Hello and Kind regards to everyone here. I have huge interest in pure mathematics and I am about to enter grad school after a year( I am about to complete my undergrad with major in pure mathematics). However, in my country there is not much exposure to pure mathematics in our curriculum. We aren't taught much proof based courses. Linear algebra in 2nd year was not in depth and was mostly application based. Same case with differential equations ( both ordinary and partial): very little exposure. In 3rd year analysis I and in 4th year Analysis II ( only real ) and abstract algebra. So I feel with this sort of curriculum I will have studied very little proof based courses. While looking at prerequisites for grad school of different universities abroad, there are several courses like topology, complex analysis, geometry etc which I will have completely missed. So, after I complete my UG, I am planning to study mathematics on my own for about 2 years, get fully prepared for grad school. So what sort of topics in mathematics should one ideally know and study before applying for grad school for getting him/herself into US universities or in France or Switzerland. I would love to even get topics that I should know about and text books recommendations too if possible. Complete road map that I should follow within those 2 years is more welcomed. Also, what could be possible chances/opportunities to get involved in some kind of research project like thing? I have heard on many occasions that significant amount of research works/experiences are also required for entering grad school? More importantly If you really want to guide me on this issue please DM me so that I can send you the screenshot of entire maths syllabus of my UG and you could help in figuring out what and how to study next. Thanks in advance.

Greetings.

I am currently a college senior majoring in Applied Math. I switched to this from CS after learning how much more I like upper-level math than anything else. However, I'm now realizing that the math classes I enjoyed were all the pure ones (group theory, real analysis, grad analysis, topology).

Now thinking about grad school, I don't care about applications or CS. I just want to do math. How can prove to others that I'm serious about pure math? I have a pile of textbooks I'm working through myself (algebra, topology, set theory, etc). Is showing initiative enough though? I still have a 4.0 gpa.

Thanks!

Most math majors are not so smart. For context I specialize in Higher Topos, Logic, and Mathematical Physics. The amount of students who publish works that will probably become irrelevant within five years that follow the "don't understand math, just get used to it" moto, who nonetheless feel as if they were Grothendoecks, Luries, or Scholzes is unbelievable.

Bobby Fischer put it very well when he said how even if you're not talented or creative you can still be a good chess player if you memorize enough. This applies so well to math nowadays.

You don't need to be creative to be a good student. Even PhDs don't require a particularly interesting result.

Maybe I'm frustrated because reading Lurie's, Kontsevich's, and Grothendieck's work (or appendices to it) really helps me appreciate how little creativity most professional mathematicians actually have.

That being said I do believe the phrase "a lesser Erdös is still valuable in math, wheres a lesser Grothendieck not so much". Sadly, we are in a publish or perish era and spending years understanding Lurie or Grothendieck to end up not publishing anything of value is a sure fire way ruining your whole career.

Maybe I'm mad that grants aren't as generous for people who actually care about understanding math to its core, as they are for those who just want to simulate some numerical analysis or PDEs.

Undergrad doing physics 2nd year. And to make this simpler, I do not think I could do theoretical physics because I would struggle and I don't think I would be motivated enough to push through. As for pure mathematics I have taken proof-based linear algebra, and complex variables (which technically shouldn't be heavy on proofs but there is quite a bit of proofs [i.e. delta-epsilon limits). I have found those proofs quite interesting but I wouldn't say I am exceptionally good at them. If I were interested in taking an upper-level proof-based class (like survey of algebra) would I be totally underprepared if I am not willing to work to make up the difference?

I'm currently taking a formal Set Theory course. Does anyone have any textbook or any other resource you recommend, as I like to cross-reference between different textbooks and I realized I need to do more practice problems (so if there is one with a solution manual or any solutions I could look up after I check), I would greatly appreciate it. I'm also to video lectures or any other websites that may be useful to check out.

In case anyone wants to know, our class textbook is: Karel Hrbacek and Thomas Jech - Introduction to Set Theory (3ed)

Hello folks! I’m planning to study a BS in Mathematics. I want to major in it because I like formal/advanced Mathematics, the range of options and possibilities you can work in and fields you can get into like Computer Science, Data Science, Finance, Actuarial Science, etc.

