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most recent 30 from mathoverflow.net 2018-07-15T13:14:40Z

Decategorification of Gaitsgory's strange functional equation?

Wed, 06/20/2018 - 10:48

Let $G$ a complex reductive algebraic group, $X$ be a smooth compact complex curve. The moduli stack $Bun_G$ of principal $G$-bundles on $X$ is generally speaking not quasi-compact (so we do not have Verdier duality). However, Drinfeld has conjectured that a different functor, $Ps\text{-}Id_{Bun_G, !}$, gives us an equivalence between the category of D-modules on $Bun_G$ and its dual $$ Ps\text{-}Id_{Bun_G, !}\colon (D\text{-}mod(Bun_G))^\vee\rightarrow D\text{-}mod(Bun_G). $$ This conjecture has been proved by Gaitsgory. As a by-product of the proof, Gaitsgory has found that there is a canonical isomorphism of functors $D\text{-}mod(Bun_M)\rightarrow D\text{-}mod(Bun_G)$ $$ Eis_!^-\cdot Ps\text{-}Id_{Bun_M, !}\simeq Ps\text{-}Id_{Bun_G, !}\cdot (CT_*)^\vee, $$ where $M$ is the Levi quotient for a parabolic $P \subset G$, $Eis_!^-$ is the geometric Eisenstein series functor for the opposite parabolic $P^-$ and $CT_*^\vee$ is the dual of geometric constant term functor. He dubbed this isomorphism 'strange functional equation'.

My question is: have decategorifications of this functional equation been studied? Does it descend to an interesting relation on the level of Hochschild cohomologies, for example?

automorphisms of a measurable space can be approximated by continuous measure preserving maps?

Wed, 06/20/2018 - 09:24

Suppose first that $D=[0,1]$ is equipped with the usual Lebesgue measure, and that $\varphi$ is a measure-preserving transformation $\varphi:D\to D$ that is bijective and whose inverse is also measure preserving (called automorphism).

Is it true that there exists a sequence of continuous measure-preserving transformations $\varphi_n:D\to D$ converging in measure to $\varphi$?

Computer program for counting graph homomorphisms

Tue, 06/19/2018 - 19:26

I would like to ask is there a computer program for counting graph homomorphisms?

Is algebraic $K$-theory a motivic spectrum?

Tue, 06/19/2018 - 12:06

I've received conflicting messages on this point -- on the one hand, I've been told that "forming a natural home for algebraic $K$-theory" was one motivation for the development of motivic homotopy theory. On the other hand, I've been warned about the fact that algebraic $K$-theory isn't always $\mathbb A^1$-local. By "algebraic $K$-theory,", I mean the algebraic $K$-theory of perfect complexes of quasicoherent sheaves (I think -- let me know if I should mean something else).

  • I'm pretty sure that Thomason and Trobaugh show that algebraic $K$-theory always (in the quasicompact, quasicoherent case) satisfies Nisnevich descent.

  • Under certain conditions, algebraic $K$-theory is $\mathbb A^1$-local and has some kind of compatibility with $\mathbb G_m$ which should make it $\mathbb P^1$-local. I think Weibel (already Quillen) calls this "the fundamental theorem of algebraic $K$-theory".

So putting this together, let $S$ be a scheme, and let $SH(S)$ be the stable motivic ($\infty$-)category over $S$.

Questions:

  1. Is algebraic $K$-theory of smooth schemes over $S$ representable as an object of $SH(S)$?

  2. How about if we put some conditions on $S$ -- say it's regular, noetherian, affine, smooth over an algebraically closed field? Heck, what if we specialize to $S = Spec(\mathbb C)$?

  3. Does it make a difference if we redefine $SH(S)$ to be certain sheaves of spectra over the site of smooth schemes affine over $S$ or something like that?

Obtaining the distribution of the First Hitting time of the Bessel Process

Tue, 06/19/2018 - 04:49

Let $(X_{t})_{t\geq 0}$ be a Bessel Process starting at $x>0$ of dimension $\delta>0$. Namely \begin{align*} X_{t}=x+W_{t}+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{X_{s}}\, ds. \end{align*} where $(W_{t})_{t\geq 0}$ is a Brownian Motion. I am interested in how to find the distribution of the Hitting Time $\tau:=\inf\{t>0\,|\, X_{t}=0\}$.

