Moduli space of almost complex structures as an algebro-geometric object

Let $ M$ be a closed real-analytic manifold of dimension $ 2n$ . Is it possible to make sense of the moduli space of real-analytic almost complex structures on $ M$ as an algebro-geometric object (probably a very non-Noetherian one)? Can this be used to gain a new perspective on, say, Fredholm-regular almost complex structures? I would not think that this would be particularly useful, but I think if this is possible it is worth doing just for the fun of it.

Maybe one terribly non-canonical way to do this is to construct it a subscheme of $ \bigcup_{p\in M}\mathrm{Mat}(2n, 2n)$ (choose a basis for the fiber of tangent bundle at some point, choose a connection to produce bases in points nearby, somehow make sense of the real-analyticity condition, and then take the vanishing scheme of the “polynomial” equation $ J^2=-\mathrm{Id}$ and hope that this can be made to work globally). Hopefully, somebody else thought of a better way.

Tensor product of compact operators on Banach modules

Let $ A$ and $ B$ be Banach algebras. Consider a right Banach $ A$ -module, $ E$ , and a right Banach $ B$ -module, $ F$ , as well as a Banach algebra morphism $ \pi\colon A\to\mathcal L_B(F)$ into the bounded $ B$ -linear operators on $ F$ . Then $ \pi$ makes $ F$ into a left Banach $ A$ -module, and the actions of $ A$ and $ B$ commute. Thus, the (completed) tensor product $ E\otimes_AF$ has the structure of a right Banach $ B$ -module.

Question: Is the tensor product of two compact operators again compact?

To be more precise, let $ F\in\mathcal K_A(E)$ and $ G\in\mathcal K_B(F)$ . Is $ F\otimes_AG\in\mathcal K_B(E\otimes_AF)$ ? As for Hilbert C*-modules, the compact operators on a Banach module are the closure of the linear span of the finite rank operators. Thus, it suffices to consider the case where $ F=F_1F_2$ with $ F_1\in\mathcal L_A(A,E)$ , $ F_2\in\mathcal L_A(E,A)$ , and where $ G=G_1G_2$ with $ G_1\in\mathcal L_B(B,F)$ , $ G_2\in\mathcal L_B(F,B)$ . However, the tensor product $ F_i\otimes_AG_i$ is not defined since $ B$ itself does not carry an $ A$ -module structure.

Motivation: In Lafforgue’s $ KK^{\mathrm{ban}}$ -theory, there is a natural map $ \Psi\colon KK^{\mathrm{ban}}(A,B)\to\mathrm{Hom}(K_0(A),K_0(B))$ which may be described as follows: Elements of $ KK^{\mathrm{ban}}(A,B)$ are represented by Banach $ (A,B)$ -bimodules, i.e. triples $ (F,\pi,T)$ with $ F$ a (graded) right Banach $ B$ -module, $ \pi\colon A\to\mathcal L_B(F)$ an even contractive Banach algebra morphism, and $ T\in\mathcal L_B(F)$ an odd operator such that $ \pi(a)(T^2-1)$ and $ [\pi(a),T]$ are compact for all $ a\in A$ . Now if $ p\in M_n(A)$ is an idempotent, one may pushforward $ p$ via $ \pi$ to obtain an idempotent $ \pi_*p\in M_n(\mathcal L_B(F))\cong\mathcal L_B(F^n)$ . Then $ \Psi[F,\pi,T]([p])=[\pi_*pF^n,(\pi_*p)T^n(\pi_*p)]\in KK^{\mathrm{ban}}(\mathbb C,B)\cong K_0(B)$ .

