Schedule of Talks: Autumn 2021

Time/Location : Seminar will be on Tuesdays 3:00-4:00pm via Zoom and occasionaly in person in Mw 154


 

TIME SPEAKER TITLE HOST
September 7 
T  
 
Michel Brion, CNRS and Institut Fourier Homomorphisms of algebraic groups: representability and rigidity Joshua
October 19 
T  
 
Padma Srinivasan (University of Georgia) Some Galois cohomology classes arising from the fundamental group of a curve Joshua
November 2 

 
Open
November 16 

 
Roberto Pirisi (University of Rome, Italy) Brauer groups of moduli of hyperelliptic curves and their compactifications, via cohomological invariants Joshua
November 30 
T  
 
Open


Schedule of Talks: Spring

Time/Location : Seminar will be on Tuesdays 3:00-4:00pm via Zoom and occasionaly in person in Mw 154


 

TIME  SPEAKER TITLE HOST
January 18 
T  
 
Fernandez Herrero (Cornell) Intrinsic construction of moduli spaces via affine grassmannians Joshua
February 1 
T  
 
Open
February 15 
T  
 
Rahul Singh (U Pittsburgh) Counting Parabolic principal G-bundles with nilpotent sections over P^1 Joshua
February 22 
T  
 
Agneet Dhillon (University of Western Ontario) The motive of Bun_G Joshua
March 8 
T  
 
Anand sawant, TIFR (Mumbai, India) Central extensions of algebraic groups through motivic homotopy theory Joshua
March 29 
T  
 
Open
April 12 
T  
 
Open
April 26 
T  
 
Open

Abstracts

(Brion Talk) The topic of the talk is the functor of homomorphisms from G to H, where G and H are algebraic groups over a field k. When G is linearly reductive and H is smooth, we show that this functor is represented by a smooth scheme M; moreover, every orbit of H acting by conjugation on M is open. (The notion of linear reductivity will be discussed in detail during the talk). This result extends a theorem of Grothendieck in which G is of multiplicative type, and has applications to algebraic group actions on projective varieties.

Zoom details for this seminar: https://osu.zoom.us/j/91057354055?pwd=RlFTTCt5SFlDS3VKV01LWnJodlkwQT09 Meeting id: 910 5735 4055, Passcode: 033969

(Srinivasan talk) We will talk about a few Galois cohomology classes naturally arising from the fundamental group of a curve.

We will first talk about the Ceresa class, which is the image under a cycle class map of a canonical algebraic cycle associated to a curve in its Jacobian. This class vanishes for all hyperelliptic curves and was expected to be nonvanishing for non-hyperelliptic curves. In joint work with Dean Bisogno, Wanlin Li and Daniel Litt, we construct a non-hyperelliptic genus 3 quotient of the Fricke-Macbeath curve with vanishing Ceresa class, using the character theory of the automorphism group of the curve, namely, PSL_2(F_8).

Time permitting, we will also talk about some Galois cohomology classes that obstruct the existence of rational points on curves, by obstructing splittings to natural exact sequences coming from the fundamental group of a curve. In joint work with Wanlin Li, Daniel Litt and Nick Salter, we use these obstruction classes to give a new proof of Grothendieck’s section conjecture for the generic curve of genus g > 2. An analysis of the degeneration of these classes at the boundary of the moduli space of curves, combined with a specialization argument lets us produce infinitely many curves of each genus over p-adic fields and number fields that satisfy the section conjecture.

(Pirisi talk) Given an algebraic variety X, the Brauer group of X is the group of Azumaya algebras over X, or equivalently the group of Severi-Brauer varieties over X, i.e. fibrations over X which are étale locally isomorphic to a projective space. It was first studied in the case where X is the spectrum of a field by Noether and Brauer, and has since became a central object in algebraic and arithmetic geometry, being for example one of the first obstructions to rationality used to produce counterexamples to Noether's problem of whether given a representation V of a finite group G the quotient V/G is rational. While the Brauer group has been widely studied for schemes, computations at the level of moduli stacks are relatively recent, the most prominent of them being the computations by Antieau and Meier of the Brauer group of the moduli stack of elliptic curves over a variety of bases, including Z, Q, and finite fields.

