Antonio Lerario-"How many zeroes of a random polynomial are real? A probabilistic approach to real algebraic geometry" Room: Room 005 In this talk I will discuss a probabilistic approach to some questions in classical real algebraic geometry, like the study of the number of real  zeros of real polynomial systems, or the study of the number and relative position of the connected components of a plane real algebraic curve. The main idea is to insist on the measure theoretic flavor of the notion of “generic” from complex algebraic geometry, by turning the space of polynomials into a probability space and studying the expectation of such topological quantities.

Andrea Cangiani-"Reliable Numerical Modelling with Partial Differential Equations" Room: Room 005 Mathematics is increasingly becoming an indispensable tool for understanding complex phenomena, ranging from the regeneration of damaged bone tissue to the drift of Earth’s continents. Thanks to modern algorithms and the growing availability of computational power, it is now possible to perform highly accurate numerical simulations of realistic models describing such processes. In this talk, I will survey recent advances in the numerical analysis of partial differential equation based models, with a particular emphasis on techniques that ensure the reliability of computations, producingnumerical solutions equipped with rigorous guarantees on their accuracy.

Barbara Fantechi Room: Room 005

Felix Rott- "Alexandrov geometry and nonsmooth Lorentzian geometry" Room: Room 005 We give a beginner friendly introduction to Alexandrov geometry. We recall some basic concepts from Riemannian geometry and give an idea of how they can be abstracted to a setting beyond manifolds. We state Toponogov's Theorem on triangle comparison and the associated globalisation theorems. We conclude with a brief outlook on how this successful theory can be applied in Lorentzian signature.

Davide Riccobelli-"Continuum Mechanics @ SISSA mathLab" Room: 7th Floor Meeting Room In this talk, we will present the main research directions in mathematical modelling and continuum mechanics developed at SISSA. A non-exhaustive list includes shape-morphing structures inspired by nature, fracture in soft materials, microswimming, active matter, and mechanobiology. We will provide an overview of current and future research projects. To offer a more detailed perspective, we will also discuss some recent results in cancer modelling and personalized medicine.

Nicola Gigli- "Introduction to optimal transport" Room: 7th Floor Meeting Room I will give an overview of the proof of the celebrated theorem by Brenier about existence of optimal maps in the Euclidean setting. Interestingly, this deep result will unwrap in front of our eyes once we identify a few key concepts, such as that of c-cyclical monotonicity and c-concavity.

Tommaso Rossi-"Regularity of sub-Riemannian geodesics" Room: 7th Floor Meeting Room Sub-Riemannian geometry is a generalization of Riemannian geometry in which a metric is defined only on a subset of preferred directions, called a distribution. For curves tangent to the distribution, one can define their length, and the curves that minimize length are called sub-Riemannian geodesics. In this talk, I will discuss the regularity of sub-Riemannian geodesics: whereas geodesics in Riemannian geometry are always smooth, sub-Riemannian geodesics are not always smooth. I will also present examples showing that sub-Riemannian geodesics can branch. This is based on joint work in progress with A. Schiavoni Piazza and A. Socionovo.

Shulamit Terracina Room: 7th Floor Meeting Room

Ilya Chevyrev-"Stochastic and rough analysis" Stochastic analysis is an area of mathematics that aims to make rigorous sense and study properties of differential equations with random functions. Such equations appear frequently in mathematical physics, finance, and biology. Random functions are often very rough and the solution theory of stochastic differential equations falls outside the scope of classical calculus, so new tools are needed to study them. In this talk, I will describe one of the tools, called rough analysis, that has been developed in the past two decades and which has significantly extended the scope of “classical” stochastic analysis.

Mikhail Bershtein- "Quantum groups" The theory of quantum groups developed in the 1980s from attempts to understand structures that appear in integrable models of quantum field theory and statistical physics.  Remarkably, almost immediately, quantum groups found applications in representation theory, low-dimensional topology, enumerative geometry, and many other areas of mathematics.  In the talk, I will review some basic ideas, examples, and applications of quantum groups.

Jacopo Stoppa-"Special Lagrangian submanifolds and the Thomas-Yau conjecture" In the first part of the talk, starting from basic notions, I will introduce special Lagrangian submanifolds, due to Harvey and Lawson. These play an important role in the calculus of variations and in string theory. Around the year 2000, R. Thomas and S.-T. Yau proposed a conjectural criterion for the existence of a special Lagrangian submanifold in a suitable fixed isotopy class. In the second part of the talk I will sketch the current viewpoint of this conjecture, due to D. Joyce and Y. Li, mentioning some positive results for a class of examples with toric symmetry.

Marcello Porta-"Quantum Statistical Mechanics" In this talk I will discuss the rigorous study of problems in quantum statistical mechanics, motivated by condensed matter physics. I will focus on the integer quantum Hall effect, a paradigmatic example of topological transport phenomenon, which motivated a lot of research both in physics and in mathematics in the last 40 years. I will give an introduction to the phenomenon, to its mathematical formulation, and to some of the mathematical methods used in its rigorous analysis.