Now I wondered if one could find a similar description when an element of $ K_0(A)$ is represented not by an idempotent over $ A$ , but rather as an element of $ KK^{\mathrm{ban}}(\mathbb C,A)\cong K_0(A)$ , i.e. by a Fredholm module $ (E,S)$ where $ E$ is a graded right Banach $ A$ -module, and $ S\in\mathcal L_A(E)$ is an odd operator satisfying $ S^2-1\in\mathcal K_A(E)$ . A first try would be to put $ \Psi[F,\pi,T]([E,S])=[E\otimes_AF,T\otimes_AS]$ , where the tensor product is built using $ \pi$ . However, one would have to prove that $ (T\otimes_AS)^2-1=(T^2-1)\otimes_A(S^2-1)$ is compact, which boils down to the question above.

About $\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$

An integral has been pushed me over the edge for several weeks. It reads as: $ $ \displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$ $

I tried to calculate the surface integral inside using spherical coordinates, but it seems that I couldn’t do any further calculation since the integrand function is something like $ $ e^{-\big(k_1(\varphi)\sin^2\theta+k_2(\varphi)\cos^2\theta+k_3(\varphi)\sin\theta\cos\theta\big)}\sin\theta .$ $ Then I tried to use variable substitution to compute, similarly, I didn’t get anything useful. I was also trying to use Maple to compute, but it didn’t work at all. My original intention is to prove that the formula $ $ \displaystyle e^{-\frac{1}{2}|x|^2}\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$ $ is bounded.

I would be grateful if you could give me a definite result.

Expected number of times of choosing a word out of a given vocabulary when words are grouped into overlapping bins

Two players (player C and player G) are playing a (modified) word guessing game. Both players share the same vocabulary $ V$ and words in $ V$ are grouped into $ K$ bins, denoted as $ b_1$ , $ b_2$ , …, $ b_{K}$ . Furthermore, we know that $ b_{i} \subset V$ , $ \cup_{i=1}^{K} b_i = V$ . Note that those bins may be overlapping and thus there may exist some case where $ b_i \cap b_j \neq \emptyset$ for $ i \neq j$ .

The interaction protocol is described as follows:

  1. Player C uniformly chooses a word $ w$ from the vocabulary $ V$ . Player G does not know which word $ w$ is.

  2. Player G chooses one bin and asks Player C whether his/her chosen word $ w$ is in the bin. If it is, the game ends. Otherwise, Player G will choose another bin.

Questions: What is the best bin choosing order and what is the expected number of times of choosing the bin, according to the best possible order?

Example:

Suppose we have a vocabulary consisting of ten words $ V = \{w_1, w_2, …, w_{10} \}$ and three bins $ b_1 = \{w_1, w_2, …, w_5\}$ , $ b_2 = \{w_6, w_7 \}$ , and $ b_3 = \{w_8, w_9, w_{10} \}$ .

One possible bin choosing order is $ b_1 \rightarrow b_3 \rightarrow b_2$ and the expected number of times of choosing the bin is $ \frac{1}{2}*1 + \frac{1}{2}*\frac{3}{5}*2 + \frac{1}{2}*\frac{2}{5}*\frac{2}{2}*3 = 1.7$ . I suspect this is the best bin choosing order but how can we prove this result?

Notes:

In this problem, we do NOT have the additional knowledge that all bins are non-overlapping, compared to a related problem.

Thanks.

About Pogorelov-Nadirashvili-Yuan’s local isometric embedding counterexample

In Pogorelov’s paper “An example of a two-dimensional Riemannian metric admitting no local realization in E3. Dokl. Akad. Nauk SSSR Tom 198(1), 42–43 (1971); English translation in Soviet Math. Dokl. 12, 729–730 (1971) 123” and Nadirashvili-Yuan’s paper “Improving Pogorelov’s isometric embedding counterexample. Calc. Var. Partial Differential Equations 32 (2008), no. 3, 319–323. 53C45”, there is one local isometric embedding counterexample for $ C^{2,1}$ Riemannian metric. As far as I know, those two papers are the only source for such counterexample for local isometric embedding.

