Natasha Morrison: Uncommon systems of equations

20th October, 2021

A system of linear equations L over 𝔽q is common if the number of monochromatic solutions to L in any two-colouring of (𝔽q)n is asymptotically at least the expected number of monochromatic solutions in a random two-colouring of (𝔽q)n. Motivated by existing results for specific systems (such as Schur triples and arithmetic progressions), as well as extensive research on common and Sidorenko graphs, the systematic study of common systems of linear equations was recently initiated by Saad and Wolf. Building on earlier work of of Cameron, Cilleruelo and Serra, as well as Saad and Wolf, common linear equations have been fully characterised by Fox, Pham and Zhao. In this talk I will discuss some recent progress towards a characterisation of common systems of two or more equations. In particular we prove that any system containing an arithmetic progression of length four is uncommon, confirming a conjecture of Saad and Wolf. This follows from a more general result which allows us to deduce the uncommonness of a general system from certain properties of one- or two-equation subsystems.

John Greenlees: The torsion Adams spectral sequence for rational torus-equivariant spectra

19th October, 2021

We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, 𝒜t(G) of finite injective dimension, a homology theory π∗𝒜t taking values in 𝒜t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence

Ext𝒜t(G)∗ , ∗ (π∗𝒜t(X), π∗𝒜t(Y)) → [X,Y]∗G

for rational G-spectra X and Y.

This approach should be viewed as an analogue of the Cousin complex in algebraic geometry. It is expected that a similar method will apply to other tensor triangulated categories with finite-dimensional Noetherian Balmer spectra.

Yakov Eliashberg: Contractibility of the space of tight contact structures on ℝ3

18th October, 2021

30 years ago I proved that any tight contact structure on the 3-sphere is diffeomorphic to the standard one. I also optimistically claimed at the same paper that similar methods could be used to prove a multi-parametric version: the space of tight contact structures on the 3-sphere, fixed at a point, is contractible. In our recent joint with N. Mishachev paper we proved this result. While the proof indeed roughly follows the strategy of my 1991 paper, it is much more involved. In particular, it uses a new criterion for tightness of a characteristic foliation on the 2-sphere, which is valid without any contact convexity assumptions.

Daniel Orr: Difference Operators for Wreath Macdonald Polynomials

17th October, 2021

very concrete auxiliary algebraic structures that were constructed in order to define them. Later, when Haiman's proof of the Macdonald positivity conjecture revolutionized the subject, the scope of Macdonald theory widened to include the geometry of Hilbert schemes of points in the plane. (For this reason, one should associate ordinary Macdonald polynomials with the Jordan quiver.)

A cyclic quiver generalization of Macdonald polynomials was born in reverse, starting with a geometric conjecture which was made by Haiman and later proved by Bezrukavnikov and Finkelberg. Thus the resulting polynomials, which are known as wreath Macdonald polynomials, arise from the geometry of cyclic quiver Nakajima varieties. Their existence relies on an elusive object known as the Procesi bundle, which is available only by deep and indirect means.

Only recently has direct understanding of wreath Macdonald polynomials begun to emerge, through methods based on the quantum toroidal algebra. In this talk, I will review the origins of (wreath) Macdonald theory and discuss new explicit results on wreath Macdonald polynomials, and anticipated applications, from joint work in progress with Mark Shimozono and Joshua Wen.

Bojko Bakalov: On the Cohomology of Vertex Algebras and Poisson Vertex Algebras

17th October, 2021

Following Beilinson and Drinfeld, we describe vertex algebras as Lie algebras for a certain operad of n-ary chiral operations. This allows us to introduce the cohomology of a vertex algebra V as a
Lie algebra cohomology. When V is equipped with a good filtration, its associated graded is a Poisson vertex algebra. We relate the cohomology of V to the variational Poisson cohomology studied previously by De Sole and Kac.

