New AMS Blog Post: Looking At the Prism

I just finished my third blog post for the AMS Grad Student Blog. It’s called Looking At the Prism, and tries to discuss the issue of bias in academia. You should read it, and then tell me what you think!

Consider this awesome interactive chart created by Ben Schmidt that allows you to see how frequently various words are used in RateMyProfessor reviews by gender and field. So for example, if you search the word “genius” we see that in every field, but especially math, it is much more frequently used in the reviews of male instructors as apposed to those of female instructors.

Well this doesn't look good...

– See more at:


First AMS Grad Student Blog

So I’ve decided to write for the AMS Grad student blog, and so I will be posting there once a month. My first post, about applying for a NSF Graduate Research Fellowship, went live today!! You should check it out:

Welcome. As this is my first post for this blog, an introduction is probably in order. I’m David, but most people call me DJ, and I am a second year graduate student at the University of Wisconsin – Madison. My research focuses on the intersection of commutative algebra, algebraic geometry, and number theory. Outside of mathematics I enjoy watching and playing sports (Go Blue!) as well as long walks on the beach. (Kidding I’d prefer to be playing volleyball on the beach.)

As Mathew mentioned in a previous blog post the National Science Foundation Graduate Research Fellowship Program (NSF-GRFP) is an amazing program that, “Recognizes and supports outstanding graduate students in NSF-supported science, technology, engineering, and mathematics disciplines who are pursuing research-based Master’s and doctoral degrees at accredited United States institutions.” [1] I have a bit of experience with the Graduate Research Fellowship (GRF) application, having applied to the NSF-GRFP twice, and am currently supported by the NSF-GRFP. Thus, since the deadline for this is fast approaching I figured I’d share a bit of advice regarding the application process. I should say that I am by no means an expert on the NSF-GRFP, and so all of this advice should be taken with a, possibly large, grain of salt.

Meeting Grotendieck – Katrina Honigs

Katrina Honigs, a post-doc at the University of Utah, recently posted a fabulous essay about the time she sought out and met Alexander Grothendieck in early 2012. You should definitely read it:

“When I looked up from my reverie, I realized that a figure with a large white beard wearing a brown robe over his clothes had appeared utterly silently quite nearby on my left. I was so startled that I felt like I was outside myself watching the scene. My eyes widened, I jumped involuntarily, my heart pounded. I thought, wildly, “Oh no, what if I have accidentally come to the home of the wrong hermit?”. But as soon as I looked at his face, I recognized it from photos I had seen. Oddly enough, it was not the more recent photos where I saw the resemblance, but to a much older black and white photo of him when he was young. “It’s you!” I managed, idiotically. Grothendieck stood impassively. In one hand, he held a short pitchfork loosely at his side. It reminded me of his doodle of devils with pitchforks around the Grothendieck–Riemann–Roch formula. His free hand rose, brandishing an admonitory finger. “Il ne faut pas entrer,” he said, advancing slowly toward me. I tried to form some sentences about visiting, but Grothendieck did not react. He continued to walk slowly toward me, wagging his finger, telling me that I shouldn’t be in here disturbing him, and asking me how I got in here.”

Hat-tip to this blog post in the Scientific American for pointing this out.

2015 AG Institute in Utah: Day 18

(For background on the conference and my time in Utah see my previous updates: Day 1,  Day 3Day 7, Day 14.)

Today marks the 18th and penultimate day of the 2015 AG Institute! This means that yesterday was essentially my last free day in Utah. Having explored the campus and gone hiking I decided to explore downtown Salt Lake more. In particular, I visited Temple Square, which is home to the Mormon Temple, the Tabernacle, and a bunch of museums dedicated to various parts of Mormonism.


Everything was immaculate and beautiful. I was also taken surprised with just how big the temple is. (Apparently this is because I’ve never seen a proper cathedral because some of the conference participants from Europe were surprised with how small it is.) I was also able to go inside the Tabernacle and see a organ recitle. That was somewhat underwhelming, but then again I am not an avid enjoyer of organ music. That said it was pretty cool to at least say I’ve been there. (Pro-tip: Don’t take pictures, even with the flash off, they don’t appreciate that.)



