Algebra Seminar in Thessaloniki:
Speaker: Κωνσταντίνος Λιάμπης (Πανεπιστήμιο Ιωάννινων).
Title: Θεωρία Τοπικοποίησης στην Ανώτερη Ομολογική Άλγεβρα και Τριγωνισμένες Κατηγορίες
Περίληψη: Στόχος της ομιλίας είναι η μελέτη της θεωρίας τοπικοποίησης των n-αβελιανών κατηγοριών και των n-τριγωνισμένων κατηγοριών. Ξεκινώντας με μία υπολογίσιμη κλάση μορφισμών S σε μία n-αβελιανή ή σε μία n-τριγωνισμένη κατηγορία, κατασκευάζουμε με καθολικό τρόπο μία τοπικοποιημένη n-αβελιανή ή μία n-τριγωνισμένη κατηγορία αντίστοιχα, όπου οι μορφισμοί της κλάσης S έχουν αντιστραφεί. Στην n-αβελιανή περίπτωση η κατασκευή της τοπικοποίησης θα γίνει σε δύο στάδια: Αρχικά, ορίζουμε την έννοια μίας ημι-n-αβελιανής (pre-n-abelian) κατηγορίας την οποία και χαρακτηρίζουμε μέσω ενός διαγράμματος που καλούμε n-διάγραμμα. Στη συνέχεια, χρησιμοποιώντας το n-διάγραμμα, για κάθε n-αβελιανή κατηγορία M και για κάθε υπολογίσιμο σύστημα μορφισμών S στην Μ, κατασκευάζουμε μία n-αβελιανή κατηγορία και έναν n-ακριβή συναρτητή, καθολικό ως προς όλους τους S-αντιστρεπτικούς n-ακριβείς συναρτητές από την M σε κάθε n-αβελιανή κατηγορία. Αντίστοιχα, στην τριγωνισμένη περίπτωση, αποδεικνύουμε ότι η τοπικοποίηση μίας n-τριγωνισμένης κατηγορίας είναι επίσης μία n-τριγωνισμένη κατηγορία, ο συναρτητής τοπικοποίησης είναι n-ακριβής και πληροί την επιθυμητή καθολική ιδιότητα. Έτσι λύνεται ικανοποιητικά το πρόβλημα της τοπικοποίησης μίας n-αβελιανής ή μίας n-τριγωνισμένης κατηγορίας.
Speaker: Δάφνη Τσίπα (Πανεπιστήμιο Αιγαίου).
Title: Γενικευμένες Baumslag-Solitar ομάδες και residual ιδιότητες
Περίληψη: Θα μιλήσουμε για τις γενικευμένες Baumslag-Solitar ομάδες οι οποίες ορίζονται ως θεμελιώδεις ομάδες γραφημάτων ομάδων όπου οι ομάδες κορυφών και ακμών είναι άπειρες κυκλικές ομαδες. Αφού τις ορίσουμε θα αναφέρουμε κάποια πρόσφατα ερευνητικά αποτελέσματα για ιδιότητες (residual nilpotency και residual finiteness p) για διάφορους τύπους υποκειμένων γραφημάτων.
Speaker: Ιορδάνης Τσελεπίδης (Münster).
Title: Το (ασθενές) Θεώρημα των Mordell-Weil
Περίληψη: Στην διάλεξη αυτή θα εστιάσουμε στο θεμελιώδες θεώρημα της Αριθμητικής των Ελλειπτικών Καμπυλών. Ειδικότερα, θα σκιαγραφήσουμε την απόδειξη του ασθενούς θεωρήματος των Mordell-Weil καθώς και το πώς αυτό μπορεί να χρησιμοποιηθεί στην απόδειξη του θεωρήματος των Mordell-Weil υπεράνω του σώματος των ρητών. Αν μας το επιτρέπει ο χρόνος, θα αναφέρουμε και κάποια πιο δύσκολα αποτελέσματα (όπως το περίφημο θεώρημα του Mazur) ή/και κάποια ανοιχτά προβλήματα.
