I am trying to determine the integrand in the equivariant Atiyah-Patodi-Singer index theorem by Donnelli ("Eta invariants for G-spaces") for spin manifolds if one only has isolated fixed points. In "Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture" by Gilkey, Leahy and Park they state (equation (1.6.20.a)) without proof or reference, that it is given by $\prod_{j=1}^k\frac{\sqrt{\lambda_j}}{(\lambda_j-1)}$, where $\lambda_j=e^{i\theta_j}$ and $\theta_j$ denotes the rotation angles at a given fixed point. I don't see how to get there and I can't find a reference. Can anyone give an explanation or a reference? Thanks!

Let $K_0(S)$ be the Grothendieck ring of $S$-varieties, where $S$ is a (smooth projective) variety over a field $k$.

How does $K_0(S)$ relate to the collection of Grothendieck rings $K_0(A_i)$ where $\{A_i\}$ is an affine cover of $S$?

What if we take an etale cover instead of a Zariski cover of $S$?

Let $R$ be a commutative ring with $1$ and $I$ be an ideal of $R$. Now let $J=\cap_{I\subseteq m\in Max(R)}m$. Set $A:=\{p\in Spec(R): I\subseteq p\}$ and $B:=\{p\in Spec(R): J\subseteq p\}$, where $Spec(R)$ and $Max(R)$ are the set of all prime and maximal ideals of $R$ respectively. When considering $Spec(R)$ with the Zariski topology, and $B$ and $A$ as two subspaces of $Spec(R)$.I want if $B$ is dense in $A$? Or in general, what kinds of topological properties hold between $A$ and $B$?

Let $(W,S)$ be a Coxeter system. One can have the *Kazhdan-Lusztig polynomial* $P_{x,\ y}(q)$.

Does $P_{x,\ y}(q)=P_{x^{-1},\ y^{-1}}(q)$ for all $x,y\in W$?

I have come across an optimization problem with the following objective function:

$$f(x_0,y_0,z_0,x_1,y_1,z_1,...,x_N,y_N,z_N) = \sum_{i=0}^N f_i(x_i,y_i,z_i, \alpha(x_{i+1}-x_i) + \beta(y_{i+1}-y_i))$$

i.e. the objective function is the sum of the functions $f_i$ that only depend on three variables $x_i,y_i,z_i$ and on a linear combination of the difference to the direct neighbors (for $x_i$ and $y_i$). So far I have tired an NLCG algorithm, but convergence is very slow.

Is there a specialized solver that can exploit the structure of the optimization problem?

I have thought about introducing new variables $s_i = \alpha(x_{i+1}-x_i) + \beta(y_{i+1}-y_i)$ and writing $$f(x_0,y_0,z_0,s_0,...,x_N,y_N,z_N,s_N) = \sum_{i=0}^N f_i(x_i,y_i,z_i,s_i)$$ because then the optimization could be conducted separately for each $f_i$ but I would have to introduce equality constraints to get back the original problem?

$f_i(x_i,y_i,z_i,s_i)$ is nonconvex (there is a $\cos(s_i)$ term in the function); however $f_i(x_i,y_i,z_i)$ is convex.

Let $X_1$ and $X_2$ be two uniformly distributed random variables on [0,1]. What is the statistical distribution of the ratio $\frac{X_1}{X_1+X_2}$ ?

Same question with 3 variables $X_1$, $X_2$ and $X_3$: what is the statistical distribution of $\frac{X_1}{X_1+X_2+X_3}$ ?

Let $A$ be finite dimensional connected algebra. A simple module $S$ is called $n$-regular in case $pd(S)=n$, $Ext_A^i(S,A)=0$ for $i=0,1,...,n-1$ and $Ext_A^n(S,A)$ being a simple $A$-left module. See for example https://www.sciencedirect.com/science/article/pii/S0001870818302809 for the relvance of 2-regular simple modules. We can assume $A$ is a quiver algebra and then being $n$-regular simply means that the injective envelope $I(S)$ of $S$ (having projective dimension $n$) occurs uniquely in the minimal injective coresolution $(I_i)$ of $A$ as a summand of $I_n$.

Questions:

In case every simple module of projective dimension $n$ (and there exists at least one such simple module) is $n$-regular, does $A$ have global dimension $n$? (In case this is false, is it true when assuming $A$ hsa finite global dimension?)

