The understatement of the century: Mathematician Dennis Gaitsgory has connected two distant fields of his discipline. For this, he is now receiving one of the most highly endowed scientific prizes.

"Fear will make the local systems submissive." So spoke Grand Moff Tarkin, the commander of the Death Star, in the 38th minute of the first "Star Wars" film. What does this sentence, in its original English, have to do with a mathematical article? Is it a nerdy "Abandon all hope" to all those who know nothing about category theory and feel at home in topological spaces?

 

Indeed, without such knowledge, it is impossible to read what Dennis Gaitsgory presented last year together with Sam Raskin of Yale University. It is nevertheless spectacular.

 

The two mathematicians thus concluded a five-part series with a total of more than 900 printed pages proving a theorem called the "Geometric Langlands Conjecture."

 

Gaitsgory, Director of the Max Planck Institute for Mathematics in Bonn since 2021, had already worked on the problem as a student in Tel Aviv. In the end, he led a nine-person team, together with his former doctoral student Raskin, they finally succeeded in proving the hypothesis. For this, Gaitsgory was awarded the Breakthrough Prize for Mathematics, endowed with a whopping three million dollars, last night in Los Angeles – and Raskin was awarded one of the six "New Horizons" prizes for young mathematicians. The Breakthrough Prizes – there are also ones for life sciences and physics – are sponsored by tech billionaires, and their presentation is modeled after the Oscars. The idea is to create publicity. But is that even possible in the case of the geometric Langlands conjecture?

 

But perhaps, with some scientific advances, it is ultimately inevitable. Anyone who takes note of news from genetic research or astrophysics today can do so because, even without the ability to read specialist literature, they have an inkling of the molecular basis of life or the possibility of a curvature of space, concepts that were no less alien to early modern people than the geometric Langlands conjecture is to most science enthusiasts today.

 

The award-winning theorem is part of a broader project, the so-called Langlands Program, one aspect of which already attracted public attention.

 

That was in 1995, when the British scientist Andrew Wiles proved Fermat's conjecture, which states that the equation x + y = z for integers x, y, and z to the power of integer n can only be fulfilled if the exponent n is greater than 2.

 

What Wiles actually proved back then was a conjecture, the correctness of which implies Fermat's conjecture. It was first proposed in 1957 by the Japanese Yutaka Taniyama and Goro Shimura, as well as the Frenchman André Weil – and it leads to something astonishing: If you consider a specific equation and insert integers with certain properties, you can determine the number of cases in which the equation holds true using a function that is familiar to so-called Fourier analysis.

 

Thus, the conjecture of Taniyama, Shimura, and Weil asserted the existence of a connection between two completely different fields of mathematics: on the one hand, number theory, which has been concerned with integers and their building blocks, prime numbers, since antiquity; and on the other, the realm of continuous functions and their decomposition into sine and cosine waves of different frequencies and amplitudes, which the French mathematician Joseph Fourier described in 1822.

 

Is there something more general behind this? This was the question Robert Langlands, a Canadian native who was then working at Princeton, asked himself in 1967, wondering whether difficult problems in number theory could be translated into less difficult problems in Fourier analysis.

 

He outlined a search for such relationships in a letter to André Weil. He wrote: "I would be happy if you would like to read this as pure speculation; if not, I'm sure you have a wastebasket handy."

 

"That was the understatement of the century," Berkeley mathematician Edward Frenkel once commented on this note. Indeed, the Langlands program developed into a veritable mathematical quest for the Holy Grail in the 1970s. "The entire number theory group in Tel Aviv was working on the original Langlands program," Dennis Gaitsgory recalled in an interview with the Frankfurter Allgemeine Zeitung about his first year as a doctoral student. "But I only heard about the geometric Langlands conjecture when Alexander Beilinson, who now teaches in Chicago, gave a lecture to us. I was overwhelmed. It was so beautiful."

 

André Weil had already had an inkling of a connection to geometry in 1940. He had seen analogies between three different materials. Thematic disciplines, including number theory and geometry, have noticed this. The latter, however, refers less to the theory of points, lines, and surfaces in ordinary space that can be described with real coordinates, but rather to so-called Riemann surfaces, which are based on an extended concept of numbers, that of complex numbers. Beilinson and his Chicago colleague Vladimir Drinfeld, among others, began searching in the 1980s for a correspondence between Riemann surfaces and a corresponding analogue to Fourier analysis. Gaitsgory and his team have now shown, in a very general way, that such a correspondence exists and what exactly it consists of.

