ V but because of signs it can not be the usual trace. Instead we can invoke the fact that the fermionic Fock space /\V is a super vector space graded by degree of tensors. We can define the Supertrace of an operator A as Tr_s (A)= Tr (A^+) - Tr (A^-) with A^+ and A^- designating the even and odd parts of A and with this understood we can write (2) as

(3) Det (1-A) =Tr_s (/\ A)

There is a similar formula for the bosonic second quantization and this time we have

(4) [Det (1-tA)]^-1 =\sum t^k Tr (S^k A)

If we put t=1 we obtain the beautiful formula

(5) [Det (1-A)]^-1 = Tr (SA)

Combining formulas (3) and (5) we obtain the boson-fermion formula

(6) Tr_s(/\ A) Tr (SA)=1

which puts in duality the exterior algebra with symmetric algebra. I have completely bypassed the convergence issues which are relevant even when H is finite dimensional since the bosonic Fock space SH is infinte-dimensional but this can be managed relatively easily....

For a particular choice of H and A, Bost and Connes in their paper `Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory. Selecta Math. (N.S.) 1 (1995), no. 3, 411--457, right in the begining show that the above formula (5) gives the Euler product formula for the zeta function (see also page 529 of this book) In fact their paper starts by quantizing the set of prime numbers by finding a natural operator whose bosonic second quantization has the set of primes as its spectrum (I leave to you as an exercise the task of finding this operator) and this is the begining of their long journey towards understanding the statistical behaviour of primes using tools of quantum mechanics and noncommutative geometry.

Another interesting issue with regard to the boson-fermion duality formula (6) is its relation with Koszul duality. The Koszul dual of the quadratic algebra /\ H is the symmetric algebra S H. I suppose it is possible to give a proof of (6) starting from Koszul duality but this will need another post and I certainly hope others who know more will jump in and enlighten us. The same with q-analogues of (6). There are many other interesting things that remain to be said things like Pfaffians as fermionic Gaussian integrals and their supersymmetric and Clifford algebra analogues, regularized determinants, etc. Hopefully others will comment on aspects of determinants and their applications in their work.

Enter Trace Unlike the determinant, trace has a nice straightforward extension to the noncommutative world. In fact trace is the real queen of noncommutative geometry! For any algebra A, commutative or not, we have a trace map

Tr: M_n(A) -> T(A)

From the algebra of n by n matrices with entries in A to the quotient of A by the linear span of commutators in A. It is, be definition, the sum of the elements on the main diagonal followed by the quotient map. It has the trace property in the sense that it is linear (over C) and satisfies T(xy)=T(yx) for all x and y. It is also easy to see that Tr is indeed the universal trace on A in the sense that any other trace tr: A ->V factors through it. This can be extended a bit. Let E be a finitely generated projective right A-module and let End_A (E) denote the algebra of A-linear maps from E to E. Then there is a trace map

Tr: End_A (E) -> T(A).

It can be defined by first embedding E into a finite and free A-module and then using the above trace. Alternatively one can use the fact that End_A (E) =E\otimes_A E* and then use the standard dual pairing between E and its dual E* to land in A and then apply the quotient map.

This allows us to extend the notion of dimension from commutative to noncommutative geometry. The word dimension is loaded with many meanings and interpretations and here we just look at one of those. Let us fix a C-valued trace tr on A. The classical formula

Dim (E) = tr (id_E)

relates the dimension of a vector space or the fiber dimension of a vector bundle to trace and is integer valued in that context. When we use this formula as the definition of the dimension of a finite projective module (aka noncommutative vector bundles) we should be prepared to see non-integral dimensions! (See page 361 of Alain's 1994 book for an example).

Remark: Apart from Hausdorff dimension, this sort of continuous dimensions were first investigated by von Neumann in a purely algebraic and synthetic manner in his book continuous geometry but then his dream of a continuous geometry was, partially, realized in his theory of von Neumann algebras. We say partially because it covered only the measure theoretic aspects of the noncommutative world. The full dream was only realized by the advent of NCG!