Given that the Bessel Process can be expressed as a time changed Brownian Motion, up to the first hitting of the boundary, is it possible to obtain the distribution of $\tau$ by utilising the Reflection Principle for Brownian Motion?

On the A+B=C conjecture

Mon, 06/18/2018 - 22:00

Could You give a comment or a reference for the conjecture as follows:

Conjecture: Let $A, B, C$ be three positive integer numbers such that $A+B=C$ with $\gcd(A, B, C) = 1$. By Fundamental theorem of arithmetic we write:

$A=a_1^{x_1}a_2^{x_2}...a_n^{x_n}$,

$B=b_1^{y_1}b_2^{y_2}...b_m^{y_m}$,

$C=c_1^{z_1}c_2^{z_2}...c_k^{z_k}$

Let $d = \min\{x_i, y_j, z_h \}$ where $1 \le i \le n, 1\ \le j \le m, 1\le h \le k$ then $$d \le 5$$

PS: I read above one hunded paper, I observed that in any case $\min\{x_i, y_j, z_h \} \le 3$?

Example 1: Ten solutions of Catalan-Fermat equation

$1^m+2^3=3^2$ for $m>6$ and

$1414^3+2213459^2=65^7$

$9262^3+15312283^2=113^7$

$7^3+13^2=2^9$

$2^5+7^2=3^4$

$3^5+11^4=122^2$

$2^7+17^3=71^2$

$17^7+76271^3=21063928^2$

$33^8+1549034^2=15613^3$

$43^8+96222^3=30042907^2$

Example 2:

$2^4.3^5.7^6+5^9.11^8=19^1.23^1.47^1.6679^1.3051977^1$

See also:

Homotopy limit of model categories in the category of categories

Mon, 06/18/2018 - 10:31

Say $$\mathcal{C'}\to \mathcal{C}\leftarrow \mathcal{D}$$ is a diagram of model categories and (e.g. Left) Quillen functors. I want to write down a (hopefully simple) model category $\mathcal{D}'$, or at least a category with weak equivalences, such that its $\infty$-categorical localization is the homotopy limit of the localizations of this diagram in the $(\infty,1)$ category of $(\infty, 1)$ categories. Is there a nice way to do this? I'm willing to impose any reasonable niceness conditions on the categories in the diagram.

About some positive elements in a group von Neumann algebra

Mon, 06/18/2018 - 09:32

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to be $\sum\overline{a_g}g^{-1}$ and for $\beta=\sum b_gg\in\mathbb C[G]$, we have the (convolution) product $\alpha\beta:=\sum a_gb_hgh$.

We call a selfadjoint elemet $\alpha\in\mathbb C[G]$ (i.e. $\alpha=\alpha^*$) golden if $a_e\in\mathbb R$ and $a_e-\sum_{g\neq e}|a_g|\geq0$. For $\beta\in\mathbb C[G]$, if it is possible to write $\beta^*\beta$ (or some of it's powers, $(\beta^*\beta)^n$) as a combination $\sum_{k=1}^nr_k\alpha_k$ where $r_k\geq0$ and $\alpha_k\in\mathbb C[G]$ is golden, for all $k=1,\dotsc,n$?

1-dimensional p-divisible groups, level structures and Cartier divisors

Mon, 06/18/2018 - 06:18

I am confused about the 1-dimensionality of $p$-divisible groups and its role in defining level structures.

Here's how I view/understand/not understand things:

If a $p$-divisible group arises from a dimension $g$ abelian variety (say over some $S$ over $\mathbb F_p$), then it is of height $2g$ and dimension at least $g$, with equality in the ordinary case.

So for $g>1$, such $p$-divisible groups are never 1-dimensional and if $g=1$, they are of height 2.

On the other hand, from what I've been reading, whenever a good notion of level structure is mentioned, the assumption is usually that the $p$-divisible group is 1-dimensional. I am still confused as to why. I understand it should be related to the fact that Cartier divisors make sense but am not entirely sure what is the ambient curve since such $p$-divisible groups do not arise from abelian varieties..