There is a crucial estimate about graph’s second derivatives: $ \min_{-c\leq t_1\leq c} h_{11}(t_1, b)\leq (M- m)\frac{b^2}{c^2}$ (see page 322 of Nadirashvili-Yuan’s paper). I can not verify this estimate.

Did anyone verify the above counterexample in details or know how to get the above estimate?

Recover a mongo database deleted by rm

My developer accidentally lost our database by a bash script, which turned out to run rm -rf /* (see this thread). Thanks to extundelete, he just recovered the /data/db/ folder:

enter image description here

However, he could not successfully load the database in Robo 3T; it looks quite empty here:

enter image description here

Here is the result of running mongod --port 27017 --dbpath /data/db --bind_ip_all:

root@iZj6c0pipuxk17pb7pbaw0Z:/data# mongod --port 27017 --dbpath /data/db --bind_ip_all  2019-03-25T08:49:52.425+0800 I CONTROL  [main] Automatically disabling TLS 1.0, to force-enable TLS 1.0 specify --sslDisabledProtocols 'none' 2019-03-25T08:49:52.452+0800 I CONTROL  [initandlisten] MongoDB starting : pid=6130 port=27017 dbpath=/data/db 64-bit host=iZj6c0pipuxk17pb7pbaw0Z 2019-03-25T08:49:52.452+0800 I CONTROL  [initandlisten] db version v4.0.7 2019-03-25T08:49:52.452+0800 I CONTROL  [initandlisten] git version: 1b82c812a9c0bbf6dc79d5400de9ea99e6ffa025 2019-03-25T08:49:52.452+0800 I CONTROL  [initandlisten] OpenSSL version: OpenSSL 1.0.2g  1 Mar 2016 2019-03-25T08:49:52.452+0800 I CONTROL  [initandlisten] allocator: tcmalloc 2019-03-25T08:49:52.452+0800 I CONTROL  [initandlisten] modules: none 2019-03-25T08:49:52.452+0800 I CONTROL  [initandlisten] build environment: 2019-03-25T08:49:52.452+0800 I CONTROL  [initandlisten]     distmod: ubuntu1604 2019-03-25T08:49:52.452+0800 I CONTROL  [initandlisten]     distarch: x86_64 2019-03-25T08:49:52.452+0800 I CONTROL  [initandlisten]     target_arch: x86_64 2019-03-25T08:49:52.452+0800 I CONTROL  [initandlisten] options: { net: { port: 27017 }, storage: { dbPath: "/data/db" } } 2019-03-25T08:49:52.455+0800 W STORAGE  [initandlisten] Detected unclean shutdown - /data/db/mongod.lock is not empty. 2019-03-25T08:49:52.455+0800 I STORAGE  [initandlisten] Detected data files in /data/db created by the 'wiredTiger' storage engine, so setting the active storage engine to 'wiredTiger'. 2019-03-25T08:49:52.455+0800 W STORAGE  [initandlisten] Recovering data from the last clean checkpoint. 2019-03-25T08:49:52.455+0800 I STORAGE  [initandlisten] 2019-03-25T08:49:52.455+0800 I STORAGE  [initandlisten] ** WARNING: Using the XFS filesystem is strongly recommended with the WiredTiger storage engine 2019-03-25T08:49:52.455+0800 I STORAGE  [initandlisten] **          See http://dochub.mongodb.org/core/prodnotes-filesystem 2019-03-25T08:49:52.455+0800 I STORAGE  [initandlisten] wiredtiger_open config: create,cache_size=256M,session_max=20000,eviction=(threads_min=4,threads_max=4),config_base=false,statistics=(fast),log=(enabled=true,archive=true,path=journal,compressor=snappy),file_manager=(close_idle_time=100000),statistics_log=(wait=0),verbose=(recovery_progress), 2019-03-25T08:49:53.090+0800 E STORAGE  [initandlisten] WiredTiger error (17) [1553474993:90473][6130:0x7f41f14f9a40], connection: __posix_open_file, 715: /data/db/WiredTiger.wt: handle-open: open: File exists Raw: [1553474993:90473][6130:0x7f41f14f9a40], connection: __posix_open_file, 715: /data/db/WiredTiger.wt: handle-open: open: File exists 2019-03-25T08:49:53.090+0800 I STORAGE  [initandlisten] WiredTiger message unexpected file WiredTiger.wt found, renamed to WiredTiger.wt.1 2019-03-25T08:49:53.689+0800 I STORAGE  [initandlisten] WiredTiger message [1553474993:689973][6130:0x7f41f14f9a40], txn-recover: Main recovery loop: starting at 4/11366912 to 5/256 2019-03-25T08:49:53.692+0800 I STORAGE  [initandlisten] WiredTiger message [1553474993:692366][6130:0x7f41f14f9a40], txn-recover: Recovering log 4 through 5 2019-03-25T08:49:53.761+0800 I STORAGE  [initandlisten] WiredTiger message [1553474993:761295][6130:0x7f41f14f9a40], txn-recover: Recovering log 5 through 5 