I don't want to say much about determinants or how linear algebra should be taught to undergraduates. This is not what this blog is all about and in fact I am not sure I am qualified enough in that regard. Others are of course most welcome to comment on all aspects of these issues as they see fit. I just wanted to mention one aspect of determinants and its relation with traces that have some implications for NCG. But first a bit of early history of determinants is perhaps in order.

Determinants have a long history going back to some work of Leibniz and the Japanese mathematician Seki Kowa, also know as Takakazu, in the 17 th century. Cramer's rule of mid 18th century seems to be the first general result on determinants. In the 19th century Sylvester suggested determinants be called `Bezoutians' in honor of Etienne Bezout (see this newly published English translation of Bezout's old text, General Theory of Algebraic Equations ). For more history and in particular to get a glimpse of what happened in the 19th century see this Wikepedia article which also cites a 3rd century BC Chinese text!

Now the point of introducing determinants was not to show that matrices (over algebraically closed fields) have eigenvalues-this came later of course. There are equally interesting applications of determinants, however (see below for just a few), and postponing a proper introduction of determinants to graduate years I don't think would be wise.

As for the existence of eigenvalues for matrices or more generally the non-emptiness of the spectrum of an element of an algebra I know at least two very general situations where one can show that the spectrum of any element is non-empty. They both work in infinite dimensional situations and neither use a determinant function (which may not exist after all). When the algebra is a complex unital Banach algebra this result is due to Gelfand. In fact the whole notion of a Banach algebra is due to Gelfand from the late 1930's who called it normed rings. This was then used in a crucial way, by Gelfand and Naimark, in the proof of their celebrated theorem on the structure of commutative C*-algebras. A second case is an algebra over an algebraically closed field where the dimension of the algebra is less than the cardinality of the field. The argument in this case is similar to the proof of the existence of eigenvalues used in the `dwd' paper mentined above. Notice that the dimension of the algebra need not be finite now. This added generality is not a luxury and comes quite handy in proving things like Hilbert's Nullstellensatz (for fields with uncountable number of elements-see the first chapter of this book). Notice that Nullstellensatz is an algebraic analogue of the Gelfand-Naimark theorem and both results are pivotal for the general philosophy of noncommutative geometry.

On the educational side, see here for a nice story on `how to compute determinants', or perhaps how not to compute a determinant! Halmos' old (1940's?) book on finite dimensional vector spaces has gone through many editions but is still highly readable and one of my favorites. It is written with a view towards functional analysis and operator theory on Hilbert space. So one learns early enough about spectral theorem, polar decomposition, and determinants which are defined using the exterior algebra and volume forms. It does not cover much in multilinear algebra though. Another favorite of mine is Manin and Kostrikin's more modern and wonderful book Linear Algebra and Geometry . It covers a lot of topics well beyond the standard stuff on canonical forms. Things such as multilinear algebra, determinants and Pfaffians, reverse triangle inequality in Minkowski space and the twin paradox, foundations of quantum mechanics, and fast multiplication algorithms. It has even some small item on Feynman rules in QFT! Another favourite of mine is Prasolov's problems and theorems in linear algebra. You should also definitely check P. Cartier's A course on determinants (in: Conformal Invariance and String Theory, 1987) for a nice and modern survey of determinants. It covers much including things like infinite dimensional Fredholm determinants, and superdeterminants.

