I've found many papers characterizing the weak lower semi-continuity of $$ \Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))dx, $$ on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$. Are there any results with regards to the same statement with $\Omega$ being all of $\mathbb{R}^d$ itself?

Here $f$ is a non-negative, Borel measurable function, which is convex in its second variable for a.e. one of its first variagbles.

Suppose that $V$ is a finite dimensional $\mathbb Q$-vector space. To each subspace $S$ of dimension $k$, we can associate the line from the origin of $\Lambda^k(V)$ through the point $s_1\wedge \ldots \wedge s_k$ where $\{s_1,\ldots,s_k\}$ is an arbitrary basis for $S$. This is essentially representing subspaces via the Plucker embedding.

Now, suppose that I have a pair of elements $\alpha \in \Lambda^k(V)$ and $\beta\in \Lambda^{k'}(V)$ corresponding to some subspaces $S$ and $S'$. These elements are given in the usual coordinates on the exterior algebra relative to some basis for $V$.

Is there a simple way to compute the element of the exterior algebra corresponding to the subspace $S+S'$?

If $S\cap S'=0$, then $S+S'$ is represented simply by $\alpha\wedge\beta$, but it's not clear how to do a similar calculation if $S$ and $S'$ intersect non-trivially

While there is a reasonably straightforwards way to do this - in particular, given $\alpha$ and $\beta$, we can find a basis for each of the $S$ and $S'$ and then, using the exterior algebra, we can relatively easily find a basis for $S+S'$ and then pass back into the exterior algebra. However, this seems rather inelegant, especially given the really nice formula that exists if the spaces intersect trivially.

It is also trivial to do this, using Hodge duality, if one were able to compute an element representing $S\cap S'$ from $\alpha$ and $\beta$. This problem seems analogous to finding a $\gcd$ of $\alpha$ and $\beta$ in $\Lambda(V)$ considered as a ring.

Mainly, I'm wondering about this since I'm trying to mechanistically do various computations on some arrangements of codimension $2$ subspaces and have found the exterior algebra to be an good tool for similar computations on arrangements of hyperplanes, but moving from codimension $1$ to codimension $2$ requires knowing how to carry out this computation in greater generality.

The following comes from some remarks of Philip Protter at page 26 of the book *Stochastic integration and Differential* equations that I have not been able to prove yet.

Let $X$ a Levy process, under a filtration satisfying the usual conditions. If $\Lambda$ is a Borel set in $\mathbb{R}$ bounded away from zero (that is $0 \notin \bar{\Lambda}$), then the jumping times

\begin{align} &T_{\Lambda}^{1} = \lbrace t \geq : \Delta X_{t} \in \Lambda \rbrace \\ &\vdots \\ &T_{\Lambda}^{n} = \lbrace t > T_{\Lambda}^{n-1} : \Delta X_{t} \in \Lambda \rbrace \end{align} are stopping times.

**My attempt**
Since the filtration satisfies the usual conditions, we only need to prove that $\lbrace T_{\Lambda} < t \rbrace \in \mathcal{F}_{t}$.

Let $\epsilon := d(0, \Lambda) >0$ and $M:= ( - \infty, - \epsilon] \cup [\epsilon, \infty)$, I am trying to prove

\begin{align} \lbrace T_{\Lambda} < t \rbrace = \left( \bigcup_{r \in [0, t) \cap \mathbb{Q}} \lbrace \Delta X_{r} \in \Lambda \rbrace \right) \cap \lbrace T_{M} < t\rbrace \end{align} If we can prove this equation, we are done. This is due to the fact that $\lbrace T_{M} \leq t\rbrace \in \mathcal{F}_{t}$ since \begin{align} \lbrace T_{M} < t\rbrace = \bigcap_{n} \bigcup_{r,s \in [0, t+1/n)\\ \vert r-s \vert < 1/n} \lbrace \vert X_{s} - X_{r} \vert > \epsilon \rbrace \end{align}

We know that the "$\supset$" is the easy part, but the "$\subset$" part is the only part that I need to prove. I was trying to prove this by contradiction, and seems that it is the best way.

If $w \in \lbrace T_{M} < t\rbrace $ and $w \notin \lbrace T_{M} < t\rbrace $ is a contradiction. This can be done using lemmas of discontinuities and the fact that $d(0, \Lambda) >0$. However the part $w \in \lbrace T_{M} < t\rbrace $ and $w \notin \left( \bigcup_{r \in [0, t) \cap \mathbb{Q}} \lbrace \Delta X_{r} \in \Lambda \rbrace \right)$ is the difficult one.