 

Just as number theory is related to Fourier analysis in certain cases, such as the Taniyama-Shimura-Weil theorem, the geometry of Riemann surfaces is also related to something in which objects can be decomposed into their components. "But that has nothing to do with Fourier anymore," explains Gaitsgory. "It's merely an analogy to this in algebra." The role of sine and cosine waves is thus assumed here by special examples from a class of objects that mathematicians call "sheaves." Only with the proof that the sheaves in question also possess the necessary properties to fulfill this task was the harvest, so to speak, reaped.

 

But with the now presented proof, Robert Langland's vision is by no means exhausted, not even for geometry. "What we considered was the so-called global unramified case," explains Gaitsgory. "There is also the ramified case. Then you have to manage to combine the two, and only then is the geometric Langlands theory complete."

 

In addition, so-called local Langlands relationships between small disks around points on Riemann surfaces and objects from number theory, the p-adic numbers [A], are being researched. Gaitsgory's institute colleague Peter Scholze, together with a French colleague, made important progress here in 2021. Scholze was supposed to receive a "New Horizons" Breakthrough Prize in 2016, but declined. Two years later, he received the Fields Medal, the highest award in mathematics, for his theory of a geometry over p-adic numbers. Furthermore, there are efforts to develop Langlands correspondences leading to quantum field theories in physics.

 

When the question arises about the practical use of his proof, however, Gaitsgory shakes his head. "That's pure mathematics," he says. If one wants to speak of utility, then it consists at most in methods developed along the way. These might be useful elsewhere, he says, but hardly directly in solving one of the great puzzles. "I don't think the proof of the geometric Langlands conjecture helps in any way with the Riemann hypothesis." A "grand unified theory of mathematics" is by no means the goal of the Langlands program either. "I don't know where that comes from," Gaitsgory comments on this phrase, which is popular in media reports. After all, the point isn't to find a mathematics that is somehow more fundamental than, say, geometry or number theory, but rather to establish connections between them, or rather to suitable analogues to Fourier analysis. Gaitsgory prefers the image of bridges being built between different continents of mathematics, connecting one to the other and back again. These bridge-building projects are awe-inspiring enough.

 

But they aren't meant to inspire fear, only to invite wonder at their potential. So why was Grand Moff Tarkin involved? "When we were working on the paper, we had named several chapters after various 'Star Wars' episodes," Gaitsgory says. "Just kidding, because I was watching 'Star Wars' with my younger son at the time." The Tarkin quote, however, does have a meaning: The central focus of this final paper, he says, was the proof of a specific equality statement. "We proved this with a trick, and it involved the behavior of a specific mathematical object, the so-called space of local systems. It was made compliant, that is, restricted so that this equality no longer had any choice. It had to be true." ULF VON RAUCHHAUPT” [B]

 

A. P-adic numbers are a system of numbers, each associated with a prime number, that extend the rational numbers. They are like the real numbers but with a different notion of "closeness" determined by the chosen prime number. Essentially, p-adic numbers are written as infinite series of digits in base p, extending to the left instead of the right like decimals.

 

1. Extension of Rational Numbers:

 

    p-adic numbers build upon the rational numbers (fractions).

    For each prime number 'p', there's a distinct field of p-adic numbers, denoted as Qp.

 

2. Different Notion of Closeness:

 

    In p-adic numbers, two numbers are considered close if their digits agree for a long way to the left (in base p).

    This contrasts with the real numbers, where closeness is based on how close the numbers are on a number line.

    For example, in 5-adic numbers, 25 is considered closer to 0 than to 10 because 25 = 5 * 5, and 5 is a "small" number in this system.

 

3. p-adic Representation:

 

    p-adic numbers are written as infinite series of the form: a₀ + a₁p + a₂p² + a₃p³ + ... , where aᵢ are integers from 0 to p-1.

 

The "digits" are to the left of the "decimal point," which is defined by the powers of p.

 

4. Applications:

 

    P-adic numbers have applications in number theory, algebra, and analysis.

    They can be used to solve problems that are difficult or impossible to solve with real numbers.

    Examples include Hensel's lemma for finding roots of polynomials and Mahler's theorem.

 

5. Key Differences from Real Numbers:

 

    P-adic numbers are not ordered like real numbers.

    They are a non-Archimedean field, meaning that they don't have the same properties as the real numbers.

 

In simpler terms:

Imagine you're looking at numbers in a different way. Instead of focusing on how close numbers are to each other on a number line (like we do with real numbers), you focus on how many digits they share in common when written in base 'p'. This leads to a different kind of "closeness" and a different way of understanding the world of numbers.

