Lectures

Core courses and Tutorials
Course 1: Computational Aspects of Burnside Algebras

1. Dilip P Patil (IISC, Bangalore)
2. M. Kreuzer (Passau, Germany) | Tutorial 1 | Tutorial 2 | Tutorial 3 | Tutorial 4 | Tutorial | Tutorial | Tutorial

Course 2: Computational Commutative and Linear Algebra

1. L. Robbiano (Genova, Italy)
2. Anna Bigatti (Genova, Italy) | Tutorial | Tutorial | Tutorial | Tutorial
3. Elisa Palezzato (Genova, Italy)

Evening/Special Lectures

1. Uwe Storch (Bochum, Germany)
On the Genera of Quadratic Algebras
Abstract: The talk reports on a joint work in progress with Guido Reissner. This work is partly motivated by an article of H. Hasse. Its objective is to develop a general factorization theory for quadratic ${\cal O}_{X}$-algebras ${\cal B}$, where $X$ is an arbitrary locally noetherian scheme. The monoid structure of the set of classes of quadratic ${\cal O}_{X}$-algebras is used. The proposed theory generalizes the well-known factorizations of the discriminant of a quadratic $\mathbb{Z}$-algebra and leads to the concept of genera of the given ${\cal O}_{X}$-algebra ${\cal B}$. The genera correspond to certain classes of etale double coverings of ${\rm Spec}\, {\cal B}$ and form, as in the classical case, an  elementary $2$-group $\,\,{\rm Gen}\,({\cal B}\,\,\vert\, {\cal O}_{X})$.
2. Gerhard Rosenberger (Hamburg, Germany)
The Tarski problems and Elementary Free Groups with consequences and applications
Abstract: Around 1945, Alfred Tarski proposed several questions concerning the elementary theory of non-abelian free groups. These remained open for more than 60 years until they were proved by O. Kharlampovich and A. Myasnikov and independently by Z. Zela. The proofs, by both sets of authors, were monumental and involved the development of several new areas of infinite group theory. In this talk we explain the Tarski problems and what has been actually proved. We then discuss the history of the solution as well as the components of the proof. We then provide the basic strategy of the proof. We finish with a brief discussion of universal and elementary free groups and describe some consequences and applications.
3. Ravi Rao (TIFR, Mumbai)
Normality of the DSER group of elementary orthogonal transformations
Abstract: Let $(Q, q)$ be a inner product space over a commutative ring $R$, and consider the Dickson-Siegel-Eichler-Roy's subgroup of the orthogonal group $O_R(Q \perp H(R)^n)$, $n \geq 1$. Weshow that it is a normal subgroup of O_R(Q \perp H(R)^n)$, for all$n$, except when$n = 2$. 4. J.K. Verma (IIT, Mumbai) On the vanishing and positivity of the coefficients of the normal Hilbert polynomials. Abstract: We shall present a partial survey of vanishing and positivity of the coefficients of the normal Hilbert polynomials of m-primary ideals in Cohen-Macaulay analytically unramified local rings. After presenting the results of Itoh and Huneke, we shall discuss solution by Goto-Hong-Mandal of the Vasconcelos conjecture about the normal Chern number and recent progress on the Itoh's conjecture about the third normal Hilbert coefficient. 5. John Abbott (Kassel, Germany) Overview of Modular Methods | Tutorial 1 | Tutorial 2 | Tutorial 3 | Tutorial 4 | Tutorial 5 Abstract: This talk will give an overview of modular methods in polynomial computations. Modular Methods play a crucial role in polynomial computations; in many cases they enable much faster computation than a "direct" method. The archetypal example is polynomial GCD: multivariate polynomials are reduced to univariate ones, and univariate polynomials over Z are reduced to univariate polynomials over a finite field. Proper use of modular methods involves handling the phenomenon of "bad primes", knowing when the "modular precision" is sufficient, and finally reconstructing the correct result. 6. Grazia Tamone (Genova, Italy) & Anna Oneto (Genova, Italy) Numerical Semigroups and applications | Tutorial Abstract: Numerical semigroup theory has interactions in many fields, as commutative algebra, algebraic geometry, combinatorics, coding theory. In this lecture we deal with local semigroup rings$R=k[[S]]$: we first recall the Wilf conjecture, then we focus on the Cohen-Macaulayness of the associated graded ring and the behaviour of the Hilbert function$H_R$. We recall the basic notions and the main known facts on these topics. Then, we show some more recent results regarding classes of semigroup rings with$H_R\$ decreasing , in particular: we show there are infinitely many one-dimensional local Gorenstein rings with decreasing Hilbert function, obtained by a technique of duplication of almost-symmetric semigroups.