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iit-jam · 7 months ago

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IIT JAM Syllabus 2025: A Comprehensive Guide

The IIT JAM (Joint Admission Test for Masters) is one of the most competitive exams for students aspiring to pursue postgraduate studies in esteemed institutions like IITs and IISc. Mathematics, being a core subject, attracts candidates with strong analytical and problem-solving skills. To excel in this exam, a thorough understanding of the IIT JAM Mathematics Syllabus 2025 is essential. This blog outlines the syllabus in detail and provides tips to help candidates prepare effectively.

Overview of IIT JAM Mathematics Syllabus 2025

The IIT JAM Mathematics Syllabus 2025 is crafted to test the candidates' knowledge of fundamental mathematical concepts covered at the undergraduate level. The syllabus is broad, covering topics such as calculus, linear algebra, differential equations, and numerical analysis. Each section focuses on key areas that are crucial for advanced studies and professional applications.

Key Topics in the Syllabus

1. Sequences and Series

This section includes the convergence of sequences and series, tests for convergence (such as comparison, ratio, and root tests), and the study of power series and their radius of convergence.

2. Differential Calculus

Candidates must understand single-variable calculus concepts like limits, continuity, and differentiability. Topics also include Taylor series, mean value theorem, and indeterminate forms. For multivariable calculus, partial derivatives, maxima, minima, saddle points, and the method of Lagrange multipliers are essential.

3. Integral Calculus

This section covers definite and indefinite integrals, improper integrals, and special functions like beta and gamma functions. The application of double and triple integrals is also emphasized.

4. Linear Algebra

A critical area of the syllabus, it focuses on vector spaces, subspaces, linear transformations, rank, nullity, eigenvalues, eigenvectors, and matrix diagonalization. Understanding the solution of systems of linear equations is vital.

5. Real Analysis

This section involves the properties of real numbers, limits, continuity, differentiability, and Riemann integration. Candidates must also be familiar with sequences, Cauchy sequences, and uniform continuity.

6. Ordinary Differential Equations (ODEs)

This includes first-order ODEs, linear differential equations with constant coefficients, systems of linear ODEs, and Laplace transform techniques for solutions.

7. Vector Calculus

Important topics include gradient, divergence, curl, line integrals, surface integrals, and volume integrals, along with Green’s, Stokes’, and Gauss divergence theorems.

8. Group Theory

The basics of groups, subgroups, cyclic groups, Lagrange’s theorem, permutation groups, and homomorphisms are covered.

9. Numerical Analysis

This section focuses on numerical solutions for non-linear equations, numerical integration and differentiation, interpolation methods, and error analysis.

Tips for Preparing the Syllabus

Understand the Weightage: Review past papers to prioritize high-scoring topics like Linear Algebra, Real Analysis, and Differential Calculus.

Strategize Your Study Plan: Divide the syllabus into manageable sections, set achievable goals, and stick to a consistent schedule.

Practice Regularly: Solve previous years’ papers and mock tests to familiarize yourself with the question patterns and improve speed.

Strengthen Fundamentals: Focus on core concepts by revisiting undergraduate textbooks and seeking clarity on challenging topics.

Leverage Online Resources: Utilize tutorials, study materials, and practice tests available online to supplement your preparation.

Conclusion

The IIT JAM Mathematics Syllabus 2025 is extensive yet well-structured, providing a clear framework for aspirants to plan their preparation. By mastering the syllabus and practicing diligently, candidates can confidently tackle the exam and achieve their dream of joining top postgraduate programs. Dedicate time, stay consistent, and focus on strengthening your mathematical foundations to excel in IIT JAM Mathematics 2025.

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positively-knotted · 10 months ago

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This all makes sense, thanks for answering!

Yeah, in the UK we do specialise earlier, mostly bc the whole undergrad is maths. So for me (although this will vary by uni):

First year of undergrad is all compulsory, 50-50 pure-applied. I covered things up to basic spectral theorems, Riemann integrals, basic PDEs, probability distributions, kinematics, multivar calc,...

Second year you start to specialise. About half is compulsory (Linear Algebra 3, Metric Spaces, Complex Analysis, Differential Equations 1), and about half is optional. Being pure-brained, I did all the pure options (Measure Theory, Topology, Rings & Modules) and Quantum, Probability, and Statistics as applied. Some lower ranked unis won't specialise until 3rd year.

Third year was all options, and you get to specialise a lot. For example, I did only geometry, algebra, and functional analysis courses, as well as auditing Further Quantum. Courses include Fundamental Group, Algebraic Curves, Galois Theory,...

Then you do a Masters, which is only 1 year in the UK. I did a dissertation (see #dissertationposting) and a bunch of courses, all relevant to topology/geometry. Courses included Homological Algebra, Low-Dimensional Topology and Lie Groups.

So by the time you start a PhD, you're already pretty specialised, and you apply to a particular supervisor to study a particular area. Depending on where you go, you might have to do courses during your first year, but they'll be "broadening courses", ie stuff adjacent to your field that you missed in undergrad. Things actually relevant to your research you'll mostly do in seminar series or reading groups. You generally take 4 years to do a PhD, starting your actual dissertation research at the end of the second year. But you'll publish small things either near the end of your first year or definitely in your second.

finally planned out my first three years of coursework

fall 2024: linear algebra, analysis I, topology

spring 2025: analysis II, algebraic topology, intro to combinatorics

fall 2025: algebra I, complex analysis, algebraic topology II

spring 2026: algebra II, graph theory

fall 2026: rings and algebras

spring 2027: combinatorics, complex analysis

this is actually above and beyond the phd course requirements, provided that i end with a B+ or higher in each one. this plan leaves me with plenty of room to retake or drop courses to take them again later though, which gives me some peace of mind.

#cognisance #maths posting

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positivelyprime · 9 months ago

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Vibes tbh.

A bit more seriously: complex analysis seems full of 'classification' type results (e.g. Liouville's theorem, Picards little/great theorem, Riemann surfaces being like one of three types, even Cauchy integral formula fits) which are more typical of an algebra style course. Of course this is a bit tongue in cheek as you do need serious analysis to obtain such results. End result is just that holomorphic functions have a great deal of structure.

Oh also maybe the fact that you can write any holomorphic function using its Laurent series which is basically just a polynomial which is basically just algebra

The two types of math are algebra and analysis. All math falls into these two categories.

Probability theory? Analysis. Category theory? Algebra. Differential geometry? Analysis. Complex analysis? Believe it or not, Algebra.