When every right simple module of projective dimension $n$ is $n$-regular (and there exists at least one such simple module) is the same true for every left simple modules of projective dimension $n$?

Let $P$ be the $1$-skeleton of a convex polyhedron fixed in $\mathbb{R}^3$, and $|P|$ the sum of the Euclidean lengths of the edges of $P$. Let $P_1, P_2, P_3$ be the perpendicular projections of $P$ onto the Cartesian coordinate planes, and $|P_i|$ the sum of the lengths of the segments of $P_i$.

For example, for the particular
placement of $P$ a unit edge-length regular tetrahedron shown below,
$|P_1|+|P_2|+|P_3|$ is nearly double $|P|=6$:

$|P_1|$ (red) $=1+\sqrt{\frac{2}{3}}+\sqrt{\frac{11}{3}}$.
$|P_2|$ (green) $=1+\sqrt{3}$.
$|P_3|$ (blue) $=3+\sqrt{3}$.
$\Sigma \approx 11.2$.

**Conjecture**. For any placement of any convex polyhedron $P$,
$|P_1|+|P_2|+|P_3| \ge |P|$, with equality uniquely achieved by the cube.

For a unit edge-length cube $P$, $|P|=12$ and $|P_i|=4$ when oriented so that each projection is a square. So I'm conjecturing that the cube hides its edges in projection more effectively than any other convex polyhedron. Can anyone see a proof or a counterexample?

I would also be interested in which orientations of the regular tetrahedron minimize $\Sigma |P_i|$.

The higher-dimensional analog could be the subject of a future post.

I wonder does the following statement holds: 1. For any sufficiently large $n$ the symmetric group $S_n$ contains at least one 2-transitive subgroup other than $S_n$ itself and $A_n$?

- Actually I'm interested in whether every symmetric group $S_n$, for sufficiently large $n$, contains a transitive subgroup whose order is divisible by $n(n-1)$. If the first statement holds then it obviously implies the second.

I would appreciate any comments on the above two questions.

Conventions: $\cong$ is homeomorphism of topological spaces and isomorphism of groups, $\equiv_G$ is the equality of two words over the generators of the group $G$. Simplicial complexes are finite.

Given two (abstract) simplicial complexes $K, L$, is it semidecidable whether their geometric realizations $|K|, |L|$ are homeomorphic, i.e. is there a computer program that, given $K, L$, eventually terminates if $|K| \cong |L|$, and never terminates if $|K| \not\cong |L|$?

If the problem is not semidecidable, then let me add an obvious continuation out of general interest. Fix a natural Gödel numbering $\mathrm{S} = f : \mathbb{N} \to \mbox{simplicial complexes}$. Let $p : \mathbb{N} \to \mathbb{N}^2$ be a natural bijection.

What is the position of the set $ E = \{ n \in \mathbb{N} \;|\; p(n) = (a,b), f(a) \cong f(b) \} $ in the lightface hierarchy of subsets of $\mathbb{N}$? Is this set in the arithmetical hierarchy?

My attempts below. I'm not an expert and these are based on a day or so of Googling, so I apologize for any misunderstandings.

It is a well-known result of Markov [Markov] that given two simplicial complexes, it is undecidable whether their geometric realizations are homeomorphic (and PL-homeomorphism is also undecidable [Poonen]).

According to Poonen [Poonen], here's an outline of the proof: There is an effective construction A that, from an f.p. group $G$ and a word $w$ over its generators, produces a simplicial complex $X_w$ such that if $w \equiv_G 1$ then $X_w \cong S^5$, while if $w \not\equiv_G 1$, then $X_w \not\cong S^5$.

This does not seem to help: Since the word problem of f.p. groups is semidecidable (uniformly in the description of the group and the word over generators), there is a uniform algorithm that, given any simplicial complex $C$ produced by construction A, solves its homeomorphism to $S^5$: enumerate all f.p. groups and words w over their generators, until you produce $C$, and semidecide $w \overset{?}{\equiv_G} 1$, which is correct assuming the previous construction A has the claimed property. (This algorithm is valid on all inputs, in the sense that it never mistakenly claims something is $S^5$ even if it is not produced by the algorithm of the previous paragraph.)