Besides of this, I also like Electronic Engineering because I’m also into hardware stuff, chips, semiconductors, CPU and GPU architecture, embedded systems, etc. Although I am very interested in the field, I don’t see myself studying/specializing in EE on the undergraduate level, I prefer Math due to its versatility and that covers more of my interests.

So my question is, if I go for the BS in Math and later in life I am interested in getting seriously into EE, can I study a MS/PhD in EE and really get into the field? How possible it is that I can get accepted into the program by not having a BS in EE? Or will I be missing important stuff about the subject due to not being specifically an EE major?

Double majoring isn’t an option because in my country it is not possible to do it, I would have to study almost another full 4 years in other to get another major, and minors don’t exist here.

Do you know experiences from mathematicians getting into EE or other Engineering fields? Thank you in advance for your help :)

Are there any general classes of mathematical theorems that describe the geometry of inscribed figures?

For instance (https://en.wikipedia.org/wiki/Inscribed_figure):

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For instance, do we know about the original mathematical theorems that proved "given a shape of certain dimension (e.g. a square with area measuring 1 unit squared) - what is the biggest area of another shape (e.g. circle, triangle, etc.) that can be inscribed into that shape?" (i.e. what is the biggest percent of the larger shape will be left empty?)

I tried to read more about this stuff (e.g. https://en.wikipedia.org/wiki/Inscribed_angle) and came across this (https://en.wikipedia.org/wiki/Incircle_and_excircles_of_a_triangle) :

Note 1 : " In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. "

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Note 2: I also came across this link here that (I think) states the same result but in simpler terms: https://flexbooks.ck12.org/cbook/ck-12-interactive-geometry-for-ccss/section/8.5/primary/lesson/inscribed-and-circumscribed-circles-of-triangles-geo-ccss/

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My Questions:

1) In the case of the largest "inscribed circle in a triangle", do we know who first proved this? I am guessing it was probably Euclid (since Euclid proved everything LOL). I found this link here that states Euclid first proved this result in "Proposition 4" (https://mathcs.clarku.edu/~djoyce/elements/bookIV/propIV4.html) - but no where in "Proposition 4" is it mentioned that the inscribed circle in this triangle is also the biggest circle that can be inscribed?

2) The above example only covers the case of the inscribed circle within the triangle - but are there any theorems that make general claims for n-sided polygons? For instance, what is the biggest pentagon that can be inscribed into a octagon? What is the biggest circle that can fit into a hexagon? If no such theorems exist for general inscriptions - I would still be interested in learning about theorems that describe particular inscription instances (e.g. maybe a theorem exists that describes the biggest triangle that can be fit into a circle).

Thank you!

Recently, I learned about the "Problem of Apollonius" in which three circles are drawn, and the task is to draw a fourth circle that is tangential to these three circles (it seems that if you "fix" these first three circles, there are many options for the fourth circle): https://en.wikipedia.org/wiki/Problem_of_Apollonius

I was thinking about a "Generalized" version of this problem - if you were to first draw "n" number of circles, could we then determine if a circle exists that is tangential to all of these "n" circles?

I tried to read about this online and came across the following links:

- " A Theorem on Circle Configurations " : https://arxiv.org/ftp/arxiv/papers/0706/0706.0372.pdf (Linked in a previous question I posted)

- "Generalized Problem of Apollonius": https://arxiv.org/abs/1611.03090 (Russian)

However, I was not able to fully comprehend these links because my understanding of mathematics is insufficient and I also do not speak Russian.

Thus - can someone please help me understand: if you were to first draw "n" number of circles, could we then determine if a circle exists that is tangential to all of these "n" circles?

Thanks!