#proof by vibes

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giantpredatorymollusk · 6 years ago

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And the mathematical models of King Krool and the beast did such fierce battle across the equation-covered table, that the constructors' pencils kept snapping. Furious, the beast writhed and wriggled its iterated integrals beneath the King's polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann's Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginning, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F1) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out, fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Δk to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets—but the beast, prepared for this, lowered its horns and—wham!!—the pencils flew like mad through transcendental functions and double eigentransformations, and when at last the beast closed in and the King was down and out for the count, the constructors jumped up, danced a jig, laughed and sang as they tore all their papers to shreds, much to the amazement of the spies perched in the chandelier-—perched in vain, for they were uninitiated into the niceties of higher mathematics and consequently had no idea why Trurl and Klapaucius were now shouting, over and over, "Hurrah! Victory!!"

The Cyberiad, Stanislaw Lem. Also I think of this whenever I'm doing Calc homework

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corporatemmorg · 3 years ago

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foxspain82 · 4 years ago

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10.4 Usubstitution Trig Functionsap Calculus

10.4 U-substitution Trig Functionsap Calculus Answers

10.4 U-substitution Trig Functionsap Calculus Pdf

10.4 U-substitution Trig Functionsap Calculus Problems

10.4 U-substitution Trig Functionsap Calculus Worksheet

Calculus II, Section 7.4, #67 Integration of Rational Functions by Partial Fractions One method of slowing the growth of an insect population without the use of pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no o spring. Let P represeent. AP Calculus AB Mu Alpha Theta Welcome to AP Calculus AB! Contact me here. Need more review? Browse the Algebra II and Pre-Calculus Tabs. AP ® Calculus AB and BC. COURSE AND EXAM DESCRIPTION. AP COURSE AND EXAM DESCRIPTIONS ARE UPDATED PERIODICALLY. Please visit AP Central. Mathematics 104—Calculus, Part I (4h, 1 CU) Course Description: Brief review of High School Calculus, methods and applications of integration, infinite series, Taylor's theorem, first order ordinary differential equations. Use of symbolic manipulation and graphics software in Calculus. Note: This course uses Maple®.

Math 104: Calculus I – Notes

Section 004 - Spring 2014

10.4 U-substitution Trig Functionsap Calculus Answers

Syllabus

Concept Videos

Skeleton NotesComplete Notes Title More Remainder 10.6, 10.9 Remainder 10.6/10.9 Series Estimation & Remainder Sections 10.8-10.10 Sections 10.8-10.10 Taylor (and Maclaurin) Series Section 10.7 Section 10.7 Power Series Introduction Section 10.6 Section 10.6 Alt. Series Test and Abs. Conv. Conv. Tests Section 10.5 Section 10.5 The Ratio and Root Tests Section 10.4 Section 10.4 The Comparison Tests Section 10.3 Section 10.3 The Integral Test Section 10.2 Section 10.2 Introduction to Series Section 10.1 Section 10.1 Sequences Section 9.2 Section 9.2 Linear Differential Equations Section 7.2 Pt 1Pt 2 Section 7.2 Separable Differential Equations Section 8.8 Section 8.8 Probability and Calculus Odd Ans. Section 8.7 Pt. 1Pt. 2Section 8.7 Improper Integrals L'Hopital Section 8.4 Pt. 1Pt. 2Section 8.4 Partial Fraction Decomposition Section 8.3 Pt. 1Pt. 2Section 8.3 Trig. Substitution Section 8.2 Pt. 1Pt. 2Section 8.2 Integrating Trig. Powers Section 8.1 Pt. 1Pt. 2 Section 8.1 Integration By Parts Section 6.6 Section 6.6 Center of Mass Section 6.4 Section 6.4 Surface Area of Revolution Section 6.3 Section 6.3 Arc Length Section 6.2Section 6.2 Volumes Using Cylindrical Shells Section 6.1 Section 6.1 Volumes Using Cross-Sections disk/washer Review Calc I Review Calc I ReviewLimit, Derivative, and Integral Area b/w CurvesArea b/w Curves Video Example U-substitution Graphs you should know

Print out the skeleton notes before class and bring them to class so that you don't have to write down https://foxspain82.tumblr.com/post/657282647494672384/achievement-unlocked-2watermelon-gaming. Hide paragraph marks in microsoft word for mac. everything said in class. If you miss anything, the complete notes will be posted after class.

10.4 U-substitution Trig Functionsap Calculus Pdf

10.4 U-substitution Trig Functionsap Calculus Problems

10.4 U-substitution Trig Functionsap Calculus Worksheet

Version #1 The course below follows CollegeBoard's Course and Exam Description. Lessons will begin to appear starting summer 2020. BC Topics are listed, but there will be no lessons available for SY 2020-2021

Unit 0 - Calc Prerequisites (Summer Work) 0.1 Summer Packet

Unit 1 - Limits and Continuity 1.1 Can Change Occur at an Instant? 1.2 Defining Limits and Using Limit Notation 1.3 Estimating Limit Values from Graphs 1.4 Estimating Limit Values from Tables 1.5 Determining Limits Using Algebraic Properties (1.5 includes piecewise functions involving limits) 1.6 Determining Limits Using Algebraic Manipulation 1.7 Selecting Procedures for Determining Limits (1.7 includes rationalization, complex fractions, and absolute value) 1.8 Determining Limits Using the Squeeze Theorem 1.9 Connecting Multiple Representations of Limits Mid-Unit Review - Unit 1 1.10 Exploring Types of Discontinuities 1.11 Defining Continuity at a Point 1.12 Confirming Continuity Over an Interval 1.13 Removing Discontinuities 1.14 Infinite Limits and Vertical Asymptotes 1.15 Limits at Infinity and Horizontal Asymptotes 1.16 Intermediate Value Theorem (IVT) Review - Unit 1

Unit 2 - Differentiation: Definition and Fundamental Properties 2.1 Defining Average and Instantaneous Rate of Change at a Point 2.2 Defining the Derivative of a Function and Using Derivative Notation (2.2 includes equation of the tangent line) 2.3 Estimating Derivatives of a Function at a Point 2.4 Connecting Differentiability and Continuity 2.5 Applying the Power Rule 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple (2.6 includes horizontal tangent lines, equation of the normal line, and differentiability of piecewise) 2.7 Derivatives of cos(x), sin(x), e^x, and ln(x) 2.8 The Product Rule 2.9 The Quotient Rule 2.10 Derivatives of tan(x), cot(x), sec(x), and csc(x) Review - Unit 2

Unit 3 - Differentiation: Composite, Implicit, and Inverse Functions 3.1 The Chain Rule 3.2 Implicit Differentiation 3.3 Differentiating Inverse Functions 3.4 Differentiating Inverse Trigonometric Functions 3.5 Selecting Procedures for Calculating Derivatives 3.6 Calculating Higher-Order Derivatives Review - Unit 3

Unit 4 - Contextual Applications of Differentiation 4.1 Interpreting the Meaning of the Derivative in Context 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration 4.3 Rates of Change in Applied Contexts Other Than Motion 4.4 Introduction to Related Rates 4.5 Solving Related Rates Problems 4.6 Approximating Values of a Function Using Local Linearity and Linearization 4.7 Using L'Hopital's Rule for Determining Limits of Indeterminate Forms Review - Unit 4

