## Past Talks

Title: Serre polynomials and geometry of character varieties
Abstract:With G a complex reductive group, let XrG denote the G-character varieties of free group Fr, of rank r, and XirrG ⊂ XrG be the locus of irreducible representation conjugacy classes. In this talk we shall present a result showing that the mixed Hodge structures on the cohomology groups of XrSLn and of XrPGLn, and on the compactly supported cohomology groups of the irreducible loci XirrSLn and XirrPGLn are isomorphic, for any n,r ∈ N. The proof uses a natural stratification of XrG by polystable type coming from affine GIT and the combinatorics of partitions. In particular, this result would imply their E-polynomials coincide, settling the question raised by Lawton-Muñoz. This is based on joint work with Carlos Florentino and Alfonso Zamora.
Date: December 20, 2022
Passcode: dGcMd15*

Speaker: Yuri Prokhorov, Steklov Mathematical Institute, Moscow State University (Russia)
Title: Finite groups of birational transformations
Abstract: First, I survey know results on finite groups of birational transformations of higher-dimensional algebraic varieties. This theory has been significantly developed during the last 10 years due to the success of the minimal model program. Then I will talk about finite groups of birational transformations of surfaces over fields of positive characteristic. In particular, I will discuss a recent result on Jordan property of Cremona groups over finite fields (joint with Constantin Shramov).
Date: November 29, 2022
Passcode: c=!*b*14

Speaker: Giorgio Ottaviani, University of Florence (Italy)
Title: The Hessian map
Abstract: In a joint work with C. Ciliberto we study the Hessian map h_{d,r} which associates to any hypersurface of degree d>=3 in P^r its Hessian hypersurface, which is the determinant of the Hessian matrix. We prove that h_{d,r} is generically finite unless h_{3,1}, and in the binary case h_{d,1} is birational onto its image if d>=5, which is sharp. We conjecture that h_{d,r} is birational onto its image unless h_{3,1}, h_{4,1} and h_{3,2}, these exceptional cases were well known in classical geometry. The first evidence for our conjecture is given by h_{3,3} (the case of cubic surfaces) which is again birational onto its image.
Date: May 10, 2022
Passcode: G?H^gJ4v

Speaker: Farbod Shokrieh, University of Washington (USA)
Title: Heights and moments of abelian varieties
Abstract: We give a formula which, for a principally polarized abelian variety $(A, \lambda)$ over a number field (or a function field), relates the stable Faltings height of $A$ with the N\'eron--Tate height of a symmetric theta divisor on $A$. Our formula involves invariants arising from tropical geometry. We also discuss the case of Jacobians in some detail, where graphs and electrical networks will play a key role. (Based on joint works with Robin de Jong.)
Date: November 1, 2022
Passcode: VJa@7#KU

Speaker: Sandra Di Rocco, KTH (Sweden)
Title: Geometry of algebraic data
Abstract: It is often convenient to visualize algebraic varieties (and hence systems of polynomial equations) by sampling. The key challenge is to have the right distribution and density in order to recover the shape, i.e the topology of the variety. Bottlenecks are pairs of points on the variety joined by a line which is normal to the variety at both points. These points play a special role in determining the appropriate density of a point-sample. Under suitable genericity assumptions the number of bottlenecks of an affine variety is finite and is called the bottleneck degree. Estimations of the bottleneck degree and certain generalizations lead to efficient sampling techniques. We will show how classical projective algebraic geometry has proven very useful in this analysis. The talk is based on joint work with D. Eklund, P. Edwards, O. Gäfvert, J Hauenstein, M. Weinstein.
Date: October 18, 2022