In a recent series of joint works with A. Di Lorenzo, we use the theory of cohomological invariants, and its extension to algebraic stacks, to completely describe the Brauer group of the moduli stacks of hyperelliptic curves, and their compactifications, over fields of characteristic zero, and the prime-to-char(k) part in positive characteristic. It turns out that the Brauer group of the non-compact stack is generated by elements coming from the base field, cyclic algebras, an element coming from a map to the classifying stack of étale algebras of degree 2g+2, and when g is odd by the Brauer-Severi fibration induced by taking the quotient of the universal curve by the hyperelliptic involution. This paints a richer picture than in the case of elliptic curves, where all non-trivial elements come from cyclic algebras. Regarding the compactifications, there are two natural ones, the first obtained by taking stable hyperelliptic curves and the second by taking admissible covers. It turns out that the Brauer group of the former is trivial, while for the latter it is almost as large as in the non-compact case, a somewhat surprising difference as the two stacks are projective, smooth and birational, which would force their Brauer groups to be equal if they were schemes.

(Herrero talk) For a projective variety X, the moduli problem of coherent sheaves on X is naturally parametrized by a geometric object M called an "algebraic stack". In this talk I will explain a GIT-free construction of the moduli space of Gieseker semistable pure sheaves which is intrinsic to the moduli stack M. This approach also yields a Harder-Narasimhan stratification of the unstable locus of the stack. Our main technical tools are the theory of Theta-stability introduced by Halpern-Leistner, and some recent techniques developed by Alper, Halpern-Leistner and Heinloth. In order to apply these results, one needs to prove some monotonicity conditions for a polynomial numerical invariant on the stack. We show monotonicity by defining a higher dimensional analogue of the affine grassmannian for pure sheaves. If time allows, I will also explain some applications of these ideas to some other moduli problems. This talk is based on joint work with Daniel Halpern-Leistner and Trevor Jones.

(Singh talk) A Higgs bundle over an algebraic curve is a vector bundle with a twisted endomorphism. An important question is to calculate the volume of the groupoid of Higgs bundles over finite fields. In 2014, Olivier Schiffmann succeeded in finding the corresponding generating function and together with Mozvogoy reduced the problem to counting pairs of a vector bundle and a nilpotent endomorphism. It was generalized recently by Anton Mellit to the case of Higgs bundles with regular singularities. An important step in Mellit’s calculations is the case of P^1 and two marked points, which allows him to relate the corresponding generating function with the Macdonald polynomials It is a natural question to generalize Mellit’s calculations to arbitrary reductive groups. In the talk we will consider the case of P^1 with two points for an arbitrary split connected reductive group G over F_q. Firstly, we give an explicit formula for the number of F_q-rational points of generalized Steinberg varieties of G. Secondly, for each principal G-bundle over P^1 , we give an explicit formula counting the number of triples consisting of parabolic structures at 0 and ∞ and compatible nilpotent sections of the associated adjoint bundle.

(Dhillon talk) The moduli stack of $G$-bundles on a curve occurs in diverse areas of mathematics. We will discuss various approaches to its motive. The talk will discuss the Aityah-Bott formula and Tamagawa numbers. In joint work with Kai Behrend we conjectured a formula for the motive in the K-ring of varieties and proved it in certain cases. Recently Benjamin Fayyazuddin-Ljungberg proved the conjecture for the symplectic group. A more refined approach has appeared in the work of Simon Pepin-Lehalleur and Victoria Hoskins which will be discussed.

(Sawant talk) Celebrated results of Steinberg and Matsumoto obtained about fifty years ago determine the universal central extension of certain algebraic groups. These results have led to a lot of interesting developments, for instance, the work of Brylinski and Deligne about determining the category of central extensions of a reductive group by K_2 in terms of certain quadratic forms. I will briefly survey these classical results and discuss how all these results can be uniformly explained and generalized using motivic homotopy theory. The main new ingredient in our work is the notion of cellular A^1-homology. The talk is based on joint work with Fabien Morel. This talk may be also of interest to those interested in motivic homotopy theory

  • Arithmetic Geometry Seminar calendar from 2020-2021
  • Arithmetic Geometry Seminar calendar from 2019-2020


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