I also spent some time exploring some of the used bookstores downtown. Sadly they didn’t have any interesting math books. (One of them literally had a shelf of random high school algebra solution books that looked to be from the last 15 years. Who in the world goes to a rare book store to buy high school algebra solution books?) That said it was nice to walk around downtown, and I ended up finding a bit of Salt Lake that didn’t feel like one big parking lot!!

Finally, I visited the olympic cauldron from the 2002 Winter Olympics. The cauldron sits next to the University of Utah’s football stadium, and it is a somewhat sad sight. There is no advertisement for it, except one column I had mistake previously mistake for the cauldron.

When you talk towards the torch you eventually find a small room attached to the football ticket office called the Olympic Visitors Center. The room is entirely empty. Even the olympic logo has been removed from the wall. It’s actually kind of sad (especially when one thinks about how much was probably spent on the olympics.)


Finally as always here are a few things I have learned about since last posting:

  • Arithmetic, Topology, and Algebra (Jesse Wolfson): Sitting in a French jail cell in 1940, Andre Weil wrote a letter to his sister, Simone Weil, in which he discussed the role of analogy in mathematics. In particular, Weil focused on the growing connections between three fields: algebra, arithmetic, and topology. This letter, which has come to be know has Weil’s Rosetta Stone, has served to inspire a lot of work in mathematics — at least indirectly — over the last 60 years. One nice way to summarize Weil’s analogy is by the following triangle: thumb_IMG_2123_1024Jesse Wolfson gave a very good talk discussing how starting on the topology corner of this triangle one can obtain arithmetic and algebro-gemoeric results. One of the interesting things Jesse Wolfson is doing is that not only is he able to obtain point counts, but he is also able to understand the Galois actions and weights!! I really like when different areas of math connect like this, and I think there is much more to said  in this realm. (For example, adding a fourth node to the triangle with motive stuff.) See his papers I & II co-authored with Benson Farb.


  • Heuristics For The Rank of An Elliptic Curve (Melanie Matchett Wood): Given an elliptic curve E defined over the rationals the rational points of E form a finitely generated abelian group i.e. E(\mathbb{Q})=\mathbb{Z}^r\oplus T where $T$ is the torsion subgroup. Since this was proved in the 1920’s by Mordell people numerous people have work to describe, which groups can appear in this context. For example, in the 1970’s Mazur proved that the torsion subgroup T is one of 15 finite groups! Understanding the rank on the other hand has proven to be more difficult. There have been numerous conjectures — parity principal, boundedness, etc. — but very little unconditional progress on any of theses. Thus, many have spent time creating heuristics for the the rank should behave.Prof. Matchett Wood’s talk focused not on a new heuristic for the rank of elliptic curves (joint work with Jennifer Park, Bjorn Poonen and John Voight.)Since we want to count elliptic curves only up to isomorphism when I say elliptic curve I mean the vanishing set of an equation of the form y^2=Ax^3+Bx+C where A, B, C\in \mathbb{Z} and if p is a prime such that p^4 \mid A then p^4 \nmid B. Let \mathcal{E} be the set of all such elliptic curves. If we are going to do some sort of counting we better be working with finite sets, and so we will partition the set of all elliptic curves by their naive heights. Recall the naive height of an elliptic curve E:=y^2=Ax^3+Bx+C is equal to the maximum of 4|A|^3 and $27 |B|^2$. (Don’t worry about the constants out front.) With all of this in mind we let:

    \mathcal{E}_{H}=\{E\in \mathcal{E} \; : \; \text{ht}(E)\leq H\}.

    The question the talk tried to answer, or provide a heuristic for answering is: What can be said about:

    \#\{E\in \mathcal{E}_{H} \; : \; \text{rk}E=r\}.

    Note that \#\mathcal{E}_{H} is equal to H^{5/6}. I will not go into the specifics of specifics of their model. However, the idea is that consider co-kernels of specific families of random integer matrices, and is motivated by the work of Bhargava et al. on heuristics for the Tate-Schaferavich group. The interesting thing is precisely what their model predicts:

    1. 50% of elliptic curves have even rank and 50% have odd rank.
    2. 50% of elliptic curves have rank 0 and 50% have rank 1.
    3. The Tate-Shaferavich group is finite 100% of the time.
    4. \#\{E \in \mathcal{E}_{H} \; : \; \text{rk}E=r\}=H^{\frac{(21-r)}{24}+o(1)}, \quad 1\leq r \leq 21.
    5. The number of elliptic curves with rank greater than or equal to 21 is finite!!!

