#geometric combinatorics

[bagalois]

day two: what inspired you to take your subjects?

how should i answer this. i'm early enough in my grad student life that i can't choose from a large variety of topics courses or other things yet, so my decisions for my classes weren't based in inspiration. i am properly motivated though.

i chose combinatorics over functional analysis because measure theory spooked out of doing more analysis... i do kind of regret not taking that class though. apparently the professor this semester is very good.

i like my combo professor - her research seems cool to me, partly because it's the only applied math in the whole department and that application is psychology. the actual math that gets done obviously will have nothing to do with psychology, but the idea that the work goes to further research in psychology is really attractive to me.

i am happy to learn more algebraic topology because of a moment of inspiration when i was a teenager. when i was 15, i was shown a ton of knot theory stuff. it was cool but the most interesting ideas went completely over my head (surfaces, invariants, etc). around then was when i resolved to do math research before i die, and the type of research i thought about was literally just geometric/algebraic topology. nowadays because im not so focused on topology anymore, it might be more accurate to say that i'm just motivated, but i think this counts!

~~~~

30 days of studying challenge from here

[kuramirocket]

Jorge Urrutia Galicia: A Mexican Pioneering Mathematician And Computer Scientist

Jorge Urrutia Galicia is a Mexican computer scientist and mathematician.

Galicia is best known for his work on geometry. He made contributions to many different areas of mathematics, including discrete geometry, discrete optimization, and computational geometry. His specialty in computational geometry has made him recognized as one of the leading researchers worldwide. His research has also focused on combinatorial optimization, which is related to combinatorial game theory.

His early works dealt with problems of separability and visibility, a field in which he is an indisputable authority. While it is clear that mathematics has always played a basic role as the underlying foundation of all technology, especially now, and in this case it is confirmed why the technological scope of Dr. Urrutia’s articles in routing is significant; suffice to mention just one: recently algorithms are being implemented based on the ideas of Dr. Urrutia, to make communication networks that can be used in case of natural disasters.

Since the end of the 20th century, he began to work on routing problems, developing algorithms for both the combinatorial and geometric problems, which literally founded a work area of great importance in its application to wireless and cellular networks. In the 21st century, Dr. Urrutia has also stood out for his numerous contributions to the study of discrete sets of points, on which he has made decisive contributions, both in their solution and formulating various variants.

Dr. Jorge Urrutia Galicia studied a bachelor’s degree in mathematics at the Faculty of Sciences of UNAM from 1971 to 1974, and a master’s and doctoral degree in mathematics at the University of Waterloo, Canada from 1976 to 1980. He has worked at the Metropolitan Autonomous University-Iztapalapa, CIMAT, Carleton University, Ottawa University from 1984-1998, where he was "full professor", and since 1998 at the Institute of Mathematics of the UNAM. On average, he teaches five courses each year (two undergraduate and three postgraduate courses).

Annually, he organizes at least two research workshops in Mexico, one of its main objectives being that its students know and work with renowned researchers and learn to collaborate with them as equals.

From 1990 to 2000, he was editor-in-chief of the journal Computational Geometry, Theory and Applications, published by Elsevier Science Publishers. He has been a member of the editorial boards of the Mexican Mathematical Society Bulletin and of Graphs and Combinatorics (Springer, and Computational Geometry: Theory and Applications (Elsevier). He was also editor of the Handbook of Computational Geometry (2000), one of Elsevier's first published handbooks.

He has published more than 270 articles in conference proceedings and research journals in mathematics and computing, which have received more than 6,000 citations, among the most important are two articles on routing in ad-hoc and wireless networks, which have received more than 2 600 citations together: “Compass Routing in Geometric Graphs” and “Routing with Guaranteed Delivery in Ad Hoc Wireless Networks.” In these investigations, Dr. Urrutia develops new strategies – highly efficient – to send information on wireless networks that take advantage of the characteristics obtained by recent technologies such as GPS, in addition to allowing them to travel through these networks effectively without having knowledge of their topology. It is worth mentioning that in 2012 he was the most cited mathematician of the UNAM.

He has given more than 40 plenary lectures at international congresses on Computational Geometry. He was editor-in-chief of "Computational Geometry, Theory and Applications" from 1990 to 2000. He has supervised more than 55 bachelor, master and doctoral theses.

In 2015, he received the "National University in Research in Exact Sciences" award at UNAM. He is is a member of the National System of Investigators, Level 3 He has organized and participated in the organizing committees of several national congresses including the "Victor Neumann-Lara Colloquium on Graphic Theory and its Applications", the "Canadian Conference on Computational Geometry", the "Japan Conference on Discrete and Computational Geometry" and the "Computational Geometry Meetings" (Spain). Oher countries where he has also participated in this way are Italy, Indonesia, Philippines, China, Canada, Peru and Argentina, as well as his home country, Mexico.

Source: (x) (x) (x)

[russianhometutors]

Teaching Astronomy Through Geometry - How It Helps?