Jiuzu Hong: Smooth Locus of Twisted Affine Schubert Varieties and Twisted Affine Demazure Modules

17th October, 2021

Let G be a special parahoric group scheme of twisted type, excluding the absolutely special case for twisted A2n. Using the methods and results of Zhu, we prove a duality theorem for general G: there is a duality between the level one twisted affine Demazure modules and function rings of certain torus fixed point subschemes in twisted affine Schubert varieties for G. Along the way, we also establish the duality theorem for untwisted E6. As a consequence, we determine the smooth
locus of any affine Schubert variety in affine Grassmannian of G, which confirms a conjecture of Haines and Richarz.

Iana Anguelova: Microlocality and Chiral Algebras

16th October, 2021

In one of my last conversations with Ben Cox, we discussed our mutual desire to work together on the axiomatic approach to multilocal and quantum chiral algebras. We both had worked already on issues related to multilocality; situations where the fields/vertex operators in question have Operator Product Expansions (OPEs) with more than the one singularity at 'z=w'. In particular, we worked together on the theory of N-point local chiral algebras, i.e., algebras that are 'complete' with respect to OPEs, and have singularities at roots of unity. But we were planning to work on the outstanding case where the OPEs have singularities at infinite multiplicative lattices. Such is the example of the Frenkel-Jing quantum vertex operators. In this talk I will discuss some problems arising in the axiomatic approach to multilocal chiral algebras, both N-point local, and quantum.

Weiqiang Wang: From Quantum Groups to iQuantum Groups

16th October, 2021

Drinfeld-Jimbo quantum groups have made major impacts on representation theory and other areas. i-Quantum groups arise from quantum symmetric pairs. We shall explain why it is natural to view i-quantum groups as a generalization of quantum groups, and then discuss some of the many new developments and applications of i-quantum groups as initiated in Huanchen Bao’s UVA dissertation.

Dijana Jakelic: On Deodhar’s Localization Functor

16th October, 2021

This talk will present several remarks on connections among Enright’s completion, Deodhar’s localization functor, Ben Cox’s early work on generalization of Deodhar’s results, and related topics.

This video was part of the Southeastern Lie Theory Workshop XII.

Milen Yakimov: Non-commutative Tensor Triangular Geometry and Support Varieties for Hopf Algebras

16th October, 2021

We will describe a theory of noncommutative tensor triangular geometry for monoidal triangulated categories. It is aimed at investigating support varieties for finite dimensional Hopf algebras via non-commutative Balmer spectra. We will state effective reconstruction theorems for these spectra and an intrinsic characterization of those categories whose support variety maps satisfy the tensor product property. As an application, we obtain a treatment of the Benson-Witherspoon Hopf algebras, which previously eluded approaches of this kind, and a proof of a recent conjecture of Negron and Pevtsova that the cohomological support maps of the Borel subalgebras of all Lusztig small quantum groups possess the tensor product property. This is joint work with Daniel Nakano (University of Georgia) and Kent Vashaw (MIT).

Chloe Papin: A Whitehead Algorithm for Generalized Baumslag-Solitar Groups

15th October, 2021

Baumslag-Solitar groups BS(p,q) =< a,t | tapt-1 = aq > were first introduced as examples of non-Hopfian groups. They may be described using graphs of cyclic groups. In analogy with the study of Out(Fn) one can study their automorphisms through their action on an "outer space". After introducing generalized Baumslag-Solitar groups and their actions on trees, I will present an analogue of a Whitehead algorithm which takes an element of a free group and decides whether there exists a free factor which contains that element.

Slava Futorny: Free Field Constructions for Affine Kac-Moody Algebras

15th October, 2021

Classical free field realizations of affine Kac-Moody algebras (introduced by M.Wakimoto, B.Feigin and E.Frenkel) play an important role in quantum field theory. B.Cox initiated the study of free field realizations for the non-standard Borel subalgebras which led to an important class of intermediate (or parabolic) Wakimoto modules. A uniform construction of such realizations will be discussed based on a joint work with L.Krizka and P.Somberg.

