Monday, May 20, 2019

Subtropical Storm Andrea (2019)

Storm Active: May 20-21

As the third week of May began, a frontal boundary moved off of the U.S. east coast. The southern end of the front stalled north of Hispaniola and formed a trough of low pressure. There the system found a relatively favorable atmosphere and marginally warm ocean temperatures, supporting some scattered storm development. Before long, a low-pressure center had developed. By May 20, the storm had a small convective shield displaced to the north and east and aircraft reconnaissance measured gale-force winds. Since the circulation was still interacting with an upper-level low to its southwest and the gale force winds were spread out from the center, the storm was classified Subtropical Storm Andrea, the first named storm of the 2019 Atlantic hurricane season.

The system moved northward that day, but began to slow down and veer eastward by the the afternoon of May 21. Meanwhile, the convection associated with the system dissipated, leaving behind just a swirl of low-level clouds to mark the center of circulation. As a result, Andrea was downgraded to a subtropical depression. Early that evening, it degenerated further into a remnant low and these remnants dissipated the next day as a new front approached. Moisture that had been associated with Andrea brought some rain showers to Bermuda on the 22nd. The formation of Andrea marked the 5th consecutive year during which a named storm formed prior to the official start of the hurricane season on June 1, surpassing the record set in 1951-4. However, short-lived weak systems such as Andrea may very well have been missed prior to the era of satellite observation.



This image shows Subtropical Storm Andrea on May 20. Also visible is the upper-level low to its southwest which helped to weaken the system.



Andrea formed in the far western Atlantic, one of the typical areas for early season cyclogenesis.

Sunday, May 19, 2019

Professor Quibb's Picks – 2019

My personal prediction for the 2019 North Atlantic hurricane season (written May 19, 2019) is as follows:

15 cyclones attaining tropical depression status,
14 cyclones attaining tropical storm status,
6 cyclones attaining hurricane status, and
3 cyclones attaining major hurricane status.

Following a fairly average hurricane season in 2018 (which nevertheless featured two devastating major hurricanes), I predict that the 2019 season will see a comparable number of cyclones, albeit with rather different areas to watch. Note that the average Atlantic hurricane season (1981-2010 average) has 12.1 tropical storms, 6.4 hurricanes, and 2.7 major hurricanes. As with any season, our prediction begins with a look at the El Niño Southern Oscillation (ENSO) index, a measure of equatorial sea temperature anomalies in the Pacific ocean that have a well-documented impact on Atlantic hurricane activity. These anomalies are currently positive, corresponding to an El Niño state, and have been since last fall. The image below (click to enlarge) shows model predictions for the ENSO index through the remainder of 2019.



In comparison to the last several years, the situation is more static: no significant change of state is expected during this year's hurricane season (though there is, of course, significant uncertainty). This state of affairs tends to suppress hurricane activity and increase the chance of cyclones in the subtropical Atlantic curving away from the North American coastline (unlike, for example, the unusual track of Hurricane Florence last year).

This is fortunate, because all indications are the subtropical Atlantic will continue to churn out named storms as it did last season. Sea surface temperatures continue to run high in the region, and El Niño effects are not as pronounced there, partially explaining why my prediction still features an above average number of storms. Other factors also somewhat offset the El Niño: ocean temperatures in the tropical Atlantic (the birthplace of most long-track hurricanes) are slightly above normal this year, a trend expected to persist over the next several months. The atmosphere has also been less dry in the region, with less Saharan dry air than in 2018 and the beginning of the 2017 season to quash developing tropical waves. Expect the tropics to be less hostile to long-track hurricane formation than last year, when all cyclones taking the southerly route dissipated upon entering the Caribbean.

My estimated risks on a scale from 1 (least risk) to 5 (most risk) for different specific parts of the Atlantic are as follows:

U.S. East Coast: 3
Though the subtropical Atlantic will be active, I predict less of a risk to the U.S. coastline, with a smaller chance of a Florence-like system this year. Though there may be a few hurricanes passing offshore, most should recurve out over open water. Bermuda, however, is at higher risk.

Yucatan Peninsula and Central America: 2
These regions may benefit the most from a persistent El Niño, with wind shear making the development of an intense hurricane in the western Caribbean difficult. Further, I expect tracks to curve northward more often than striking Central America directly. Later season cyclones originating in the monsoonal gyre near Panama may pose the primary threat, and these tend to be principally rainmakers.