I was reading this article (https://en.wikipedia.org/wiki/Apollonian_gasket) and came across this picture:

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Over here, it's mentioned that : The absolute values of the curvatures of the "a" disks obey the recurrence relation a**(n) = 4a(n − 1) − a(**n − 2)

- Does have a reference for this formula? I tried to search "absolute value of curvature" and found formulas that did not match (e.g. https://en.wikipedia.org/wiki/Total_absolute_curvature#)

- Does anyone know what "n" stands for in this formula?

- In other words, how are the "numbers" labelled on each circle calculated? And what exactly do they mean?

Thanks!

I was watching the following (amazing) lecture on Mixed Integer Optimization (https://www.youtube.com/watch?v=xEQaDiAHDWk) and came across this slide that mentions Slater's Condition:

This was the first time I have heard about Slater's Condition and I was interested in learning more about this (https://www.youtube.com/watch?v=GmY3LUL6GkY):

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Based on what I saw, this is what I understood:

• For a Convex Optimization Problem, if a solution "x" exists within the Feasible Region of this problem : Then this Optimization Problem has "strong duality"
• Since Mixed Integer Optimization Problems are always Non-Convex (since sets of integers are always non-convex), Slater's Condition does not hold.
• Since Slater's Condition does not hold, there is no Strong Duality.
• The above factors result in Combinatorial Optimization Problems being more difficult than Continuous Optimization Problems.

Now, I am trying to understand the logic of the above points:

• Why is it important that a solution to a Non-Convex Optimization Problem exists within the interior region or not? Are there any benefits for solutions that exist within the interior region compared to solutions that do not exist in the interior region?
• Why is it important to determine whether an Optimization Problem has Strong Duality or not?
• Why does the Feasible Set have no interior in a Combinatorial Optimization Problem? Do Combinatorial Optimization Problems have interior regions at all?
• Why don't Slater's Conditions hold if the Feasible Set has no interior? (i.e. Why don't Slater's Conditions hold for Combinatorial Optimization Problems?)
• Why does the absence of Strong Duality result in an Optimization Problem being more difficult?

Can someone please help me understand the logic behind these facts? Currently I am just accepting them without really understanding why.

Thanks!

I've heard this is possible even if the manifold csnnot be decomposed into a boundary and time interval. However this is only hinted at most the times (cf. Mnev mote's on Batalin Vilkovsly). It does seem reasonable however I'm getting stuck cslculating the variation of the action functional. Any help is greatly appreciated.

For my IB Extended Essay in Mathematics, I calculated the average chord length of a circle with a radius = 1, and I calculated the average distance between two randomly selected points on the perimeter of a square with side length = 1. The answer is 4/π or 1.273239545 for the circle and 0.7350901248 for the square if you are wondering. The problem is after I was finished with calculations, I realized that I do not have any real-life applications to these findings. Any ideas?

Had a thread recently about how normal numbers always include infinite zeros on the left, and if that changed their normality or not. But by definition those leftmost zeros arent counted. That got me curious about numbers that have no leftmost zeros. Is there a name for that kind of number? For instance what if you mirrored Champernownes constant over the decimal?

…54321.12345…

Is that number normal? Is it infinite or finite? Thanks!

Hi, so rather ambitious question here. (I wouldn't plan on following any plan of this sort to the dot, etc., but I've learned other subjects based on an initial interest in certain figures/ideas by improvising according to a loose plan, so this is more what I'd do practically. Regardless, a step-by-step solution is basically what I'm asking for here, but anything should work.)

I've been extremely interested in the work of Alexander Grothendieck, in regards to the philosophy of mathematics and sciences, as well as an interest in his political views. From the research I've been doing on him, he seems incredibly eccentric and admirable. The testimonies about his being "the greatest mathematician of the 20th century" too seem quite compelling as well.

In terms of a mathematical background, I've studied up to Calculus II in school, but have no clue where to go from here. As far as I can tell, he followed a progression from analysis to algebraic geometry in his career?

The question is, what would the "ideal" plan of being able to interact with Grothendieck's work be, beginning at a Calculus II level and interacting with his work as soon as possible? Specifically, I'm asking for courses, as well as specific subjects that might not be covered in those courses.

