- Vladyslav Zveryk - Kirillov Algebras I
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The talk will be the first in the series of upcoming talks of mine aimed to present my research at IST Austria. I will start with a short overview of the whole project, followed by an introduction to the theory of vertex algebras. The main example will be the vertex algebra associated to an affine Kac-Moody algebra, which will play a crucial role in the future talks.
- Lorenzo Guerrieri - Gorenstein licci ideals of deviation 2
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I will describe some old results towards the classification of licci Gorenstein ideals of deviation 2, following the PhD thesis of Elias Lopez.
- Vladyslav Zveryk - Kirillov Algebras II
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The talk will be the second in the series of talks aimed at presenting my research at IST Austria. I will give an introduction to the theory of vertex algebras, starting with their definition and then deriving their main properties. Attendance at the previous talk is not required.
- Tymoteusz Chmiel - TBA
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TBA
- Maciej Dołęga - Rationally weighted b-Hurwitz numbers via W-algebras
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Weighted Hurwitz numbers were introduced by Harnad and Guay-Paquet as an object covering a wide class of Hurwitz numbers of various types. A particularly strong property of Hurwitz numbers is that they are widely governed by topological recursion (TR) of Chekhov--Eynard--Orantin. The program of understanding how TR can be used to compute various Hurwitz numbers was carried over the last two decades by considering each case separately, and finally the general case of rationally-weighted Hurwitz numbers was proved recently by Bychkov--Dunin-Barkowski--Kazarian--Shadrin. We are going to discuss a more general case of weighted $b$-Hurwitz numbers. We show that their generating function can be associated with the Whittaker vectorfor a W-algebra of type A. In particular it satisfies W-constraints that are the Airy structure -- an algebraic reformulation of the concept of TR due to Kontsevich--Soibelman. Our result gives a new explanation of remarkable enumerative properties of Hurwitz numbers and extends it to the $b$-deformed case. This is a joint work with Nitin Chidambaram and Kento Osuga.
- Jakub Koncki - Nakajima's creation operators and the Kirwan map
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The Hilbert scheme of points in the affine complex plane is a smooth variety. Several descriptions of its cohomology groups are known. One may use Białynicki-Birula decomposition, Nakajima creation operators, or the Kirwan map. I will present a relation between the last two of the mentioned methods. I will describe the action of Nakajima's creation operators on the characteristic classes of the tautological bundle. This is joint work with Magdalena Zielenkiewicz.
- Anna Szumowicz - Bounding Harish-Chandra characters
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Let $G$ be a connected reductive algebraic group over a $p$-adic local field $F$. We study the asymptotic behaviour of the trace characters $\theta _{\pi}$ evaluated at a regular element of $G(F)$ as $\pi$ varies among supercuspidal representations of $G(F)$. I give the sketch of the proof that for $G$ semisimple the trace character is uniformly bounded on $\gamma$ under the assumption, which is believed to hold in general, that all irreducible supercuspidal representations of $G(F)$ are compactly induced from an open compact modulo center subgroup of $G(F)$. If time allows I could also discuss progress in finding explicit bounds on the trace characters.
- Oana Veliche - On the minimal free resolution of the residue field
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In a paper from 1968, Golod proved that the Betti sequence of the residue field of a local ring attains the upper bound given by Serre if and only if the homology algebra of the Koszul complex of the ring has trivial multiplications and trivial Massey operations. This is the origin of the notion of Golod ring.
Using the Koszul complex components as building blocks Golod also constructed a minimal free resolution of the residue field of a Golod ring.
With Van Nguyen, we extend this construction for an arbitrary local ring, up to homological degree five, and explicitly show how the multiplicative structure of the homology of the Koszul algebra is involved, including the triple Massey products. The talk will illustrate this construction and various consequences of it.
- Pedro Macias Marques - Reducible families of Artinian Gorenstein algebras
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We study local Artinian Gorenstein (AG) algebras and consider the set of Jordan types of elements of the maximal ideal, i.e. the partition giving the Jordan blocks of the respective multiplication map. In joint work with Tony Iarrobino, we construct examples of families Gor(T) of local AG algebras with given Hilbert function T, and use obstructions that the symmetric decomposition of the associated graded algebra of an AG algebra A imposes on the Jordan type of A to study their irreducible components.