Guide Tame Algebras: Some Fundamental Notions

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Introduction

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A field with irreducible specializations is called Hilbertian. The numerous applications of this theorem make the question of under what conditions an extension of a Hilbertian field is again Hilbertian intersting. It turns out the the most difficult part is separable algebraic extensions.

In this talk I shall discuss a solution of the conjecture using Galois representations. The classical discriminant is a polynomial in the coefficients of f that vanishes precisely when the form f is degenerate. More generally one considers the discriminant of compact complete intersections. Then I will describe one of the modern reincarnations, in the study of compact varieties with non-isolated singularities. The discriminant of the transversal singularity type consists of the points of singular locus where the transversal type degenerates.

Postdoc Seminar "Representation theory"

In many cases its cohomology class can be computed by reduction to the classical discriminant. Vladimir Bavula conjectured that the same is true over the fields with finite characteristic. It turned out that exactly one half of his conjecture is correct. Uri Onn Title: Introduction to representation zeta functions Abstract. This is an introductory talk to the subject of representation growth and representation zeta functions of groups.

Tensors: Algebra-Computation-Applications (TACA)

The main focus will be on arithmetic groups and p-adic analytic groups. I will describe some recent results in the area and open problems. A modular conjecture due to Kemper-Kording-Malle-Matzat-Vogel-Wisse will be also considered and, if time permits, also similar results by Benson-Carlson regarding the cohomology ring of finite groups. Alexander S. Sivatski St. Petersburg Electrotechnical University Title: Central simple algebras of exponent 2 and index 8, and divided power operations Abstract.

This talk can be viewed as an extension of the colloquium talk on December 4. A few open related questions are posed. Let A be a biquaternion algebra a tensor product of two quaternion algebras over a field F of characteristic dierent from 2. A decomposition of A into a tensor product of two quaternion algebras is not unique, and there is no canonical one.

However, it turns out that any two decompositions of A can be connected by a chain of decompositions in which neighboring ones do not dier "too much". In fact there is an analogue of the chain lemma for a quaternion algebra. Any two biquaternion decompositions of A are equivalent to one another, and can be connected by a chain of length 3. Moreover, this bound is strict, i.

Tame algebras: some fundamental notions

We will outline the structure of the full proof of Belov's theorem that the polynomial identities of an affine PI-algebra over a commutative Noetherian ring are finitely based. More details of the proof are to be given in Prof. Margolis' seminar on representation theory. We discuss this conjecture, classify the rigid triples of primes for simple algebraic groups, and present a result stating that the conjecture holds in many cases. No matter how you color the natural numbers RED and BLUE there will be arithmetic sequence that is, numbers equally spaced of length that is all the same color.

In a series of lectures I will present the following: I Warmup: No matter how you 2-color the lattice points of the plane there will be a monochromatic square. This is folklore. They have been reduced quite a bit over the years- we will discuss this. Rado's theorem generalizes this and gives a condition about which types of equations it holds for.

Everyone sees all numbers but their own. They want to communicate as few bits as possible. How well can they do? Come and find out! Why this sequence? Can we replace d, 2d, 3d, We use it to get graphs of large chromatic number AND large girth. I have been looking at ways to cut them down OR look at variants where the bounds are more reasonable. Have I made progress? Daniel Lenz Title: Order based constructions of groupoids from inverse semigroups Abstract.

We discuss how the universal groupoid of an inverse semigroup introduced by Paterson can be obtained by a simple order based construction. Along the way one obtains canonically a reduction of this groupoid. In the case of inverse semigroups arising from graphs respectively, tilings this reduction is the graph groupoid introduced by Kumjian et al respectively, the tiling groupoid of Kellendonk. We discuss some topological features of this reduction as well as the structure of its open invariant sets.

Since Magnus it has been well known that one-relator groups have a decidable word problem.


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However, solvability of the word problem in one- relator monoids is far from being completely studied. Only few examples of inverse monoids with solvable word problem are known. Recently, the solvability of the word problem in inverse monoids with a single sparse relator has been announced by Hermiller, Lindblad and Meakin.

We consider certain one-relator inverse monoids. In our attempt to solve the word problem, we rely on the result of Ivanov, Margolis and Meakin which states that the word problem for the inverse one-relator monoid is decidable if the membership problem for the corresponding prefix monoid is decidable. Thus, we first solve the membership problem for the prefix monoid and then apply the theorem to solve the word problem.

Our methods involve van Kampen diagrams and word combinatorics. Borel Abstract. Given an element P X,Y, Assuming that K is an arbitrary field of characteristic different from 2, we prove that if P is not an identity in sl 2,K , then this map is dominant for any Chevalley algebra g. This result can be viewed as a weak infinitesimal counterpart of Borel's theorem on the dominance of the word map on connected semisimple algebraic groups.

As in the group case, the proof is based on a construction of division subalgebras due to Deligne and Sullivan. We also prove that for the Engel monomials [[[X,Y],Y], We also discuss consequences of these results for polynomial maps of associative matrix algebras.

Bandman, N. Gordeev, and E. In commutative algebra, complete intersection rings are the next best thing after regular rings. Such rings have a structural description, but they also have two homological characterizations: in the seventies Gulliksen characterized complete intersection local rings by the growth rate of their homology; more recently Benson and Greenlees characterized such rings by the existence of a certain structure on their derived categories.

These homological characterizations can be easily adapted for the cochain-algebras of connected spaces, where the coefficients are in some prime field.