'Topics on Fano manifolds'

1. Fano variety in general: definition and first examples
2. Surfaces
2.1 Sketches on Enriques-Kodaira classification
2.2 Classification of delPezzo surfaces
2.3 Projections and ancestral examples
3. Dimension 3
3.1 Mukai biregular classification (vector bundle methods)
3.2 10 families of prime Fano threefolds of index 1. The yoga of key varieties. Link to K3 and canonical curves
3.3 Fano 3-folds of higher index or higher picard rank
3.4 Q-Fano threefolds: state of the art (Tom&Jerry reprised, etc)
3.5 (extra) Rationality properties and Torelli problem
4. Dimension 4 and beyond
4.1 K"uchle classification of Fano 4-folds of index 1 from homogeneous vector bundles
4.2 Classification of Fano of higher index
4.3 delPezzo manifolds and Mukai manifolds
5. Extra
5.1 Fano varieties with special properties. Fano of K3 type. Link with hyperk"ahler geometry
 

Introduction to numeration systems Number expansions in lattices
Reading course

1. Fundamentals of number expansions, mathematical background
2. Decision and classification problems, algorithmical complexity
3. Fast computations: expansivity, congruences
4. Discrete dynamic, attractors and periodic elements 5. Optimizing by basis transformation 6. Generalized binary number systems 7. Construction problems, open questions 8. Block diagonal systems, simultaneous systems 9. Applications
 
Literature:
S. Akiyama , T. Borbély , H. Brunotte , A. PethÅ‘ , J. M. Thuswaldner:
On a generalization of the radix representation -- a survey (2004) IN ”HIGH PRIMES AND MISDEMEANOURS: LECTURES IN HONOUR OF THE 60TH BIRTHDAY OF HUGH COWIE WILLIAMS”, FIELDS INSTITUTE COMMUCATIONS
 
A. Kovács papers in
http://compalg.inf.elte.hu/~attila/Publications.html
 

Mini-course: Polyhedral structures in algebraic geometry

Abstract: Algebraic geometry studies the zero locus of polynomial equations connecting the related algebraic and geometrical structures. 
In several cases, nevertheless the theory is extremely precise and elegant, it is hard to read in a simple way the information behind such structures. A possible way of avoiding this problem is that of associating to polynomials some polyhedral structures that immediately give some of the information connected to the zero locus of the polynomial. In relation to this strategy I will introduce Newton-Okounkov bodies and Tropical Geometry, underlying the connection between the two theories.
I will conclude stating a recent result in collaboration with E. 
Postinghel, where the interplay of tropical geometry and Newton-Okounkov bodies gives a flat degeneration for Mori Dream Spaces.
 

Time Series Analysis

1. We will recall the basic principles of applied and numerical Fourier analysis: Fourier series and transform, energy and power spectrum, mutual and autocorrelation and their numerical computation.
2. Impulse and harmonic response of a system.
3. Filtering of a time series.
4. Time series as sampling of a continuous signal.

Foundational themes

Some themes of the last century research on foundations of logic and mathematics will be considered under the point of view of the current logical research: 1. Incompleteness theorems. 2.  Constructive omega-rule. 3.  Ordinal numbers and dilators

Lie Symmetries of Differential and Difference Equation

1. Lie symmetries of differential equatios and their extensions and generalizations
2. Lie point symmetries of Difference Equations; their derivation and their applications.
3. From Point Symmetries to Generalized Symmetries for Difference Equations
4. Generalized Symmetries from the Integrability of Difference Equations
5. Formal Symmetries and Integrable Lattice Equations

Advanced Graphics for Scientific Data

We will introduce some usefull tools for Scientific Data visualization, starting from GNUplot and MATLAB for drawing basic 1D and 2D functions, then moving to Paraview for exploring more complex features of 3D data sets

Cubic hypersurfaces
(reading course)

Topics on cubics and rationality
 to be chosen in the following list:
1. The Century of cubic surfaces  
2. Basic history of rationality and cubics 
2.1. XIX Century: the Lueroth problem  
2.2. Cubics and unirationality  
2.3. XX Century: Fano on cubic threefolds  
2.4. XX Century: unirational non rational threefods  
2.5. Early XXI Century: Vosin on stable rationality  
3. Basic algebraic geometry of cubics  
3.1. The Jacobian ideal of a hypersurface  
3.2. The polar map of a cubic  
3.3. The Fano variety of lines  
3.4. Nodal cubic threefolds  
4. Which cubics are rational?  
4.1. Cubic threefolds and rationality  
4.2. Conic bundles and stable rationaliy  
4.3. Which cubics are rational? 
5. The examples of rational cubic fourfolds  
5.1. Morin and Fano  
5.2. Cubics as Pfaffians  
5.3. Rational cubics and OADP varieties  
5.4. Hassett’s quadric bundles 
5.5. Togliatti primal  
6. General geography of cubic fourfolds  
6.1. Hodge-Noether-Lefschetz theory 
6.2. The Fano variety of lines F (X ) 
6.3. Beauville-Donagi construction  
6.4. Hassett’s theory and main conjectures  
7. Geography of the rational locus  
7.1. Working with the Fano variety F (X )  
7.2. Some divisors and some special surfaces  
7.3. Cubic fourfolds and K3 moduli spaces  
7.4. Further work and examples 
 

K-THEORY IN CONDENSED MATTER PHYSICS

In the last decades, solid state physics has witnessed a plethora of phenomena that have a topological origin: from the pioneering works of Thouless and collaborators to explain the quantization of the transverse conductivity in the quantum Hall effect, to the thriving field of topological insulators and superconductors. This field of research rapidly attracted the attention of mathematical physicists as well.
 
The methods involved in the theoretical understanding of these phenomena, at least in the one-particle approximation, rely on the theory of vector bundles and K-theory. The course will therefore be divided in two parts:

• the first part of the course will cover topics from differential topology and geometry, in particular the classification of vector bundles, their invariants, and K-theory;
• the second part will be devoted to the physical applications, discussing the periodic table of topological insulators, and the relation of their topological labels with quantum transport.

Some basic differential geometry and a first course in the mathematics of quantum mechanics will be assumed as prerequisites.
 
Depending on the inclinations, background and interests of the students, and if time permits, more advanced topics could also be covered, including for example:

• K-theory for C*-algebras and applications to disordered topological insulators;
• obstruction theory and constructive algorithms for Wannier functions;
• universality of the Hall conductivity with respect to weak interactions via renormalization group methods.