Renowned researchers in their fields, supported by the “Scientist and Young Scientist of Our State” programs from FAPERJ, present the concepts behind the studies they are developing

Como a matemática ajuda a precificar ações na bolsa de valores? O que ela tem a ver com campanhas de vacinação durante uma pandemia, ou com a forma como os computadores interpretam letras, sons e imagens? Para mostrar que a matemática está por trás de decisões que impactam a vida cotidiana, pesquisadores da Escola de Matemática Aplicada da Fundação Getulio Vargas (FGV EMAp) gravaram uma série de vídeos em que explicam conceitos matemáticos utilizados nas pesquisas que desenvolvem com o apoio da Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ). 

As bolsas, concedidas pela FAPERJ nas modalidades Cientistas e Jovens Cientistas do Nosso Estado, apoiam projetos de pesquisa coordenados por cientistas vinculados

Claudio Struchiner, Maria Soledad Aronna, Jorge Poco, Alberto Paccanaro, Roger Behling, and Yuri Saporito are FAPERJ scholarship holders | Photo: FGV EMAp

How does mathematics help determine stock prices on the stock market? What does it have to do with vaccination campaigns during a pandemic, or with the way computers interpret letters, sounds, and images? To show that mathematics is behind decisions that impact everyday life, researchers from the School of Applied Mathematics at Fundação Getulio Vargas (FGV EMAp) recorded a series of videos in which they explain the mathematical concepts used in the research they develop with support from the Carlos Chagas Filho Foundation for Research Support of the State of Rio de Janeiro (FAPERJ).

The grants, awarded by FAPERJ through the “Scientist and Young Scientist of Our State” programs, support research projects coordinated by scientists affiliated with institutions in the state of Rio de Janeiro. Below is a summary of the presentations:

Mathematics in the Financial Market

In the video, researcher Yuri Saporito presents an overview of how mathematics is present in virtually every aspect of the financial market, helping to formalize it and increase its effectiveness.

During the presentation, the FGV EMAp researcher demonstrates how optimization, statistics, probability, stochastic processes, and differential equations are used to solve problems such as asset pricing, the functioning of automated investment algorithms, and the structuring of stock portfolios. One of the highlights of the lecture is a detailed explanation of interest rate modeling, based on a simple concept: people prefer to consume today rather than in the future. This preference leads to the creation of interest as a form of compensation. The researcher explores how compound interest works and goes further into the idea of continuously compounded interest.

The stock market is also a central topic of the presentation, especially when Saporito explains the two most common types of orders: limit orders and market orders.

The modeling of stock prices and the study of their returns introduce the audience to Modern Portfolio Theory (MPT), developed by Harry Markowitz. The video shows, in a didactic way, how to calculate the expected risk and return of a portfolio with two assets, taking into account variance, covariance, and correlation.

To explain more advanced concepts, such as derivative pricing, Saporito uses a playful example: a casino roulette wheel. He compares a derivative to a ticket whose value depends on the outcome of the roulette spin. The “fair price” of this ticket is determined by a fundamental principle in finance: the no-arbitrage principle — in other words, the price must equal the cost of building a portfolio that replicates exactly the payout promised by the ticket.

Researcher Yuri Saporito used a roulette analogy to explain the concept of derivative pricing | Image: Screenshot

Decision-Making in Times of Crisis: Lockdowns, Vaccination, and Contact Tracing Researcher Claudio Struchiner demonstrates how mathematical models are powerful allies when it comes to shaping public policies, forecasting scenarios, and measuring the impact of actions such as lockdowns, vaccination, and contact tracing.

The video begins with an explanation of how epidemic outbreaks emerge: a new pathogen (such as a virus) comes into contact with a susceptible population, triggering the spread of contagion. Many factors influence this process, from the pathogen's origin and mode of transmission to human behavior, weather conditions, and individuals' immune responses.

A central concept is the basic reproduction number (R₀), which represents the average number of new infections caused by a single infected individual. The higher this number, the greater the disease's spreading potential. This value depends on variables such as the infectious period, transmission efficiency, and population susceptibility. However, Struchiner explains, not every epidemic follows a simple pattern. COVID-19, for example, showed complex dynamics, with seasonal trends and successive waves, requiring mathematical models capable of adapting to varying scenarios over time.

On the other hand, the FGV EMAp researcher notes that analyses related to how vaccines should be distributed across a population rely on more detailed models, since vaccines act in different ways: some block infection, others reduce transmission, and some only prevent severe symptoms.

The professor also explains how mathematics is used to evaluate the cost-benefit of interventions during an epidemic, helping to balance the impact of potential social distancing measures.

A chart presented during Professor Claudio Struchiner’s lecture shows how the SIR model describes the progression of an epidemic over time by dividing the population into susceptible (S), infected (I), and recovered (R) individuals | Image: Screenshot

The Mathematics Behind Epidemics Researcher Maria Soledad Aronna begins with a real case from London in 1854 to explain how mathematics became an essential tool for understanding, predicting, and controlling epidemics. At the time, a cholera outbreak had struck the British capital, and physician John Snow, using a map and Voronoi diagrams, was able to demonstrate how the disease was spreading: the cases were concentrated around the Broad Street water pump, indicating that cholera was transmitted through water.

The researcher also recalls the work of British doctor Ronald Ross, who, in the early 20th century, mathematically demonstrated how controlling the mosquito population could reduce the incidence of malaria. “Ross was one of the first to use compartmental models to describe the interaction between human hosts and vectors, and for that contribution, he received the Nobel Prize in Medicine,” says Aronna.