Any hint will be welcome.

The so-called orbital integral problem "amounts to determining the value of a function on $G$ at $e$ in terms of its integrals over (generic) conjugacy classes" (Helgason).

I suppose my question is quite vague, but why do we only care about the value at the identity? A function is surely not explicitly given by its value at the identity alone (even a smooth, compactly supported one).

Let $\rho$ be a continuous, finite dimensional complex representation of the Galois group $\operatorname{Gal}(\overline{F}/F)$, for $F$ a $p$-adic field. Is there a general notion of an Artin conductor of $\rho$?

If $\rho$ is a character of $\operatorname{Gal}(\overline{F}/F)$, then the conductor of $\rho$ is defined as a sum involving the higher ramification groups of a finite Galois extension of $F$ through which $\rho$ factors.

Is there moreover a definition of the Artin conductor of $\rho$ when $\rho$ is a representation of the local Weil group $W_F$?

A basic PDE I would like to understand much better is the viscous Hamilton-Jacobi equation, such as: \begin{equation*} u - \epsilon \Delta u + H(Du) = f(x) \end{equation*} or \begin{equation*} u_{t} - \epsilon \Delta u + H(Du) = f(x) \end{equation*} with or without boundary conditions in the stationary case, or the Cauchy problem in the time-dependent case. I'm interested in the case when $\epsilon > 0$.

Very general viscosity solutions theory implies these equations have continuous solutions under mild assumptions on $H$ and $f$. However, my understanding is the Laplacian term should give us much better regularity than just continuity.

This is a relatively basic example and quite well-motivated from the point of view of stochastic control theory, but nonetheless I'm having trouble finding a down-to-earth reference that shows how to establish regularity for these equations without throwing in the kitchen sink. (In other words, I'm looking for a reference at the level of lecture notes so that I can avoid a little longer wading through Gilbarg-Trudinger or the parabolic equivalent.). It would be particularly nice if the reference in question used a fixed point theorem argument to get existence and regularity simultaneously, but I'm open to an alternative approach.

Is anyone aware of lectures notes that explain how to establish regularity for these equations? Alternatively, are there papers where this is explained in a compact way? My complaint as a student here is this is touched on only very briefly in Evans (in the discussion of fixed point theorems) and the more advanced textbooks on this strike me as extremely dense and somewhat old-fashioned. I may as well start working my way into those books, but if I can get a head start with something more concrete it would be nice.

I asked a related question here on MO without any answers yet.

The question is in the title - give an example of a convex $n$-gon that cannot be subdivided into $k>1$ congruent convex polygons. Even better, give a family that solves this for all combinations of $(n,k)$.

Intuitively, any generic $n$-gon should work, but the crux is in the details - I am very curious about what methods one can use to rigorously prove that a subdivision is impossible.

There are of course many variations, e.g., drop the convex restriction, and remove the restriction that the pieces are polygons.

It's well know that it is surprisingly difficult to prove that $\mathbb{R}^n$ and $\mathbb{R}^m$ are not homeomorphic for $n\neq m$. Commonly proofs go through Brouwer's fixed point theorem, which is 'computably false' for dimensions greater than one: let $K$ denote the computable real numbers. For $n>1$, there are computable functions from $\left( [0,1]\cap K \right)^n$ to itself with no computable fixed point.

That by itself may not seem very bad, since the function may still extend to a continuous function on $[0,1]^n$ and have an incomputable fixed point, but a corollary of this is that there is a computable retraction of $\left( [0,1]\cap K \right)^n$ onto its boundary. Such a function clearly can't be extended to a continuous function on $[0,1]^n$. So we can see that the topological behavior of $K^n$, even when restricted to computable functions, is very different from the topological behavior of $\mathbb{R}^n$.

On the other hand, $K$ is homeomorphic to $\mathbb{Q}$ (although not computably so) and $\mathbb{Q}^n$ is homeomorphic to $\mathbb{Q}^m$ for any $n$ and $m$, so $K^n$ is homeomorphic to $K^m$ for any $n$ and $m$.

So the question is: Is there a computable homeomorphism between $K^n$ and $K^m$ for some $n\neq m$? If there is one I would assume we need $n,m>1$.

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?

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 $\varphi$?

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

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:**

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

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 $S = Spec(\mathbb C)$?

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?

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?

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:**

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.

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 $k=1,\dotsc,n$?

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.

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.

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 !

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)?