In the 19th century when people wanted to prove that the sum and product of algebraic numbers is again an algebraic number they would use determinants. Nowadays of course this is done using vector spaces and the formula dim_E K=(dim_E F)( dim_F K) for field extensions E C F C K. But imagine you really want to find a polynomial with rational coefficients P such that P (a+b)=0, or P(ab)=0, assuming a and b are algebraic. What would you do? Here is a modern adaptation of the classical method to find such a polynomial P quickly. Notice that a complex number is algebraic iff it is the eigenvalue of a matrix with rational coefficients. If a and b are eigenvalues of A and B, then a+b is an eigenvalue of I \otimes B + A\otimes I (remember the Hamiltonian of a combined system of two particles in QM?) and ab is of course an eigenvalue of A \otimes B. We should then just compute the characteristic polynomials of these matrices which is pretty straightforward. A similar proof applies to show that if a and b are algebraic integers then so are ab and a+b.....

Determinants and Physics: Pauli's exclusion principle in quantum mechanics can be formulated mathematically as saying that if the Hilbert space of states of a fermion is H then the Hilbert space of states of a pair of fermions should be H /\H, the exterior product of H with itself. By the same principle the Hilbert space of n fermions should be the n-th exterior power of H, /\^n H. For bosons, on the other hand, the appropriate n particle Hilbert space is the n-th symmetric power S^n H. Now, mandated by the special theory of relativity, quantum field theory and the second quantization of fields tell us that the number of particles can not remain constant and so, in the case of fermions, we go over to the so called fermionic Fock space

/\ H = C+ H + H/\H + H/\H/\H +..........

An operator A: H -> H induces an operator

/\ A : /\ H -> /\ H, (/\ A) (v_1 /\v2 ....../\v_n)= Av_1 /\Av_2 ..../\Av_n

Similarly for bosons the apprpriate Hilbert space is the bosonic Fock space

SH =C+ H + S^2 H + S^3 H +......

with the associated operator

SA: SH -> SH

Now if H is n-dimensional, /\^n V is one dimnsional and we see that the exterior algebra gives us a formula/definition for the determinant in terms of trace:

Det (A) = Tr (/\^n A)

We see a direct link here between the exterior algebra, fermions, and determinants. The above formula is the beginning of a series of formulas relating the determinant and trace. For example the beautiful MacMahon Master Theorem states that

(1) Det (1+tA) =\sum t^k Tr (/\^k A)

and if we put t=-1 we obtain

(2) Det (1-A) =\sum (-1)^k Tr (/\^k A)

To prove this one can first assume A is diagonal(izable) in which case the proof is really easy and then use the fact that diagonalizable matrices are dense among all matrices, plus continuity and invariance under conjugation of both sides. There are of course more algebraic proofs that work over any commutative ring.

We would like to write the RHS of (2) as trace of /\A acting on /\ V but because of signs it can not be the usual trace. Instead we can invoke the fact that the fermionic Fock space /\V is a super vector space graded by degree of tensors. We can define the Supertrace of an operator A as Tr_s (A)= Tr (A^+) - Tr (A^-) with A^+ and A^- designating the even and odd parts of A and with this understood we can write (2) as

(3) Det (1-A) =Tr_s (/\ A)

There is a similar formula for the bosonic second quantization and this time we have

(4) [Det (1-tA)]^-1 =\sum t^k Tr (S^k A)

If we put t=1 we obtain the beautiful formula

(5) [Det (1-A)]^-1 = Tr (SA)

Combining formulas (3) and (5) we obtain the boson-fermion formula

(6) Tr_s(/\ A) Tr (SA)=1

which puts in duality the exterior algebra with symmetric algebra. I have completely bypassed the convergence issues which are relevant even when H is finite dimensional since the bosonic Fock space SH is infinte-dimensional but this can be managed relatively easily....

For a particular choice of H and A, Bost and Connes in their paper `Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory. Selecta Math. (N.S.) 1 (1995), no. 3, 411--457, right in the begining show that the above formula (5) gives the Euler product formula for the zeta function (see also page 529 of this book) In fact their paper starts by quantizing the set of prime numbers by finding a natural operator whose bosonic second quantization has the set of primes as its spectrum (I leave to you as an exercise the task of finding this operator) and this is the begining of their long journey towards understanding the statistical behaviour of primes using tools of quantum mechanics and noncommutative geometry.

