Maryam Mirzakhani & Forbes, Teichmüller Space, WeChat - Fully Hyperbolic Neural Networks

Hi Readers,

I had a super-busy day at work yesterday and wrote a blog at 1 AM. But, today I have taken a leave to spend some time to chill out.  

--- One of the reasons I speak about Mirror Symmetry and Maryam Mirzakhani is this Forbes article- https://www.forbes.com/sites/startswithabang/2017/08/01/maryam-mirzakhani-a-candle-illuminating-the-dark/?sh=5aa7449836c1 

Sometimes Forbes writes really cool articles right.. I hope they get right influencers to spread awareness of such brilliant articles through influencer marketing while understanding stubborn & non-stubborn players in social networks with opinion mining - Ref - https://arxiv.org/abs/1609.03465 (it would be cool & optimum to spread awareness of math articles through math based influencer marketing right?)  

--- One of the brilliant papers by Maryam Mirzakhani is Volume Growth on Teichmuller Space which speaks about topological entropy of the geodesic flow. (September 20, 2011- Lattice point asymptotics and volume growth on Teichmüller space Jayadev Athreya, Alexander Bufetov, Alex Eskin, Maryam Mirzakhani) Kind of cool paper to study Geodesic CNNs (Geodesic Convolutional Neural Networks bit more)...  

--- The paper Hyperbolic spaces in Teichmüller Space by Christopher J. Leininger and Saul Schleimer explains the way hyperbolic space almost-isometrically embeds into the Teichmüller Space (  https://arxiv.org/abs/1110.6526 ) kind of very smart inspiration from Mirzakhani's paper.. 

--- which is further adopted in Hyperbolic Neural Networks by Octavian-Eugen Ganea ( https://arxiv.org/abs/1805.09112 )which focuses on hyperbolic embeddings - Möbius gyrovector spaces with the Riemannian geometry of the Poincaré model of hyperbolic spaces which helps to embed sequential data and perform classification in the hyperbolic space (for further Poincare Glove as well - HyperE at Stanford)  

While this paper cites Sanja Fidler's paper on Order Embeddings, One of the best papers which has recently come out is by WeChat - Fully Hyperbolic Neural Networks .. one of the smartest point in the design being.. Fully Hyperbolic Attention Layer - https://arxiv.org/pdf/2105.14686.pdf

1.Yang-Mills, 2.Job Postings 3.Calabi-Yau 4.EconoPhysics, Divergent Recovery, TopologyGAN, Ricci Flow, Hodge theory

Hi Readers,

Yang Mills - First of all, lot of people must have seen my old orchestration on Yang Mills & Social Collider - Kdnuggets (and Neural Networks & Ricci Flow my old blog) - Here comes the absolutely brilliant paper on Yang Mills almost depicting the same - Emergent Attention on Yang-Mills Space of Connections - https://lukepereira.github.io/notebooks/documents/2021-moduli-attention/main.pdf (from Universidade Federal Fluminense .... Brazil & India is where so much of mathematical innovation is happening right ) Btw, you should watch my Ricci Flow video on Community detection & clustering to before reading this paper - https://www.youtube.com/watch?v=z_pjsJisdHQ 

Job Postings - This is an absolutely brilliant paper by Pratik Kothari on
Job Postings and Aggregate Stock Returns A much needed paper in today's time especially when CNBC announced Stock futures bounce as investors assess start of new quarter, bond market recession ( https://www.cnbc.com/2022/03/31/stock-market-futures-open-to-close-news.html ) 
- https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3223141

Calabi Yau - While I am impressed with the Inception Neural Network for Complete Intersection Calabi-Yau 3-folds - inspired by Google’s Inception model to compute the Hodge number I am still waiting for application on DeepSwarm in Calabi Yau. 

EconoPhysics - Dr. Gita Gopinath mentions Divergent Recovery in her IMF blog. How to make it optimum can be a very cool question to solve through EconoPhysics?

Most of you might have read my previous blog on Ricci Curvature & Market Failure ( https://researchcircle.blogspot.com/2022/02/job-posting-paper-ricci-flow-economics.html ). And some of you might have seen my recent tweet on Hodge Theory & Trading Networks in the evening. ( https://twitter.com/Prasad_Kothari/status/1509634908125335559 )

The following figure is from Sandhu's paper (w.r.t. Average Ricci curvature over a 15-year span of the S&P 500 - Choosing a window of T = 22 days )

I have mentioned portrait divergence in the following orchestration to compare sub-topologies between the economic & trade networks.  