Unit 5 - Analytical Applications of Differentiation 5.1 Using the Mean Value Theorem 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points 5.3 Determining Intervals on Which a Function is Increasing or Decreasing 5.4 Using the First Derivative Test to Determine Relative Local Extrema 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema 5.6 Determining Concavity of Functions over Their Domains 5.7 Using the Second Derivative Test to Determine Extrema Mid-Unit Review - Unit 5 5.8 Sketching Graphs of Functions and Their Derivatives 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative (5.9 includes a revisit of particle motion and determining if a particle is speeding up/down.) 5.10 Introduction to Optimization Problems 5.11 Solving Optimization Problems 5.12 Exploring Behaviors of Implicit Relations Review - Unit 5

Unit 6 - Integration and Accumulation of Change 6.1 Exploring Accumulation of Change 6.2 Approximating Areas with Riemann Sums 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation 6.4 The Fundamental Theorem of Calculus and Accumulation Functions 6.5 Interpreting the Behavior of Accumulation Functions Involving Area Mid-Unit Review - Unit 6 6.6 Applying Properties of Definite Integrals 6.7 The Fundamental Theorem of Calculus and Definite Integrals 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation 6.9 Integrating Using Substitution 6.10 Integrating Functions Using Long Division and Completing the Square 6.11 Integrating Using Integration by Parts (BC topic) 6.12 Integrating Using Linear Partial Fractions (BC topic) 6.13 Evaluating Improper Integrals (BC topic) 6.14 Selecting Techniques for Antidifferentiation Review - Unit 6

Unit 7 - Differential Equations 7.1 Modeling Situations with Differential Equations 7.2 Verifying Solutions for Differential Equations 7.3 Sketching Slope Fields 7.4 Reasoning Using Slope Fields 7.5 Euler's Method (BC topic) 7.6 General Solutions Using Separation of Variables 7.7 Particular Solutions using Initial Conditions and Separation of Variables 7.8 Exponential Models with Differential Equations 7.9 Logistic Models with Differential Equations (BC topic) Review - Unit 7

Unit 8 - Applications of Integration 8.1 Average Value of a Function on an Interval 8.2 Position, Velocity, and Acceleration Using Integrals 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts 8.4 Area Between Curves (with respect to x) 8.5 Area Between Curves (with respect to y) 8.6 Area Between Curves - More than Two Intersections Mid-Unit Review - Unit 8 8.7 Cross Sections: Squares and Rectangles 8.8 Cross Sections: Triangles and Semicircles 8.9 Disc Method: Revolving Around the x- or y- Axis 8.10 Disc Method: Revolving Around Other Axes 8.11 Washer Method: Revolving Around the x- or y- Axis 8.12 Washer Method: Revolving Around Other Axes 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled (BC topic) Review - Unit 8

Unit 9 - Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC topics) 9.1 Defining and Differentiating Parametric Equations 9.2 Second Derivatives of Parametric Equations 9.3 Arc Lengths of Curves (Parametric Equations) 9.4 Defining and Differentiating Vector-Valued Functions 9.5 Integrating Vector-Valued Functions 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions 9.7 Defining Polar Coordinates and Differentiating in Polar Form 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve 9.9 Finding the Area of the Region Bounded by Two Polar Curves Review - Unit 9

Unit 10 - Infinite Sequences and Series (BC topics) 10.1 Defining Convergent and Divergent Infinite Series 10.2 Working with Geometric Series 10.3 The nth Term Test for Divergence 10.4 Integral Test for Convergence 10.5 Harmonic Series and p-Series 10.6 Comparison Tests for Convergence 10.7 Alternating Series Test for Convergence 10.8 Ratio Test for Convergence 10.9 Determining Absolute or Conditional Convergence 10.10 Alternating Series Error Bound 10.11 Finding Taylor Polynomial Approximations of Functions 10.12 Lagrange Error Bound 10.13 Radius and Interval of Convergence of Power Series 10.14 Finding Taylor Maclaurin Series for a Function 10.15 Representing Functions as a Power Series Review - Unit 8

Version #2 The course below covers all topics for the AP Calculus AB exam, but was built for a 90-minute class that meets every other day. Lessons and packets are longer because they cover more material.

Unit 0 - Calc Prerequisites (Summer Work) 0.1 Things to Know for Calc 0.2 Summer Packet 0.3 Calculator Skillz

Unit 1 - Limits 1.1 Limits Graphically 1.2 Limits Analytically 1.3 Asymptotes 1.4 Continuity Review - Unit 1

Unit 2 - The Derivative 2.1 Average Rate of Change 2.2 Definition of the Derivative 2.3 Differentiability (Calculator Required) Review - Unit 2

Unit 3 - Basic Differentiation 3.1 Power Rule 3.2 Product and Quotient Rules 3.3 Velocity and other Rates of Change 3.4 Chain Rule 3.5 Trig Derivatives Review - Unit 3

Unit 4 - More Deriviatvies 4.1 Derivatives of Exp. and Logs 4.2 Inverse Trig Derivatives 4.3 L'Hopital's Rule Review - Unit 4

Unit 5 - Curve Sketching 5.1 Extrema on an Interval 5.2 First Derivative Test 5.3 Second Derivative Test Review - Unit 5

Unit 6 - Implicit Differentiation 6.1 Implicit Differentiation 6.2 Related Rates 6.3 Optimization Review - Unit 6

Unit 7 - Approximation Methods 7.1 Rectangular Approximation Method 7.2 Trapezoidal Approximation Method Review - Unit 7

Unit 8 - Integration 8.1 Definite Integral 8.2 Fundamental Theorem of Calculus (part 1) 8.3 Antiderivatives (and specific solutions) Review - Unit 8

Unit 9 - The 2nd Fundamental Theorem of Calculus 9.1 The 2nd FTC 9.2 Trig Integrals 9.3 Average Value (of a function) 9.4 Net Change Review - Unit 9

Unit 10 - More Integrals 10.1 Slope Fields 10.2 u-Substitution (indefinite integrals) 10.3 u-Substitution (definite integrals) 10.4 Separation of Variables Review - Unit 10

Unit 11 - Area and Volume 11.1 Area Between Two Curves 11.2 Volume - Disc Method 11.3 Volume - Washer Method 11.4 Perpendicular Cross Sections Review - Unit 11

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drunk-math · 7 years ago

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Let'sh shay there ain't no 'nfinny!

Imagine there's no heaven, it's easy if you try... take a look at Carathéodory's notion of entropy and it'll tell you something about the nature of consciousness. Anyway.