Another interesting issue with regard to the boson-fermion duality formula (6) is its relation with Koszul duality. The Koszul dual of the quadratic algebra /\ H is the symmetric algebra S H. I suppose it is possible to give a proof of (6) starting from Koszul duality but this will need another post and I certainly hope others who know more will jump in and enlighten us. The same with q-analogues of (6). There are many other interesting things that remain to be said things like Pfaffians as fermionic Gaussian integrals and their supersymmetric and Clifford algebra analogues, regularized determinants, etc. Hopefully others will comment on aspects of determinants and their applications in their work.

Enter Trace Unlike the determinant, trace has a nice straightforward extension to the noncommutative world. In fact trace is the real queen of noncommutative geometry! For any algebra A, commutative or not, we have a trace map

Tr: M_n(A) -> T(A)

From the algebra of n by n matrices with entries in A to the quotient of A by the linear span of commutators in A. It is, be definition, the sum of the elements on the main diagonal followed by the quotient map. It has the trace property in the sense that it is linear (over C) and satisfies T(xy)=T(yx) for all x and y. It is also easy to see that Tr is indeed the universal trace on A in the sense that any other trace tr: A ->V factors through it. This can be extended a bit. Let E be a finitely generated projective right A-module and let End_A (E) denote the algebra of A-linear maps from E to E. Then there is a trace map

Tr: End_A (E) -> T(A).

It can be defined by first embedding E into a finite and free A-module and then using the above trace. Alternatively one can use the fact that End_A (E) =E\otimes_A E* and then use the standard dual pairing between E and its dual E* to land in A and then apply the quotient map.

This allows us to extend the notion of dimension from commutative to noncommutative geometry. The word dimension is loaded with many meanings and interpretations and here we just look at one of those. Let us fix a C-valued trace tr on A. The classical formula

Dim (E) = tr (id_E)

relates the dimension of a vector space or the fiber dimension of a vector bundle to trace and is integer valued in that context. When we use this formula as the definition of the dimension of a finite projective module (aka noncommutative vector bundles) we should be prepared to see non-integral dimensions! (See page 361 of Alain's 1994 book for an example).

Remark: Apart from Hausdorff dimension, this sort of continuous dimensions were first investigated by von Neumann in a purely algebraic and synthetic manner in his book continuous geometry but then his dream of a continuous geometry was, partially, realized in his theory of von Neumann algebras. We say partially because it covered only the measure theoretic aspects of the noncommutative world. The full dream was only realized by the advent of NCG!

## Wednesday, June 6, 2007

### On the Chicago conference

Hi guys, sorry it took me so long to decide to post something. I was at the Chicago conference and I agree that it was a very interesting event. Unfortunately, I did not go to all the talks (I am a bit overloaded with conferences as the moment, so - much as I would like it - I never manage to attend more than a couple of talks a day), so I can only give a very partial overview of what was happening there. One talk I very much regretted missing was the one Sasha Goncharov gave, which everyone said was great. If someone would be so kind as to give me a quick summary of it, I'd love to hear more about it. Another one I would have been really interested in but I also missed due to having miscalculated the time (shame on me) was the one Kevin Costello gave on renormalization. In both cases, I very much hope that papers will be soon available on the archive (keep an eye on that). I missed many other talks I would have really liked to attend, which I will not try to list here.

Now for something I did see: David Ben-Zvi's talk was very inspiring. It was based on the joint work with David Nadler, who gave one more talk on the subject. The paper is on the archive at arXiv:0706.0322. The main idea is to use loop space geometry to build a parallel between the category of equivariant D-modules on flag varieties associated to reductive groups and categories of equivariant coherent sheaves on Steinberg varieties, via a version of S^1 localization. I am not doing justice to this really interesting work with these few words, but please do look at the paper on the eprints, and since you are at it take a look also at the work of Ben-Zvi and Thomas Nevins (also a speaker at the conference and a former classmate of mine in graduate school).