Reference - 

1. Symmetry perception with spiking neural networks -  https://www.nature.com/articles/s41598-021-85232-3

2. Comparing methods for comparing networks -  https://www.nature.com/articles/s41598-019-53708-y

3. Ricci Curvature-Based Semi-Supervised Learning on an Attributed Network - https://ui.adsabs.harvard.edu/abs/2021Entrp..23..292W/abstract

4. Controlling Entropy via Discrete Ricci Flow Over Networks - https://arxiv.org/abs/1910.04560

The most interesting thing which I have not mentioned in this solution (apart from the code) is Thermodynamic Entropy in Quantum Statistics for Stock Market Networks from Shanghai University which talks about - Asian Financial Crisis (what could have been cool is use of HodgeNet (Graph Neural Networks for Edge Data) in it - https://ieeexplore.ieee.org/document/9049000

Edge entropy distribution of network structure before, during, and after the 1997 Asian financial crisis. (a)-(e) Bose-Einstein statistics. (d)-(f) Fermi-Dirac statistics.

  

Mirror Symmetry, Swarm Intelligence, Calibi Yau, GAN, SNNs, Econophysics

Hey Readers,

I am back again with a new blogpost on Saturday morning 5:30 AM. I got lot of inquiries about Dennis Sullivan and his amazing victory on Abel Prize. One of the best papers I have read by Dennis Sullivan is Circle Packing (off course my NLP, Reinforcement Learning, Ant Colony Optimization, Profiling etc. based prediction on last 50 years of math research papers is taking shape with Bhargav Bhatt's Hodge Theory, Maryna Viazovska's sphere packing etc. while off course Terence Tao has just published another paper on Sphere packing.. Sphere packing seems trend of the year while I am already thinking Duck pack algorithm — A new swarm intelligence algorithm for route planning based on imprinting behavior).. 

I also used to write a lot about Econophysics, String Theory (M-Theory etc. as I was big fan of Stephen Hawkings and probably had entire Brief History of Time by heart that time) Hodge Theory, Hodge Decomposition of information flow in networks & its impact on markets. 

Perelman's 2008 paper The entropy formula for the Ricci flow and its geometric applications says that he did not know enough about Blackholes & (pseudo)-riemannian geometry, Ricci Flow, Black Hole Thermodynamics, etc https://arxiv.org/pdf/math/0211159.pdf

Hence, I thought a perfect way to apply Perelman's work was AI, Data Science (please see my Ricci Flow- https://www.youtube.com/watch?v=z_pjsJisdHQ&t=26s ) & Econophysics using black hole thermodynamics etc. 

I see few papers published on this topic these days -

1. Trading networks and Hodge theory - https://iopscience.iop.org/article/10.1088/2399-6528/abd1c2/pdf

2. Semiparametric Estimates of Monetary Policy Effects: String Theory Revisited - https://economics.mit.edu/files/13030

3. Ricci Solitons & Black Holes - http://dspace.rri.res.in/bitstream/2289/3857/2/Chapter_7.pdf

4. Entropy of Reissner–Nordström 3D Black Hole in Roegenian Economics - https://www.semanticscholar.org/paper/Entropy-of-Reissner%E2%80%93Nordstr%C3%B6m-3D-Black-Hole-in-Udriste-Ferrara/fb2052f27e6700fed05ae259d63232639b465643

 

I used to write a lot about swarm intelligence (ant colony optimization etc.) on Google Plus, G-spaces long time back on this blog to give mathematical meaning to Google Plus (Hodge Cycle, Supersymmetric Yang Mills, Social Collider etc.) & also give some movie meaning to it. Also, I wrote a lot about Ricci Flow (Used to prove Poincare Conjecture) -

1. Meet the parents, Circle of Trust and Google Plus - https://www.kdnuggets.com/2013/10/circle-trust-google-plus.html

2. Yang-Mills: A million dollar connection between Twitter and quantum physics - https://www.kdnuggets.com/2013/11/yang-mills-million-dollar-connection-twitter-quantum-physics.html

3. A lot about Ricci Flow, Group theory, Quantum Groups (remember Sir Atiyah's 1984 paper with TIFR which I cited few blogposts back)

4. I also wrote a lot about applications of deep learning & swarm intelligence to solve mathematical conjectures 

Hence the new orchestration which Cao & Zhao might like (as I see their paper on Calabi Yau, Machine Learning and Convolutional Neural Networks .. I would like to see possible use of Geodesic Convolutional Neural Networks) on Yang Mills which also uses Maryam Mirzakhani's concept on Mirror Symmetry in Spiking Neural Networks... 