PS: Continuing the game of fun things you find in a business school I present you with ethics contests! This raises so many questions: What exactly is an ethics contest? Does the need for an ethics contests suggest a problem with business culture / business school? How would one cheat in an ethics contest?


Fun fact, Spencer Eccles is accused of having helped relatives of IOC members get jobs in the lead up to the bidding process for the 2002 Olympics.

Link Dump

There have been a bunch of really good articles on math related stuff. A lot of these have been mentioned on other blogs — see the hat tips — but I want to collect them all here so I don’t forget about them. Also who knows maybe not everyone reading this has seen them yet. So here are a bunch of links:

  • The Singular Mind of Terry Tao (NYT):  This might be the best profile of a mathematician done by a popular media outlet in a long time. The author — Gareth Cook — does an great job not falling into the standard tropes regarding mathematicians i.e. crazy genius, lone genius, etc. Moreover even his praise of Tao as a genius is tempered by his discussion of how part of this is Tao’s ability/desire to collaborate with others. Overall the piece is a phenominal and humanizing to someone who is very well one of the best mathematicians working today. That said I think my favorite part of the the article is how it captures the process of doing mathematics research:

    “The steady state of mathematical research is to be completely stuck. It is a process that Charles Fefferman of Princeton, himself a onetime math prodigy turned Fields medalist, likens to ‘‘playing chess with the devil.’’ The rules of the devil’s game are special, though: The devil is vastly superior at chess, but, Fefferman explained, you may take back as many moves as you like, and the devil may not. You play a first game, and, of course, ‘‘he crushes you.’’ So you take back moves and try something different, and he crushes you again, ‘‘in much the same way.’’ If you are sufficiently wily, you will eventually discover a move that forces the devil to shift strategy; you still lose, but — aha! — you have your first clue.

    As a group, the people drawn to mathematics tend to value certainty and logic and a neatness of outcome, so this game becomes a special kind of torture. And yet this is what any ­would-be mathematician must summon the courage to face down: weeks, months, years on a problem that may or may not even be possible to unlock. You find yourself sitting in a room without doors or windows, and you can shout and carry on all you want, but no one is listening.”

    Hat tip to Quomodocumque — who also has a quote in the article.

  • Empowering Who(m)? The Challenge of Diversifying Mathematics: Prof. David Kung, a former UW Madison Ph.D., discusses the state of diversity in mathematics (hint: it’s not great), what he thinks the causes of the lack of diversity might be, and what he thinks can be done to change this. For some reason WordPress or YouTube won’t let me embed this. Hat tip to Mathbabe.
  • Tohoku: Prof. Rick Jardine gives a very nice overview of both the content and impact of Grothendieck’s famous Tohoku paper. (Here is an English translation.) For those unfamiliar with this paper it was Grothendieck’s recast of homological algebra in his style, and served as the starting point for much of his future work on algebraic geometry:

    “Grothendieck’s Tōhoku paper was the start of a long period of axiomatic machine building…The machines themselves are context independent combinatorial constructions. They go anywhere. And they are applicable in multiple parts of the mathematical sciences, including traditional mathematical areas, but also in other disciplines.”

    Hat tip to Not Even Wrong.

2015 AG Institute in Utah: Day 14

(For background on the conference and my time in Utah see my previous updates: Day 1,  Day 3, Day 7.)

Week two of three is now officially in the books, and I have exactly seven six days left in Salt Lake. The second week was a little more laid back with a bit less exploring. Instead I spent most of my free time playing basketball at the gym, talking to other participants, and studying. Oh and on Wednesday a couple other Wisconsin grad students and I found a decent bar downtown!!

2015-07-22 14.42.33

First good beer in SLC.

2015-07-22 18.02.56

I wonder how well Mormons take this…

2015-07-22 18.02.13

I like the Food Pairings…

Friday turned out to be Pioneer Day, which celebrates the arrival of the Mormons in the Salt Lake Valley. According to a local it is bigger than the 4th of July, which based on the number of things that were closed on a Friday just might be true. (As an aside can anyone else think of another local holiday — state or city/town — that is widely observed and results in things shutting down?)



