*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 ‘Recent progress on the effective Mordell problem’

**Minhyong Kim**, University of Ottawa

**Minhyong Kim**

**9 December 2020**

*Abstract:* In 1983, Gerd Faltings proved the Mordell conjecture stating that curves of

genus at least two have only finitely many rational points. This can be understood as the statement that most polynomial equations (in a precise sense) f(x,y)=0 of degree at least 4 have at most finitely many solutions. However, the effective

version of this problem, that of constructing an algorithm for listing all rational

solutions, is still unresolved. To get a sense of the difficulty, recall how long it

took to prove that there are no solutions to x^n+y^n=1 other than the obvious ones. In this talk, I will survey some of the recent progress on an approach to this problem that proceeds by encoding rational solutions into arithmetic principal bundles and studying their moduli in a manner reminiscent of geometric gauge

theory. Watch the recording.

##### SMRI Algebra & Geometry Online ‘Coherent categorification of quantum loop sl(2)’

**Peng Shan**, Tsinghua University

**Peng Shan**

**26 October 2020**

*Abstract:* We explain an equivalence of categories between a module category of quiver Hecke algebras associated with the Kronecker quiver and a category of equivariant perverse coherent sheaves on the nilpotent cone of type A. This provides a link between two different monoidal categorifications of the open quantum unipotent cell of affine type A₁, one given by Kang–Kashiwara–Kim–Oh–Park in terms of quiver Hecke algebras, the other given by Cautis–Williams in terms of equivariant perverse coherent sheaves on affine Grassmannians. The first part of the talk will be devoted to introduction to quiver Hecke algebras and categorification of quantum cluster algebras. The main result will be explained in the second part. This is a joint work with Michela Varagnolo and Eric Vasserot. Watch the recording.

##### SMRI Algebra & Geometry Online ‘Stability conditions and automata’

Anthony Licata, Australian National University

**21 October 2020**

*Abstract:* Autoequivalences of triangulated categories are an interesting and understudied class of groups. In large part due to the development of the theory of Bridgeland stability conditions, there are suggestive parallels between these groups and mapping class groups of surfaces. The goal of this talk will be to explain how some of the geometric group theory which appears in the study of mapping class groups also arises in the study of triangulated autoequivalences.

##### SMRI Algebra & Geometry Online ‘Harmonic analysis on GLₙ over finite fields’

**Shamgar Gurevich**, University of Wisconsin, Madison

**Shamgar Gurevich**

**8 October 2020**

*Abstract:* There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio: Trace(ρ(g)) / dim(ρ), for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G. Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms”, which is (since the 60s) the main organization principle, and is based on the (huge collection) of “Large” representations. This talk will discuss the notion of rank for the group GLₙ over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.

This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried with Steve Goldstein (Madison) and John Cannon (Sydney).

##### SMRI Algebra & Geometry Online ‘Tate’s thesis in the de Rham setting’

**Sam Raskin**, University of Texas at Austin

**Sam Raskin**

**4 October 2020**

*Abstract:* This is joint work with Justin Hilburn. We will explain a theorem showing that D-modules on the Tate vector space of Laurent series are equivalent to ind-coherent sheaves on the space of rank 1 de Rham local systems on the punctured disc equipped with a flat section. Time permitting, we will also describe an application of this result in the global setting. Our results may be understood as a geometric refinement of Tate’s ideas in the setting of harmonic analysis. They also may be understood as a proof of a strong form of the 3d mirror symmetry conjectures in a special case. Watch the recording.

##### SMRI Algebra & Geometry Online ‘On the derived category of the Iwahori–Hecke algebra’

**Eugen Hellmann**, University of Münster

**Eugen Hellmann**

**9 December 2020**

*Abstract:* In this talk I will state a conjecture which predicts that the derived category of smooth representations of a p-adic split reductive group admits a fully faithful embedding into the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We will make the conjecture precise in the case of the principal block of GLn and relate it to the construction of a family of representations on the stack of L-parameters that interpolates a modified version of the local Langlands correspondence. The existence of this family is suggested by the work of Helm and Emerton–Helm. I will explain why the derived tensor product with this “Emerton–Helm family” should realize the expected embedding of derived categories and discuss some explicit examples. Watch the recording.

##### SMRI Algebra & Geometry Online ‘Incompressible symmetric tensor categories’

**Victor Ostrik**, University of Oregon

**Victor Ostrik**

**25 June 2020**

*Abstract:* This talk is based on joint work with Benson and Etingof. We say that a symmetric tensor category is incompressible if there is no symmetric tensor functor from this category to a smaller tensor category. Our main result is a construction of new examples of incompressible tensor categories in positive characteristic. Watch the recording.

##### SMRI Algebra & Geometry Online ‘Boundary conditions and hamiltonian actions in geometric Langlands’

**David Ben-Zvi**, University of Texas at Austin

**David Ben-Zvi**

**3 June 2020**

*Abstract:* I will discuss some of the structures suggested by the physics of supersymmetric gauge theory of relevance to the geometric Langlands program. The discussion will include defects of various dimensions in field theory, with an emphasis on the role of boundary conditions and how they lead to the suggestion of a duality between hamiltonian actions of Langlands dual groups. Watch the recording.

##### SMRI Algebra & Geometry Online ‘Introduction to derived categories of coherent sheaves’

**Tom Bridgeland**, University of Sheffield

**Tom Bridgeland**

**26 February – 11 March 2020**

*Abstract:* These lectures will cover some basic results about derived categories of coherent sheaves (e.g. the structure of the derived category of a curve, Fourier-Mukai transforms and how to construct them, tilting bundles, auto-equivalence groups, perhaps spaces of stability conditions). I will assume that the audience is vaguely familiar with the definition of derived and triangulated categories, but I will spend a fair amount of time in the first few lectures trying to give some intuitive feel for these general constructions, and explaining how one makes calculations in practice. I will also need to assume some familiarity with basic algebraic geometry and sheaf theory.