Onto the greatest bugaboo of mathematics, infinity. Essentially, in the universe, there's no such thing as infinity, and yet so much mathematics today is about the various subtleties of it. Consider in particular the infamous axiom of choice - if all sets are finite, then the axiom of choice is trivial. Of course it's ridiculous to believe that all sets are finite, as ridiculous as it is to believe that the set of all sets does or does not contain itself.

So consider two paradoxes, Russell's and Galileo's. Galileo who's worshipped by proddies for poking the holy bear (...well, I guess the holy bear per se'd be further East) until he bit came up with the mindblowing paradox that there are infinite even numbers and infinite odd numbers, and came to the conclusion that the notions of greater than and less than make no sense for infinite sets. Should've been left there, but someone had to cant his diagonal Canticle over Canticles and make manifest the notion of multiple infinities and therefore of infinite sets.

"But wait!" you say, "weren't there always infinite sets? What about Euclid's proof that the primes were infinite? Hell, what about the very notion of natural number?" Well, yes, but are those sets? That's where Russell's paradox comes in. The notion of a set as "a bunch of shit" doesn't work, because then you could have all the shit that doesn't have its own shit, which both does and doesn't have its own shit. (Incidentally, while the doctor was a woman - both large and small D going into 2018 - the barber was definitely a man. Making the barber a woman misses the point.) Although... actually, maybe the barber sort of is a woman? Maybe.

Basically, what I'm saying is woman = class. And the first thing you need to know about classes, i.e., women, is that they don't exist. I mean, nothing in math quite exists, but these especially don't exist, and their nonexistence is critical to understanding them. These objects, such as the universe of discourse, are not in the universe of discourse, and therefore can't be discussed. What's the universe of discourse? It'd be the set of all sets, but that doesn't exist, so it's the class of all sets, which also doesn't exist, and since it doesn't exist, it's a class.

Why doesn't it exist? Because the only way you can make sense of sets is by saying that, for any set and coherently expressed property, if you have a set, you have a subset consisting of those elements that have that property (even if there aren't any, in which case it's the empty set, and that's fine). If the property is not containing itself, then the set of all sets would have that property iff it didn't have that property. That's Russell's paradox, and that means that a distinction has to be made between sets, which do exist, and classes, which don't. In particular, you cannot be allowed to make the statement, even in the negative, "this class is a member of..."

So why should infinite sets be allowed? Be rid of the axiom of choice and the axiom of infinity; replace the latter with an axiom of finiteness and an axiom of empty set. This will give you Peano arithmetic, and it will make the notion of countability nonsense. What Cantor diagonals would reveal has already been revealed by Galileo. But then what about sets on the real line? Because even restricted to rationals, let alone proper superclasses, that would fail the axiom of finiteness.

So how to define finiteness? The normal definition involves a bijection with a cardinal set, the normal definition of which (a sort of canonical set of a given natural size) essentially is the axiom of infinity, so that's right out. Easiest to axiomatize would be Tarski finiteness, which means that, given a set and a set of subsets of that set such that for every pair of elements one contains the other, then one of those elements is contained in no other. (I'm really tempted to try to write that in symbolic language, but I will have behaved myself so well so far.) Normally, this doesn't imply finiteness in the first sense without the axiom of choice (which is obviously right out), but ZF alone - without infinity or regularity - is enough to show it implies Dedekind finiteness (no proper subset has a surjection onto the whole set, or equivalently, no subset has a bijection to the natural numbers), and that Dedekind finiteness of the set of subsets of the set of subsets is enough to imply natural-number finiteness; therefore, if every set is Tarski finite, then the power set of the power set stupid double subset thing I just mentioned of a given set must be Tarski finite, and therefore Dedekind finite, so every set is natural-number finite. (The definition of the natural numbers - which don't comprise a set - will be a bit ad hoc.)

(This is where that entire Banach-Tarski rant from the last post went.)

Of course the theme of this post gets rid of that paradox, so forget the entire last paragraph. In fact, it collapses any number of paradoxes into what amount to one mega-paradox with fewer metamathematical consequences. I don't know what those are, but what they are they really aren't, are they? After all, the probability of the destruction of mathematics is impossible to assess (any truly rigorous definition of the set of definable numbers would do it - although there are many contingent ones, even contingent ones that can be without contradiction established by fiat to be, per Skolem's paradox - but that lies behind a wall of universal Skolemization) - and perhaps all probabilities are impossible to assess, with such a recasting of set theory. I'm not the first person to suggest this - Kronecker being the most celebrated - although perhaps the first so incoherently (I expect this post, indeed, the entirety of this blog, is just the right combination of informed and erroneous to be infuriating to every level of mathematical acumen), and there are untold hurdles, one of which I've just mentioned. So let's go back to that thing I harp on all the time - PNT. Let's build from the ground up, sort of, in an unholy fusion of Euclidean geometry and set theory applied to modern analysis and number theory. That might be getting me where I'm going. The Lebesgue integral certainly relies on infinite sets as a part of measure theory, so best to stick with the Riemann.

Of course, that brings up the notion of how to define limit. In ZF, it's impossible to prove without countable choice that the epsilon-delta definition of limit is equivalent to the existence of a sequence that comes to the limit in question. In this universe, the latter definition is incoherent, since no infinite sequence exists at all. The former can be restricted harmlessly to the rational numbers, which raises the question of what exactly the limit is; the limit is the rule, which is finite. Similarly, the twofold - two is less than infinity ("less than" expressed as an ordering of the set of cardinalities with at least one infinite cardinality as an element) - composition of limits (one of those limits being a series) in the definition of the integral is a composition of rules. (This computational approach makes me essentially the set-theoretic equivalent of the "nullity" guy, by the way. Well, that's not fair; he got better grades.)

Now, recall the "evidence." Now, it might seem shocking to try to work with the Euler product, in either form, without infinite sets (the reals themselves - Dedekind cuts - are infinite sets), but an infinite sequence, speaking informally, need not imply an infinite set, as long as the rule is finite. What's generated, then, provided the series is convergent, is a Dedekind cut, which would be a proper class under the axiom of Tarski finiteness. However, that's not really important, but rather, what's important is the rule, and manipulations of this rule under the arithmetic operations by a sort of composition; a rigorous definition not found in this post would most likely come from the theory of computation.

I remember how much easier I found vector calc than I did linear algebra. This was because in vector calc, I had already guessed most of the basic operations from simply generalizing the rules from one-dimensional high school calculus. At that age, I couldn't wrap my head around the fact that this clearly wasn't enough. To some extent, modern set theory's treatment of analysis is a formalization of this misapprehension, so that it ceases to be a misapprehension. Let's take the alternative perspective, Turing standing on Peano's shoulders rather than Zermelo's for the hypercomputability hierarchy to replace Gödel's. Let's enhance our confusion to create a grand certainty.