Caribbean Islands: 4
With the main development region (MDR) of the tropical Atlantic more favorable this year, the Caribbean is unlikely to continue the reprieve last year that followed arguably its worst season of all time (2017). Early season storms are still likely to fizzle out due to El Niño-related shear, but a wetter atmosphere suggests that tropical disturbances will have to be watched carefully. This includes a greater possibility of tropical cyclogensis in the Caribbean itself.

Gulf of Mexico: 3
Sea temperatures are consistently higher in the Gulf this year than they have been recently, especially near the Florida gulf coastline, but conditions here overall are a mixed bag. A strong jet stream across the continental U.S. will support more severe thunderstorms over land this summer, but this actually may work against cyclones thriving in the region. Balancing these factors yields an average risk, though this overall rating is a combination of a higher-than-normal risk in the eastern Gulf and a lower-than-normal risk farther west.

Overall, I expect the 2019 hurricane season to feature close-to-average activity. Nevertheless, this is just an informal forecast. Individuals in hurricane-prone areas should always have emergency measures in place. For more on hurricane safety sources, see here. Remember, devastating storms can occur even in otherwise quiet seasons.

Sources: https://www.cpc.ncep.noaa.gov/products/analysis_monitoring/lanina/enso_evolution-status-fcsts-web.pdf, https://www.cpc.ncep.noaa.gov/products/CFSv2/CFSv2seasonal.shtml, https://www.ospo.noaa.gov/Products/ocean/sst/anomaly/

Wednesday, May 15, 2019

Hurricane Names List – 2019

The name list for tropical cyclones forming in the North Atlantic basin for the year 2019 is as follows:

Andrea
Barry
Chantal
Dorian
Erin
Fernand
Gabrielle
Humberto
Imelda
Jerry
Karen
Lorenzo
Melissa
Nestor
Olga
Pablo
Rebekah
Sebastien
Tanya
Van
Wendy

This list is the same as the list for the 2013 season, with the exception of Imelda, which replaced the retired name Ingrid.

Tuesday, May 7, 2019

The abc Conjecture: Applications and Significance

This is the third part of a three-part post concerning the abc conjecture. For the first, see here.

The first post in this series presented some explanation as to why the abc conjecture seems like a reasonable attempt to mathematically codify a big idea. This idea is that the prime factorization of a sum of two numbers should not really relate to those of the individual numbers. Equivalently, it says that if we see an equation like 3 + 53 = 27, we should think of it as a "rare event" or "coincidence" that big powers of small primes are related in this way. The second post provided some examples and numerical evidence rigorous version of the conjecture. To review, this states that

The abc Conjecture: For any ε > 0, no matter how small, for all but finitely many equations of the form a + b = c where a and b are relatively prime, rad(abc)1 + ε > c.

Again, the radical rad(n) of an integer n is the product of its distinct prime factors. However, none of what has been discussed so far constitutes a mathematical proof that the abc conjecture is true or false.

In 2012, the Japanese mathematician Shinichi Mochizuki shocked the mathematical community by publishing, out of the blue, what he claimed was a proof of the abc conjecture. However, the initial excitement at this announcement was quickly replaced by confusion; almost no one was able to decipher the tools used in the proof, which totaled over 500 pages in length! Mochizuki, working in isolation for years, had built up a brand new mathematical formalism which he called "Inter-Universal Teichmüller Theory" that was bizarre and unfamiliar to other researchers. The language and notation (an sample of which is provided in the screenshot below) seemed alien, even to mathematicians!



Moreover, he refused to publicly lecture on the new material, instead only working with a few close colleagues. The combination of the length and inscrutability of the proof with his unwillingness to elucidate it discouraged people from attempting to understand it. In the years since the proof was published, skepticism has mounted concerning the proof's validity. While a small group of mathematicians defend it, a majority of the mathematical community thinks it is unlikely that the proof is valid. For now, the abc conjecture remains effectively open.

Nevertheless, it is certain that attempts to prove the conjecture will continue. It has a number of useful applications that would solve a myriad of other mathematical problems, should it be true. To illustrate the power of the abc conjecture, we give one famous example of an application: Fermat's Last Theorem.