I am familiar with Katz-Mazur's definition of level structure/full set of sections, I understand Drinfeld modules and the notion of level structure (in that case the Drinfeld module is 1-dimensional over the base so Cartier divisors make sense etc). I am however confused how this all relates among each other... in the etale case a level structure seems to be a choice of isomorphism with the constant group scheme, but then there is also a notion of level structure for formal $p$-divisible groups, and I've usually interpreted (maybe erroneously?) "formal" roughly as being "connected"?; but then over a perfect base (say a perfect field) there are no sections and then any level structure is trivial?.. I really hope this brief rambling exposes to an expert where my confusion is..

As an example of an explicit question, on page 20 of https://arxiv.org/pdf/1005.2558.pdf to a Drinfeld level structure $\varphi:\mathbb F_p^d\to X_0[p]$ a filtration is defined by the equality of divisors $[H_i]=\displaystyle\sum_{x\in\text{span}(e_1,\ldots,e_i)}[\varphi(x)]$. Q: Where do these divisors "live" and where exactly is 1-dimensionality used?

Any illuminating comments/answers would be greatly appreciated.

Thank you.

Edit: By dimension of a $p$-divisible groups I mean the (locally constant) rank of the Lie algebra (such as in Messing, or page 59 of Harris-Taylor "The Geometry and Cohomology of Some Simple Shimura Varieties"). In particular the height can be any integer $h\ge1$. The arxiv paper referenced above works with higher height 1-dimensional groups, which shouldn't arise from abelian varieties, yet speaks of Cartier divisors, hence my confusion as I don't know on which ambient scheme these live on.

Closures of torus orbits in flag varieties

Sun, 06/17/2018 - 20:37

Consider the Lie group $G=SL_n(\mathbb C)$ with Borel subgroup $B$ and maximal torus $T\subset B$. I'm interested in the (Zariski) closures of $T$-orbits in the flag variety $F=G/B$.

Now, as far as I can tell, for a generic point in $F$ this closure is the toric variety associated with the permutahedron. Further, let us choose an element of the Weyl group $w$ and let $X_w\subset F$ be the corresponding Schubert variety. Then for a generic point in $X_w$ the closure seems to be the toric variety associated with the convex hull of vertices of the permutahedron corresponding to $w'\ge w$ with respect to the Bruhat order, i.e. the convex hull of the weight diagram of the corresponding Demazure module in an irrep with a regular highest weight.

Questions.
1) Is this last description accurate?
2) My main question. Are, in fact, all orbits of this form? More precisely, let $S$ be the set of all points in $F$ that are generic in some Schubert variety in the above sense. Is it then true that any point outside of $S$ can be mapped to a point in $S$ by the action of the Weyl group?

References to literature are much appreciated.

Can we recover a sheaf from its original presheaf on a basis

Sun, 06/17/2018 - 20:12

This question might be completely totological (I apologize in advance if it is the case):

suppose that we are given two sheaves $\mathcal{F}, \mathcal{G}$ of Abelian groups on a topological space $X$, and denote by $F, G$ their underlying respective presheaves (that is their images via the inclusion functor $Sh(X) \hookrightarrow Psh(X)$). Suppose that $F$ and $G$ agree on a basis of topology. Does this imply that the sheaves agree ?

I know that, in general, the data of a presheaf on a basis doesn’t suffice to recover the sheaf, but here it is the presheaf coming from a sheaf, so I have somehow the feeling that it is different. Am I missing something ?

Thanks a lot !

Compact generation of the category of D-modules on moduli stack of principal bundles for algebraic groups?

Sun, 06/17/2018 - 19:32

Let $k$ be an algebraically closed field of characteristic 0. Let $X$ be a connected smooth complete curve over $k$. Consider the moduli stack $\mathrm{Bun}_G$ of principal $G$-bundles on $X$ for connected reductive group $G$.

Geometric Langlands conjecture states (among other things) that the DG category $D(\mathrm{Bun}_G)$ of D-modules on $\mathrm{Bun}_G$ should be equivalent to the DG category of ind-coherent sheaves on the moduli stack of $\check{G}$-local systems with singular support contained in the global nilpotent cone. Our ability to work with $D(\mathrm{Bun}_G)$ in practice depends on it admitting a set of compact generators.