Yuri Manin gave the introductory talk of the workshop on his work with Dennis Borisov (math/0609748) on internal cohomomorphisms for operads. A broad context envisioned for dealing with symmetries and moduli problems in noncommutative geometry. Starting with internal cohomomorphisms of associative algebras, which was the way Manin approached quantum group symmetries of noncommutative spaces in his noncommutative geometry and his quantum groups books. From that starting point the theory is developed to include the kind of operadic constructions (as functors of labelled graphs) that are essential in the theory of moduli spaces of curves. This is an operadic version of noncommutative geometry designed to carry over moduli problems and combine them with Hopf algebra symmetries.

Another talk I enjoyed was the one given by Spencer Bloch. That's work in progress, so once again keep an eye on upcoming stuff (probably on Bloch's webpage if not on the archive). His current work is related to "Feynman motives" namely motives associated to graph hypersurfaces, that are meant to realize Feynman integrals as periods. On this you can look at the very nice Takagi Lectures on Bloch's webpage as well as the famous Bloch-Esnault-Kreimer paper . In the Chicago talk he described the role of certain compactifications due to Betsvina-Feighn of Out(F_n) in studying resolutions of singularities for graph hypersurfaces and the relation between the Connes-Kreimer Hopf algebra and Kontsevich's graph homology, as well as on the lifting of the CK Hopf algebra at the motivic level, a theme already discussed in his Takagi Lectures.

Jonathan Block gave a talk on a derived categories framework for spectral triples in terms of curves DGA's. This looks like a very promising viewpoint, especially in applications of spectral triple to algebro-geometric or number-theoretic contexts. This framework is the basis of an upcoming series of papers by the author, of which the first two are available on the archive as math/0509284 and math/0604296. These are meant to provide a setting where Mukai transform and Baum-Connes conjecture naturally interact.

Another really nice talk I attended was that given by Sasha Polishchuk on solutions to the associative Yang-Baxter equations (and relations between these and quantum and classical Yang-Baxter equations) obtained from elliptic curves and degenerations thereof. You can find some of this in the paper math/0612761.

Masha Vlasenko gave a nice talk on her recent number theoretic work on the "Eisenstein cocycle" (see math/0611214). Her previous work on theta functions of noncommutative tori with real multiplication is very interesting too (see math/0601405).

Among other talks I attended there was a nice one by Voronov on duality in graph homology, which gives a very nice identification between Koszul duality for operads and Verdier duality for constructible sheaves on spaces of graphs, in particular for the case of "Outer space" X_n/Out(F_n) with X_n the space of metric graphs with markings and the case of moduli spaces of curves, realized via moduli spaces of ribbon graphs.

The talk was based on joint work of Voronov with Lazarev available as math/0702313.

Victor Ginzburg talked about Calabi-Yau algebras (see math/0612139) in relation to Kai Behrend's perfect obstruction theory (related to Behrend's nice paper on Donaldson-Thomas invariants math/0507523) and the relation of CY algebras to quiver representations. There were some nice examples like Heegaard splittings of 3-manifolds, quantum del Pezzo surfaces and McKay correspondence in dimension 3.

I won't comment on the other talks. In fact, I apologize to all the speakers I mentioned here for the inaccuracies and outright mistakes I made in reporting on their talks. I also apologize to all the speakers whose talks I missed or attended but did not mention here. I just wanted to give a brief feeling of the general flavor of the conference. It would be very nice if other people who attended it would like to complement this very partial report with something more accurate.

Now for something I did see: David Ben-Zvi's talk was very inspiring. It was based on the joint work with David Nadler, who gave one more talk on the subject. The paper is on the archive at arXiv:0706.0322. The main idea is to use loop space geometry to build a parallel between the category of equivariant D-modules on flag varieties associated to reductive groups and categories of equivariant coherent sheaves on Steinberg varieties, via a version of S^1 localization. I am not doing justice to this really interesting work with these few words, but please do look at the paper on the eprints, and since you are at it take a look also at the work of Ben-Zvi and Thomas Nevins (also a speaker at the conference and a former classmate of mine in graduate school).