Astronomy and geometry may seem to be two distinct disciplines, but they share a deep connection going back to the ancient times. Teaching Astronomy Through Geometry not only helps in increasing understanding but also brings to life the intricate dance of celestial bodies in the universe. This way of exploring the cosmos is really engaging while developing the essential mathematical skills.

The Interplay Between Astronomy and Geometry

Geometry, the science of shapes, sizes, and spaces, underlies the understanding of any astronomical concepts. It was used by the ancient astronomers - Ptolemy and Copernicus - for mapping the heavens and for explaining the motion of the planets and stars. Thus, the student, using these geometric concepts, is able to visualize the space relationships of celestial objects in a better way to gain comprehensive knowledge of astronomical phenomena.

For example, draw the phases of the Moon. Using elementary geometric shapes, such as circles and spheres, students can understand how the Moon's position relative to the Earth and Sun causes its phases. In this manner, abstract concepts may become more accessible and believable to learn.

Using Geometry to Teach Astronomy

Orbits and Ellipses: Kepler's laws of planetary motion are used to describe the orbits planets make around the sun in elliptical paths. The drawing and interpretation of ellipses will teach students about distance variations between celestial bodies and the Sun, reinforcing their knowledge in geometry concerning conic sections.

Angular Measurements: To understand the size and distance of heavenly objects often requires angular measurements. As a branch of mathematics, geometry allows students to calculate angles, which plays a necessary role in calculating the apparent size of the Moon or the Sun from Earth. It applies to practical situations like navigation and positioning satellites.

Triangulation: This geometric technique is applied in the determination of distances of distant stars and planets. Students form triangles between known points to compute unknown distances-a technique that astronomers have used for thousands of years.

Coordinate Systems: The celestial sphere has coordinate systems comparable to latitude and longitude on the Earth. Star and planet positioning on the celestial sphere gives pupils practice with their geometric grid skills and experience with coordinate transformation.

Correct Solutions Aid For Discrete Math

Discrete math is another area of mathematics that really benefits from the logical thinking developed through geometry and astronomy. Most problems in discrete math require exact, step-by-step solutions, just like solving geometric problems in astronomy.

Logical Reasoning: Studying geometric proofs in astronomy leads to logical reasoning. This is an important characteristic of discrete mathematics, where one has to go step by step logically to find the right solution. For example, proving that an orbit possesses certain properties involves similar deductive reasoning as when proving a theorem in discrete math.

Concepts of star charts and constellations are found to be in association with graph theory. The stars can be well thought to be as vertices, and the edges will be the lines connecting them. Correct Answers Help For Discrete Math helps to understand difficult concepts of combinatorics and graph theory to students.

Set Theory and Functions: The classification of celestial objects into sets, for example, planets, stars, and galaxies, introduces students to set theory. Functions, like distance versus luminosity, are also central in astronomy and discrete math.

Conclusion

Teaching astronomy through geometry is a multidimensional method. Through this, not only will the universe become accessible to more people, but mathematical skills, particularly in terms of discrete math, will be enhanced. By relating these disciplines, the student gets a general understanding of how they work together and can be more proficient for further study in mathematics and sciences. Correct Answers Help For Discrete Math often arise out of the bright, logical thought developed through the geometric exploration of the universe and, therefore, is an extremely valuable educational approach.

[avocodedigital]

Exploring New Horizons in Math - AI's Role

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Unlocking Mathematical Mysteries with AI

In a world increasingly influenced by technological innovation, the intersection of math and artificial intelligence (AI) offers an enticing preview of how future discoveries may unfold. Among the luminaries illuminating this path is Terence Tao, a renowned mathematician whose insights into this evolving synergy underscore the transformative potential of AI in mathematical research.

The Convergence of Two Giants: Mathematics and AI

The realm of mathematics has long thrived on human intellect—puzzles being solved through sheer mental rigor and boundless curiosity. However, as we step into an era defined by digital prowess, AI emerges as a potent catalyst in propelling mathematical inquiry into new dimensions. This transformative alliance amplifies the capacity for discovery, offering tools and methodologies that reimagine traditional problem-solving paradigms.

Enhanced Computational Power: The advent of AI has ushered in a revolutionary phase of computational speed, enabling mathematicians to process complex problems with unprecedented efficiency.

Advanced Pattern Recognition: Machine learning algorithms are adept at identifying patterns invisible to the unaided eye, allowing researchers to uncover insights that may otherwise remain hidden.

Collaborative Synergy: The interplay between human intuition and machine precision fosters a collaborative environment, where each enhances the other, leading to novel theoretical breakthroughs.

Terence Tao, with his profound grasp of mathematical intricacies, proves to be an instrumental figure in exploring these convergences. His thoughtful engagement with AI offers a glimpse into a future where the boundaries of mathematical exploration are redrawn.