Drinfeld and Gaitsgory have shown that for $X$ and $G$ satisfying the conditions above, the category $D(\mathrm{Bun}_G)$ is compactly generated. This poses a natual question: is the category $D(\mathrm{Bun}_G)$ compactly generated for any connected affine algebraic group $G$? My question is: has there been any progress on this question since Drinfeld--Gaitsgory (or maybe people have constructed counterexamples)?

Definitions I don't find

Sun, 06/17/2018 - 19:05

I am reading an article and I didn't find some of the definitions of the terms the author use in the article. If someone know I will appreciate ! What is the prime-to-$p$ part of a torsion subgroup on a module ? (I know what his a torsion subgroup but prime-to-$p$ part ? ) And what is a pro-$p$ quotient ? Thanks !

Given a representation-infinite algebra, when is every AR component infinite?

Sun, 06/17/2018 - 18:44

Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. The Auslander-Reiten quiver $\Gamma_A$ of $A$ is a means of presenting the category of finitely generated right $A$-modules. The Auslander-Reiten quiver is a locally finite quiver whose vertices are indecomposable modules (up to isomorphism) and whose arrows are irreducible morphisms between indecomposable modules. It is given the structure of a translation quiver via the Auslander-Reiten translate $\tau$.

Two distinct vertices/modules of $\Gamma_A$ are said to belong to the same component if there exists a finite path/composition of irreducible morphisms between them. If $A$ is representation-finite (i.e. the category of finitely generated right $A$-modules contains finitely many non-isomorphic indecomposable objects), then $\Gamma_A$ consists of finitely many components, which each have finitely many vertices. However if $A$ is representation-infinite, then necessarily $\Gamma_A$ is infinite.

It is easy to construct an example of a representation-infinite algebra $A$ for which there exists a finite component of $\Gamma_A$. For example, consider the algebra $KQ/I$, where $KQ$ is the path algebra of the quiver $$ Q\colon \; 1 \begin{smallmatrix} \alpha \\ \rightarrow \\ \rightarrow \\ \beta \end{smallmatrix} 2 \begin{smallmatrix} \gamma \\ \rightarrow \\ \color{white} \gamma \end{smallmatrix} 3 \begin{smallmatrix} \delta \\ \rightarrow \\ \color{white} \delta \end{smallmatrix} 4$$ and $I$ is the ideal generated by the set $\{\beta\gamma, \gamma\delta\}$. This is representation-infinite because it contains the Kronecker quiver as a subquiver. But the relation $\gamma\delta$ 'cuts off' the right-hand side of the quiver, which means we have a finite Auslander-Reiten component $$\begin{matrix} S(4) & & & & S(3) \\ & \searrow & & \nearrow & \\ & & P(3)=I(4) & & \end{matrix}$$ (which is essentially $\mathrm{mod}\;K\mathbb{A}_2$) in $\Gamma_A$, where $S(v)$, $P(v)$ and $I(v)$ are the simple, indecomposable projective, and indecomposable injective modules corresponding to the vertex $v$ in $Q$ respectively. (One can also use the fact that this is a string algebra to easily compute $\Gamma_A$.)

My question is this: When is it the case that for a representation-infinite algebra $A$, every component of $\Gamma_A$ contains infinitely many vertices? Is this for example true if the algebra is self-injective? (I imagine the proof of the latter question is straightforward if true, but please point me to a reference if one exists.)

Upward confluence in the interaction calculus

Sun, 06/17/2018 - 17:29

The lambda calculus is not upward confluent, counterexamples being known for a long time. Now, what about the interaction calculus? Specifically, I am looking for configurations $c_1$ and $c_2$ such that $\exists c: c_1 \rightarrow^* c\ \wedge\ c_2 \rightarrow^* c$, but $\nexists c': c' \rightarrow^* c_1\ \wedge\ c' \rightarrow^* c_2$.

If a triangle can be displaced without distortion, must the surface have constant curvature?

Sun, 06/17/2018 - 17:17

Suppose $S$ is a surface, a Riemannian manifold in $\mathbb{R}^3$. Let $T$ be a geodesic triangle on $S$: a triangle whose edges are geodesics. If $T$ can be moved around arbitrarily on $S$ while remaining congruent (edge lengths the same, vertex angles the same), does this imply that $S$ has constant curvature?