Yuri Manin gave the introductory talk of the workshop on his work with Dennis Borisov (math/0609748) on internal cohomomorphisms for operads. A broad context envisioned for dealing with symmetries and moduli problems in noncommutative geometry. Starting with internal cohomomorphisms of associative algebras, which was the way Manin approached quantum group symmetries of noncommutative spaces in his noncommutative geometry and his quantum groups books. From that starting point the theory is developed to include the kind of operadic constructions (as functors of labelled graphs) that are essential in the theory of moduli spaces of curves. This is an operadic version of noncommutative geometry designed to carry over moduli problems and combine them with Hopf algebra symmetries.

Another talk I enjoyed was the one given by Spencer Bloch. That's work in progress, so once again keep an eye on upcoming stuff (probably on Bloch's webpage if not on the archive). His current work is related to "Feynman motives" namely motives associated to graph hypersurfaces, that are meant to realize Feynman integrals as periods. On this you can look at the very nice Takagi Lectures on Bloch's webpage as well as the famous Bloch-Esnault-Kreimer paper . In the Chicago talk he described the role of certain compactifications due to Betsvina-Feighn of Out(F_n) in studying resolutions of singularities for graph hypersurfaces and the relation between the Connes-Kreimer Hopf algebra and Kontsevich's graph homology, as well as on the lifting of the CK Hopf algebra at the motivic level, a theme already discussed in his Takagi Lectures.

Jonathan Block gave a talk on a derived categories framework for spectral triples in terms of curves DGA's. This looks like a very promising viewpoint, especially in applications of spectral triple to algebro-geometric or number-theoretic contexts. This framework is the basis of an upcoming series of papers by the author, of which the first two are available on the archive as math/0509284 and math/0604296. These are meant to provide a setting where Mukai transform and Baum-Connes conjecture naturally interact.

Another really nice talk I attended was that given by Sasha Polishchuk on solutions to the associative Yang-Baxter equations (and relations between these and quantum and classical Yang-Baxter equations) obtained from elliptic curves and degenerations thereof. You can find some of this in the paper math/0612761.

Masha Vlasenko gave a nice talk on her recent number theoretic work on the "Eisenstein cocycle" (see math/0611214). Her previous work on theta functions of noncommutative tori with real multiplication is very interesting too (see math/0601405).

Among other talks I attended there was a nice one by Voronov on duality in graph homology, which gives a very nice identification between Koszul duality for operads and Verdier duality for constructible sheaves on spaces of graphs, in particular for the case of "Outer space" X_n/Out(F_n) with X_n the space of metric graphs with markings and the case of moduli spaces of curves, realized via moduli spaces of ribbon graphs.

The talk was based on joint work of Voronov with Lazarev available as math/0702313.

Victor Ginzburg talked about Calabi-Yau algebras (see math/0612139) in relation to Kai Behrend's perfect obstruction theory (related to Behrend's nice paper on Donaldson-Thomas invariants math/0507523) and the relation of CY algebras to quiver representations. There were some nice examples like Heegaard splittings of 3-manifolds, quantum del Pezzo surfaces and McKay correspondence in dimension 3.

I won't comment on the other talks. In fact, I apologize to all the speakers I mentioned here for the inaccuracies and outright mistakes I made in reporting on their talks. I also apologize to all the speakers whose talks I missed or attended but did not mention here. I just wanted to give a brief feeling of the general flavor of the conference. It would be very nice if other people who attended it would like to complement this very partial report with something more accurate.