In [Lazarus] I find the following slightly different take: Given f.p. groups $G$, $H$ we produce combinatorial manifolds $X_G, X_H$ such that $X_G \cong X_H$ if and only if $G \cong H$.

This again does not seem to help: Isomorphism of f.p. groups is semidecidable, so again given two simplicial complexes produced by this method, simply enumerate pairs of f.p. groups, and once you find groups mapping to those complexes, semidecide their isomorphism.

Two simplicial complexes are called combinatorially equivalent if they have subdivisions that are isomorphic (bijection on vertices that induces a bijection on faces). This is semidecidable, just guess subdivisions and check isomorphism for each.

- This does not seem to help: A famous example of Milnor [Milnor] shows that there are examples of pairs of homeomorphic simplicial complexes which are not combinatorially equivalent.

If $f : |K| \to |L|$ is continuous, then there are subdivisions $K', L'$ of $K, L$ respectively, and a simplicial map $f' : K' \to L'$ such that $f'$ is homotopic to $f$ [Ranicki]. It follows that homotopy equivalence is semidecidable.

- This does not seem to help: there exist simplicial complexes which are homotopy equivalent but not homeomorphic, one example being $\{\{0\}\}$ and $\{\{0\}, \{1\}, \{0,1\}\}$.

It seems at least that $E$ is in $\Sigma^1_1$, i.e. lightface analytic. Just guess a rational approximation to a homeomorphism (and modulus of continuity?) I don't know much about levels this high, so my intuition may be off here. (In any case this is a bit too high, and seems to kind of defeat the point of combinatorial representations of topological spaces.)

About my application: I am dealing with a class of topological spaces that contains all finite simplicial complexes, so I can stop trying to semidecide their homeomorphism if the above problem is not semidecidable. I do not have a direct use for a semidecidability result, but by now I would be quite interested in this as well. Results that are intrinsically about PL-homeomorphism / simplicial homotopy&homology are not particularly useful to me because I don't know what the extension of those would be for my superclass, as it contains also e.g. Stone spaces so simplex embeddings are not very useful.

[Markov]: Markov, Andreï Andreyevich. "Insolubility of the problem of homeomorphy." In Dokl. Akad. Nauk SSSR, vol. 121, no. 195, p. 8. 1958.

[Poonen]: Poonen, Bjorn. "Undecidable problems: a sampler." Interpreting Gödel: Critical Essays (2014): 211-241.

[Lazarus]: Lazarus, Francis, and Arnaud de Mesmay. "Undecidability in Topology." (2017).

[Milnor]: Milnor, John. "Two complexes which are homeomorphic but combinatorially distinct." Annals of Mathematics (1961): 575-590.

[Ranicki]: Ranicki, A. A., A. Casson, D. Sullivan, M. Armstrong, C. Rourke, and G. Cooke. "The Hauptvermutung Book." Collection of papers by Casson, Sullivan, Armstrong, Cooke, Rourke and Ranicki, K-Monographs in Mathematics 1 (1996).

Let $\varphi\!:\!S\to R$ be a homomorphisms of $K$-algebras for some field $K$. Let $\{a_{\lambda}\}_{\lambda}$ be a family of ideals of $S$.

Is there some "natural" assumption on $\varphi$ to guaranty that $$ \cap_{\lambda} a_{\lambda}^e = (\cap_\lambda a_\lambda)^e, $$ where $\_^e$ denotes the extension to $R$.

The inclusion $\cap_{\lambda} a_{\lambda}^e \supseteq (\cap_\lambda a_\lambda)^e$ is always satisfied. Which conditions on $\varphi$ (as general as possible) guaranty that the converse inclusion is also satisfied?

For example, if $R= S\otimes_K T$ for some $K$-algebra $T$ and $\varphi(s)=s\otimes 1$. Let's check it.

The ring $R$ is a free $S$ module via $\varphi$. Moreover, given a $K$-basis $\{t_l\}_l$ of $T$, the set $\{1\otimes t_l\}_l$ is a $S$-base of $R$. So, given an ideal $I\subseteq S$ and $r\in R$ with $r=\sum_l s_l(1\otimes t_l)$, then $r\in I^e$ if and only if $s_l\in I$ for all $l$. Hence, if $r\in \cap_\lambda a_\lambda^e$, then $s_l\in a_\lambda$ for all $l$ and $\lambda$, that is $s_l\in\cap_\lambda a_\lambda$ for all $l$ and then $r\in (\cap_\lambda a_\lambda)^e$.