The talks this week focused on derived algebraic geometry, mirror symmetric, tropical geometry and so I was pretty far outside of my comfort zone. Thus, these summaries are going to be a bit shorter than previously. However, here are a few things I found interesting/learned:

  • Uniform Bounds on Rational Points (Joseph Rabinoff): Given a curve C over a the rational numbers how many rational points does C have? Of course this question, and many similar questions, have been intensely studied by numerous people over the years. In his talk Joseph Rabinoff, speaking about joint work with Eric Katz and David Zureick-Brow, discussed recent work on this question. In particular, he presented the following really cool theorem:

    Theorem (KRZ-B): If C is a smooth curve of genus g over \mathbb{Q} satifying some additional conditions then:

    \#C(\mathbb{Q})\leq 76g^2-82g+22.

    Notice that in this theorem the bound only depends on the genus of the curve, nothing else! The precise statement of their theorem, which is vastly more general can be found in their paper. In the paper they also address a similar question called the Mumford-Manin conjecture. Once again providing uniform bounds for a class of specific curves.

  • Generalized Fields (Jacob Lurie): The idea behind recent Breakthrough Prize winner Jacob Lurie’s three plenary lectures is that algebraic topology seems to provide a way to generalize many of the algebraic structures we know and love. To illustrate this concept Jacob spent his lectures focusing on the example of generalized fields. A generalized field is an associative ring spectrum E such that every module over E^*(\{x\}) is free. (A ring spectrum, at least in my mind is a cohomology theory, which spits out rings. Although by Brown Representability you might be supposed to think about this as a sequence of topological spaces. Not really sure on this…) This definition is intended to mimic definition of skew field as an associative ring all of whose modules are free.

    One can also transfer other algebraic notions from algebra to algebraic topology; often to surprising results. For example, you can define the characteristic of a generalized field, and instead of there being just two flavors — zero and p — there turns out to be generalized fields of intermediate characteristics. Jacob went on to discuss how one can do other algebraic things over these fields of intermediate characteristic — representation theory & roots of unity — and how the results differ from the results we know over regular fields.

    All of this was very interesting and exciting, but did leave me somewhat unsure of whether the analogy between algebra and topology is a deep connection or simply a useful mental crutch for understanding these new ideas. (Possibly both??) That said this almost certainly stems from a lack of understanding on my part. Regardless his talks were extremely enjoyable and down to Earth. I would high recommend them if only as a glimpse to a new area of math that seems to potentially have a lot of promise. You can find his slides here, and videos of the talks should be posted shortly.


Tom Bridgeland also gave a series of excellent talks on stability conditions, but I am not sure I understood them quite well enough to write about. So if you’re interested check out his slides and video when posted.

PS: The afternoon talks are hosted in the business school, and on Friday I found something called the “Leadership Lounge” which appears to be a lounge for business (grad) students complete with multiple ping pong tables and a foozball table.



I am starting to think the lives of a math grad student and a business grad student are very different… (Also what exactly makes a lounge “leadership”?)

2015 AG Institute in Utah: Day 7

(As has consistently been the case this post is one day late, and today is actually Day 8. For background on the conference and my time in Utah see: Day 1 and Day 3.)

Today marks one week into the three week mega conference in algebraic geometry! Things have begun to settle in a bit, and this weekend was a nice break from the usual five+ hours of talks a day. That said the weird thing about this conference is that since it is 3 weeks long people are always coming and going. It’s almost as if the conference is actually three miniature conferences in one.

2015-07-16 17.57.41

This weekend has also given me a bit more time to explore Salt Lake City. Here are a couple things I have noticed:

  • Everything in this city feels fair away, and not just because I am living at the furtherest edge of the city on the foothills. (It is actually about a 3 mile walk from my place to the edge of what appears to be the start of the city.) I mean that the blocks seem really long and streets very wide, and so getting from place to place takes a while. Someone told me that I am not crazy and Salt Lake does in fact have abnormally large block and wide streets, and apparently this dates back to the city’s founding and Brigham Young. Apparently Brigham Young wanted each block to be exactly 10 acres with streets being something like 100 feet wide. He thought each family should essentially have a small farm or something. While this might have been great back in the 19th century when Salt Lake was being founded it seems less ideal in the 21st century. As a plus Brigham Young did have a pretty sensical way to name streets as everything is essentially labeled as if it were on the cartesian plane.
  • In addition, to the large blocks Salt Lake also seems really spread out — did I mention I am like 3 miles to the start of down town. Luckily there is a pretty nice public transportation system featuring buses (haven’t used them) and a train (have used it). The train is incredibly nice. It is quite, smooth, and pretty clean. Moreover it covers a large amount of the city. (This is how I’ve been getting into the city.) That said it does have a few huge down sides: 1) It is like 25-3oish minutes between trains and 2) The train stops at 11pm on Friday nights!! Really 11pm on a Friday! Does Salt Lake have any nightlife? All of these things combined have meant that for the most part going out at night is a huge chore and has generally not seemed worth the effort.
2015-07-18 18.28.38

Strange things you see from the train.