From these stories, the professor introduces the concept of a mathematical model: a simplified representation of reality capable of analyzing scenarios, testing hypotheses, and simulating intervention strategies. According to the researcher, one of the most well-known and widely used models in epidemiology is the SIR model, which divides the population into three groups — Susceptible, Infected, and Recovered — and describes the dynamics of a disease based on two key parameters: the transmission rate and the recovery rate.

“With this type of model, it’s possible to estimate how many people might get sick, calculate the epidemic’s peak, plan hospital bed usage, and even evaluate the impact of public policies such as school closures or insecticide use,” explains Aronna.

During the presentation, Aronna emphasizes how mathematical models remain crucial for guiding decisions in public health crises. They help assess the risk of outbreaks, simulate vaccination strategies (by age, region, or priority groups), calculate the spatial spread of diseases, and compare different control measures — from isolation to biological control, including actions like quarantine or vaccination. “More than just equations, these models represent decisions that save lives,” summarizes the professor.

The pattern revealed helped prove that the disease was transmitted through water, not air — a milestone in the history of epidemiology and the application of mathematics to public health | Image: Screenshot

Computer Science Is for Everyone In his presentation, researcher Jorge Poco introduces the world of Computer Science as a field that goes far beyond simply knowing how to use computers. It’s a discipline aimed at designing programs, software, and most importantly, solving real-world problems. “The applications are vast: from Artificial Intelligence to Data Science, from Cybersecurity to video games,” says the FGV EMAp professor. According to him, computing is an accessible and transformative science, rooted in a deep understanding of how computers process and represent information.

Poco explains how we communicate with computers and how they interpret numbers, letters, images, or sounds using sequences of just two digits — the binary system. Using examples like converting the decimal number 123 into binary, he illustrates how this system is based on powers of 2.

To allow computers to handle text (letters, punctuation, symbols, and emojis), encoding systems such as ASCII are used, assigning a number to each character (for instance, the letter "A" corresponds to the number 65). These numbers are then converted into binary. However, ASCII has limitations in representing characters from different languages or more complex graphic elements, which led to the creation of the Unicode system, capable of encompassing thousands of symbols from various cultures and languages.

Videos, in turn, rely on the RGB (Red, Green, Blue) system, where each color displayed on screen is formed by a combination of three numerical values indicating the intensity of red, green, and blue in each pixel. Thus, a digital image can be understood as a table of pixels, in which each cell holds a trio of numbers.

Another key concept presented in the lecture is that of algorithms. According to the researcher, algorithms are used for everything: organizing data, performing calculations, recognizing patterns, and making decisions. He explains that an algorithm is a sequence of well-defined instructions to solve a problem. To illustrate this idea, he uses the metaphor of a cake recipe — which, like an algorithm, requires following specific steps in a precise order.

The professor also provides an example using a phone book search. A linear search, though simpler, is less efficient than a binary search, which is more complex but significantly faster. This highlights the importance of algorithm efficiency and its impact on the time required to solve problems — especially as the problem size increases.

Artificial intelligence, data science, cybersecurity, and gaming are key areas of interest in Computer Science | Image: Screenshot

A Mathematical Application in Image Contraction Researcher Roger Behling invites us to explore Banach’s Fixed Point Theorem, a concept typically taught in the final years of a mathematics degree. However, Behling introduces it to the public using a simple example: what happens when you place a reduced version of an image on top of the original one?

According to Behling, Banach’s Theorem states that a point remains in the same position even after a transformation is applied. To make this abstract concept more accessible, the researcher explains that if you take a picture and create a smaller version of it — not necessarily proportionally scaled — and then overlay the reduced image on top of the original, there will be at least one common point between the two.

Building on this idea, the researcher explains how the transformation can be described mathematically using linear functions on a Cartesian plane. The fixed point can then be found algorithmically, by repeatedly applying the transformation until the image “settles” on a single point.

At the end of the presentation, the researcher emphasizes that this visual experience makes a deep mathematical result more accessible. “Even if the transformation isn’t perfectly proportional, the fixed point will be there, hidden within the layers of the image — and mathematics guarantees its existence,” he states.

Researcher Roger Behling illustrates Banach’s Fixed Point Theorem using overlapping images | Image: Screenshot

Artificial Intelligence: How Computers Learn Researcher Alberto Paccanaro guides us through a lecture on artificial intelligence (AI), a field that has been studied since the 1960s.

According to the researcher, early approaches to AI focused on logical reasoning — machines that applied “if-then” rules to simulate experts in fields such as medicine or technical diagnostics. Over time, it became clear that this strategy was limited, especially for tasks that humans perform intuitively, like recognizing faces, understanding voices, or distinguishing handwritten numbers. That’s when, according to Paccanaro, AI began drawing inspiration from the human brain, giving rise to artificial neural networks. Machine learning occurs when connections are optimized based on examples: the system receives an input, compares it to the expected output, and adjusts the weights to reduce future errors — a process known as supervised learning.

“Mathematics is the foundation that allows us to describe, train, and improve these models, while computer science provides the tools to implement and deploy them,” the researcher emphasizes.

During the lecture, Paccanaro references the film WarGames (1983) as a fictional portrayal of a computer that "learns" on its own. The FGV EMAp professor also demonstrates current applications of AI — in self-driving cars, machine translation systems, soccer-playing robots, and virtual assistants.

Sharing his own journey as a researcher, Paccanaro explains that machine learning is being used to solve complex problems in biology, medicine, and pharmacology — from drug discovery to understanding genetic networks.

For the professor, the future is promising for those who want to work with AI, but he emphasizes that the field's rapid advancement brings important ethical concerns. “This includes worries about biases in the data that can lead to skewed outcomes in AI systems,” Paccanaro notes.

A slide presented by Professor Alberto Paccanaro illustrates the concept of “Learning Capacity” in artificial neural networks, highlighting the inspiration drawn from biological neurons | Image: Screenshot