SYZ conjecture, Maryam Mirzakhani - AI, Geodesics, Riemannian surfaces, Dennis Sullivan

Hey readers,

Everyone must have seen my last comment on Andrica's conjecture & Firoozbakht's conjecture and how looking at prime numbers as dynamical system is a smarter way to solve this. I also have a way to look at it further in terms of How to use GANs - Generative Adversarial Networks (with optimum convergence while leveraging Nash entropy) to solve PDEs in Dynamical & Chaotic systems generated by prime numbers.

There was this beautiful paper from Maryam Mirzakhani (field prize winner 2014) - Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces (2005) - https://www.math.stonybrook.edu/~mlyubich/Archive/Geometry/Teichmuller%20Space/Mirz3.pdf

A lot of my discussions during my evening time in 2010 happened on how to apply her paper in AI. So, I came up with an orchestration in 2011 on Geodesics CNN on Riemannian Surface (I used to write a same blog by the name of DataHulk - which I deleted in 2017 and restarted this year).

Here is a very interesting paper which got published in Switzerland on the same topic - Geodesic convolutional neural networks on Riemannian manifolds - https://arxiv.org/pdf/1501.06297.pdf . The most interesting part of this paper was heat diffusion on manifolds, Heat kernel signature etc.

This was followed by my 2 lectures & video on

1. Ricci Flow (heat diffusion equation with derivative form) in Networks ( https://www.kdnuggets.com/2014/05/poincare-conjecture-perelman-topology-social-networks.html )

2. Some part of my lecture in SDSU which focused on Pruning of Neural Networks using Ricci Flow after which University of Cambridge published a new paper on 
RicciNets: Curvature-guided Pruning of High-performance Neural Networks Using Ricci Flow - https://arxiv.org/abs/2007.04216

3. You might like my video on Ricci Flow & its use in use in community detection  - https://www.youtube.com/watch?v=z_pjsJisdHQ&t=23s  

Here is one way to look at SYZ conjecture -

By the way you might like the old paper - THE HOMOLOGY THEORY OF THE CLOSED GEODESIC PROBLEM MICHELINE VIGUE-POIRRIER & DENNIS SULLIVAN - https://www.math.stonybrook.edu/~dennis/publications/PDF/DS-pub-0034.pdf  ( Dennis Sullivan won Abel prize on 24th March for amazing circle packing work )

Edited - Zhiren Wang, associate professor of mathematics at Penn State, has been award the 11th Brin Prize in Dynamical Systems MAY 24, 2022 

PK the genius, Misinformation detection in networks including Vimeo, TopologyGANs, HodgeNet, Duck Swarm intelligence

Hey readers, 

Happy Sunday morning! A lot of my readers liked my blogpost on Econophysics, Ricci Flow and Gita Gopinath's post on Divergent Recovery ( https://researchcircle.blogspot.com/2022/02/ricci-flow-geomstats-lie-groups-on.html ) 

Also, Lot of my readers liked my video on Ricci Flow Animation - 

https://www.youtube.com/watch?v=z_pjsJisdHQ&t=16s 

Also lot of people liked my previous blog about Duck Swarm Intelligence, Ricci Flow (my tweet few months back on TopologyGANs & HodgeNet).. 

Few years back I met Raghuram Rajan at Chicago Booth to discuss his book & I also told him that I am working on new & innovative approach to detect misinformation over networks (I know there are lot of approaches but nothing like this one & yes this is different from the one I wrote about Inorganic Social Network Growth through Ant Colony Optimization in my old DataHullk - researchcircles blog.. I will write more about it in my next one). 

Every social network including Vimeo faces a problem of Misinformation. Here is my new orchestration to detect misinformation in social networks (I have mentioned about Hodge Decomposition of information Flow over narrow or complex networks) 

- 

Ricci Flow, Market Fragility, Market Topology, Influencers & Game Theory, Ramanujan graph+Block Sparse Neural Networks, Hodge decomposition of information flow, RNNs

Hi Readers,

I know lot of readers missed me for last few days. I have been really busy with projects & proposals. 

But, here continues my blog to teach everyone new ways of using Hamilton & Perelman's Ricci Flow since 2009 ( Please refer - https://www.kdnuggets.com/2014/05/poincare-conjecture-perelman-topology-social-networks.html )

Also, refer to my video on Ricci Flow & Community Detection - https://www.youtube.com/watch?v=z_pjsJisdHQ&t=12s 

1. When I was in Chicago, I wrote a lot about Ramanujan Graph in the night but never published, 

2. I used to write a lot about G-spaces on this datahulk blog in 2011 also lot about Ricci Curvatures, market topology, 

3. Stubborn & Non-Stubborn players in Game theory and social networks to identify right influencers

4. Hodge Decomposition of information flow 

5. Ricci Curvatures of network graph, network topology, Ricci Solitons

Here is my new orchestration (which kind of predicted Google stock going down on Friday & Monday) - 

Firoozbakht's conjecture, BDS conjecture & deep learning, Lie Groups, Statistical Riemannian Framework, Ramanujan Machine..