  • The landscape in Salt Lake is amazing with mountains on all sides providing great views. However, the city itself — or at least the bits I’ve been to — are not that pretty. Maybe it is due to the fact that all of the roads seem to be 8 lanes wide and every business has a huge parking lot, but walking downtown is just not enjoyable.
Salt Lake City From Campus

Salt Lake City From Campus

Finally here again are some of the interesting things I learnt / heard during talks on Day 4 and 5. (Once again a quick disclaimer that there are most likely errors in what follows and those are due entirely to myself.)

  • Non-Negativity and Sums of Squares (Greg Smith): Given a real polynomial f\in \mathbb{R}[x_0,\cdots,x_n] we say that:
    • f is non-negative iff f(x)\geq0 for all $latex x\in S:=\mathbb{R}^{n+1}.
    • f is a sum of squares iff f=g_1^2+\cdots+g_s^2 for some $;atex g_i\in S$.

    Having made these definitions there are two fairly obvious, but important, observations one can make: 1) the definition of non-negative only makes sense for real polynomials and 2) a sum of squares is clearly non-negative. In light of observation 2) one might ask when the converse holds i.e. when is a non-negative polynomial necessarily a sum of squares. It turns out one of math’s most influential question asker — David Hilbert — asked essentially this question as his 17th question back in 1900. As is the case with many of Hilbert’s problems this one was solved in the form of the following theorem due to E. Artin:

    Theorem (E. Artin): A polynomial f\in S is non-negative if and only if there exists h_1,\ldots, h_s, g_1,\ldots,g_r\in S such that:

    f(h_1^2+\cdots+h_s^r)=g_1^2+\cdots+g_r^2 \quad \quad (1).

    While this theorem nicely answers the question at hand, it does so in an entirely non-constructive way — the proof apparently uses the axiom of choice — and so one could wonder if there is an effective version. That is if f is a non-negative polynomial then is there an upper bound on how large the degree of the h_i we might need to check for the above theorem. It turns out a 2014 paper of Lombardi, Perrucci, and Roy provides such a bound:

    Theorem (L,P,R): If f\in S is non-negative polynomial such that \deg(f)=2j then there exists h_1,\ldots, h_s, g_1,\ldots,g_r\in S satisfying (1) such that:

    \deg(h_i)\leq 2^{2^{2^{j^{4^{n+1}}}}}.

    Ok so that bound while theoretical effective is in practice no where near useful. Moreover it would seem mildly ridiculous if that bound was actually sharp. In his talk Greg Smith discussed how this result can be improved.

    The general idea behind Smith and collaborators’ — whose names I cannot find in my notes — work is to utilize the power of algebraic geometry. Most of the work done on this question to this part goes via the use of semi-algebraic sets and not varieties. However, in many ways varieties are much better understood, and more intrinsic, than semi-algebraic sets. Thus, the idea is try and rephrase this question in terms of varieties and then leverage the machinery that is algebraic geometry.

    In order to do this Smith et al. first have to rephrase the problem in terms of algebraic geometry. From now on X\subset \mathbb{P}^n_{\mathbb{C}} is will be a real projective sub-variety. By real sub-variety we mean that X is defined by real polynomials. Moreover we require that X(\mathbb{R}) be Zariski dense in X. While this last condition might seem restrictive it is actually equivalent to X having a smooth real point. Finally let R be the coordinate ring associated to X. With this set up we say f\in R_{2j} is:

    • non-negative iff its evaluation at each point in X(\mathbb{R}) is greater than zero.
    • a sum of squares iff there exist g_1,\ldots, g_r\in R_{j} such that f=g_1^2+\cdots g_2^2.