### Report on recent conferences in NCG

I have some feeling of guilt for being late in reporting on the two recent conferences I (partially) attended: one in Vanderbilt, the other in Chicago, and for indulging in lighter stuff like reporting on the (very nice) conference on Pauli and Jung... Sorry, but that was easier.

As for the conference in Vanderbilt the slides are available on line, as well as the list of talks. I'll try to write something on that conference first and will report on the Chicago conference later. Unfortunately I was there only very briefly and it would be great if somebody who attended the whole meeting would volunteer to write a report! So far we got very little help, with the notable exception of David Goss, and we are badly in need for spontaneous contributions.

It is of course impossible to report on all of the talks, and one can at best give some "impressionist" summary of some of the talks that happened to trigger some interaction with one's mathematical preoccupations of the time. This generates a very personal view, and not mentioning a talk should, of course, not be taken as an offense.

Thus after these preliminaries I'll just begin. One of the "raison d'être" of the Vanderbilt school, this year, was to try and bridge the gap between quantum groups and NCG. Both the classes of Christian Kassel and the classes of Masoud Khalkhali and Atabey Kaygun gave the basics in the two subjects with as a possible goal the Hopf cyclic theory--the noncommutative analogue of the Chern-Weil theory-- in relation with quantum groups. There are plenty of open questions and problems to work on in that area (Hopf cyclic + quantum groups) and we hope that some of the students will be ready to work on these.

As it turned out an important puzzle in the interaction between quantum groups and NCG, that remained obscure for quite some time got completely resolved recently and opens up a new area of interaction. One natural set-up for the analogue of Riemannian geometry in NCG is the paradigm of "spectral triples" (A,H,D) where A is the algebra (of coordinates on the NC space) and D is the "Dirac operator", both concretely represented in the same Hilbert space H. The basic requirement is that the resolvent of D is compact, while the commutator [D,a] of D with elements of A is bounded. These two requirements create a tension which allows to measure distances in the corresponding NC space.

Now it has remained for long quite unclear whether this paradigm would work for the homogeneous spaces of compact quantum groups. The first guess is to take simply the q-analogue of the ordinary Dirac operator. This does not work because the eigenvalues of this q-analogue grow geometrically so that the space looks 0-dimensional, in contradiction with the positive dimension of the undeformed maximal torus in the q-group.

Together with Gianni Landi we proposed at some point, by analogy with a particular class of deformations, to replace the q-analogue by its isospectral counterpart, hoping that this would suffice to eliminate the above dimension obstruction. Then followed a rather epic story starting with a no-go theorem and then a first breakthrough by two Indian mathematicians, Partha Sarathi Chakraborty, Arupkumar Pal who managed to construct a spectral triple on the q-group SU(2)_q that had the hoped for boundedness and regularity. It was a very ingenious construction, all the more because it was

**not**a deformation of the commutative spectral triple! In fact the obtained NC-geometry is sufficiently esoteric that I got cold sweat trying to compute what the local index formula was amounting to in that case. One extremely interesting feature is that whereas "locality" has the usual straightforward meaning in the commutative case, the above geometry of Chakraborty and Pal teaches us that in the NC-case "locality" can mean that you are able to strip all the formulas from the irrelevant "details" that only modify them by very small operators.The next breakthrough was done in a paper by Ludwik Dabrowski, Giovanni Landi, Andrzej Sitarz, Walter van Suijlekom and Joseph C. Varilly who managed to show that there was a natural isospectral deformation of the classical Dirac operator with all the required properties (including the bi-invariance under the q-group action on itself!). Of course SU(2) is very important but the issue of finding the analogue for arbitrary compact q-groups (still in the Lie category) remained opened, and has now been beautifully settled by Sergey Neshveyev, and Lars Tuset who construct the isospectral Dirac in general, with the hoped for regularity properties. This opens up the question of computing the local index formula in that generality, the pseudo-differential calculus and the analogue of the cotangent space. There are also potential links with Hopf-cyclic and finally with the general theory of deformations.

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