Hey Readers,

It's 4:30 AM Saturday night. And I am at it again. A blog I restarted which I used to write 12 years back to motivate new thinking in data, math and AI. The paper I like the most these days is Job Postings and Aggregate Stock Returns - https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3223141

1. If I ever write my next book, there will be special mention of Iranian mathematician Farideh Firoozbakht - almost an unsung hero of math - She studied pharmacology and later mathematics at the University of Isfahan, and later taught mathematics at that university (Talk about reinventing the career). 

Extremely famous for her Firoozbakht's conjecture (pattern in prime numbers in simple words) which was stated in 1982 - this conjecture achieved some fame 

- in 2011 when I wrote about it along with new orchestrations on dynamical systems (Naive Entropy, Nash Entropy etc), 

- also on Maryam Mirzakhani's work on topology, geometry and dynamical systems (specially her work on Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces - https://www.math.stonybrook.edu/~mlyubich/Archive/Geometry/Teichmuller%20Space/Mirz3.pdf And when I also wrote about possibility of GeoDesic Convolutional Neural Networks - Geodesic CNNs & 

- & on Laura De Marco's work on dynamical systems my datahulk blog & 

in 2015 when Alexei Kourbatov published a paper on it in International Mathematical Forum 10 (2015), 283–288. Verification of the Firoozbakht conjecture for primes up to four quintillion, (unproven till now and has a very strong affiliation with Riemann Hypothesis).

I have a very strong viewpoint that Andrica's conjecture & Firoozbakht's conjecture and how looking at prime numbers as a dynamical system can have a smarter way to solve this. I also have a way to look at it further in terms of How To use GANs - Generative Adversarial Networks (with optimum convergence while leveraging Nash entropy) to solve PDEs in Dynamical & Chaotic systems generated through prime numbers. Anyway, more to come later... some surprises in store along with my Ricci Flow Video..  

Firoozbakht passed away in 2019 without many people knowing it... Almost an unsung hero who could have been celebrated everywhere. I always found it bit disappointing that Terence Tao mentioned Cramér conjecture in his blog but not Firoozbakht's conjecture. But, hey, everyone makes their own choice (that's why Thaler's choice architecture right??? right????) 

2. Another person who is doing some cool work is Nina Miolane - Amazing work on Statistics on Lie groups (& abelian groups): A need to go beyond the pseudo-Riemannian framework - https://aip.scitation.org/doi/abs/10.1063/1.4905963

Again, I want to highlight 1984 paper by Sir Atiyah, TIFR and Sir Atiyah's Student George Lusztig and his work on Quantum Groups. I know Nina's GeomStats is focus on Biomedical Topology but there is one more way to grow GeomStats in topology

i.e. growing GeomStats like Ramanujan Machine where AI would solve complex group theory problems trained on Lusztig's group theory paper including lie groups, abelian groups, Weyl groups, Chevalley groups, semisimple p-adic groups, quantum groups etc..

Btw, The first suggestion I gave to Ramanujan Machine when I became their advisor was this paper on BDS conjecture & deep learning by Laura Alessandretti - Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer - https://arxiv.org/abs/1911.02008

Ricci Flow, Geomstats, Lie Groups on Julia & EconoPhysics (and belated Happy Valentines day)

Hey Readers,

Again... thanks for reading my blog. I have been super-busy with lot of work. But,  30,000 views in single night is pretty cool. It is 9 PM in the evening, I just came from my walk in Maryland and I am on a mission to motivate everyone to learn more about data, AI, Math, Topology and so on. I should give more guest lectures when I get time.. but, for now please have a look at wonderful lecture by Geometric Statistics in Machine Learning GeomStats with Nina Miolane - https://www.youtube.com/watch?v=3yNInHuIWlY&t=1251s 

Hence, the next blogpost in the night on Ricci Flow (the topological concept that Grigori Perelman used to prove Poincare Conjecture) and Network Geometry.

1. Network geometry and market instability -https://royalsocietypublishing.org/doi/10.1098/rsos.201734

This paper talks about Time series of log-returns over a 32-year period (1985–2016) with network of stocks with 

1. Ollivier–Ricci (ORE), 
2. Forman–Ricci (FRE), 
3. Menger–Ricci (MRE), 
4. Haantjes–Ricci (HRE) 
5. Algebraic topological aspects, such as the homology groups and Betti numbers

minimum risk Markowitz portfolio of all the stocks, to better understand tipping points, systemic risk and resilience in financial networks, and enable us to develop monitoring tools required for the highly interconnected financial systems and perhaps forecast future financial crises and market slowdowns with use of Python package NetworkX (Yeah.. I used to use a lot of R, SAS, SQL, Cytoscape, Python in 2006 but Julia became my favorite language in 2012).