    As a quick reality check you should think why the definition of non-negativity makes sense. Remember f is not necessarily a function and we are working with projective points. In this set up they are able to obtain the following theorem:

    Theorem (Smith, et al.): Let X\subset \mathbb{P}^n be a curve of degree d and arithmetic genus p_a. If $f\in R_2j$ is non-negative then there exists a sum of squares $h\in R_2k$ such that $fh$ is a sum of squares such that:

    k\leq \max\{d-n+1, \text{ceil}(2p_ad^{-1})\}.

    Notice that this bound is way way more reasonable looking that the bound found by Lombardi, Perrucci, and Roy. Moreover Smith, et al. have been able to show that their bound is sharp!! Also notice that the bound is entirely independent of f and depends solely on the geometry of X!!! Sadly they are not able to extend this result to higher dimensions. However, they do know any optimal bound in higher dimensions cannot be independent of f.

    The idea behind the proof seemed a bit complicated, and I don’t want to go into the sketch presented in the talk. However, I will say this it utilizes everyone’s second favorite type of cone. (For those who didn’t know what that I was assured it was spectrahedral cones.

  • Geometry and Syzygies of Curves (Robert Lazarsfeld): Robert Lazarsfeld also gave an amazing talk highlighting the connected between the geometry of a curve and its syzygies. The general set up is very similar to that of Gavril Farkas’s talk (see Day 3). Let C be a smooth curve of genus g\geq2 and L a line bundle on C such that \deg(L)=d>>0. Now |L| defines an embedding of C into \mathbb{P}^{d-g}, and we will let S=\text{Sym} H^0(L) be the coordinate ring associated to C. Given another line bundle B on C we define R(B, L) to the S-module:

    R(B, L)=\bigoplus_{m\geq0} H^0(C, B+mL).

    If we let $B=K_c$ be the canonical canonical class it turns out R(K_C, L) is finitely generated as an S-module by H^0(C, K_C). In fact, the multiplication map \mu: H^0(C, K_c)\otimes H^0(C, L)\rightarrow H^0(C, K_C+L) is surjective. Taking the kernel of this map we get the following complex:

    S(-1)^N\rightarrow S^g\rightarrow R(K_C, L) \rightarrow 0.

    Thus, the question becomes: When is the above complex exact? Put differently when is the above a linear presentation for R(K_C, L)? It turns out that Mark Green gave an answer to this question:

    Theorem (Green): If C is a smooth projective curve of genus g and L is a line bundle on C such that \deg(L)>>0 then:

    S(-1)^N\rightarrow S^g\rightarrow R(K_C, L) \rightarrow 0.

    is exact if and only if C is not hyper-elliptic.

    The question Lazarsfeld went on to discuss when does this pattern continue? That is if E_{-}\rightarrow R(B, L)\rightarrow 0 is a free resolution of R(B, L) when can we guarantee that E_{p}=S(-p)^{n_p} for some range of $p$? With this question in mind we say that B satisfies M_l iff E_p=S(-p)^{n_p} for all 0\leq p \leq l.

    With all of this set up the main conjecture that Lazarsfeld discussed was the following:

    Conjecture: If C is a smooth projective curve of genus g and L is a line bundle on C such that \deg(L)>>0 then K_C fails M_l if and only if \text{gon}(C)\leq l+1.

    Recall that a curve C has gonality at least l if there is a degree l+1 map from C to \mathbb{P}^1. Thus, this conjecture explicitly connects the geometry of a curve to the its syzygies. In this light it has a very similar flavor to the Green-Lazarsfeld Secant Conjecture discussed in my post on Day 3.

    This conjecture turns out to be true!! In fact a stronger conjecture relating the vanishing of syzygies to the ampleness of the line bundle of interest is true!! However, this post is already getting long so I will spare righting about that. As Lazarsfeld explains it the proofs of these conjectures were hanging in plain sight, however, they only were noticed following Voisin’s work on Green’s Conjecture.

    Overall this talk was awesome, and I am really hoping to learn more about this area at some point in the near future. In fact I found it so exciting I purchased this book on Koszul cohomology in algebraic geometry. As a final note one subtle thing regarding all of these theorems is that the degree of the embedding line bundle needs to sufficiently positive. However, from the talk it wasn’t clear what sufficiently positive means. Is there an explicit bound above, which these theorems work, or is it just an eventually type statement?

Well that’s all for now. Stay tune for another update either Wednesday or Thursday!!