AI as a Co-Investigator in Mathematical Exploration

With AI's capabilities extending far beyond mere calculation, its role as a co-investigator in mathematics becomes increasingly apparent. Through deep learning and neural networks, AI systems emulate aspects of human learning, making significant inroads into pattern detection and theoretical generalization—an area of great interest to Tao and his contemporaries. Consider the breakthroughs in topology, number theory, or combinatorics, where AI has played a critical role:

Data Analysis in Topology: By managing and analyzing vast datasets, AI tools can enhance our understanding of geometric structures and facilitate the development of complex models.

Number Theory Advances: AI-driven algorithms provide fresh perspectives on prime distribution or predictive models for solving longstanding conjectures.

Optimizing Combinatorial Problems: New methods in AI allow for more efficient solutions in combinatorial optimization, proving invaluable in logistics, network design, and beyond.

These examples illustrate how AI not only complements but extends the reach of mathematical exploration, a notion that Tao himself emphasizes in his interaction with AI systems.

The Human Element: Mathematicians and AI

While machines master massive calculations, it is the human element—intuition, creativity, and the knack for theoretical innovation—that remains irreplaceable. Mathematicians such as Terence Tao play a pivotal role in directing AI applications toward meaningful and insightful questions, ensuring that technology serves as a conduit for profound intellectual exploration rather than an end to itself. Tao's approach to AI underscores a partnership model, where the machine's learning capabilities are leveraged to enhance human problem-solving without supplanting it. In this framework, AI is not merely an adjunct but a collaborator, augmenting the capacity of mathematicians to address and unravel complex equations.

Educational Implications: A New Dawn for Mathematical Learning

The interweaving of AI into mathematics extends beyond research into the educational sphere, reshaping how mathematical concepts are taught and comprehended:

Interactive Learning Platforms: AI-powered tools provide interactive simulations that allow students to visualize and interact with mathematical concepts in real-time, enhancing comprehension.

Personalized Learning Experiences: AI systems can adapt to individual learning paces, offering customized problem sets that address unique learner challenges and bolster conceptual understanding.

Research and Scholarship Facilitation: AI aids students and early-career researchers in accessing vast repositories of mathematical literature, promoting deeper engagement with the subject matter.

Terence Tao, a fervent advocate for mathematical education, acknowledges the role of AI in democratizing access to knowledge, particularly as a means of fostering a new generation of problem solvers equipped to tackle the grand challenges of our time.

Looking Ahead: The Future Trajectory of Math and AI

As we look to the future, the symbiotic relationship between mathematics and AI promises to redefine disciplinary boundaries. The potential for collaborative breakthroughs that push beyond current theoretical limitations is vast. More importantly, this partnership underscores a broader cultural shift—embracing collaboration not only between scholars and machines but across disciplinary lines, promoting holistic progress. Terence Tao's vision for AI in mathematics accentuates an era where human curiosity is coupled with machine precision, leading to unprecedented advancements. By exploring the boundless possibilities of this new frontier, we can envisage a future rich in mathematical discovery and innovation. As humanity stands on the brink of technological transformation, the fusion of AI and math is set to light the way forward, guiding us toward untapped territories filled with wonder and promise. Want more? Join the newsletter: https://avocode.digital/newsletter/

[xenopoem]

Fractal Structures and Spallation

The depiction of "linguistic comets smashing through the archetypal layers" described by Andrew C. Wenaus is related to the concept of fractals. Fractals are geometric structures that exhibit self-similarity, with the same patterns repeating at every scale. The way language repeatedly collides and continuously generates new patterns demonstrates the infinite complexity of fractals. This is also consistent with the mathematical model of spallation (fragmentation and reformation caused by collisions), which explains how matter splits and takes on new forms under certain conditions.

Ritual and Semiotic Acts

The ritualistic semiotic acts central to the work are related to combinatorics and information theory in mathematics. Combinatorics studies how different combinations can be generated from a set of elements. This is suitable for mathematically modeling how ritualistic acts and symbols can hold different meanings.

From the perspective of information theory, rituals and semiotic acts can be seen as processes of encoding and decoding information. Shannon's information theory emphasizes how information is transmitted and interpreted during the sending and receiving of messages. Ritualistic acts serve as concrete examples of how information is transformed in this process.

Monads and Post-Human Identity

The concept of the monad, originating from Leibniz's philosophy, is mathematically interpreted as a unit or fundamental element. Monadic thinking is viewed as individual units possessing unique completeness while being part of a whole. Mathematically modeling this involves set theory and topology to explore the relationship between the whole and its parts.

The neuronoid strategies pointed out by Germán Sierra are related to mathematical models of neural networks and machine learning. These models explain how human cognition and identity are simulated through data and algorithms. Neuronoid strategies play a crucial role in exploring post-human identity because they efficiently process complex data patterns and generate new meanings.