From the geometric point of view it is clear that the corresponding map $f\!:\! X\to Y$, where $X=Spec(R)$ and $Y=Spec(S)$, has to be surjective. Also flatness looks a reasonable assumption and then $\varphi$ is faithfully flat. But with this two assumptions on $\varphi$, I am not able to fine neither a proof that $\cap_{\lambda} a_{\lambda}^e \subseteq (\cap_\lambda a_\lambda)^e$ nor a counterexample. Hence, I am not sure whether "faithfully flat" is the assumption on $\varphi$ that I am looking for or not, but I think it is. (In the example $R=S\otimes T$, $\varphi$ is faithfully flat).

Facts about faithfully flat homomorphisms that could be useful are:

- $\varphi$ is injective, so $S$ is a subring of $R$.
- For every ideal $I\subseteq S$, $I=I^e\cap S$.
- For every prime ideal $p\subseteq R$ the ideal $p\cap S$ is a prime ideal of $S$.

Any suggestion or comment would be highly appreciated.

Given $N$ boxes with the same capacity $C$, I toss coins into the boxes uniformly, one by one. When any one of the boxes is full, the sum of the coins in all boxes is denoted $S$. How to compute the probability density function of $S(N,C)$?

a) Do you think f(x) is continuous at x = 0? Why or why not?

b) What is f(0)?

c) What is equation image indicator? Be sure to justify your answer; you may include a table or graph to help with your justification.

d) Do your answers to b and c support your answer to a? If not, what do you conclude? If so, prove that your answer is correct.?

E.) A student in your class missed a day in class during continuities. Please explain to him what continuities with some images.

We can obtain the Jones polynomial by the Temperly-Lieb algebra and the HOMFLYPT polynomial from the Hecke algebra. Were there attempts to categorify the algebras itself and obtain the Khovanov homology or HOMFYLPT homology from there? When googling, one can find a lot of papers containing certain categorifications of algebras but I find it hard to pinpoint which of these arise most naturally regarding my question.

Recall that the Stiefel-Whitney classes of a smooth manifold are defined to be those of its tangent bundle - this definition doesn't extend to topological manifolds as they don't have a tangent bundle. Wu's theorem states that for a closed smooth manifold, $w = \operatorname{Sq}(\nu)$. The expression $\operatorname{Sq}(\nu)$ makes sense for a closed topological manifold and therefore serves as a definition for the Stiefel-Whitney classes on such a manifold.

Recall that if $M$ is a closed smooth $n$-dimensional manifold, then $w_n(M)$ is equal to the mod $2$ reduction of $e(M)$, see Corollary 11.12 of Milnor and Stasheff's *Characteristic Classes*. In particular, the Stiefel-Whitney number $\langle w_n(M), [M]\rangle$ is the mod $2$ reduction of the Euler characteristic. Is this still true for closed topological manifolds?

Let $M$ be a closed topological $n$-dimensional manifold. If $w_n(M)$ is the top Stiefel-Whitney class of $M$, as defined above, is the Stiefel-Whitney number $\langle w_n(M), [M]\rangle$ the mod $2$ reduction of $\chi(M)$?

Given a random $d$-regular graph on $n$ nodes, what is the expected number of common neighbors between two nodes?

I don't know if it is as simple as just assuming that each neighbor of the first node has a $\frac{d}{n}$ probability of being a neighbor of the second, as the set of $d$-regular graphs on $n$ nodes is difficult to construct.

**I originally asked this on Mathematics Stack Exchange but realized it might be better to ask it here.**

The question is mostly related to homogenization theory in mathematical physics.

$\textbf{Background}$

I will describe the periodic case first and then generalize it to the random case later.

Let's fix a domain $\Omega \subset \mathbb{R}^d$. Define $\Omega_\varepsilon$ to be a domain with holes of radius $a_\varepsilon$ cut out periodically with period $\varepsilon$. Take a look at the picture for reference.

Now consider the laplacian on this cut out domain as follows: \begin{equation*} \begin{split} -\Delta u_\varepsilon&=f \hspace{1em} \text{ in } \Omega_\varepsilon\\ u_\varepsilon&=0 \hspace{1em} \text{ on } \partial \Omega_\varepsilon \end{split} \end{equation*} Not that the boundary conditions also imply that the solution is zero on the holes.