B. Das Understatement des Jahrhunderts: Der Mathematiker Dennis Gaitsgory hat zwei weit entfernte Felder seines Fachs verbunden. Dafür erhält er jetzt einen der höchstdotierten Wissenschaftspreise. Frankfurter Allgemeine Zeitung; Frankfurt. 07 Apr 2025: 12. 

Ar Zelenskio režimas Ukrainoje galėtų subyrėti per porą dienų?

 

Žinoma. Sudėtingos sistemos subyra palaipsniui, o paskui - staiga (Hemingway'aus žodžiais tariant). Geras pavyzdys – katastrofa Afganistane, valdant JAV prezidentui Bidenui, kai valdžią staiga perėmė Talibanas. Ar ES karo kurstytojai tai supranta ir blokuoja taiką, kad sukeltų problemų D. Trumpui, kai Zelenskį chaose nužudytų koks nors dronas? Gali būti.

Could Zelensky's regime in Ukraine disintegrate in a couple days?

 Of course. Complicated systems disintegrate gradually, then suddenly (Hemingway's phrase). A good example is the catastrophe in Afghanistan under US president Biden, with sudden takeover by Taliban. Do EU warmongers understand this and block peace to make trouble for D. Trump with Zelensky killed in chaos by some drone? It could be.

Analyst on tax reform: it's a scam and confiscation of a third of your assets - how it will happen

 

"When it comes to the proposed tax reform, the public’s attention is usually limited to higher taxes on real estate and people who earn a lot of money.

 

However, practically nothing is said about the fact that if the current model is adopted, everyone may have to pay huge amounts. For example, what will happen when you sell inherited real estate and a third of the amount will have to be paid to the state as personal income tax (GPM) on extremely high income.

 

About this in the show "Verslo požiūris" with Šarūnas Andriukaitis - Sutkus, risk analyst, and Gintaras Verbickaite, head of the "Vienaragiai LT" association.

 

"When talking about tax reform, we cannot look only at wages, because the changes will affect all income. After selling assets, successfully earned from investments, creating additional income from individual activities in addition to hired work, it is very easy to exceed the maximum taxable amount. The important thing is that income generation per year is assessed, but how long it took to generate that income is not assessed. For example, waiting for shares to rise in price. This will affect many more people than the 40 thousand that politicians are talking about," G. Verbickaitė argues and says that business is not against contributing to defense. The corporate tax rate for business was already increased by one percent in the fall, and it is planned to increase it further. No one objects to this. However, the general package of tax changes is causing discussions.

 

“Tax changes are important for business, because they are directly related to the best employees and managers. We will punish the most qualified people who have international experience. These changes will make Lithuania unattractive and uncompetitive,” says G. Verbickaitė and predicts that our country is the second largest in the EU in terms of population taxation. The tax burden on employees is already huge. It is also necessary to consider the fact that additional benefits, such as health insurance for employees, will also be taxed. Due to the taxation of individual activities, everyone will also feel the increase in the cost of services.

 

“The reform should be called what it is. It is a fraud and the confiscation of a third of your assets.

 

I can’t find another word. It is covered up under the guise of solidarity and defense, although penies will be shed for defense, and almost everyone will give up a third of their assets. One example is inherited real estate. Small and medium-sized businesses that do not use various tax avoidance schemes will suffer very badly. The richest thousand people will not feel it, because they have already put together the necessary schemes,” says Š. Andriukaitis-Sutkus.

 

The analyst also gives an example. A person created a small business in the region with an authorized capital of 100 thousand euros. Over fifteen years, he managed to accumulate a profit of 1.5 million euros and is selling the company for 3 million. According to the current scheme, income will be assessed as follows: 3 million minus 100 thousand euros, which is formally the acquisition value of the shares. Taxes 36 percent or a million euros.

 

“A normal question arises, why do I need it? Why invest and create value, if you can simply withdraw dividends and pay 15 percent.

 

Private owners will disappear from Lithuania, who will establish formal holdings somewhere in Luxembourg and thus avoid taxes,” explains Š. Andriukaitis-Sutkus and says that this is the problem, that no tax problems are being solved and all the loopholes are left. For example, farmers, real estate developers. Nothing changes for them. The same for investors with larger investment accounts when money is withdrawn.