[jcmarchi]

AI at the International Mathematical Olympiad: How AlphaProof and AlphaGeometry 2 Achieved Silver-Medal Standard

New Post has been published on https://thedigitalinsider.com/ai-at-the-international-mathematical-olympiad-how-alphaproof-and-alphageometry-2-achieved-silver-medal-standard/

AI at the International Mathematical Olympiad: How AlphaProof and AlphaGeometry 2 Achieved Silver-Medal Standard

Mathematical reasoning is a vital aspect of human cognitive abilities, driving progress in scientific discoveries and technological developments. As we strive to develop artificial general intelligence that matches human cognition, equipping AI with advanced mathematical reasoning capabilities is essential. While current AI systems can handle basic math problems, they struggle with the complex reasoning needed for advanced mathematical disciplines like algebra and geometry. However, this might be changing, as Google DeepMind has made significant strides in advancing an AI system’s mathematical reasoning capabilities. This breakthrough is made at the International Mathematical Olympiad (IMO) 2024. Established in 1959, the IMO is the oldest and most prestigious mathematics competition, challenging high school students worldwide with problems in algebra, combinatorics, geometry, and number theory. Each year, teams of young mathematicians compete to solve six very challenging problems. This year, Google DeepMind introduced two AI systems: AlphaProof, which focuses on formal mathematical reasoning, and AlphaGeometry 2, which specializes in solving geometric problems. These AI systems managed to solve four out of six problems, performing at the level of a silver medalist. In this article, we will explore how these systems work to solve mathematical problems.

AlphaProof: Combining AI and Formal Language for Mathematical Theorem Proving

AlphaProof is an AI system designed to prove mathematical statements using the formal language Lean. It integrates Gemini, a pre-trained language model, with AlphaZero, a reinforcement learning algorithm renowned for mastering chess, shogi, and Go.

The Gemini model translates natural language problem statements into formal ones, creating a library of problems with varying difficulty levels. This serves two purposes: converting imprecise natural language into precise formal language for verifying mathematical proofs and using predictive abilities of Gemini to generate a list of possible solutions with formal language precision.

When AlphaProof encounters a problem, it generates potential solutions and searches for proof steps in Lean to verify or disprove them. This is essentially a neuro-symbolic approach, where the neural network, Gemini, translates natural language instructions into the symbolic formal language Lean to prove or disprove the statement. Similar to AlphaZero’s self-play mechanism, where the system learns by playing games against itself, AlphaProof trains itself by attempting to prove mathematical statements. Each proof attempt refines AlphaProof’s language model, with successful proofs reinforcing the model’s capability to tackle more challenging problems.

For the International Mathematical Olympiad (IMO), AlphaProof was trained by proving or disproving millions of problems covering different difficulty levels and mathematical topics. This training continued during the competition, where AlphaProof refined its solutions until it found complete answers to the problems.

AlphaGeometry 2: Integrating LLMs and Symbolic AI for Solving Geometry Problems

AlphaGeometry 2 is the latest iteration of the AlphaGeometry series, designed to tackle geometric problems with enhanced precision and efficiency. Building on the foundation of its predecessor, AlphaGeometry 2 employs a neuro-symbolic approach that merges neural large language models (LLMs) with symbolic AI. This integration combines rule-based logic with the predictive ability of neural networks to identify auxiliary points, essential for solving geometry problems. The LLM in AlphaGeometry predicts new geometric constructs, while the symbolic AI applies formal logic to generate proofs.

When faced with a geometric problem, AlphaGeometry’s LLM evaluates numerous possibilities, predicting constructs crucial for problem-solving. These predictions serve as valuable clues, guiding the symbolic engine toward accurate deductions and advancing closer to a solution. This innovative approach enables AlphaGeometry to address complex geometric challenges that extend beyond conventional scenarios.

One key enhancement in AlphaGeometry 2 is the integration of the Gemini LLM. This model is trained from scratch on significantly more synthetic data than its predecessor. This extensive training equips it to handle more difficult geometry problems, including those involving object movements and equations of angles, ratios, or distances. Additionally, AlphaGeometry 2 features a symbolic engine that operates two orders of magnitude faster, enabling it to explore alternative solutions with unprecedented speed. These advancements make AlphaGeometry 2 a powerful tool for solving intricate geometric problems, setting a new standard in the field.

AlphaProof and AlphaGeometry 2 at IMO

This year at the International Mathematical Olympiad (IMO), participants were tested with six diverse problems: two in algebra, one in number theory, one in geometry, and two in combinatorics. Google researchers translated these problems into formal mathematical language for AlphaProof and AlphaGeometry 2. AlphaProof tackled two algebra problems and one number theory problem, including the most difficult problem of the competition, solved by only five human contestants this year. Meanwhile, AlphaGeometry 2 successfully solved the geometry problem, though it did not crack the two combinatorics challenges

Each problem at the IMO is worth seven points, adding up to a maximum of 42. AlphaProof and AlphaGeometry 2 earned 28 points, achieving perfect scores on the problems they solved. This placed them at the high end of the silver-medal category. The gold-medal threshold this year was 29 points, reached by 58 of the 609 contestants.