Back to the point. To recap, the "evidence" is that if you take the logarithmic derivative of the zeta function you get the logarithm of each prime in turn divided by one minus that prime to the negation of the parameter, which can alternatively be expressed, per Euclid, as the logarithm of that prime multiplied by the sum from zero to infinity of one over the powers of the prime in question to the parameter (remember, though, that this is a finite rule as opposed to an infinite set). Considering that, all terms being positive, this converges absolutely, the terms can be rearranged; note that the terms that are generated by the rule are exactly fractions with prime powers raised to the parameter in the denominator with the log of the base in the numerator, so let's put them in the order of the prime powers.

Before we go on l let's go back to that sentence, "considering that this converges absolutely, the terms can be rearranged." Remember when we all learned that in high school? But the concept of a permutation on an infinite set is taken as read there, so it's imperative to prove the equivalent principle again in this new framework. By the definition of limit, we can always run the partial long enough that it'll be within some given distance of the sum. In that, provided we have some idea where the terms are going, there must be a maximum destination, which must be equal or greater. Therefore, it must include all the terms, and more. If they're all positive, then this can only be larger. Therefore, it's true of the absolute value. Term-by-term summation following from that of the partials and the arbitrarily low upper bound on the tail, you get from there and the convergence of a subseries to the case of absolute convergence.

Anyway, back to the "evidence." From here it's pretty clear. You can make this sum by layering up integrals that start at each prime power, which will each be the parameter times the log (when there is one) divided by the index to the power of the parameter plus one. So moving the sum inside the integral (since it's absolutely convergent - same argument as before, only this time with Riemann sums slid in) you'll get the second Chebyshev function divided by the parameter of the integral to the power of one plus the parameter of the zeta function, all multiplied by the parameter of the zeta function. If you plug in one, the derivative of the log of the zeta function blows up, and so this integral blows up as well - which is what you'd expect if the second Chebyshev function asymptotically approached identity, because then you'd be dividing identity by the square to get the reciprocal so that would blow up. You'd also expect, then, that subtracting the reciprocal would cause it not to blow up, and in fact this would imply the asymptotic approach. The suggestion of this comes from the fact that if you multiply the zeta function in the log by one less than the parameter, the derivative manifestly converges.

So that's the "evidence," but it's not the proof because we can't obviously move the limit inside, not even with the normal machinery of set theory/analysis. So why can we move the limit inside? The short and incoherent answer is because there are no zeroes on the edge of the critical strip. The proof of this I won't restate - it's just algebra and trig - but the connection still isn't inherently obvious. From there, what's left is the Wiener-Ikehara theorem, which it's even more imperative to view now in terms of Fourier analysis, that is to say, constructing a function from an uncountable accumulation of sinusoids. Might be some kinks to work out there in the absence of infinite sets.

So basically, as I said before, the actual proof is based on the notion of an "approximation of unity," a family of functions of integral one that approach zero bar an infinite isolated point. (Again, none of these concepts exist, but speaking in shorthand, that's what it is.) Multiplying this by offsets of another function allow you to show that that function approaches a constant, and in this case to show that the second Chebyshev function approaches the identity. The fact that this approaches a limit comes from the equation discussed before.

Now, so far, I'm just drunkenly ruminating on cud already so thoroughly chewed. What's interesting, though, is that not only are all these limits determined by finite rules, but that they themselves are their solutions, and these rules for arithmetic operations axiom schemata, rather than inferences. The question, then, isn't whether they follow, but whether they're eliminable in whatever context is important to our purposes. And that's what brings us back to Euclid.

Remember, the challenge there is to build from the sparse postulates of his geometry and "number theory," to which not even he can hold himself entirely, to get to such a radical conclusion regarding the prime numbers. In his terms, as before, it can be expressed only in terms of the harmonic series - at least without great difficulty. New chapters might be introduced bringing it to modern terms (i.e., at the very least, terms in which RH in prime-number form would be meaningful) without a modern notion of infinity, but these would be lengthy chapters. For now, let's go with the prime counting function and the harmonic series. I said then that we would need definite error bounds, but let's replace that with "as close as we like."

Let's work backwards. We first need to start with the fact that an increasing (non-strictly) function that's convergent when you add it up (at integers because no infinity - note that this is an integral of a function of countable range in the ordinary paradigm) subtracting identity and dividing by the square must approach in ratio identity. This follows from the divergence of the harmonic series in effect - if it didn't approach identity in ratio, then the ratio would either approach a number greater or less than one, or it would diverge, being unable to oscillate due to the function's monotonicity. If it did those things, the function wouldn't converge.

From there, two things remain to get us to "as close as we like." If only I could remember what they were. (I suppose I should mention on this point that my hippocampus is a pickled seahorse on a toothpick.) I suspect they're to show that the increasiness is finite for positive parameters and that it reflects the limit. The former is easy enough to show from series (to wit, by saying that it holds for any power, however small, the concept being expressed easily enough geometrically), the latter coming from Fourier analysis. In any case, really, there's no real logic to this other than what I mentioned before, only the relevance of it to this notion of mathematics without infinite sets.

So let's not fuck around. Fourier analysis, the analysis of periodic functions on the real line expressed as the integral of an uncountable set of sine waves, without infinite sets. How? Well, not at all, really, but basically by geometry, obviously. Sine waves, after all, come from sines, the string of a bow the chord, the musical sense of "chord" after all coming from the chord a string made against a lyre (as far as you know), and Euclid knew the law of cosines in a geometric, very Alexandrian form (propositions twelve and thirteen of book II).

So let's go back to the very beginning of this blog. The normal distribution. Now, the slovenly proof I gave then doesn't really show anything and isn't the normal (so to speak) approach anyway. A better notion of probability comes through set theory and measure theory, the latter of which certainly doesn't work as normally understood in a finitist paradigm.

So you'll remember I explained (ish) why a series of fair coinflips should approach the standard normal, but I didn't explain probability beyond that at all. So what is probability? Well, you take a set, a subset of that set's power set closed under complementation (relative to the base set) and countable union, and a function from the latter to [0, 1] such that the empty set maps to zero, the union of countable disjoint sets maps to the sum of those sets' image (implying the union of countable non-disjoint sets is less than or equal to the sum of those sets' image), and the whole shebang maps to one. That's a probability space. Now, if the set has to be finite, you're golden to model, e.g., cards or dice, but you're SOL if you want to model a dartboard, unless you feel like working it out at the quantum level, but even there there may or may not be infinities where probability is concerned. Therefore, under this paradigm, the base set and space would have to be understood in terms of the rules that generate them - but aren't they anyway?

So with that in mind, the notion of approaching a probability space via a series of fair coinflips is analogous to the notion of approaching a real, computable or uncomputable, by a Turing-equivalent machine. Now, I didn't bring up Turing machines (exactly) above because even in this paradigm uncomputable numbers can be defined, such as the sum of two to the power of the negations of successive busy beaver numbers.

I've been writing this for months now, and I'm sure it's wicked self-contradictory and flows like tar, but fuck it, it's going up.