Umberto Hryniewicz: Results on abundance of global surfaces of section

15th October, 2021

One might ask if global surfaces of section (GSS) for Reeb flows in dimension 3 are abundant in two different senses. One might ask if GSS are abundant for a given Reeb flow, or if Reeb flows carrying some GSS are abundant in the set of all Reeb flows. In this talk, answers to these two questions in specific contexts will be presented. First, I would like to discuss a result, obtained in collaboration with Florio, stating that there are explicit sets of Reeb flows on S3 which are right-handed in the sense of Ghys; in particular, for such a flow all finite (non-empty) collections of periodic orbits spans a GSS. Then, I would like to discuss genericity results, obtained in collaboration with Colin, Dehornoy and Rechtman, for Reeb flows carrying a GSS; as a particular case of such results, we prove that a C∞-generic Reeb flow on the tight 3-sphere carries a GSS.

Congling Qiu: Modularity and Heights of CM cycles on Kuga-Sato varieties

14th October, 2021

We study CM cycles on Kuga-Sato varieties over X(N) via theta lifting and relative trace formula. Our first result is the modularity of CM cycles, in the sense that the Hecke modules they generate are semisimple whose irreducible components are associated to higher-weight holomorphic cuspidal automorphic representations of GL2(ℚ). This is proved via theta lifting. Our second result is a higher weight analogue of the general Gross-Zagier formula of Yuan, S. Zhang and W. Zhang.

This is proved via relative trace formula, provided the modularity of CM cycles.

Louis Rowen: Finitely generated axial algebras

14th October, 2021

This lecture is a continuation of the general talk given at the Drensky conference last month, on axial algebras, which are (not necessarily commutative, not necessarily associative) algebras generated by semisimple idempotents. After a review of the definitions, we investigate the key question, being, "Under what conditions must an axial algebra be finite-dimensional?" Krupnik showed that 3 idempotents can generate arbitrarily large dimensional associative algebras (and thus infinite-dimensional algebras via an ultraproduct argument), so some restriction is needed. We consider 'primitive' axes, in which the left and right eigenspaces having eigenvalue 1 are one-dimensional.

Hall, Rehren, Shpectorov solves obtained a positive answer for commutative axial algebras of 'Jordan type' λ ≠ 1/2, although the proof relies on the classification of simple groups and the given bound of the dimension is rather high. Gorshkov and Staroletov provided a sharp bound for 3-generated commutative axial algebras of 'Jordan type'. Our objective in this project is give a non-commutative version and indicate how to investigate 4-generated commutative axial algebras of 'Jordan type', in terms of the regular representation.

Our method is to build an associative algebra from the adjoint algebra of A, which has a strictly larger dimension which nevertheless also is finite-dimensional.

Camillo De Lellis: What is the h-principle?

11th October, 2021

The honest answer to the question is that I actually do not know. I will therefore rather talk about several famous examples that are widely called 'h-principle results' and try to explain some of the ideas behind the ones I am most familiar with.

Andrew Hanlon: Tropical Lagrangian sections and Looijenga pairs

11th October, 2021

We will discuss the first steps in an approach to proving homological mirror symmetry for Looijenga pairs through tropical Lagrangian sections. Namely, we will see how to construct these Lagrangian sections from tropical data corresponding to line bundles on the mirror and include them in a version on the Fukaya-Seidel category. Moreover, the Lagrangian Floer coholomogy of certain sections corresponds with integral points of polytopes that encode theta functions on the mirror.

Rima Chatterjee: Cabling of knots in overtwisted contact manifolds

8th October, 2021

Knots associated to overtwisted manifolds are less explored. There are two types of knots in an overtwisted manifold – loose and non-loose. Non-loose knots are knots with tight complements whereas loose knots have overtwisted complements. While we understand loose knots, non-loose knots remain a mystery. The classification and structure problems of these knots vary greatly compared to the knots in tight manifolds. Especially we are interested in how satellite operations on a knot in overtwisted manifold changes the geometric property of the knot. In this talk, I will discuss under what conditions cabling operation on a non-loose knot preserves non-looseness.

Alexei Myasnikov: On the Andrews-Curtis conjecture

8th October, 2021

I am going to talk about the group-theoretic aspects of the Andrews-Curtis conjecture, some recent results, and some old. From my viewpoint the Andrews-Curtis conjecture is not just a hard stand-alone question, coming from topology, but a host of very interesting problems in group theory.