**Title**: Decision problems in Algebra

**Abstract:** In many situations, it is possible to provide algorithms that decide whether or not a given problem has a solution. We will present some historically important decision problems, discuss the existence of such algorithms, and present a general algorithm that comes across in several distinct areas of algebra.

**Professor**: António Malheiro, DM-FCT-UNL

**Title**: Statistics and Data Visualization in Big Data Era

**Abstract**: The expression Big Data can generically be understood as referring to large volume sets of data, possibly greatly varying in type and/or with a quick production flow, such that the traditional applications for data processing are no longer suitable. This type of data and the concepts built around them are transforming the information processing and analysis procedures in all scientific domains, e.g., from finance engineering to biomedical sciences.

**Session 1 : **Introduction to Big Data and Data Visualization.

**Professor**:João Moura Pires, DI-FCT-NOVA

Let’s talk about what is and is not Big Data: how do we arrive here and why is it different from before? What are the most promising opportunities and some of the most important challenges? What is data visualization and what is its main role in the new world of Big and Small Data? Foundations of (Interactive) Data Visualization and why it is important to study and understand Data Visualization. Few examples of modern Data Visualization.

**Session 2**: Turning Big Data into Small Data

**Professor**: Regina Bispo, DM-FCT-NOVA

The complexity and/or the dimension associated with Big Data usually makes dimensionality reduction techniques a necessary tool before any other additional statistical procedure is performed. Principal Component Analysis (PCA) has therefore been increasingly used in the context of multivariate statistical analysis and reduction of unsupervised dimensionality of Big Data, with the main objective of finding a smaller subspace which retains as much information as possible.

**Session 3**: Bayesian Statistics and Machine Learning

**Professor**: Isabel Natário, DM-FCT-NOVA

In machine learning, methods and algorithms that use past data about a system are used to learn about that system, in an automatic way. That is crucial to extract useful information from large data sets. Machine learning is based on the use of statistical models that make predictions about the system in the future, being the Bayesian statistics paradigm particularly adequate for this purpose, due to its sequential and adaptive genesis that allows the incorporation of the new data that is being made available.

**Title**: An Automatic Classification Model

**Abstract**: Artificial Intelligence (AI) is a buzzword nowadays but about the contribution of Mathematics to this area of knowledge little is said. Many of models and methods behind AI algorithms are mathematical. In this module we will talk about a classification model in Machine Learning.

**Session 1**: Introduction to optimization

In this first session we will present some optimization notions. We will try to introduce the concepts using simple examples first that will help to understand more formal definitions.

**Session 2**: Support Vector Machine

We will describe this well-known method for data classification. Although it involves simple concepts, it has shown to be very efficient in practice.

**Session 3**: Classification for a set of real data

Using Matlab or Octave (freeware) we will work with a set of real data and apply the concepts learned in the previous sessions.

**Professors, students and alumni involved**:

Paula Amaral (DM-FCT-NOVA)

Paula Couto (DM-FCT-NOVA)

Rui Rodrigues (DM-FCT-NOVA)

Rui Malha (PhD Student in Mathematics, DM-FCT-NOVA)

Gonçalo Arsénio (Master student, DEE-FCT-NOVA)

**Title**: What’s an Actuary?

**Abstract:** "What is an actuary?" - this is a question that all actuaries had to answer somewhere in their lifetime. In very general terms, an actuary is a mathematician, a statistician, and a risk analyst. In this course, we will cover the general aspects of the actuarial profession, and discuss some possible career opportunities. In addition, we will teach you how to "get your hands on data " and, using the Portuguese official Mortality Table, calculate death and survival probabilities for different ages and terms.

**Professors:** Gracinda Rita Guerreiro (DM-FCT-UNL), Lourdes Afonso (DM-FCT-UNL), Pedro Corte Real (DM-FCT-UNL)

**Title**: One Introduction to commutative Banach algebras

**Abstract:** This mini-course is a short introduction to the theory of commutative Banach algebras. We start with the definition of a unital Banach algebra, further we discuss the notion of the spectrum of an element of a Banach algebra, and study basic properties of maximal ideals and multiplicative linear functionals in commutative Banach algebras. The course culminates with one of the gems of Mathematics of XX century - the Gelfand theory of commutative Banach algebras. We illustrate its power by presenting a short proof given by Israel Gelfand (1941) of the "1/f theorem" of Norbert Wiener (1932)..

**Professor**: Oleksiy Karlovych (DM-FCT-UNL)

**Title**: Analysis and Mathematical Finance

**Session 1**: A general overview on stock price models, option pricing and optimization problems in Finance.

**Professor**: Fernanda Cipriano, DM-FCT-NOVA

**Abstract:** We briefly describe the modern stock price models in Finance, which correspond to solutions of linear stochastic differential equations driven by Wiener or Lévy processes. We introduce the notion of portfolio and wealth process. Using the replication strategy, we deduce the theoretical price of European, Barrier, Asian and American options. Given a general utility function for investors, we formalize the portfolio problem, as an optimal control problem.

**Session 2**: Statistical methods for Financial models.

**Professor**: Pedro Mota, DM-FCT-NOVA

**Abstract:** In this course we discuss several continuous time stochastic processes that are important for financial data modelling. Since the data analysis is a fundamental tool in the improvement of the financial models, we focus in some statistical concepts and methods needed to deal with the data and calibrate the models.

**Session 3**: Numerical methods for option pricing problems

**Professor**: Nuno Martins, DM-FCT-NOVA

**Abstract: **Options are a contract between two parties about trading assets at a certain future time, this trading often involves the computation of a fair price of the underlying assets premium (market price). Usually, these computations are based on sophisticated models that can only be solved numerically. For instance, to price an American put option, we are asked to solve a free boundary partial differential equation. In recent years, fast and accurate numerical methods have become essential tools for option pricing and more generally for mathematical finance. In this presentation, we will discuss some numerical methods for pricing two types of options: European and American.

**Session 4**: Measures of risk management

**Professor**: Marta Faias, DM-FCT-NOVA

**Abstract: **Decision-making under risk is one of today's greatest challenges and the financial markets provide one of the contexts where risk is particularly crucial. In this course we first give an overview of the different types of financial risk, like credit risk or market risk and then we present the modern risk measures and discuss the mathematical tools involved in its conception.

**Session 5**: Portfolio optimization

**Professor**: Diogo Pereira, student at DM-FCT-NOVA

**Abstract: **The portfolio optimization is one of the fundamental problems in Mathematical Finance. In this course, we focus on the Robert Merton (Nobel prize in 1997 with M. Scholes) portfolio optimization problem for the Black-Scholes model, maximizing utility from terminal wealth. In addition, we discuss some limitations of the Black-Scholes model to describe accurately the reality of financial markets. For instance, it doesn’t capture the volatility smile and skew feature of implied volatilities. In order to solve these problems, we allow the volatility of the stocks to be stochastic, then we address the optimal portfolio problem for the so-called stochastic volatility models.