*This seminar series is co-presented by SMRI. Algebra seminars (2022-) and former SMRI Algebra and Geometry Online (SAGO) seminars are specialised research talks by international researchers in algebra and geometry. *

##### SMRI Algebra & Geometry Online Seminar ‘Blowup formulas for nilpotent sensitive cohomology theories’

**Shane Kelly**, **Tokyo Institute of Technology**

**Shane Kelly**

**2 December 2021**

*Abstract:* This is joint work in progress with Shuji Saito. Many cohomology theories of interest (l-adic cohomology, de Rham cohomology, motivic cohomology, K-theory…) have long exact sequences associated to blowups. Such a property can be neatly encoded in a Grothendieck topology such as the cdh-topology or the h-topology. These topologies appeared in Voevodsky’s proof of the Bloch-Kato conjecture, and more recently in Beilinson’s simple proof of Fontaine’s CdR conjecture, and in Bhatt and Scholze’s work on projectivity of the affine Grassmanian.

A feature of these topologies which often turns out to be a bug is that the associated sheaves cannot see nilpotents. In this talk I will discuss a modification which can see nilpotents, and which still has long exact sequences for many blowups. Read the notes.

*Biography: *Shane Kelly is an Associate Professor at Tokyo Institute of Technology. His research area is algebraic K-theory and motivic homotopy theory, and more recently he is interested in applications to representation theory. His graduate studies were mostly based in Paris; in 2012 he received a PhD jointly from Université Sorbonne Paris Nord and The Australian National University under the joint supervision of Cisinski and Neeman, respectively.

##### SMRI Algebra & Geometry Online Seminar ‘On the group completion of the Burau representation’

**Jack Morava**, Johns Hopkins University

**Jack Morava**

**11 November 2021**

*Abstract:* On fundamental groups, the discriminant ∏i≠k(zi – zk) ∈ \mathbb {C} × of a finite collection of points of the plane defines the abelianization homomorphism

sending a braid to its number of overcrossings less undercrossings or writhe. In terms of diffeomorphisms of the punctured plane, it defnes a kind of `invertible topological quantum field theory’ associated to the Burau representation, and in the classical physics of point particles the real part of its logarithmic derivative is the potential energy of a collection of Coulomb charges, while its imaginary part is essentially the Nambu-Goto area of a loop of string in the three-sphere. Its higher homotopy theory defines a very interesting a double-loop map \mathbb {Z} × Ω2S3 → 𝒫ic(S0) to the category of lines over the stable homotopy ring-spectrum, related to Hopkins and Mahowald’s exotic (E2) multiplication on classical integral homology, perhaps related to the `anyons’ of nonclassical physics. Watch the recording.

*Biography:* Jack Johnson Morava, of Czech and Appalachian descent, studied under Eldon Dyer and Sir Michael Atiyah, graduating with a PhD from Rice University in 1968, followed by an Academy of Sciences postdoc in Moscow with Yuri Manin and Sergei Novikov. He joined the Johns Hopkins faculty in 1979 where he was involved in the Japan-US mathematical institute, and from roughly 2003 to 2010 he worked half-time on the DARPA FunBio initiative. He retired in 2017 to live with his anthropological linguist wife in Charlottesville, Virginia and get some work done.

##### SMRI Algebra & Geometry Online Seminar ‘Symplectic duality and (generalized) affine Grassmannian slices’

**Joel Kamnitzer**, **University of Toronto**

**Joel Kamnitzer**

**21 October 2021**

*Abstract:* Under the geometric Satake equivalence, slices in the affine Grassmannian give a geometric incarnation of dominant weight spaces in representations of reductive groups. These affine Grassmannian slices are quantized by algebras known as truncated shifted Yangians. From this perspective, we expect to categorify these weight spaces using category O for these truncated shifted Yangians.

The slices in the affine Grassmannian and truncated shifted Yangians can also be defined as special cases of the Coulomb branch construction of Braverman-Finkelberg-Nakajima. From this perspective, we find many insights. First, we can generalize affine Grassmannian slices to the case of non-dominant weights and arbitrary symmetric Kac-Moody Lie algebras. Second, we establish a link with modules for KLRW algebras. Finally, we defined a categorical g-action on the categories O, using Hamiltonian reduction. Watch the recording.

*Biography:* Joel Kamnitzer is a Professor of Mathematics at the University of Toronto. His research concerns complex reductive groups and their representations, focusing on canonical bases, categorification, and geometric constructions. His 2005 Ph.D. thesis from UC Berkeley focused on the study of Mirkovic-Vilonen cycles in Affine Grassmannians. He received the 2011 Andre Aisenstadt Prize, a 2012 Sloan Research Fellowship, a 2018 E.W.R. Steacie Memorial Fellowship, a 2018 Poincare Chair, and the 2021 Jeffrey-Williams Prize.