Next Leap: Natural Language for Math Challenges

AlphaProof and AlphaGeometry 2 have showcased impressive advancements in AI’s mathematical problem-solving abilities. However, these systems still rely on human experts to translate mathematical problems into formal language for processing. Additionally, it is unclear how these specialized mathematical skills might be incorporated into other AI systems, such as for exploring hypotheses, testing innovative solutions to longstanding problems, and efficiently managing time-consuming aspects of proofs.

To overcome these limitations, Google researchers are developing a natural language reasoning system based on Gemini and their latest research. This new system aims to advance problem-solving capabilities without requiring formal language translation and is designed to integrate smoothly with other AI systems.

The Bottom Line

The performance of AlphaProof and AlphaGeometry 2 at the International Mathematical Olympiad is a notable leap forward in AI’s capability to tackle complex mathematical reasoning. Both systems demonstrated silver-medal-level performance by solving four out of six challenging problems, demonstrating significant advancements in formal proof and geometric problem-solving. Despite their achievements, these AI systems still depend on human input for translating problems into formal language and face challenges of integration with other AI systems. Future research aims to enhance these systems further, potentially integrating natural language reasoning to extend their capabilities across a broader range of mathematical challenges.

[positively-knotted]

I'm sure all of this is already in the notes but this is such a crazy poll to me:

No geometry? One of the most fundamental areas of maths and one that is actually taught at highschool?

Measure theory being separated from analysis

Also graph theory from combinatorics

Linear algebra winning? Tbh if I was making this poll I would probably not have included it

"Abstract algebra" I think I know what you mean but I'm also not really sure

The brave (but maybe correct) choice of No Stats

Actually just, no applied at all. Wild. Although it is harder to split up I suppose

Mostly for the exercise of creating a poll like this, here would be my version. Feel free to vote in it (who doesn't love a poll) but I'm not trying to supplant op:

"Hard" analysis is classical analysis, analysis of PDEs, dynamical systems, etc. "Soft" analysis is functional analysis, operator algebras, etc.

RIP numerical analysis you did not fit, but I also doubt anyone misses you. I would also have wanted to split geometry and topology up but with only 12 options it seems a bit unnecessary.

That maths poll was interesting but I'd love to hear from people who chose to study maths at a higher level. No "i hate maths" option.

[mathsminds]

Feature of Math

Mathematics

Content of Mathematics

Course Overview:

This course aims to provide a comprehensive understanding of fundamental mathematical concepts and techniques. Through a combination of theory, problem-solving exercises, and practical applications, students will develop critical thinking skills and mathematical proficiency necessary for success in higher-level mathematics and related fields.

Module 1: Number Systems

Understanding the properties of real numbers

Integers, rational numbers, irrational numbers, and their properties

Introduction to complex numbers and their operations

Exploring number patterns and sequences

Module 2: Algebraic Expressions and Equations

Simplifying algebraic expressions

Solving linear and quadratic equations

Factoring polynomials and solving polynomial equations

Graphing linear and quadratic functions

Module 3: Functions and Relations

Understanding the concept of a function

Identifying types of functions: linear, quadratic, exponential, logarithmic, etc.

Analyzing graphs of functions and their transformations

Solving systems of linear equations and inequalities

Module 4: Geometry

Exploring geometric shapes and properties

Understanding angles, lines, and polygons

Calculating area, perimeter, and volume of geometric figures

Introduction to trigonometry: sine, cosine, tangent, and their applications

Module 5: Probability and Statistics

Understanding basic concepts of probability

Calculating probabilities of events and outcomes

Introduction to descriptive statistics: mean, median, mode, and range

Analyzing data sets and making statistical inferences

Module 6: Calculus

Introduction to limits and continuity

Understanding derivatives and their applications

Calculating rates of change and optimization problems

Introduction to integrals and their applications in finding area and volume

Module 7: Discrete Mathematics

Exploring combinatorics and counting principles

Introduction to sets, relations, and functions

Understanding logic and proof technique

Exploring graph theory and its applications

Module 8: Mathematical Modeling

Understanding the process of mathematical modeling

Formulating mathematical models for real-world problems

Analyzing and interpreting mathematical models

Evaluating the effectiveness and limitations of mathematical models

Module 9: Applications of Mathematics

Exploring interdisciplinary applications of mathematics in science, engineering, finance, and other fields

Case studies and real-world examples demonstrating the relevance of mathematical concepts

Ethical considerations and implications of mathematical applications

Module 10: Review and Final Assessment

Reviewing key concepts and techniques covered in the course

Solving comprehensive problem sets and practice exam

Final assessment covering all topics and skills learned throughout the course.

[jhavelikes]

check whether a proof assistant can do differential topology. Many people still think that formal mathematics are mostly suitable for algebra, combinatorics, or foundational studies. So we chose one of the most famous examples of geometric topology theorems associated to tricky geometric intuition: the existence of sphere eversions. Note however that we won’t focus on any of the many videos of explicit sphere eversions. We will prove a general theorem which immediately implies the existence of sphere eversions.