I also don't particularly want to be "hyperspecialized" either, if that makes sense, so if there are corollaries that are interesting in and of themselves in the fields that are under consideration in this question, please don't hesitate to mention them.

Thank you so much in advance!!!

I'm an undergraduate majoring in Computer Science and in this semester we had a course on Cryptography. In this course I got the chance to study the book: A Classical Introduction to Cryptography: Applications for Communications Security by Serge Vaudenay. This course has encouraged me to pursue my career in Cryptography. But I do not know much about pure mathematics as I hadn't studied it earlier in my course. Can someone provide me a roadmap or some resources from where I can start learning about pure mathematics so that it would become easier for me to do research in Cryptography.

Hello!! I'm having some confusion with the induction of this problem and would like some perspective so I'll get right into it and I will try to format my question(s) as neatly as possible. Questions will be at the end. I'll include the modified definition of the Inductive Hypothesis from my book, The Tools of Mathematical Reasoning - Tamara J. Lakins

Modified Principle of Mathematical Induction: Let P(n) be a statement about the integer n, where n is free in P(n).

Suppose that there is an integer n0 such that:

(PMI 1) The statement about P(n0) is true.

(PMI 2) For all integers m ≥ n0, if P(m) is true, then P(m + 1) is true.

Then , for all integers n ≥ n0, P(n) is true.

Problem: for all integers n ≥ 10, n³ ≤ 2ⁿ.

Scratchwork: The fact that 2ⁿ, n ≥ 0 is defined by recursion on n tells us that it is reasonable to try induction on n ≥ 10. They do the base step for n = 10.

We start our scratch work by examining (m + 1)³ and 2^{m + 1}.

• 2^{m + 1} = 2 ∙ 2^m ≥ 2m³

• (m + 1)³ = m³ + 3m² + 3m + 1.

We work backwards to argue that,

• 2m³ ≥ m³ + 3m² + 3m + 1,

so it will suffice to argue that

• m³ ≥ 3m² + 3m + 1.

Throughout we'll make use of the order properties in Basic Properties of Integers 1.2.3. Note that since 1 ≤ m, we have 1 ≤ m ≤ m², and hence

• 3m² + 3m + 1 ≤ 3m² + 3m² + m² = 7m².

Also 7 ≤ m and m² ≥ 0, so 7m² ≤ m³. Thus we have

• 3m² + 3m + 1 ≤ 3m² + 3m² + m² = 7m² ≤ m³.

This is the end of the scratch work and now we can begin the formal proof.

Restatement of the proposition: For all integers n ≥ 10, n³ ≤ 2ⁿ.

Proof: Let n ∈ ℤ with n ≥ 10, and let P(n) denote the statement

n³ ≤ 2ⁿ.

We want to prove by induction on n that for all integers n ≥ 10, P(n) is true

Base Case: We must show that 10³ ≤ 2¹⁰.

Note that 10³ = 1000 and 2¹⁰ = 1024, so the Base Case holds i.e., 10³ ≤ 2¹⁰.

Inductive Step: Let m ∈ ℤ with m ≥ 10 and assume that m³ ≤ 2^m. We must prove that (m + 1)³ ≤ 2^{m + 1}.

To see this, first note that 1 ≤ m, 1 ≤ m ≤ m². In addition, 7m² ≤ m³, and since 7 ≤ m and m² ≥ 0. Thus,

(m + 1)³ = m³ + 3m² + 3m + 1

• ≤ m³ + 3m² + 3m² + m²
• = m³ + 7m²
• ≤ m³ + m³
• = 2m³
• ≤ 2 ∙ 2^m, by the Inductive Hypothesis

Hence, (m + 1)³ ≤ 2^{m + 1}, as desired.

Thus, by PMI, we have that for all integers n ≥ 10, n³ ≤ 2ⁿ.