##### SMRI Algebra & Geometry Online Seminar ‘Solving semidecidable problems in group theory’

**Giles Gardam**, University of Münster

**Giles Gardam**

**5 October 2021**

*Abstract:* Group theory is littered with undecidable problems. A classic example is the word problem: there are groups for which there exists no algorithm that can decide if a product of generators represents the trivial element or not. Many problems (the word problem included) are at least semidecidable, meaning that there is a correct algorithm guaranteed to terminate if the answer is “yes”, but with no guarantee on how long one has to wait. I will discuss strategies to try and tackle various semidecidable problems computationally with the key example being the discovery of a counterexample to the Kaplansky unit conjecture. Watch the recording.

*Biography: * Giles Gardam is a research associate at the University of Münster working in geometric group theory. He studied mathematics and computer science at the University of Sydney, receiving his Bachelor’s degree in 2012, and completed his doctorate at Oxford in 2017. He was then a postdoc at the Technion before starting at Münster in 2019.

##### SMRI Algebra & Geometry Online Seminar ‘The singularity category of C^*(BG) for a finite group G’

**John Greenlees**, Warwick University

**John Greenlees**

**16 September 2021**

*Abstract:* The cohomology ring H^*(BG) (with coefficients in a field k of characteristic p) is a very special graded commutative ring, but this comes out much more clearly if one uses the cochains C^*(BG), which can be viewed as a commutative ring up to homotopy. For example C^*(BG) is always Gorenstein (whilst this is not quite true for H^*(BG)).

This leads one to study C^*(BG) as if it was a commutative local Noetherian ring, though of course one has to use homotopy invariant methods. The ring C^*(BG) is regular if and only if G is p-nilpoent and so it is natural to look for ways of deciding where C^*(BG) lies on a the spectrum between regular and Gorenstein rings. For a commutative Noetherian ring, one considers the singularity category D_{sg}(R) (the quotient of finite complexes of finitely generated modules by finitely generated projectives). This is trivial if and only if R is regular, so is the appropriate tool. The talk will describe how to define this for C^*(BG), show it has good basic properties and describe the singularity category in the simplest case it is not trivial (when G has a cyclic Sylow p-subgroup). Watch the recording.

##### SMRI Algebra & Geometry Online Seminar ‘A singular Coxeter presentation’

**Hankyung Ko**, Uppsala University

**26 August 2021**

*Abstract:* A Coxeter system is a presentation of a group by generators and a specific form of relations, namely the braid relations and the reflection relations. The Coxeter presentation leads to, among others, a similar presentation of the (Iwahori-)Hecke algebras and the Kazhdan-Lusztig theory, which provides combinatorial answers to major problems in Lie theoretic representation theory and geometry. Extending such applications to the `singular land’ requires the singular version of the Hecke algebra. Underlying combinatorics of the singular Hecke algebra/category comes from the parabolic double cosets and is the first step in understanding the singular Hecke category. In this talk, I will present a Coxeter theory of the parabolic double cosets developed in a joint work with Ben Elias. In particular, I will explain a generalization of the reduced expressions and describe the braid and non-braid relations. Watch the recording.

##### SMRI Algebra & Geometry Online Seminar ‘Schubert polynomials, the inhomogeneous TASEP, and evil-avoiding permutations’

Lauren Williams, Harvard University

**5 October 2021**

*Abstract:* The totally asymmetric simple exclusion process (TASEP) was introduced around 1970 as a model for translation in protein synthesis and traffic flow.

It has interesting physical properties (e.g. boundary-induced phase transitions) and also beautiful mathematical properties. The inhomogeneous TASEP is a Markov chain of weighted particles hopping on a ring, in which the probability that two particles interchange depends on the weight of those particles. If each particle has a distinct weight, then we can think of this as a Markov chain on permutations. In many cases, the steady state probabilities can be expressed in terms of Schubert polynomials. Based on joint work with Donghyun Kim. Watch the recording.

*Biography: * Lauren Williams is the Robinson professor of mathematics at Harvard and the Seaver Professor at the Harvard Radcliffe Institute. Her research is in algebraic combinatorics. Williams received her BA in mathematics from Harvard College in 2000, and her PhD from MIT in 2005. Subsequently, she was a postdoc at UC Berkeley and Harvard, then a faculty member at UC Berkeley from 2009 to 2018, before returning to Harvard in 2018. She is the recipient of a Sloan Research Fellowship, an NSF CAREER award, the AWM-Microsoft research prize, and is an Honorary member of the London Mathematical Society.

##### SMRI Algebra & Geometry Online Seminar ‘Tits groups of Iwahori-Weyl groups and presentations of Hecke algebras’

**Xuhua He**, Chinese University of Hong Kong

**Xuhua He**

**5 August 2021**

*Abstract:* Let G(ℂ) be a complex reductive group and W be its Weyl group. In 1966, Tits introduced a certain subgroup of G(ℂ), which is an extension of W by an elementary abelian 2-group. This group is called the Tits group and provides a nice lifting of W. In this talk, I will discuss a generalization of the notion of the Tits group T to a reductive p-adic group G. Such T, if exists, gives a nice lifting of the Iwahori-Weyl group of G. I will show that the Tits group exists when the reductive group splits over an unramified extension of the p-adic field and will provide an example in the ramified case that such a Tits group does not exist. Finally, as an application, we will provide a nice presentation of the Hecke algebra of the p-adic group G with ln-level structure. This talk is based on the recent joint work with Ganapathy. *Watch the recording*.