Speaker: Andreas Hayash (Massachusetts, Amherst)
Title: Quiver Zastava spaces and quantum groups
Speaker: Tiago Cruz (Stuttgart)
Title: Ringel self-duality via relative dominant dimension
Abstract: Quasi-hereditary algebras provide an abstract framework for the homological structure of Schur algebras and the BGG category O of a semi-simple Lie algebra and they always appear in pairs via Ringel duality. In this talk, we discuss a generalisation of dominant dimension using relative homological algebra. This homological invariant is compatible with the tools from integral representation theory and it increases our understanding of classical dominant dimension. In particular, this homological invariant provides tools to deduce that quasi-hereditary covers formed by quasi-hereditary algebras with a simple preserving duality with large enough dominant dimension also appear in pairs. As an application, we give a new proof of Ringel self-duality of the blocks of the BGG category O of a complex semi-simple Lie algebra.
Speaker: Christos Aravanis (Sheffield)
Title: Derived categories of sheaves and Hopf algebras
Abstract: The aim of this talk is to discuss a Hopf algebra object in the derived category of coherent sheaves of a smooth projective variety. The motivation comes from the mathematical study of a three dimensional topological quantum field theory based on a hyperkahler manifold. This is joint work with Simon Willerton.
Speaker: Laertis Vaso (Uppsala)
Title: Higher dimensional AR-theory through Nakayama algebas
Abstract: In this talk I will start by recalling two important notions from representation theory of finite dimensional algebras: the Auslander-Reiten quiver and the global dimension of an algebra. I will explain how one may generalise the very well understood case of representation-finite algebras of global dimension one to n-representation-finite algebras of higher global dimension. Throughout I will present a few tools that one may use to construct n-representation finite algebras using examples coming from Nakayama algebras.
Speaker: Aristides Kontogeorgis (Athens)
Title: Συζυγίες του κανονικού ιδεώδους και αυτομορφισμοί
Abstract: Θα δώσουμε μία περιγραφή της ομάδας αυτομορφισμών μιας αλγεβρικής καμπύλης ως αλγεβρικό σύνολο της προβολικής γραμμικής ομάδας βασισμένοι στο θεώρημα Petri. Στην συνέχεια θα ορίσουμε δράση της ομάδας αυτομορφισμών πάνω στις ελάχιστες επιλύσεις του κανονικού ιδεώδους.
Speaker: Ioannis Dokas (Athens)
Title: Quillen-Barr-Beck συνομολογία για την κατηγορία των restricted Lie αλγεβρών
Abstract: Στο πρώτο μέρος της ομιλίας παραθέτουμε στοιχεία από την θεωρία των restricted Lie αλγεβρών. Στο δεύτερο μέρος, ακολουθώντας την θεωρία (συν)-ομολογιών των Quillen-Barr-Beck για μία αλγεβρική κατηγορία, ορίζουμε συνομολογία για την κατηγορία RLie_k των restricted Lie αλγεβρών πάνω σ’ενα σώμα k χαρακτηριστικής p\noteq 0. Ειδικότερα, προσδιορίζουμε για L στην RLiek την κατηγορία των Beck-L-modules και την αβελιανή ομάδα Derp(L, M) των Beck derivations, όπου M είναι ένα Beck-L-module. Κατόπιν, ορίζουμε ομάδες συνομολογίων H^*_{RLie}(L, M). Η συνομολογία H^*_{RLie}(L, M) ταξινομεί γενικευμένες επεκτάσεις restricted Lie αλγεβρών που δεν ταξινομούνται από την θεωρία συνομολογιών του Hochschild για τις restricted Lie άλγεβρες.
Speaker: Alexis Kouvidakis (Crete)
Title: Cubic threefolds (Τριδιάστατες κυβικές υπερεπιφάνειες)
Abstract: Στην ομιλία θα δώσουμε μια σύντομη ιστορική επισκόπιση τού προβλήματος τού Lüroth, τήν αλγεβρική και γεωμετρική διατύπωσή του, και θα δούμε τό ρόλο που έπαιξαν οι τριδιάστατες κυβικές υπερεπιφάνειες στην εύρεση αντιπαραδείγματος στη διάσταση 3. Κατόπιν θα παρουσιάσουμε γεωμετρικές ιδιότητες των τριδιάστατων κυβικών υπερεπιφανειών και τής αντίστοιχης επιφάνειας τού Fano, τά σημεία τής οποίας παραμετρούν τίς ευθείες που περιέχονται στις παραπάνω υπερεπιφάνειες.