#infinity #set theory #probability #gambling #computability #prime number theorem

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jonathankatwhatever · 5 years ago

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Describing the parasitic function. It exists because it’s now a series of steps, so it can be treated as a step function and assigned montonic character, which allows metered counts. And it becomes alive by virtue of Cantor and Leibniz, who define the oscillation across interpolative infinities, which invokes the scale and growth constants. Scale is determined by the ‘delta sum’, which to me means the additive difference that proceeds as alternating Riemann integrals, meaning inside and outside alternation, where the limit of alternation defines the depth of connection as it disappears to a point in the distance. As in, the nth decimal taken at any point measures zK because the actual square in the distance conveys this specific value, which reduces to an extremely specific chain that draws an nth tight focus ring. That is, the value in the distance is a grid square and it counts in zK because the real combines x,yK. Those two sentences are mixed together.

That makes a ring in complex space to that many steps. This has an implication from the past: remember the crazy idea that information would transfer over depth to a limit? This means beyond that limit constructs whatever occurs outside toward the view across the internal space. This is true when the limit is connection, which is the alternation of the oscillation.

Each time a doubt pops up, it gets crushed by something powerful. That’s new.

Working on information beyond connection. This is layering for process. We’ve done that so time to revisit.

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dhyeyaiasnoida-blog · 5 years ago

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crospibakon · 6 years ago

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This is an integral called the Bose integral. Notice that the integral is taken with respect to t, but there is still an x in the power of t. The process for solving this integral is fairly straightforward, once you see it. Step 2 is just getting the exponentials in the form of an infinite geometric series. Step 4 changes the infinite geometric series into sigma notation. The integral in Step 7 is the definition of the Gamma Function and the infinite sum in Step 9 is the definition of the Riemann Zeta Function. Both of these functions are quite interesting in their own right and they come together to form this elegant integral quite nicely.

#math #mathematics #integral #integration #gamma #zeta function

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academyofengineers · 5 years ago

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craigbrownphd-blog-blog · 7 years ago

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Some Irresistible Integrals, Computed Using Statistical Concepts

#ICYDK: Below are a few integrals that you won't find in textbooks. Solving them is a good exercise for college students with some advanced calculus training. We provide the solution, as well as a general framework to compute many similar integrals. Maybe this material should be part of the standard math curriculum. Here, p, q, r are positive real numbers, with q larger than p. The Gamma symbol represents the gamma function. It is possible that these results are published here for the first time. These are known as Frullani integrals, although the ones mentioned here are not covered by Frullani's theorem, nor any recent generalization that I am aware of. Indeed, AI-based automated integration platforms such as WolframAlpha can not find the exact value (only an approximation) while they are able to compute standard Frullani integrals exactly. My approach to derive the exact values is different from the classical approaches, as it relies on the statistical concept of expectation, possibly leading to interesting areas of research. How to compute such integrals? These integrals are a particular case of the following main result, proved in the next section: where g(x) / x tends to 1 as x tends to infinity, and f is a bounded function with a finite expectation. Some additional conditions may be required, for instance the fact that there is no singularity point in the above quotient, and that g(x) has a lower bound that is strictly positive. The expectation of f, also called average value, is defined as For instance, if f(x) = sin(SQRT(x)), then the expectation exists, and it is equal to E(f) = 2 / Pi. (Prove it!) The main result introduced at the beginning of this section, is rather intuitive but needs great care to prove it rigorously, including correctly stating the required assumptions on f and g to make it valid. Some cases might require working with non-Riemann integrals. Here we only provide the intuitive explanation. Proof of the main result (sketch) Here p, q and n are integers, with q greater than p. We are interested in the case where n tends to infinity. We approximate integrals using the Euler-Maclaurin summation formula. The approximations below become equalities as n tends to infinity. We used the classic approximation of the harmonic series to make the logarithm terms appear. Note that for large values of k, g(k) is asymptotically equal to k. This was one of the requirements for the formula to be valid. We also have: Using the change of variable y = x / q in the first integral, and y = x / p in the second integral, we obtain: Let us remark that: * q / g(qy) is asymptotically equivalent to 1 / y (for large values of y) * p / g(py) is asymptotically equivalent to 1 / y * both integrals diverge, so the impact of small values of y eventually vanishes in each integral separately * the difference between the two integrals converges In view of this, we have: This concludes the proof. Related Problems * Four Interesting Math Problems * Curious Mathematical Problem * Two Beautiful Mathematical Results - Part 2 * Two Beautiful Mathematical Results * Number Theory: Nice Generalization of the Waring Conjecture * Yet Another Interesting Math Problem - The Collatz Conjecture To not miss this type of content in the future, subscribe to our newsletter. For related articles from the same author, click here or visit www.VincentGranville.com. Follow me on on LinkedIn, or visit my old web page here. DSC Resources * Book and Resources for DSC Members * Comprehensive Repository of Data Science and ML Resources * Advanced Machine Learning with Basic Excel * Difference between ML, Data Science, AI, Deep Learning, and Statistics * Selected Business Analytics, Data Science and ML articles * Hire a Data Scientist | Search DSC | Find a Job * Post a Blog | Forum Questions https://goo.gl/RXZFDe

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thetwomeatmeal · 8 years ago

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bernoulli would’ve won

Today I learned that Bernoulli numbers are p-adic integrals. TeX-heavy so click to read on.

First of all, Bernoulli numbers are traditionally defined as coefficients of the Taylor series $$ \frac{s}{e^s - 1} = \sum_{k=0}^\infty B_k\frac{s^k}{k!}. $$ We have $$B_0 = 1, B_1 = -\frac{1}{2}, B_2 = \frac{1}{6}, B_3 = 0, B_4 = \frac{1}{30}, $$ and $B_{2n+1} = 0$ for $n \ge 1$. I don’t have a cute slogan for what they mean, because they’re one of these sequences that shows up all throughout math: in calculating sums of powers of natural numbers, in the Euler-Maclaurin summation formula, and in Kummer’s half-proof of Fermat’s Last Theorem, for starters. One of my favorite theorems, Frank Adams’ computation of the image of the J homomorphism in the homotopy groups of spheres, uses the denominators of the Bernoulli numbers in its statement, apparently because they satisfy a certain divisibility criterion, though topologists to this day are looking for deeper reasons. The numerators, meanwhile, are more of a mystery, and it’s an open problem whether infinitely many primes $p$ divide the numerator of $B_{2p}$.

On the other hand, there’s a $p$-adic integral called the Volkenborn integral, which I learned about from Robert’s A Course in p-Adic Analysis. If $f$ is a continuous function on $\mathbb{Z}_p$, the Volkenborn integral is defined as a limit of Riemann sums where you sample at integer points: $$\int_{\mathbb{Z}_p} f(x)\,dx = \lim_{n \to \infty} \sum_{i=0}^{p^n - 1} \frac{f(i)}{p^n}. $$ Note that the factor of $1/p^n$ is the measure of the disk around $i$ of radius $|p^n|$, so that $\mathbb{Z}_p$ itself has measure 1; it’s a weird feature of $p$-adic analysis that these measures get larger as the disks get smaller.