*Biography:* Xuhua He is the Choh-Ming Professor of Mathematics at the Chinese University of Hong Kong. He works in pure mathematics. His research interests include Arithmetic geometry, Algebraic groups, and representation theory. He received his Bachelor’s degree in mathematics from Peking University in 2001 and a Ph.D. degree from MIT in 2005 under the supervision of George Lusztig. He worked as a member at the Institute for Advanced Study during 2005-2006 and Simons Instructor at Stony Brook University during 2006-2008. He worked at the Hong Kong University of Science and Technology during 2008-2014 as an assistant Professor and associated Professor, and then moved to the University of Maryland during 2014-2019 as a Full Professor of Mathematics before joining CUHK in 2019. He received the Morningside Gold Medal of Mathematics in 2013, the Xplorer Prize in 2020 and is an invited sectional speaker of the International Congress of Mathematicians in 2018.

##### SMRI Algebra & Geometry Online Seminar ‘Root components for tensor product of affine Kac-Moody Lie algebra modules’

**Shrawan Kumar**, University of North Carolina

**Shrawan Kumar**

**5 August 2021**

*Abstract:* Let G(ℂ) be a complex reductive group and W be its Weyl group. In 1966, Tits introduced a certain subgroup of G(ℂ), which is an extension of W by an elementary abelian 2-group. This group is called the Tits group and provides a nice lifting of W. In this talk, I will discuss a generalization of the notion of the Tits group T to a reductive p-adic group G. Such T, if exists, gives a nice lifting of the Iwahori-Weyl group of G. I will show that the Tits group exists when the reductive group splits over an unramified extension of the p-adic field and will provide an example in the ramified case that such a Tits group does not exist. Finally, as an application, we will provide a nice presentation of the Hecke algebra of the p-adic group G with ln-level structure. This talk is based on the recent joint work with Ganapathy. Read the notes. Watch the recording.

##### SMRI Algebra & Geometry Online Seminar ‘Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution’

**Ulrich Thiel**, University of Kaiserslautern

**Ulrich Thiel**

**8 July 2021**

*Abstract:* Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4 – the symplectically primitive but complex imprimitive groups – and 10 exceptional groups up to dimension 10, for which it is still open. Recently, we have proven that for all but possibly 39 cases in the remaining infinite series there is no symplectic resolution. We have thereby reduced the classification problem to finitely many open cases. We do not expect any of the remaining cases to admit a symplectic resolution. This is joint work with Gwyn Bellamy and Johannes Schmitt.

##### SMRI Algebra & Geometry Online Seminar ‘Geometric Satake over KU’

**Gus Lonergan**, A Priori Investment Management

**Gus Lonergan**

**24 June 2021**

*Abstract:* We describe a K-theoretic version of the equivariant constructible derived category. We state (with evidence!) a ‘geometric Satake’ conjecture relating its value on the affine Grassmannian to representations of the Langlands dual group. Watch the recording.

*Biography: *Gus Lonergan is the Chief Mathematician at A Priori Investment Management LLC. He was previously a L.E. Dickson Instructor in the mathematics department at the University of Chicago. He is interested in representation theory.

Lonergan completed his PhD at MIT under Roman Bezrukavnikov; the thesis was an attempt to understand mod p phenomena in algebraic topology in the context of geometric representation theory. He attended Cambridge University for his undergraduate and master’s degree. He plays a little music on the side.

##### SMRI Algebra & Geometry Online Seminar ‘Advertising symmetric quivers and their representations’

Magdalena Boos, Ruhr University Bochum

**17 June 2021**

*Abstract:* We introduce the notion of a symmetric quiver as provided by Derksen and Weyman in 2002 and discuss symmetric degenerations in this context (which correspond to orbit closure relations in the symmetric representation variety). After motivating our particular interest in the latter by presenting connections to group actions in algebraic Lie Theory, we explain our main questions: are symmetric degenerations induced by “usual” degenerations in the representation variety of the underlying quiver? We look at (counter)examples and recent results. This is joint work with Giovanni Cerulli Irelli. Watch the recording.

##### SMRI Algebra & Geometry Online Seminar ‘Representation theory of exceptional Lie superalgebras’

**Shun-Jen Cheng**, Institute of Mathematics, Academia Sinica

**Shun-Jen Cheng**

**6 May 2021**

*Abstract:* In the first half of the talk we shall introduce the notion of Lie superalgebras, and then give a quick outline of the classification of finite-dimensional complex simple Lie superalgebras. In the second part of the talk we shall discuss the representation theory of these Lie superalgebras and explain the irreducible character problem in the BGG category. Our main focus will be on our computation of the irreducible characters for two of the exceptional Lie superalgebras. This part is based on recent joint works with C.-W. Chen, L. Li, and W. Wang. Watch the recording.