*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. *

##### Algebra Seminar ‘Homological comparison of resolution and smoothing’

**Will Donovan**, Tsinghua University

**Will Donovan**

**23 September 2022**

*Abstract:* A singular space often comes equipped with (1) a resolution, given by a morphism from a smooth space, and (2) a smoothing, namely a deformation with smooth generic fibre. I will discuss work in progress on how these may be related homologically, starting with the threefold ordinary double point as a key example. Watch the recording. Read the notes.

*Biography:* Will Donovan is currently an associate professor at Yau MSC, Tsinghua University, Beijing. He is also a member of the adjunct faculty at BIMSA, Yanqi Lake, Huairou, Beijing and a visiting associate scientist at Kavli IPMU, University of Tokyo. He received his PhD in Mathematics in 2011 from Imperial College London. His interests are algebraic geometry, noncommutative geometry, representation theory, string theory and symplectic geometry.

**SMRI Algebra & Geometry Online Seminar** ‘From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra’

**Monica Vazirani**, University of California, Davis

**14 April 2022**

*Abstract:* Young diagrams and standard tableaux on them parameterize irreducible representations of the symmetric group and their bases, respectively. There is a similar story for the double affine Hecke algebra (DAHA) taking periodic tableaux, or for the rational Cherednik algebra (a.k.a. rational DAHA) with appropriate modifications. This construction of the basis makes use of an alternate presentation of the rational DAHA and the basis diagonalizes the action of its Dunkl-Opdam subalgebra. We make use of the combinatorics to construct explicit maps between standard modules parameterized by hooks, thus recovering the BGG resolution of the simple module parameterized by the trivial hook.

We can also describe this simple module using the geometry of parabolic Hilbert schemes of points on plane curve singularities. The “tableau” basis that diagonalizes the Dunkl-Opdam subalgebra is the basis of equivariant homology that comes from torus fixed points. Watch the recording.

This is joint work with Eugene Gorsky and José Simental.

*Biography: *Monica Vazirani is a professor at UC Davis. She received her PhD from UC Berkeley, after which she had an NSF postdoc she spent at UC San Diego and UC Berkeley, as well as postdoctoral positions at MSRI and Caltech. Dr. Vazirani’s research interests center on the representation theory of algebras related to the symmetric group and how to express algebraic phenomena via the combinatorics of partitions, tableaux, crystal graphs and parking functions.

**SMRI Algebra & Geometry Online Seminar** ‘What is deterministic amplification?’

**Clément Canonne**, University of Sydney

**Clément Canonne**

**12 April 2022**

*Abstract:* Suppose we want to solve a given task (say, a decision problem) and have a randomised algorithm for it which is correct; but only with some non-trivial probability, for instance .51. We would like to “amplify” this probability of success to an arbitrarily small amount, as close to 1 as possible: how to do this? And, more importantly, how to do this using as little extra randomness as possible?

I will first discuss why one would want to do this, then how to achieve it naively, and — quite surprisingly — how we can do much better than this naive approach using expander graphs. Watch the recording.

*Biography:* Clément Canonne is a Lecturer in the School of Computer Science of the University of Sydney; he obtained his Ph.D. in 2017 from Columbia University, before joining Stanford as a Motwani Postdoctoral Fellow, then IBM Research as a Goldstine Postdoctoral Fellow. His main areas of research are distribution testing (and, broadly speaking, property testing) and learning theory; focusing, in particular, on understanding the computational aspects of learning and statistical inference subject to various resource or information constraints.

**SMRI Algebra & Geometry Online Seminar** ‘Optimal regularity of mapping class group actions on the circle’

**Sang-hyun Kim**, Korea Institute for Advanced Study

**2 March 2022**

*Abstract:* We prove that for each finite index subgroup H of the mapping class group of a closed hyperbolic surface, and for each real number r>1 there does not exist a faithful C^r-action (in Hölder’s sense) of H on a circle. For this, we partially determine the optimal regularity of faithful actions by right-angled Artin groups on a circle. (Joint with Thomas Koberda and Cristobal Rivas). Watch the recording.

*Biography: *Sang-hyun Kim works at Korea Institute for Advanced Study as Professor in the School of Mathematics since 2019. Before this, he worked at Seoul National University, KAIST, Tufts University, the University of Texas at Austin and MSRI. He received Ph.D in 2007 at Yale University under the supervision of Andrew Casson. His research interests focus on the interplay between geometric group theory and low–dimensional topology, particularly motivated by right-angled Artin groups and manifold diffeomorphism groups. He was selected as the Scientist of the Month by the Korean Ministry of Science and ICT in 2020.