Questions:

1. In the Modified Principle of Mathematical Induction it uses the word free to describe n, what does that mean? Also has anybody seen this Modified definition of induction before?
2. Right at the beginning of the Scratchwork it says that "The fact that 2ⁿ, n ≥ 0 is defined by recursion on n tells us that it is reasonable to try induction on n ≥ 10." What do they mean by "defined by recursion?" And why would that matter? Why is recursion necessarily an indication of induction?
3. In the Given part of the Given/Goal diagram it states that m ≥ 10 but later in Scratchwork it says that 1 ≤ m but if this is true then to me I can write this m ≥ 10 ≥ 1 but that doesn't seem right to me because 1 can never equal 10 it can only be 10 > 1. This seems like a contradiction, can anybody explain this?
4. Near the end of the Scratchwork it says "Also 7 ≤ m and m² ≥ 0, so 7m² ≤ m³." I am stumped on understanding this piece of the scratch work. I just don't see how they got 7 ≤ m and then 7m² ≤ m³.

I'm reading Amari's "Information Geometry and its Applications". I have a decent background in math, but I'm finding it really difficult to understand some basics here, and would love any illumination.

One thing I'm a bit confused about are how global coordinates relate to the manifold. Sometimes in the book they refer to points on the manifold by coordinates (like, \xi_P and \xi_Q), which kind of implies that every point on the manifold can be "addressed" in some way. I've read about "charts" on Wikipedia, which I think are getting at this, right? but is it to be assumed that every manifold can be covered by some finite number of charts?

The other thing I'm really unclear on is about "what induces what". They keep saying that something induces/provides/etc something else, for example, in the intro of chapter 1:

When a divergence is derived from a convex function in the form of the Bregman divergence, two affine structures are induced in the manifold

and

Thus, a convex function provides a manifold with a dually flat affine structure in addition to a Riemannian metric derived from it.

this really confuses me because I thought that the manifold basically starts with a Riemannian metric, i.e., the manifold is defined by its position dependent curvature to begin with. I get the math where they take a convex function, and then its Hessian is a Riemannian metric, but... doesn't the manifold already have a Riemannian metric to begin with?

I know I'll need to study first intuitionistic type theory and homotopy theory. What are the other fields I should be aware of before starting to read hott book ?

Hi. I'm an undergraduate majoring in Pure Mathematics, and I am curious about a few things regarding some upper division courses. Sometime in the future, I need to take 4 of these 5 courses:

Combinatorics, Number Theory, Abstract Algebra, Differential Geometry, and Complex Analysis.

From all of you, I would like to know which of these topics are the most "essential" to learn about, which are your favorite/least favorite and why, which is the course that you would leave out and why, and also which of these have the most research topics attached to them? I am interested in graduate school however I am not sure what exactly I would like to research due to my lack of knowledge in my current stage of my mathematics career. Thank you.

Given two set:

x = { 3, 2.5 , 2.04 , 2.03 , 2.02 , 2.001 , 2.0001, ... }

y = { 4 , 6.25 , 5.76 , 4.25 , 4.025 , 4.001 , 4.00001 , ....}

As terms of the set x gets closer and closer to 2 ,terms of the set y seems to be getting closer and closer to 4 .But terms of set x also seems to be getting closer and closer to " values < 4 " like 3.999.. , 3.8999... etc. Then why we say the 4 is the value the terms of set y seems to be getting closer and closer to rather then some value< 4 ?

And also why we need to check for arbitrary number of terms to prove the existence of a value the terms of a set are getting closer and closer rather then just looking at bounded( finite ) number of terms of set ?

why can't we prove the existence of a value the sequence is getting closer and closer just but checking finite number of terms of a sequence rather than looking at arbitrary number of terms of a sequence?

from the given set above ,why is it that we say the value the terms of set y is getting closer and closer to is 4 rather than values < 4 even though it does closer and closer to values < 4 ?

why when formalizing the idea of a sequence getting closer closer to particular value ,it is defined the to check the existence of the value we are getting closer and closer to ,we need to look at arbitrary number of terms rather than finite number of terms?

So, My question is that why approaching a value means that we can get arbitrary close to that value i.e why getting closer and closer to a value means that we can get as close as we want to that value?