 

“The taxation that has existed so far has not been ideal, but at least it was average in terms of size in the EU. In addition, when 15 percent is left for dividends, it will stop any investments, especially in regions where there are many owners who already want to sell the businesses they have created,” explains Š. Andriukaitis-Sutkus.

 

G. Verbickaite assures that experience shows that the higher the taxes, the more complicated they are, which only increases the shadow that Lithuania has already managed to reduce.

 

Š. Andriukaitis-Sutkus says that this will affect not only entrepreneurs, but also all people who create and inherit property.

 

The analyst says that we are a frontline state and we need to attract funds for the defense structure, which we will build with borrowed funds, and for servicing. Why should foreigners invest in us, who compete for workers in the global market, and here they cannot pay normally, offer shares, from which they can also earn money.

 

"People forget that our demographic situation is one of the worst in the EU, and now we want to take everything away from people who can create more, in order to divide it among who is not clear. We need to start tax reform from a different angle. At the beginning, we need to reduce the number of activities and equalize rates. They can be not only increased, but also reduced. Where is the logic that a person who is engaged in individual activity will pay 36 percent, and a business 29 percent? Especially since some activities are such that the law does not allow them to be converted into a company."

Analitikas apie mokesčių reformą: tai apgavystė ir trečdalio jūsų turto konfiskacija – kaip tai nutiks

"Kalbant apie siūlomą mokesčių reformą viešumoje paprastai apsiribojama didesniais mokesčiais nekilnojamam turtui ir ypatingai daug uždirbantiems žmonėms.

Tačiau praktiškai nieko nekalbama apie tai, kad jei bus priimtas dabartinis modelis, tai didžiules sumas gali tekti sumokėti kiekvienam. Pavyzdžiui, kas bus kai parduosite paveldėtą nekilnojamąjį turtą ir trečdalį sumos reikės sumokėti valstybei kaip GPM nuo ypatingai didelių pajamų.

Apie tai laidoje „Verslo požiūris“ su Šarūnu Andriukaičiu – Sutkumi, rizikos analitiku ir Gintare Verbickaite, „Vienaragiai LT“ asociacijos vadove.

„Kalbėdami apie mokesčių reformą negalime žiūrėti vien tik į darbo užmokestį, nes pakeitimai palies visas pajamas. Pardavus turtą, sėkmingai uždirbus iš investicijų, sukūrus papildomas pajamas iš individualios veiklos šalia samdomo darbo labai nesunku ribas peržengti didžiausią apmokestinamą sumą. Svarbu tai, kad vertinamas pajamų gavimas per metus, bet nevertinama, kiek laiko užtruko tas pajamas sugeneruoti. Pavyzdžiui, laukti kol akcijos pabrangs. Tai palies kur kas daugiau žmonių nei 40 tūkst., apie kuriuos kalba politikai“, – dėlioja G. Verbickaitė ir sako, kad verslas nieko prieš prisidėti prie Gynybos. Verslui jau rudenį procentu padidintas pelno mokesčio tarifas, planuojama jį dar didinti. Dėl to niekas neprieštarauja. Tačiau diskusijų kelia bendras mokesčių pakeitimų paketas.

„Verslui svarbu mokesčių pakeitimas, nes tai tiesiogiai susiję su geriausiais darbuotojais ir vadovais. Bausime kvalifikuočiausius žmones, kurie turi tarptautinę patirtį. Šie pakeitimai darys Lietuvą nepatrauklia ir nekonkurencinga“, – sako G. Verbickaitė ir prognozuoja, kad mūsų šalis pagal gyventojų apmokestinimą yra antra pagal dydį ES. Mokestinė našta darbuotojams ir taip jau didžiulė. Reikia įvertinti ir tai, kad bus apmokestinamos ir papildomos naudos, tokios kaip sveikatos draudimas darbuotojams. Dėl individualios veiklos apmokestinimo visi pajus ir paslaugų brangimą.

„Reformą reikia pavadinti tuo, kuo ji yra. Tai apgavystė ir trečdalio jūsų turto konfiskacija. Kito žodžio nerandu. Prisidengiama solidarumo ir gynybos skraiste, nors gynybai nubyrės ašaros, o trečdalį turto atiduosite beveik visi. Vienas iš pavyzdžių paveldėtas nekilnojamas turtas. Labai skaudžiai nukentės smulkus ir vidutinis verslas, kurie nesinaudoja įvairiomis mokestinio apėjimo schemos. Tūkstantis turtingiausių žmonių to nepajus, nes jau susidėlioją reikalingas schemas“, – dėsto Š. Andriukaitis-Sutkus.