One of the first equations we considered in this series was x2 + y2 = z2, which relates the side lengths of right triangles. This equation has infinitely many solutions, namely 32 + 42 = 52, 52 + 122 = 132, etc. Fermat's Last Theorem states that if we raise the exponents from 2 to any higher power, there are no solutions in the positive integers. That is, x3 + y3 = z3, x4 + y4 = z4, and so on are not satisfied by any x, y, and z > 0. Famously claimed by Pierre de Fermat in the 17th century, this problem remained unsolved for centuries. In 1985, when the abc conjecture was first stated, it remained open.

So let us assume that we have (somehow) proven the abc conjecture, and were interested in Fermat's Last Theorem. The first thing to note about the equation xn + yn = zn is that if we had a solution for this equation, we could always find one for which xn and yn were relatively prime. This is because if they have a common prime factor, so must zn, and we can cancel this factor (raised to the nth power) from both sides. Therefore, we have arrived at a situation in which we can apply the abc conjecture. The radical of xn, for any n, is at most x since multiplying x by itself does not introduce any more prime factors that were not already there. Hence rad(xnynzn) = rad(xn)rad(yn)rad(zn) ≤ xyz < z3. Therefore, for ε > 0, we have that

rad(xnynzn)1 + ε < (z3)1 + ε = z3 + 3ε.

On the other hand, applying the conjecture to this triple, we have that for ε > 0,

rad(xnynzn)1 + ε > zn

in all but finitely many cases. Since we can choose ε to be any positive number, we can make it small enough so that 3 + 3ε < 4 (e.g. if ε = 0.1). Then if n ≥ 4, the two inequalities above directly contradict each other. Since the top one always holds and the bottom holds in all but finitely many cases, we conclude that there can be at most finitely many exceptions to Fermat's Last Theorem when n ≥ 4.

So the abc conjecture does not quite imply Fermat's Last Theorem, but it comes very close. If, in addition, we knew just a bit more about how the exceptional abc triples behaved, we could manually verify that there are no counterexamples to Fermat's Last Theorem for n ≥ 4. Interestingly, this argument does not say anything about the n = 3 case, that is, about the non-existence of solutions to x3 + y3 = z3. This special case, however, had already been proven by Euler in the mid-1700s.

Of course, the abc conjecture remains unproven, while Fermat's Last Theorem was finally proven by Andrew Wiles in 1995. This was done by entirely different means. Nevertheless, this serves as a relatively simple example of how the conjecture can prove results about Diophantine equations without invoking very difficult mathematics. Another example of a consequence is the following statement, sometimes called Pillai's conjecture:

Conjecture: Every natural number k occurs only finitely many times as the difference of two perfect powers.

For example, the special case k = 1 is the subject of Catalan's conjecture, and states that xp - yq = 1 has only one solution: 32 - 23 = 1. This was proven by Preda Mihăilescu in 2002 (again by very different means from those above and from Wiles' methods), but the general case remains unsolved. If we knew for a fact that the abc conjecture were true, we would be able to prove this result by a very similar argument to the one given above for Fermat's Last Theorem (the reader is encouraged to try this!). Note that Pillai's conjecture also implies that the original equation that motivated the abc conjecture, namely y2 = x3 + k, also has only finitely many solutions (for fixed k). This is the result David Masser and Joseph Oesterlé sought on their way to first formulating the statement.

These examples start to indicate how important the abc conjecture is to the study of Diophantine equations; if it were proven, it would resolve many different problems that are currently treated separately in a single stroke. Even reproving known results in a new and simple way would be greatly beneficial to the theory, since a set of tools that could prove abc would help to unify disparate parts of number theory. As a result, mathematicians will doubtlessly continue work toward solving the conjecture and probing the most fundamental structure of numbers.

Sources: http://projectwordsworth.com/the-paradox-of-the-proof/, Shinichi Mochizuki: Inter-Universal Teichmüller Theory I: Construction of Hodge Theaters, http://mathworld.wolfram.com/PillaisConjecture.html, https://rlbenedetto.people.amherst.edu/talks/abc\_intro14.pdf, Brian Conrad: The abc Conjecture, 12 sep 2013.