I realize this is a naive question. If $S$ has constant curvature, then $T$ can be moved around without distortion. I would like to see reasoning for the reverse: If $T$ can be moved around while maintaining congruence, then $S$ must have constant curvature. What is not clear to me is how to formalize "moved around."

Steiner's inequality reference request

Sun, 06/17/2018 - 15:27

I remember seeing somewhere that for every connected compact set $\Omega$ in $\mathbb{R}^2$ with piecewise $C^1$ boundary we have $$A(\Omega_r)\leq A(\Omega)+L(\partial \Omega)r+ \pi r^2,$$ where $$\Omega_r=\{x\in \mathbb{R}^2: d(x,\Omega)\leq r\},$$ $A$ denotes the area and $L$ the length. I tried to find a reference for this inequality, but I only found Steiner's formula which states that equality holds when $\Omega$ is convex. Can someone please give me a reference for that inequality?

Classifying stacks and semidirect product

Sun, 06/17/2018 - 15:11

I am trying to understand the behaviour of quotient stacks under the semidirect product.

Let $G\rtimes H$ be an algebraic group acting on a scheme $X$, over a field $k$. Assume that $G$ acts trivially on $X$. The canonical map $[X/G\rtimes H]\to [X/H]$ is a $G$-gerbe, and it admits a section $s:[X/H]\to[X/G\rtimes H]$ induced by the inclusion $H\subseteq G\rtimes H$. Is $s$ a $G$-torsor? By this I mean the following:

for any scheme $T$ and any map $T\to [X/G\rtimes H]$, the induced map $T\times_{[X/G\rtimes H]}[X/H]\to T$ is a $G$-torsor.

The map $s$ is clearly representable, so at least the question makes sense. Objects of $T\times_{[X/G\rtimes H]}[X/H]$ are $H$-torsors $P$ with an $H$-equivariant map to $X$ such that $P\times^H(G\rtimes H)$ is isomorphic to a fixed $G\rtimes H$-torsor (with compatibility of the maps to $X$), but I cannot define a $G$-action on this set.

Added: consider a map $S\to [X/G\rtimes H]$, corresponding to a torsor $P$. We have $S\rtimes_{[X/G\rtimes H]}X=P$, and therefore $S\times_{[X/G\rtimes H]}[X/H]=P/H$. This certainly becomes isomorphic (as a scheme) to $G$ after passing to a cover of $S$, but I don't see how one can define an action on it to make it a $G$-torsor.

Spectral algebraic geometry vs derived algebraic geometry in positive characteristic?

Sun, 06/17/2018 - 14:58

Let $R$ be a commutative ring. Then there is a forgetful functor from the $\infty$-category of simplicial commutative $R$-algebras to the $\infty$-category of connective $E_{\infty}$-algebras over $R$. It's well-known that this functor admits left and right adjoint. Moreover, it's an equivalence if $R$ contains the field of rational numbers $\mathbb{Q}$. If I understand correctly, this means that SAG and DAG are (more or less) equivalent over such $R$.

On the other hand, if $R$ does not contain $\mathbb{Q}$ (say $R$ is a finite field), then the forgetful functor above is not necessarily an equivalence. This implies that SAG and DAG are very different in this situation. I wonder if there are any examples of problems where using SAG (resp. DAG) is more appropriate? Such examples would help to illustrate the difference between SAG and DAG in positive characteristic.

P.S.: related but not identical questions: $E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras, Why do people say DG-algebras behave badly in positive characteristic?

Which proportion of elliptic curves over Q is proved to satisfy BSD conjecture?

Sun, 06/17/2018 - 14:06

In 2014, Bhargava and other authors proved that more than 66 % of all elliptic curves defined over $ \mathbb{Q} $ satisfy the Birch and Swinnerton-Dyer conjecture. Tonight I stumbled upon an article of Kozuma Morita (arXiv:1803.11074) claiming to prove that all elliptic curves with complex multiplication satisfy this conjecture, entailing a solution of the congruent number problem through Tunnell's theorem.

Assuming such a result holds, which lower bound to the proportion of rational elliptic curves satisfying BSD conjecture could be reached ?

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