Analitikas pateikia ir pavyzdį. Žmogus sukūrė regione mažą verslą, kurio įstatinis kapitalas 100 tūkst. eurų. Per penkiolika metų pavyko sukaupti pelno už 1,5 mln. eurų ir įmonę parduoda už 3 milijonus. Pagal dabartinę schemą, pajamos bus vertinamos taip: 3 milijonai minus 100 tūkst. eurų, kiek formaliai yra akcijų įsigijimo vertė. Mokesčiai 36 proc. arba milijonas eurų.

„Kyla normalus klausimas, kam man to reikia. Kam investuoti ir kurti vertę, jei galima tiesiog išsiiminėti dividendais ir mokėti po 15 proc. Lietuvoje išnyks privatūs savininkai, kurie steigs formalius holdingus kur nors Liuksemburge ir taip išvengs mokesčių“, – aiškina Š. Andriukaitis-Sutkus ir sako, jog tai ir yra problema, kad nesprendžiamos jokios mokestinės problemos ir paliekamos visos landos. Pavyzdžiui, ūkininkai, nekilnojamojo turto vystytojams. Jiems niekas nesikeičia. Tas pats investuotojams, turintiems didesnes investicines sąskaitas kai bus išimami pinigai.

„Iki šiol buvęs apmokestinimas nebuvo idealus, tačiau bent jau buvo vidutinis pagal dydį ES. Be to, kai paliekama 15 proc. dividendams, tai sustabdys bet kokias investicijas ir ypatingai į regionus, kur daug savininkų, jau norinčių parduoti kurtus verslus“, – aiškina Š. Andriukaitis-Sutkus.

G. Verbickaite tikina, kad patirtis rodo, kad tuo didesni mokesčiai ir jie sudėtingesni, tai tik didina šešėlį, kurį Lietuvoje jau buvo pavykę sumažinti.

Š. Andriukaitis-Sutkus sako, kad tai palies ne tik verslininkus, bet ir visus žmones, kurie kuria, paveldi turtą.

Analitikas sako, kad mes esame pafrontės valstybė ir turime pritraukti lėšų gynybos struktūrai, kurią pasistatysime iš skolintų lėšų, aptarnavimui. Kodėl pas mus turėtų investuoti užsieniečiai, kurie dėl darbuotojų konkuruoja globalioje rinkoje, o čia negali normaliai mokėti, siūlyti akcijų, iš kurių prieaugio irgi gali užsidirbti.

„Žmonės pamiršta, kad mūsų demografinė situacija viena prasčiausių ES ir dabar norime iš žmonių, kurie gali daugiau sukurti, viską atimti, kad padalintume neaišku kam. Mums mokesčių reformą reikia pradėti ne nuo to galo. Pradžioje reikia sumažinti veiklų skaičių, sulyginti tarifus. Juos galima ne tik didinti, bet ir mažinti. Kur logika, kad žmogus, kuris versis individualia veikla mokės 36 proc., o verslas 29 proc.? Tuo labiau, kad dalis veiklų tokios, kad jų keisti į UAB neleidžia įstatymas. Pasikartosiu, tai tiesiog trečdalio turto konfiskavimas“, – dėlioja Š. Andriukaitis-Sutkus.

Analitikas sako, kad dar vienas mitas, jog Lietuva šalis, kuris per biudžetą perskirsto vieną mažiausių BVP dalį. Realiai mes atitinkame EBPO vidurkį – 41 proc. Izraelis iki karo Gazoje perskirstydamas 38 proc. sugebėjo gynybai duoti 5,5 proc., švietimui 8 proc. Mes atitinkamai 3 ir 5 proc.

„Akivaizdu, kad vietos efektyvumui, kaip tinkamai naudoti lėšas, turime daugybę“, – dėsto Š. Andriukaitis-Sutkus ir sako, kad pradžioje reikia užkaišyti daugybę skylių. Pavyzdžiui, žemės ūkio bendrovės, verslo liudijimai. Daugybė veiklų, kurias reikia naikinti ir kitas niuansas – sulyginti pajamų tarifus. Jei Individualiai veiklai 10 proc. per mažai, bet jų negalima staiga kelti iki 36 proc. Viešajame sektoriuje 25 proc. daugiau darbuotojų nei vidutiniškai EBPO.

„Pas mus labai daug pagaminta įvairių lengvatų, kurios atsirado įgyvendinant rinkiminius pažadus“, – dėlioja Š. Andriukaitis-Sutkus.”