There’s an “indefinite sum” operator $S$, defined on the natural numbers by $$Sf(n) = \sum_{i=0}^{n-1} f(i),$$ and extended to $\mathbb{Z}_p$ by continuity. We see that $$\int_{\mathbb{Z}_p} f(x)\,dx = \lim_{n \to \infty} \frac{Sf(p^n)}{p^n} = (Sf)’(0),$$ i.e., the integral is also a derivative.

Now the trick is to take the function $f(x) = (1 + t)^x$, where $t$ is some sufficiently small $p$-adic number, and write it two different ways. On the one hand, this function has an expansion in terms of binomial coefficients, $$(1 + t)^x = \sum_{k=0}^\infty t^k\binom{x}{k}.$$ This is obviously true if $x$ is a natural number, if we take $\binom{x}{k} = 0$ for $x < k$, and it’s true in general because the binomial coefficient functions are $p$-adically continuous. They’re also well-behaved for indefinite sum: $$S\binom{x}{k} = \binom{x}{k+1}.$$ It follows that $$\int (1 + t)^x\,dx = \frac{d}{dx}\left(\sum_{k=0}^\infty t^k\binom{x}{k+1}\right)\big\vert_{x=0}.$$ But $\binom{x}{k+1} = \frac{x}{k+1}\binom{x-1}{k}$, so its derivative at 0 is $$\frac{1}{k+1}\binom{-1}{k} = \frac{1}{k+1}\cdot\frac{(-1)(-2)\dotsm(-k)}{(k)(k-1)\dotsm (1)} = \frac{(-1)^k}{k+1}.$$ Therefore, $$\int (1+t)^x\,dx = \sum_{k=0}^\infty t^k \frac{(-1)^k}{k+1} = \frac{1}{t}\log(1+t).$$ You can take my word for it (I can talk about it separately, if people care) that this logarithm makes $p$-adic sense for small $t$.

Let’s put $s = \log(1+t)$. We’ve just proved that $$\int e^{sx}\,dx = \frac{s}{e^s - 1},$$ and the right-hand side is the function whose Taylor series gave us the Bernoulli numbers! Now, again taking my word for it, we can write $$e^{sx} = \sum_{k=0}^\infty \frac{s^k x^k}{k!},$$ and the convergence is nice enough that we can swap the sum and integral sign. We get $$\sum_{k=0}^\infty \frac{s^k}{k!} \int_{\mathbb{Z}_p} x^k\,dx = \sum_{k=0}^\infty B_k \frac{s^k}{k!},$$ or $$ \int_{\mathbb{Z}_p} x^k\,dx = B_k.$$

A number of nice properties of the Bernoulli numbers follow from this theorem plus some mild $p$-adic integration theory. For instance, the odd Bernoulli numbers vanish because for these $k$, $x^k$ is an odd function with zero first derivative. (The problem with $B_1$ is that the integral is really symmetric under the involution $x \mapsto -1-x$, and it’s not quite translation-invariant, so reflecting about 0 ends up introducing an error term of $-f’(0)/2$.) There’s also a nice bound on the size of the integral, $|(Sf)’(0)|$, in terms of the coefficients of the polynomial $Sf$, which says in this case that $pB_k \in \mathbb{Z}_p$. This means that the denominators of the Bernoulli numbers are squarefree -- probably not obvious from the definition! The full Clausen-von Staudt theorem -- that the primes dividing the denominator of $B_k$ (for $k$ even) are just those $p$ such that $p-1$ divides $k$ -- isn’t much harder.

Of course, the most amazing part of this theorem is that we can define these rational numbers with a $p$-adic integral that just happens to not depend on $p$. The first thing I did when I read this was to look for an analogous description in terms of an archimedean integral. Using Cauchy’s residue theorem, you can calculate Bernoulli numbers as contour integrals: $$ \frac{2\pi i B_k}{k!} = \oint_{|z| = 1} \frac{z}{e^z - 1} \frac{1}{z^{k+1}}\,dz.$$ I don’t think this is a great parallel to the $p$-adic formula, and it’s not quite as shocking, either, seeing as you could do the same thing for any function that’s analytic in a neighborhood of 0. However, I’m tempted by the fact that both the residue theorem and the Volkenborn integral show a certain unity between differentiation and integration...

#p-adic spirits #mathema #number theory

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mathematicianadda · 5 years ago

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Moonlighting https://ift.tt/eA8V8J

In the last week of August, I attended for the first time a virtual conference. This was the 2020 Ural Workshop on Group Theory and Combinatorics, organised by Natalia Maslova at the Ural Federal University in Yekaterinburg and her colleagues. The conference was held as a Zoom meeting, and ran with only one hitch. As fate would have it, it was Natalia’s talk that was disrupted by a technological failure, so she started ten minutes late and had to talk fast. My co-author and St Andrews student Liam Stott was talking in the other parallel session immediately afterwards, so I switched as quickly as I could, only to find that the chair of that session had started him early (I assume the previous speaker hadn’t shown up), and he was three-quarters of the way through his talk already. Fortunately I knew what he was talking about!

Yekaterinburg is four hours ahead of St Andrews, so we had a week of very early rising; we had lunch at 10am, and were finished for the day (in both senses) by 2pm most days.

There were some very enjoyable talks, and as usual I can only mention a few. Cheryl Praeger talked about totally 2-closed finite groups. A permutation group G is 2-closed if every permutation preserving all the G-orbits on ordered pairs belongs to G. Cheryl and her colleagues call an abstract group 2-closed if every faithful transitive permutation representation of it is 2-closed. These groups were first studied by Abdollahi and Arezoomand, who found all nilpotent examples; with Tracey they subsequently found all soluble examples. Now this team augmented by Cheryl has considered insoluble groups. At first they found none, but they found that in fact six of the 26 sporadic simple groups (the first, third and fourth Janko groups, Lyons group, Thompson group, and Monster) are totally 2-closed. Work continues.

We had a couple of plenary talks about axial algebras; Sergey Shpectorov and Alexey Staroletov explained what these things (generalised from the Griess algebra for the Monster) are, and what the current status of their study is.

Greenberg’s Theorem states that any finite or countable group can be realised as the automorphism group of a Riemann surface, compact if and only if the group is finite. Gareth Jones talked about this. The proof, he says, is very complicated. He gave a new and much simpler proof; it did less than Greenberg’s Theorem in that it only works for finitely generated groups, but more in that the Riemann surface constructed is a complex algebraic curve over an algebraic number field.

