An introduction to representation theory of groups

Wednesday 30 April 2014 h. 14:30, room 2BC30
Martino Garonzi (Padova, Dip. Mat.)
"An introduction to representation theory of groups"

Label the faces of a cube with the numbers from 1 to 6 in some order, then perform the following operation: replace the number labeling each given face with the arithmetic mean of the numbers labeling the adjacent faces. What numbers will appear on the faces of the cube after this operation is iterated many times? This is a sample problem whose solution is a model of the application of the theory of representations of groups to diverse problems of mathematics, mechanics, and physics that possess symmetry of one kind or another.
In this introductory talk I will present the tools from representation theory needed to solve this problem. I will also point out the connection with harmonic analysis by expressing Fourier analysis as an instance of representation theory of the circle group (the multiplicative group of complex numbers with absolute value 1) and by stating a version of Heisenberg's uncertainty principle for finite cyclic groups.

Geometric modeling and splines: state of the art and outlook

Wednesday 9 April 2014 h. 14:30, room 2BC30
Michele Antonelli (Padova, Dip. Mat.)
"Geometric modeling and splines: state of the art and outlook"

We will give an introductory presentation of the research field of geometric modeling and its applications, with specific attention to the use of splines for the representation of curves and surfaces.
In particular, we will start by introducing basic notions of geometric modeling leading up to the definition of splines, which are piecewise functions with prescribed smoothness at the locations where the pieces join. Splines will be exploited for the representation of parametric curves and surfaces, and we will present their application in the context of computer-aided geometric design for shape description by means of approximation and interpolation methods. Finally, we will discuss some open problems in this topic and we will sketch some recent approaches for addressing them.

Solving PDEs on surfaces with radial basis functions: from global to local methods

Thursday 10 April 2014 h. 15:30, room 2BC30
Grady B. Wright (Department of Mathematics - Boise State University - USA)
"Solving PDEs on surfaces with radial basis functions: from global to local methods"

Radial basis function (RBF) methods are becoming increasingly popular for numerically solving partial differential equations (PDEs) because they are geometrically flexible, algorithmically accessible, and can be highly accurate. There have been many successful applications of these techniques to various types of PDEs defined on planar regions in two and higher dimensions, and to PDEs defined on the surface of a sphere. Originally, these methods were based on global approximations and their computational cost was quite high. Recent efforts have focused on reducing the computational cost by using "local" techniques, such as RBF generated finite differences (RBF-FD).
In this talk, we first describe our recent work on developing a new, high-order, global RBF method for numerically solving PDEs on relatively general surfaces, with a specific focus on reaction-diffusion equations. The method is quite flexible, only requiring a set of "scattered" nodes on the surface and the corresponding normal vectors to the surface at these nodes. We next present a new scalable local method based on the RBF-FD approach with this same flexibility. This is the first application of the RBF-FD method to general surfaces. We conclude with applications of these methods to some biologically relevant problems.
This talk represents joint work with Ed Fuselier (High Point University), Aaron Fogelson, Mike Kirby, and Varun Shankar (all at the University of Utah).

Seminario Dottorato: Shape optimization and polyharmonic operators

Wednesday 26 March 2014 h. 14:30, room 2BC30
Davide Buoso (Padova, Dip. Mat.)
"Shape optimization and polyharmonic operators"

We will start by introducing the general shape optimization problem, giving motivations for its importance in applications.
Then we will turn to the problem of shape optimization for eigenvalues of elliptic operators (in particular, poyharmonic operators), which has regained popularity since 1993 with the paper by Buttazzo and Dal Maso. We will recall the most important classical results, giving the main ideas behind the proofs, together with the last ones.
Finally, we will move our attention to the problem of criticality with respect to shape deformations for eigenvalues of polyharmonic operators. After explaining the techniques involved, we will provide a characterization of criticality and show that balls are always critical.

Seminario Dottorato: Market models with optimal arbitrage

Wednesday 12 March 2014 h. 14:30, room 2BC30
Chau Ngoc Huy (Padova, Dip. Mat.)
"Market models with optimal arbitrage"

In this talk, we will introduce basic notions on financial mathematics, classical no arbitrage theory and some results on markets with arbitrage. We present a systematic method to construct market models where the optimal arbitrage strategy exists and is known explicitly.

Seminario Dottorato: Minimum time function for linear control systems

Wednesday 26 February 2014 h. 14:30, room 2BC30
Nguyen Van Luong (Padova, Dip. Mat.)
"Minimum time function for linear control systems"

In this talk, we will first introduce basic notions on linear control systems and minimum time function for linear control systems. We will end with some recent results on regularity of minimum time function for linear control systems.

Seminario Dottorato: Reachability problems via level set approach

Wednesday 12 February 2014 h. 14:30, room 2BC30
Athena Picarelli (ENSTA ParisTech and INRIA Saclay-Ile de France)
"Reachability problems via level set approach"

Given a controlled dynamical system, the characterization of the backward reachable set, i.e. the set of initial states from which it is possible to reach a given target set, can be very interesting in many applications. However realistic models may involve some constraints on state and/or control variables (for taking into account physical or economical constraints, obstacles, etc.) and this can make the characterization of this set much more complicated.
After an introduction to the notion of backward reachability in the deterministic as well in the stochastic framework, aim of the talk is to present a technique, based on a level set approach, for characterizing and numerically computing the reachable set also if state constraints are taken into account.

Seminario Dottorato: An introduction to geometry over the field with one element

Wednesday 29 January 2014, h. 14:30, room 2BC30
Federico Bambozzi (Padova - Dip. Mat.)
"An introduction to geometry over the field with one element"

In this talk we first give a brief overview of the motivations behind the research on geometry over the field with one element. We then show one possible way to define affine schemes over the field with one element in analogy with the classical theories of algebraic varieties over the complex numbers and of schemes by Grothendieck. We end the talk by giving some examples of schemes over F_1.

Seminario Dottorato: An introduction to stochastic ergodic control

Wednesday 15 January 2014, h. 15:15, room 1BC45
Marco Cirant (Padova - Dip. Mat.)
"An introduction to stochastic ergodic control"

In this talk we give an introduction to stochastic ergodic control problems, where an agent aims at minimizing a long-time average cost by controlling his own state.
We will show, through a toy example, the main features of the problem and how it is possible to produce an optimal control by solving a suitable elliptic nonlinear partial differential equation.
In the final part of the talk we will explore briefly how the minimization problem for a single agent can be considered more in general for a continuum of identical agents. This research field is called Mean Field Games and has attracted the experts' attention in the last ten years.

Seminario Dottorato: An introduction to the Clemens-Schmid exact sequence

Wednesday 18 December 2013, h. 14:30, room 2BC30
Genaro Hernandez Mada (Padova - Dip. Mat.)
"An introduction to the Clemens-Schmid exact sequence"

In this talk, we give a very elementary introduction to the Clemens-Schmid exact sequence. In a classical setting, this is a topological result about certain families of complex varieties or manifolds. Therefore, the purpose of the talk is to explain all the concepts involved.
If time allows, we will introduce the elements to understand in which cases we can obtain an arithmetic version of this result.

Rif. int. C. Marastoni, T. Vargiolu

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