Speaker: Sotiris Karanikolopoulos (Athens)
Title: Automorphisms of Curves and Weierstrass semigroups
Abstract: Let X be a projective nonsingular algebraic curve defined over an algebraically closed field of positive characteristic p. I consider Harbater-Katz-Gabber covers: these are p-group Galois covers from X to P^1 with only one fully ramified point. This simple construction provide us with examples of curves with "huge" number of automorphisms; they are important because of their applications over finite fields and the Harbater-Katz-Gabber compactification theorem for Galois actions on complete local rings. The sequences of the jumps of the higher ramification filtrations of their Galois group are related to the Weierstrass semigroup of the global cover at the ramified point. I will compute these jumps in terms of the pole numbers and, if time permits, give some applications for curves with zero p-rank: maximal curves and curves that admit a "big action".
Speaker: Eirini Chavli (Stuttgart)
Title: Complex reflection groups, braid groups, and Hecke algebras
Abstract: Two decades ago, Broué, Malle and Rouquier published a paper in which they associated to every complex reflection group two objects which were classically associated to real reflection groups: a braid group and a Hecke algebra. Their work was further motivated by the theory, developed together with Michel, of “Spetses”, which are objects that generalise finite reductive groups in the sense that their associated Weyl groups are complex reflection groups. In this talk we explain these two objects and we willfocus on the properties and representation theory of the Hecke algebras.
Speaker: Vassilis Metaftsis (Samos)
Title: On the linearity of groups
Abstract: Linearity is a rare property of groups, very hard to detect. In the present talk we give some early results concerning the linearity of groups and we show how the linearity of right-angled Artin groups help us to prove this property for more general families of groups.
Speaker: Julian Külshammer (Uppsala)
Title: Gorenstein homological algebra for generalised species
Abstract: Over an algebraically closed field, every finite dimensional algebra is Morita equivalent to the quotient of a path algebra of a finite quiver by an admissible ideal. For perfect fields, the role of the finite quiver is played by a species. Recently, with the purpose of categorification of cluster algebras, a generalisation of this concept has been studied by Geiss, Leclerc, and Schröer. In this talk we study the Gorenstein homological algebra of such algebras as originated in commutative algebra. We generalise work of Ringel and Schmidmeier on almost split sequences in monomorphism categories to this context. This is based on joint work with Nan Gao, Sondre Kvamme, and Chrysostomos Psaroudakis.
Speaker: Georgios Dalezios (Copenhagen/Murcia)
Title: Singularity categories and singular equivalences
Abstract: The singularity category of a ring was introduced by Buchweitz as a categorical measure of its singularities. In this talk, after explaining the basic aspects of the theory, we will discuss some old and new results on singular equivalences of Morita type, in the context of finite dimensional algebras. We will also discuss generalizations of certain results on singular equivalences of self-injective algebras in the context of Gorenstein algebras.
Speaker: Christos Tatakis (Ioannina)
Title: The structure of complete intersection graphs and their planarity (joint work with A. Thoma)
Abstract: We study the complete intersection property of the graph ideals I_G. In general, the graph ideal I_G is complete intersection if and only if it can be generated by h binomials, where h=m-n+1 if G is a bipartite graph or h=m-n if G is not a bipartite graph. The answer is known in the case of bipartite graphs, i.e. graphs with no odd cycles. In the last years, several useful partial results have been proved and they provide key properties of complete intersection graph ideals .
We focus on the general case, where G is a random graph and we present a structural theorem which gives us necessary and sufficient conditions in which the toric ideal I_G is complete intersection. Moreover, we characterize the complete intersection graphs which are planar.
17.30 - 18.15: William Crawley-Boevey (Bielefeld)
Title: My Struggle with the Deligne Simpson Problem
Abstract: The Deligne Simpson problem (DSP) arises in the classification of linear ODEs in the complex domain, but it is elementary to state: given k conjugacy classes in GL(n,C), determine whether or not there are matrices in these classes with no common invariant subspace and product equal to the identity. Work I did on quiver algebras led to a conjectural answer for the DSP, I proved one direction in 2006 with P. Shaw, and although I announced a proof of the other direction, it never got written up. In this talk I will explain the background to the DSP and discuss recent work with A. Hubery aimed at completing the other direction.