2019-03-25T08:49:53.826+0800 I STORAGE  [initandlisten] WiredTiger message [1553474993:826192][6130:0x7f41f14f9a40], txn-recover: Set global recovery timestamp: 0 2019-03-25T08:49:53.859+0800 I RECOVERY [initandlisten] WiredTiger recoveryTimestamp. Ts: Timestamp(0, 0) 2019-03-25T08:49:53.863+0800 E STORAGE  [initandlisten] WiredTiger error (17) [1553474993:863179][6130:0x7f41f14f9a40], WT_SESSION.create: __posix_open_file, 715: /data/db/_mdb_catalog.wt: handle-open: open: File exists Raw: [1553474993:863179][6130:0x7f41f14f9a40], WT_SESSION.create: __posix_open_file, 715: /data/db/_mdb_catalog.wt: handle-open: open: File exists 2019-03-25T08:49:53.863+0800 I STORAGE  [initandlisten] WiredTiger message unexpected file _mdb_catalog.wt found, renamed to _mdb_catalog.wt.1 2019-03-25T08:49:53.877+0800 I CONTROL  [initandlisten] 2019-03-25T08:49:53.877+0800 I CONTROL  [initandlisten] ** WARNING: Access control is not enabled for the database. 2019-03-25T08:49:53.877+0800 I CONTROL  [initandlisten] **          Read and write access to data and configuration is unrestricted. 2019-03-25T08:49:53.877+0800 I CONTROL  [initandlisten] ** WARNING: You are running this process as the root user, which is not recommended. 2019-03-25T08:49:53.877+0800 I CONTROL  [initandlisten] 2019-03-25T08:49:53.877+0800 I CONTROL  [initandlisten] ** WARNING: This server is bound to localhost. 2019-03-25T08:49:53.877+0800 I CONTROL  [initandlisten] **          Remote systems will be unable to connect to this server. 2019-03-25T08:49:53.877+0800 I CONTROL  [initandlisten] **          Start the server with --bind_ip <address> to specify which IP 2019-03-25T08:49:53.878+0800 I CONTROL  [initandlisten] **          addresses it should serve responses from, or with --bind_ip_all to 2019-03-25T08:49:53.878+0800 I CONTROL  [initandlisten] **          bind to all interfaces. If this behavior is desired, start the 2019-03-25T08:49:53.878+0800 I CONTROL  [initandlisten] **          server with --bind_ip 127.0.0.1 to disable this warning. 2019-03-25T08:49:53.878+0800 I CONTROL  [initandlisten] 2019-03-25T08:49:53.878+0800 I CONTROL  [initandlisten] 2019-03-25T08:49:53.878+0800 I CONTROL  [initandlisten] ** WARNING: soft rlimits too low. rlimits set to 3824 processes, 65535 files. Number of processes should be at least 32767.5 : 0.5 times number of files. 2019-03-25T08:49:53.895+0800 I STORAGE  [initandlisten] createCollection: admin.system.version with provided UUID: 07c86964-7065-424b-ad94-81a3388b41c1 2019-03-25T08:49:53.910+0800 I COMMAND  [initandlisten] setting featureCompatibilityVersion to 4.0 2019-03-25T08:49:53.921+0800 I STORAGE  [initandlisten] createCollection: local.startup_log with generated UUID: 70822443-d50b-41af-b5ce-1a51e96e4be3 2019-03-25T08:49:53.937+0800 I FTDC     [initandlisten] Initializing full-time diagnostic data capture with directory '/data/db/diagnostic.data' 2019-03-25T08:49:53.940+0800 I NETWORK  [initandlisten] waiting for connections on port 27017 2019-03-25T08:49:53.957+0800 I STORAGE  [LogicalSessionCacheRefresh] createCollection: config.system.sessions with generated UUID: ce139bc0-906a-427a-8e57-4aa42b061e6a 2019-03-25T08:49:53.974+0800 I INDEX    [LogicalSessionCacheRefresh] build index on: config.system.sessions properties: { v: 2, key: { lastUse: 1 }, name: "lsidTTLIndex", ns: "config.system.sessions", expireAfterSeconds: 1800 } 2019-03-25T08:49:53.974+0800 I INDEX    [LogicalSessionCacheRefresh]     building index using bulk method; build may temporarily use up to 500 megabytes of RAM 2019-03-25T08:49:53.976+0800 I INDEX    [LogicalSessionCacheRefresh] build index done.  scanned 0 total records. 0 secs 2019-03-25T08:50:51.611+0800 I NETWORK  [listener] connection accepted from 127.0.0.1:59894 #1 (1 connection now open) 2019-03-25T08:50:51.612+0800 I NETWORK  [conn1] received client metadata from 127.0.0.1:59894 conn1: { application: { name: "MongoDB Shell" }, driver: { name: "MongoDB Internal Client", version: "4.0.7" }, os: { type: "Linux", name: "Ubuntu", architecture: "x86_64", version: "16.04" } } 