Introduction

[isomorphismes]

The demands of university teaching, addressed to students … with [only] a modest (and frequently less than modest) mathematical baggage, led me to … start from an intuitive baggage common to everyone, independent of any technical language used to express it, and anterior to any such language—it turned out that the geometric and topological intuition of shapes, particularly two-dimensional shapes, formed such a common ground. These themes can be grouped under the general names “topology of surfaces” or “geometry of surfaces”, … the main emphasis being on the … combinatorial aspects which form the most down-to-earth technical expression of them—and not on the differential, conformal, Riemannian, holomorphic, [Kähler, contact, symplectic, Moishezon] aspects—and from there on to ℂ algebraic curves.

Alexandre Grothendieck, 1988, in a letter to

translated by Michael Barr

[co-xeter-group]

"Office Hours with a Geometric Group Theorist" edited by Matt Clay and Dan Margalit, chapter 13 "Coxeter Groups" by Adam Piggott

Humphreys "Reflections and Coxeter Groups"

Davis "The Geometry and Topology of Coxeter Groups"

Björner and Brenti "Combinatorics of Coxeter Groups"

Bourbaki has an interesting approach to the proof of the classification of finite coxeter groups IMO. Let me know if you want me to find it.

Start with the first reference IMO. Humphreys and Björner and Brenti are undergrad appropriate. I found Davis a bit challenging. DM me if you want help getting the books.

None of these focus on the tesseract specifically - as far as I know it really is best understood as the member of the infinite family B_n. I'm not aware of any special 4 dimensional topology stuff happening here since its a ball topologically.

haiii ^-^, I love seeing other mathematicians on tumblr. Cool name btw

about that, how much do you know about the symmetry group of tesseracts (or could you direct me to some good sources)?

Thank you!

What you're looking for is the study of Coxeter groups and Dynkin diagrams.

https://en.m.wikipedia.org/wiki/Coxeter_group

The tesseract is Coxeter group B4, but if you want a "general" study of it, this is the one you're looking for.

[red-catmander]

Big into the idea that weavers, by merit of their whole deal being 'picking apart reality then knitting it in my design', share a lot of guiding principles as followers of the eternal alchemy even if they don't actively believe so when Jozra 'If You Leave Your God Unattended I Will Slay It' Emberthroat returns to Lion's Arch after season 4 she has to dodge some extremely devout asura who keep trying to stealthily induct her as a nun

[thousandmaths]

Parking Functions 201: The Shi Arrangement

As this suggests, this post is a continuation on the recent parking functions post, as well as the even-more recent Shi arrangements post. It is also based on Nathan Williams’s talk “Parkour: Inverting Parking Zeta” at the CRM Equivariant Combinatorics workshop.

[ I don’t plan on writing a whole post about Williams’ talk, but, unsurprisingly, it was closely related to his MCC talk from about a month prior. ]

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Pak-ing Functions*

Recall that the Shi arrangement $\text{Shi}(n)$ is a union of hyperplanes in $n$-dimensional space whose equations are $x_i-x_j=a$, where $a$ is either $0$ or $1$. Using a standard trick we can draw these pictures in $n-1$ dimensions, so that, for instance, we get this picture when $n=3$:

At the end of the Shi arrangements post, we observed that this hyperplane arrangement cuts the plane into $16=(3+1)^{3-1}$ regions, which we recognized as the number of length-$3$ parking functions.

It would be really nice if this extended to a full geometric interpretation of parking functions. In other words, we are hoping for three things:

The Shi arrangement always cuts its ambient $n$-dimensional space (or $n-1$ dimensional, if you prefer) into $(n+1)^{n-1}$ regions.

There were a systematic way to label the regions uniquely by parking functions.

Proofs of both of 1 and 2.

All three of these things turn out to be possible! In this post we will ignore the third point, since the proofs are surprisingly intricate. Moreover, we will ignore the first point, since describing the labeling is sufficient (modulo the proof) to conclude that there are as many regions as there are parking functions, which is $(n+1)^{n-1}$, as we want.

The labeling was supposedly discovered one night when Igor Pak (pronounced pahk), after a couple of beers, returned to the question of enumerating the regions of the Shi arrangement. He experimented with some things with $\text{Shi}(3)$ until he stumbled upon a guess that used all sixteen of the parking functions, and then proceeded to try $\text{Shi}(4)$, where his guess used all $121$ of the parking functions. Excited, he sent an email to Richard Stanley asking if he had actually resolved the question.

The email must have confused Stanley somewhat, because $(4+1)^{4-1}=125$, not $121$... In his talk, Williams quipped that it was “the first bijection discovered between two finite sets of different sizes.” :D 

But despite the rocky start, Pak’s guess turned out to be largely correct, and the labeling that we will describe in this post is nowadays called the Pak-Stanley labeling.