Misha Volkov gave a beautiful talk about synchronizing automata. He began with the basic stuff around the Černý conjecture, which I have discussed before, but added a couple of things which were new to me: a YouTube video of a finite automaton taking randomly oriented plastic bottles on a conveyor belt in a factory and turning them upright; and the historical fact that the polynomial-time algorithm for testing synchronization was in the PhD thesis of Chinese mathematician Chung Laung Liu (also transliterated as Jiong Lang Liu), two years before the Černý conjecture was announced. Then he turned to new results, and showed that, with only tiny changes (allowing the automaton to have no transition for some state-symbol pair, or restricting the inputs from arbitrary words to words in a regular language) the synchronization problem can jump up from polynomial to PSPACE-complete!

Alexander Perepechko gave a remarkable talk, connecting the Thompson group T, the Farey series, automorphism groups of some affine algebraic surfaces, and Markov triples, solutions in natural numbers to the Diophantine equation x2+y2+z2 = 3xyz. (There is a long-standing conjecture that a natural number occurs at most once as the greatest element in some such triple. The sequence of such numbers begins 1, 2, 5, 13, 29, 34, 89, …. I will not attempt to explain further.)

Rosemary became the fourth author of the “diagonal structures” quartet to talk about that work, which I discussed here. She concentrated on the heart of the proof, the first place in the work where the remarkable appearance of algebraic structure (a group) from combinatorial (a Latin cube with a mild extra hypothesis) appears. Without actually describing how the hard proof goes, she explained the context and ideas clearly. I think this ranks among my best work; and all I did, apart from the induction proof of which Latin cubes form the base, was to insist to my co-authors that a result like this might just be possible, and we should go after it.

One of my early heroes in group theory was Helmut Wielandt; his book on permutation groups was my first reading as a graduate student. Danila Revin gave us a Wielandt-inspired talk. Wielandt had asked, in Tübingen lectures in the winter of 1963-4, about maximal X-subgroups of a group G, where X is a complete class of finite groups (closed under subgroups, quotients and extensions). Let kX(G) be the number of conjugacy classes of maximal X-subgroups of G, Wielandt said that the reduction X-theorem holds for the pair (G,N) if kX(G/N) = kX(G), and holds for a group A if it holds for (G,N) whenever G/N is isomorphic to A. Wielandt asked for all A, and then all pairs (G,N), for which this is true; this is the problem which Danila and his co-authors have now solved.

(I hope Danila will forgive me an anecdote here. At an Oberwolfach meeting in the 1970s, one of the speakers told us a theorem which took more than a page to state. Wielandt remarked that you shouldn’t prove theorems that take more than a page to state. Yet the solution to his own problem took nearly ten pages to state. I think this is inevitable, and simply teaches us that finite group theory is more complicated than we might have expected, and certainly more complicated than Michael Atiyah expected. Indeed, in the very next talk, Chris Parker told us about work he and his colleagues have done on subgroups analogous to minimal parabolic subgroups in arbitrary groups. This is intended as a contribution to revising the Classification of Finite Simple Groups, and they hoped to show that with an appropriate list of properties only minimal parabolics in groups of Lie type and a few other configurations could arise; they obtained the full list and were rather dismayed by its length, which would make the applications they had in mind very difficult.)

Among other fun facts, I learned that the graph consisting of a triangle with a pendant vertex is called the paw in Yekaterinburg, but is the balalaika in Novosibirsk.

On the last day of the seven-day meeting, we had two talks on dual Seidel switching, by Vladislav Kabanov and Elena Konstantinova, who were using it and a more general operation to construct new Deza graphs and integral graphs.

After a problem session, the conference ended by a virtual tour of Yekaterinburg (or Sverdlovsk, as it was in Soviet times), covering the history, architecture and economics, and illustrated by photographs and historical documents; the tour guide was Vladislav’s daughter.

Life was made more difficult and stressful for me because I was doing something which would have been completely impossible in pre-COVID times: I spent some time moonlighting from the Urals conference to attend ALGOS (ALgebras, Graphs and Ordered Sets) in Nancy, France, a meeting to celebrate the 75th birthday of Maurice Pouzet, which I didn’t want to miss. Many friends from a different side of my mathematical interests were there; as well as Maurice himself, Stéphan Thomassé, Nicolas Thiéry, Robert Woodrow, Norbert Sauer, and many others.

The three hours’ time difference between Yekaterinburg and Nancy meant that there was not too much overlap between the two meetings, so although I missed most of the contributed talks in Nancy, I heard most of the plenaries.

Stéphan Thomassé talked about twin-width, a new graph parameter with very nice properties. Given a graph, you can identify vertices which are twins (same neighbours) or nearly twins; in the latter case, there are bad edges joined to only one of the two vertices; the twin-width is the maximum valency of the graph of bad edges. Bounded twin-width implies bounded treewidth (for example) but not conversely; a grid has twin-width 4. Graphs with bounded twin-width form a small class (at most exponentially many of them), and, remarkably, it is conjectured that a converse also holds.

Jarik Nešetřil and Honza Hubička talked about EPPA and big Ramsey degrees respectively; I had heard these nice talks in Prague at the MCW, but it was very nice to hear them again.

Norbert Sauer talked about indivisibility properties for permutation groups of countable degree. I might say something about this later if I can get my head around it, but this may take some time. In particular, Norbert attributed a lemma and an example to me, in such a way that I was not entirely sure what it was that I was supposed to have proved! (My fault, not his – it was the end of a long day!)

Nicolas Thiéry gave a very nice talk on the profile of a countable relational structure (the function counting isomorphism types of n-element induced substructures), something to which Maurice Pouzet (and I) have given much attention, and on which there has recently been a lot of progress. (I discussed some of this progress here, but there has been more progress since.) In particular, structures whose growth is polynomially bounded are now understood, due to Justine Falque’s work, and for primitive permutation groups there is a gap from the all-1 sequence up to growth 2n/p(n), where p is a polynomial, thanks to Pierre Simon and Sam Braunfeld.

Unfortunately the conference was running on BigBlueButton, some conference-enabling software which I had not encountered before but which is apparently popular in France. I am afraid that it was simply not up to the job. The second day of the conference saw some talks and sessions abandoned, because speakers could not connect; I could sometimes not see the slides at all, and the sound quality was terrible. I discovered that one is recommended to use Chrome rather than Firefox, and indeed it did work a little better for me, but not free of problems. On this showing I would not recommend this system to anyone.

In particular, a beautiful talk by Joris van der Hoeven was mostly lost for me. I couldn’t see the slides. Joris’ explanations were perfectly clear, even without the visuals, but sometimes I lost his voice as well. The talk was about the connections between different infinite systems: ordinals, Hardy fields, and surreal numbers. In better circumstances I would have really enjoyed the talk.

I hasten to add that the problems were completely ouside the control of Miguel Couceiro, the organiser, and marred what would have been a beautiful meeting.

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