You can also check out few packages such as 

1. TheanoGeometry ( https://arxiv.org/abs/1712.08364 )- Riemann, Ricci and scalar curvature & geodesics  
2. Geomstats ( https://arxiv.org/abs/2004.04667 )   
3. McTorch  - McTorch, a manifold optimization library for deep learning ( https://arxiv.org/abs/1810.01811 ) 
4. Pymanopt: A Python Toolbox for Manifold Optimization using Automatic Differentiation ( https://arxiv.org/abs/1603.03236 )
5. Topological Entropy for Geodesic Flows under a Ricci Curvature condition: Jacobi field, Ricci curvature, topological entropy, tangent bundle, Negative Ricci curvature, isometry group and Injectivity radius - ( https://www.ams.org/journals/proc/1997-125-06/S0002-9939-97-03780-5/S0002-9939-97-03780-5.pdf )
6. And how can I forget my new favorite Julia for manifold Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds - a fast and easy to use library of Riemannian manifolds and Lie groups - ( https://arxiv.org/pdf/2106.08777.pdf ) 

Ricci Flow, Economics & a brilliant blog by Gita Gopinath

Hey readers,

It's 2:40 AM on Saturday night. Thanks for reading my DataHulk blog. I was glad to see 20,000 views in a single night on my last post - Yang Mills, Vector bundles, Quantum Information Geometry & Fisher-Bures Adversary Graph Convolutional Networks etc. And Yes, I noticed Terence Tao wrote his blog on the same day talking about Sphere Pinching paper (Smart people think alike I guess) - https://terrytao.wordpress.com/2022/02/08/perfectly-packing-a-square-by-squares-of-nearly-harmonic-sidelength/ 

I used to write this blog 12 years back to motivate everyone to learn math, data & AI folks. By the way, I have an exciting blogpost today which might change the way some of my followers look at economics (I know.. I know.. I promised to write about Spin Neural Networks, diffusion and capsule networks in my last blog.. which I will write later). 

1. Ricci Flow & Economics (And why Gita Gopinath's sentence in her blog is genius) - 

Here is an interesting paper followed by 2013-14 Ricci Flow & Poincare Conjecture in topology of Social Networks & Information Geometry article ( https://www.kdnuggets.com/2014/05/poincare-conjecture-perelman-topology-social-networks.html ).... Ricci curvature: An economic indicator for market fragility and systemic risk - https://www.researchgate.net/publication/303600815_Ricci_curvature_An_economic_indicator_for_market_fragility_and_systemic_risk which is an analysis based on geometric feature extraction on network data of daily returns from a set of stocks & market topology comprising the Standard and Poor’s 500 (S&P 500) over a 15-year span to highlight the fact that corresponding changes in Ricci curvature, is negatively correlated to increases in network fragility. 

network, is negatively correlated to increases in network fragility. To illustrate this insight, we examine daily

returns from a set of stocks comprising the Standard and Poor’s 500 (S&P 500) over a 15-year span to highlight

the fact that corresponding changes in Ricci curvature constitute a financial “crash hallmark.”

  

Average Ricci curvature over a 15-year span of the S&P 500 - Choosing a window of T = 22 days, we see that curvature captures several financial crashes and show that, on average, market behavior is fragile.

2. Gita Gopinath - 

Here is an interesting & brilliant blog (April 2021) by Gita Gopinath on divergent recovery -
Managing Divergent Recoveries - https://blogs.imf.org/2021/04/06/managing-divergent-recoveries/ speaking about 

1. Financial risks, 

2. Financial stability risks using macro-prudential tools, 

3. Cross-country gaps, global poverty reduction, 

4. Emerging markets, 

5. Divergent recovery paths, 

6. High degree of uncertainty and developing economies 

7. International liquidity

8. Liquidity protection

9. Debt restructuring 

10. financial stability risks using macro-prudential tools

11. Withdrawn loan payments, firm insolvencies

12. Cross-border profit shifting

which reminded me of network entropy, nodal entropy, geodesics curvature & divergence mentioned in the Sandhu's paper - Ricci curvature & Ricci Flow: An economic indicator for market fragility and systemic risk and my KDnuggets article on which intended to perform Ricci Flow on Social network analysis & information economics.