Does anyone understand what’s happening? does anyone know how to recover this mongodb?

Non-Compact Example where Putinar’s Positivstellensatz Fails

One way to state Putinar’s Positivstellensatz is as follows: a compact set polynomial inequalities $ \mathcal{P} = \{P_1(x) \geq 0, \ldots, P_m(x) \geq 0\}$ is unsatisfiable if and only if there exists sum of squares polynomials $ Q_0(x), \ldots, Q_m(x) \in \Sigma^2$ such that $ -1 = Q_0 + \sum_{i \in [m]} Q_i(x) P_i(x)$ . I was wondering if an explicit non-compact set of polynomial inequalities is known on which the generalization of Putinar’s Positivstellensatz where we drop the compactness requirement is known to fail? Thanks for the help.

Chirality and Anti-Chirality of links in 3 and in 5 dimensions

We know there is a chiral knot which is a knot that is not equivalent to its mirror image. It is well known in the mathematical field of knot theory: https://en.wikipedia.org/wiki/Chiral_knot

enter image description here

My first question is about

(1) the literature and the References on

  • the chirality of link in 3 dimensions

  • the chirality of link in 5 dimensions

What are some good text/Refs on these chiralities of link of 1-submanifolds in 3 dimensions? addressed somewhere in the literature?

Are these chiralities of link of 2-submanifolds and 3-submanifolds in 5 dimensions be addressed somewhere in the literature?

For example, let me consider a 5-sphere $ S^5$ . Let me define a new quartic link Q of 5-dimensions in $ S^5$ : such that $ \Sigma^3_{W_{{(i)}}},\Sigma^3_{W_{{(ii)}}},\Sigma^3_{W_{{(iii)}}}$ are 3 sets of 3-submanifolds, while the $ \Sigma^2_U$ is a 2-surface. Let $ V^4_{W_{{(i)}}}, V^4_{W_{{(ii)}}}, V^4_{W_{{(iii)}}}, V^3_U$ be their Seifert volumes in one higher dimensions.

