Algebraic geometry is the study of solutions to polynomial equations, and it is one of the most classical but at the same time very active areas in mathematics. In particular, many surprising connections to other subjects have been established in recent years. In this course, I will give a concrete introduction to the basic ideas and concepts of algebraic geometry. I will do that by using as a thread the geometry of the maximum likelihood method, which is fundamental in modern statistics and machine learning, and appears in particle physics as well. An important part will be played by explicit computations with software. 

Prerequisites: calculus and linear algebra.

Spectral graph theory reveals the properties of a graph when it is represented as a matrix, e.g., its adjacency matrix or Laplacian matrix, whose characteristic polynomial, eigenvalues, and eigenvectors of these matrices can be studied to understand the graph. The increasing interest in graph signal processing and the ubiquity of graph neural networks have made spectral graph theory critical in applied mathematics and machine learning. In the Spectral Graph Methods and Machine Learning mini-course, we will introduce spectral graph theory through the basics of graph signal processing and demonstrate its natural connections with classical harmonic analysis. They will discuss current theoretical and applied questions. In particular, there will be computational experimentation, including graph image processing and graph neural networks. Finally, time permitting, we will introduce recent applications of graph signal processing to signal simplicial complexes, perhaps eliciting connections with Topological Data Analysis. 

Prerequisites: TBA

In this course, we shall describe basic mathematical tools which serve as foundations for quantum computations.

Prerequisites: TBA

The aim of this mini-course is to introduce topological data analysis (TDA) and its main algorithm, persistent homology (PH). PH is a computational tool used to extract a topology-based fingerprint from complex data, that relies on the computation of topological features such as connected components, loops, and voids. PH has been used with increasing frequency and success to quantify the structure of complex data. Among other applications, PH has enabled novel insights in cancer studies, pathology, evolution, ecology and material science. In the first half of the course, we will introduce the mathematical concepts of simplicial complexes and simplicial homology. We will then talk about how to construct a filtration of simplicial complexes from data, and we will introduce PH and its properties. Time permitting, we will look into PH computations and what we can learn from them, and we will discuss recent applications. 

Prerequisites: linear algebra.

Complex Analysis is the study of complex numbers and functions of complex numbers. It is a fertile area of pure mathematics which exhibits many interesting results. It is a source of powerful techniques which are widely applied in sciences.

The complex numbers can be identified with the points in the Euclidean plane. Here, we give basic definitions and theorems from plane topology that are useful in complex function theory. In particular, topology of the complex plane shall be discussed.