Now as we take $\varepsilon \to 0$ and if we have

$$ a_\varepsilon= \begin{cases} exp(\frac{-C_0}{\varepsilon^2}), \text{ if } d=2\\ C_0\varepsilon^{\frac{d}{d-2}}, \text{ if } d>2 \end{cases} $$

Then we have that $u_\varepsilon$ converges weakly to $u$ in $H^1_0(\Omega)$ where $u$ satisfies the following equation

\begin{equation*} \begin{split} -\Delta u+\mu u&=f \hspace{1em} \text{ in } \Omega\\ u&=0 \hspace{1em} \text{ on } \partial \Omega \end{split} \end{equation*}

Here $\mu$ is a specific constant which can be worked out explicitly. The interesting thing is that there is an extra term which appears to be coming out of nowhere. More on this can be found in the paper https://link.springer.com/chapter/10.1007/978-1-4612-2032-9_4.

Or here https://www.math.u-bordeaux.fr/~cprange/documents/coursEDMI2016_lecture2.pdf

**What is also interesting is that if we play around with $a_\varepsilon$, and consider $a_\varepsilon$ to be smaller than the scaling provided above then the limiting function solves the usual laplacian equation without the $\mu$. If $a_\varepsilon$ is larger then the limiting function is zero.**

Since this stuff is mostly qualitative it's an interesting question to see what are the quantitative rates for the convergence and what is the best Sobolev space in which we get convergence.

We recently considered this problem of quantifying this periodic case and a random case with appropriate assumptions which roughly guarantee us that we would not have a lot of holes in a specific region via a large deviation bound assumption(**think of poisson point cloud as a representative example**).

We were able to prove that even in the random case, under the same scaling we get the strange term. Though in this case, it will be a random variable. And if the scaling is changed then we get the Laplace's equation or the zero function a.s.

**The same question can be seen as a random walk and the holes to be traps, i.e the random walker dies when it hits the boundary of the hole.**

**Question**

Now let's consider the Schrodinger's equation with random potential on $\Omega$. The random potentials are supported on the random holes and are such that if an electron gets in, it essentially stays trapped(I'm thinking of an infinite potential well).

**Now intuitively, a similar trichotomy for the limiting function(if it exists!) should hold ie: diffusion, Brinkmann type equation and localization.**

I was wondering:

If this is at all related to the mighty Anderson Localization problem?

If so, would it be interesting to see if such a limiting object exists and if so, how fast does the convergence occur?

I know how to compute the $\delta$-hyperbolicity of a $2D$ poincaré disk of radius 1, but I was wondering how to generalize such computations to:

- Higher dimensions
- Poincaré disk with radius $r$ and conformal factor $\lambda^r_x=1/(1-\Vert x\Vert^2/r^2)$
**A cartesian product of several poincaré disks/balls**

The third point is the most important.

Intuitively, I think 1) $\delta$ should be independent of the dimension, 2) increasing the radius $r$ should increase $\delta$ because it makes the space look more Euclidean, and 3) I have no idea.

If I have a bilinear form $$x^TAy,$$ which I then compare to a bilinear form $$x^T(I-uu^T)^TA(I-vv^T)y,$$ where $||u||_2 = ||v||_2 = 1$, do there exist simple conditions such that the second form is bounded by the first? $A$ can be assumed to be PSD, if it helps.

My vague intuition is that because $(I-uu^T)$ is still PSD but upper bounded by the identity these conditions should exist, but I haven't been able to find any.

Problem Statement: I would like to find out a generic way of maximizing the sum of modulo 2 additions of binary numbers.

Please see a sample problem, here the variables a0,a1,a2........a10 are binary numbers which can take values 0 and 1.

Here '+' denotes modulo 2 addition.

Our goal is to to find out the values of a0,a1a,a2......a10 such that sum of A0,A1,A2,A3,A4,A5 is maximum.

A0 = a0+a1+a2a+a3+a4;

A1 = a6+a7+a1+a3+a4;

A2 = a10+a9+a8+a7+a2;

A3 = a10+a9+a8+a7+a5;

A4 = a1+a10+a5+a4+a2;

I want a generic algorithm for this irrespective of the the number of equations or the variables associated with each equation.