There can be a link invariant defined in thie manner: $ $ { \#(V^4_{W_{{(i)}}}\cap V^4_{W_{{(ii)}}}\cap V^4_{W_{{(iii)}}}\cap V^3_U)\equiv\text{Q}^{(5)}(\Sigma^3_{W_{{(i)}}},\Sigma^3_{W_{{(ii)}}},\Sigma^3_{W_{{(iii)}}},\Sigma^2_U)} $ $

I am suspecting there could be a opposite chirality of this invariant defined as $ \overline{\text{Q}^{(5)}}$ as follows: $ $ { \#(V^4_{W_{{(ii)}}}\cap V^4_{W_{{(i)}}}\cap V^4_{W_{{(iii)}}}\cap V^3_U)+\dots \equiv\overline{\text{Q}^{(5)}}(\Sigma^3_{W_{{(ii)}}},\Sigma^3_{W_{{(i)}}},\Sigma^3_{W_{{(iii)}}},\Sigma^2_U)} $ $

(2) Do similar chirality and anti-chirality of link invariants in 3 dimensions, happening for example to Borromean rings? Or other Brunnian links? Examples and References are welcome.

Calculating the number of solutions of integer linear equations

Let $ N$ be a natural number. Consider the following set of matrices whose entries are non-negative integers: $ $ X_N:=\{(c_{ij})_{i,j=1}^4\in M_4(\mathbb{Z}_{\geq 0})| \sum_j c_{1j} = \sum_i c_{2j}, \sum_i c_{i1}=\sum_i c_{i2}, \sum_{i,j}c_{ij} = N\}.$ $ Write $ f(n) = |X_N|$ .

Question: Is there an algorithm / theorem which gives an explicit formula for $ f(n)$ as a function of $ N$ ? It should be polynomial. Without the constraints about the sum of the elements of the first two columns is equal and the sum of the elements of the first two rows is equal, this will just be $ \binom{N+15}{15}$ which is a polynomial of degree 15. I am looking for a similar formula for the number of elements of $ X_N$ .

Exercise on formal group laws over an algebraically closed field

There is an exercise in Weinstein’s notes on Lubin–Tate theory, namely show that there is a unique (up to isomorphism) one-dimensional formal group law of given finite height $ h$ over an algebraically closed field of characteristic $ p$ . A hint is to show that for the Diedonne module $ M$ of such a formal group law $ \mathcal{F}$ we have $ F^h(M)=pM$ .

I think that to prove the latter statement we need the following facts:

  • by definition, there is a power series $ g(X)\in W(\bar{F_p})[[X]]$ such that $ [p]_{\mathcal{F}}X=g(F^h(X))$ and such that $ g'(0)\neq 0 (\mathrm{mod}\, p)$ . I believe this means that we can find another power series $ f\in W(\bar{F_p})[[X]]$ such that $ g\circ f=1(\mathrm{mod}\,p)$ .
  • power series comparable $ \mathrm{mod}\,p$ induce the same map on $ H^1_{dR}$ .
  • reparametrization by a power series from $ W(\bar{F_p})[[X]]$ preserves the class of closed/exact forms.

I am not sure how to proceed from here. One problem is that Frobenius does not really induce a linear map between Diedonne modules but a semi-linear map (maybe here we should use that the ground field is algebraically closed, so $ g(F^h(X))=F^h(g_1(X))$ where $ g_1$ is a power series whose coefficients are the inverse images of the coefficients of $ g$ under the lift of $ \bar{F_p}$ -Frobenius to $ W(\bar{F_p})$ ). Could somebody give a detailed proof so that a novice would understand?