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Parking Functions, Geometrically

When $n=3$, the Pak-Stanley labeling is described by this picture, which is quite busy, but we will spend a fair bit of time analyzing:

[ Full disclosure: This is not the same picture that you will see in most standard references on the subject. But after redrawing the damn thing about four times because of stupid mistakes, I can say with complete confidence that it is correct. ]

The first thing I want to do is make a note about all the scribbles which are not labels. In between each set of parallel lines, on one side, there is an expression like $y-z$. This is meant to remind you that these two lines are actually the hyperplanes $y-z=0$ and $y-z=1$. The ones and zeros in the boxes attached to the ends tell you whether it’s the $=0$ or $=1$ situation. Finally, the bolded dot corresponds to the origin.**

Now that you know what to ignore, you should convince yourself that the $16$ labels on the diagram are, indeed, the parking functions of length $3$.

From here, we start to pencil in the parking functions, and we will want to do this “layer by layer”. You can see that I have shaded the innermost triangular region dark, and the bigger triangle a bit lighter. Finally, there is the unshaded outer layer, which consists precisely of those regions which are unbounded; they go off to infinity in some direction.

Once we have the layers, we can get to work. For the single (dark) triangle in the innermost layer, we label it $111$; that’s the easy one.

The middle layer consists of three triangles, all of which share a side with the dark triangle. This side lies on some $x_i-x_j=a$; since there are only three of them, let’s write them down:

$x-y=0$

$y-z=0$

$x-z=1$.

When we cross one of these hyperplanes from the dark triangle into a lighter one, we will be adding one to some entry of the list $111$. Which entry we increment depends on whether we’re crossing an $=0$ or an $=1$, and which variable has the minus sign:

Crossing $x-y=0$ (that is, $x_1-x_2=0$) adds one to the first coordinate since $x$ has the plus sign: $111\mapsto 211$.

Crossing $y-z=0$ (that is, $x_2-x_3=0$) adds one to the second coordinate since $y$ has the plus sign: $111\mapsto 121$.

Crossing $x-z=1$ (that is, $x_1-x_3=1$) adds one to the third coordinate since $z$ has the minus sign: $111\mapsto 112$.

The outer, unshaded layer is dealt with in the same way: every time we cross $x_i-x_j=0$, we add one to the $i^\text{th}$ entry of the list, and every time we cross $x_i-x_j=1$, we add one to the $j^\text{th}$ entry of the list.

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Doubts and Extensions

It is worth noting that it’s not totally clear that all the labels we generate in the procedure above are parking functions. Why, for instance, does $133$ never show up? Or $222$? The answers to these questions, as far as I’ve seen, exist but are not terribly enlightening. 

Of course, the answer relies on the combinatorics of the Shi arrangement. Somehow the setup guarantees that if you always go “out one layer” for each move, you lock yourself out of the regions of the arrangement which might increment the wrong number. But to say anything more specific at an intuitive level is not something I know how to do.

In particular, you may recall that in addition to $\text{Shi}(n)$ we also defined $\text{Shi}(n,k)$, a larger arrangement that contains $\text{Shi}(n)=\text{Shi}(n,1)$, e.g. $\text{Shi}(3,2)$ looks like this:

Nothing stops us from playing the same sort of hyperplane-crossing game here (see the footnote for how the layers are defined), perhaps increasing the $i^\text{th}$ when $a\leq 0$ and the $j^\text{th}$ coordinate when $a>0$. Calculating a few of these labels quickly shows that you get many lists which are not parking functions.

However, to the extent that these extended Shi arrangements are important, we may want to look for commonalities in the labels. Without going through any calculations, let me tell you the punchline. Every one of the regions in $\text{Shi}(n,k)$ are labeled by a $k$-parking function: a list of $n$ numbers which, when rearranged in increasing order, has that the $i^\text{th}$ entry is at most $k\cdot i$. 

Moreover, there are exactly as many $k$-parking functions as there are regions in the extended Shi arrangement. Counting these also gives rise to a very nice formula: $(kn+1)^{n-1}$. The analogies between $k$-parking functions and ordinary parking functions are much deeper, and we could spend 

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Footnote: Higher Dimensions

When $n>3$, then even the reduced-by-one-dimension picture of the Shi arrangement cannot be drawn in the plane, and so the notion of a “layer” becomes a lot less visually intuitive. In this case, we generally will have lots of layers, any they become a little trickier to define. One way to do it is by making a further subdivision of the unbounded regions, based on how many (linearly independent) directions in which they are unbounded. For instance, in the picture above,

the six regions labelled with a parking function having one $1$, one $2$, and one $3$, go off to infinity in two directions, whereas

the other six regions only go off to infinity in one direction.