18.30 - 19.00: Kostas Karagiannis (Thessaloniki)
Title: On the canonical embedding of the Kummer-Artin-Schreier-Witt family of curves
Abstract: Arithmetic Geometry, the discipline that bridges Algebraic Geometry and Number Theory, mainly studies algebraic curves over fields of prime characteristic p>0. Technical difficulties led Serre, Tate and Grothendieck to develop lifting techniques to fields of characteristic 0, over which curves are better understood. Applying these techniques, Oort-Sekiguchi-Suwa (1989) and Bertin-Mézard (2000), unified Artin-Schreier theory and Kummer theory, by introducing an appropriate family of curves over the ring of Witt vectors. Using results of Karanikolopoulos-Kontogeorgis (2014) on the sheaf of holomorphic differentials Ω, we study the family's induced canonical embedding into projective space. In particular, in joint work with professors Charalambous and Kontogeorgis, we give explicit generators for the embedding's defining ideal, combining lifting techniques with elements of combinatorial commutative algebra and the theory of Gröbner bases.
19.00 - 19.30: Ioannis Zachos (Michigan)
Title: Introduction to Local Models
Abstract: In this talk we will try to define the local models and give a small sketch of the historical development of the theory. The importance of the local models lies in the fact that under some assumptions they model the singularities that arise in the reduction modulo p of Shimura varieties. The Shimura varieties are often moduli spaces of abelian varieties with additional structure. Understanding the structure of local models will help us to have a better understanding of the moduli spaces of the abelian varieties with some additional structure. In this talk we will try to give most of the background needed in order to understand the above mathematical objects. Time permitted, we will state some open problems.
19:45 - 20:15: Alexandros Grosdos (Osnabrück)
Title: Moment Ideals of Mixture Distributions
Abstract: Moments are quantities that measure the shape of statistical or stochastic objects and have recently been studied from an algebraic and combinatorial point of view. We start this talk by introducing (local) mixture distributions and their moment ideals. We explain how mixing distributions on the statistical side corresponds to taking secants of the algebraic varieties on the geometric one and we compute generators for the ideals involved. Furthermore, we apply elimination theory and Prony's method in order to do parameter estimation, and showcase our results with an application in signal processing. A main goal of this talk is to highlight the natural connections between algebraic statistics, geometry, combinatorics and applications in analysis throughout the talk.
Speaker: Julia Sauter (Bielefeld)
Title: Going relative with faithfully balanced modules
Abstract: We briefly revisit faithfully balanced modules for finite-dimensional algebras and the well-known correspondences they induce (for (co)-generators and (co)tilting modules). Relative homological algebra (RHA) replaces the usual exact structure on finitely generated modules with another one which is easily defined - this has been introduced by Auslander and Solberg. We explain how the previously mentioned correspondences interplay with RHA and their relative versions. This is work in progress with Biao Ma.
Speaker: Chrysostomos Psaroudakis (Thessaloniki)
Title: Explicit right adjoints between homotopy categories
Abstract: Let T be a triangulated category with coproducts and let X be a set of compact objects. Then X generates a certain t-structure, and in particular describes explicitly a left adjoint to the inclusion of the coaisle. Unfortunately, it does not make much sense to consider the naive dual of this setup; cocompact objects rarely appear in categories which occur naturally. Motivated by this, we introduce a weaker version of cocompactness called 0-cocompactness, and show that in a triangulated category with products these objects cogenerate a t-structure. As an application, we provide explicit right adjoints between certain homotopy categories. This is joint work with Steffen Oppermann and Torkil Stai.
Speaker: Steffen Oppermann (Trondheim)
Title: Auslander-Reiten quivers and higher dimensional analogs
Abstract: In my talk I will first discuss some classical concepts in representation theory of finite dimensional algebras. In particular we will look at Auslander Reiten quivers, and what information these quivers contain. I will then focus on a more recent twist: the generalization of Auslander Reiten theory to a “higher dimensional” version, and discuss some aspects that generalize nicely (as well as possibly some that don’t).
Speaker: Torkil Stai (Trondheim)
Title: Triangulated categories and orbit categories
Abstract: The concept of a triangulated category comes up everywhere, in particular in mathematics! Our primary objective is to provide some very basic intuition for this notion, with an emphasis on connections and similarities with the more familiar abelian categories. For instance, taking subcategories and quotients of triangulated categories is straightforward. A much more mysterious construction is that of an orbit category of a triangulated category with respect to an automorphism; there is no a priori reason why this gadget should be triangulated. The secondary aim of this talk is to explain how algebraists have coped with this nightmare over the last decade. Time permitting, I will report on ongoing work with Steffen Oppermann on the topic.