We then declare that the regions which are unbounded in the most number of directions are in the outermost layer, so we will label them last. Then, recursively define layers further and further inward by saying that, 

if we can get from a region to the outermost layer by crossing only one hyperplane, this region is in the second-outermost-layer

(note: no “diagonal” crossings allowed, e.g. moving from the 321 region to the 111 region through the “corner”)

if we can get from a region into the second-outermost-layer by crossing only one hyperplane, this region is in the third-outermost-layer,

and so on.

You can see that this description works perfectly well in the $n=3$ case as well: we then have four layers, of which the innermost is still the dark shaded triangle and the next-innermost consist of the three lightly-shaded ones.

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[ * I apologize for this joke, which is almost as bad as Theo’s, but at least the pronunciation is correct :P ]

[ ** or, more accurately, the line $x=y=z$ which comes “out of the page”; remember this is supposed to be in 3D, after all ]

[max1461]

@toastymath

Thought I'd make this its own post: where does computational homology sit with respect to the rest of homology theory (is that the term?)? All I know about homology is the basics of simplicial homology, which I've mostly forgotten by now anyway. But your description of computational homology makes it sound appealing to me, especially the part about working with abstract n-cells rather than actual geometric objects (I find combinatorics much more comfortable than geometry...). Would something like this book be more-or-less accessible to me with this background?

[threequarkvolume]

ASIF one believed in spherical packing of our smallest knowledgable region of scale, the quark, into a hexagon direction in space much like Computer Gaming Programmers learnt was a better system for "Civilization". Euclid and probability with such a basic grid is historic of course .

Then geometrically it is easy to show that combining ANY SOLUTION for a quark rules for a single hexagon can give EXACTLY the same number to fill their fractal scale shell for THREE quarks grouped as below. The Papyrus rules of Euclid would be that everything can bounce flat, In PAPER TIME (physics="a single quark" (3d=a voxel, 2d="hexagon location") has a shell of 6 NEIGHBORS . And those below do so also at 6 neighbors. I am claiming stable as the "spheres" in "Sphere packing" are stable as quarks in this universe to sustain fitting together as hadrons.

All space is a sea of quarks that spin faster or slower as the waves of string theory energy pour together or apart through time. We are moving either relative to some historical point in spacetime or as the location in a god wave of energy through the voxels of the universe. Horseman speak for science.

Called "Combinatorics".

Projected as 44, they are 22 in the two rows.

Matter and/or Antimatter Rows

26 does equal my 22 "small quark hexagons permutations" plus the 4 for being a puzzle piece through time or voxel if computered.

ASIDE 26 = 22 + 4

26 is long know as the number of the maximum caculable number of possible dimensions Mathematically and socially through the Industrial Age to unify the world to need the sanity of a 26 letter alphabet.

In this collection there are Six dual spacetime voxels

Dual = Matter on one side of the puzzle piece and Anti Matter is the other side of the puzzle piece.

Both sides of the Singularity Puzzle Piece

This one called a singularity is simply in a single "Physics World Flat Earth Argumentative Position of a Papy Rus"

And it is going to just be a set of voxels that are stable and in every "the center of the star". Whether We make a movie or picture or actually measure reality, In reality We might not call spacetime a "voxel at frame N" ...

the possible positions that three quarks situated in the position of a local minimum or as MATH on graph papayrus as a LOCATION.

Neutrons are a tensor combination of fractal proportions 1:2 with the neutron being fine and stable if moving ASIF just a hexagon voxel which can go on as light inowhatever string theory loop is tracking through a protons forward velocity.

Neutron tensor pair out of step with stablility=gravity,teleport=hex directions object at location flows to next.

Fractals work simply like adding. Believe in some Number. Each addition to this number is another shell. And the fractal line is rotating up to three amounts per unit time, but as a protector of children and adults from the danger of RELIGIONS=CULTS of historical sadness not in the angle of 90 or perpendicular. But with the geometry which that cult SCIENCE of ours just ignores because it has REDUNDANCIES.

MAN/WOMAN

GRAVITY/ELEVATOR

TELEPORT LEFT / TELEPORT RIGHT

I understand i worked at SLAC.

I understand I might just be some upgraded T2.

I understand what a lifetime for here

I understand physicsForum banned me for asking if they were ready to just trust Quarks as an AXIOM (Or they do have the right to disallow me as a internet user based on law). In which case I see things as if America built Guantanamo Bay That was Originally paid to Try to hold me. But they are just getting ready I suppose.

[maaarine]

MBTI & Academia Terence Tao: INTP

“Terence Chi-Shen Tao FAA FRS (born 17 July 1975) is an Australian-American mathematician who has worked in various areas of mathematics.

He currently focuses on harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, compressed sensing and analytic number theory.

As of 2015, he holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles.

Tao was a recipient of the 2006 Fields Medal and the 2014 Breakthrough Prize in Mathematics. (…)

Tao exhibited extraordinary mathematical abilities from an early age, attending university-level mathematics courses at the age of 9.”

Sources: video, wiki/Terence_Tao