Algebraic Geometry and Algebraic KTheory
May 23rd  25th, 2022
Washington University in St. Louis
Speakers
 Roya Beheshti (Washington University in St. Louis)
Moduli spaces of rational curves on del Pezzo surfaces and Fano threefolds
I will talk about several results on the geometry of moduli spaces of rational curves on del Pezzo surfaces and Fano threefolds with regard to the Geometric Manin's Conjecture. The key question that I will discuss is: what can be said about the number of irreducible components of the moduli space as the anticanonical degree of the curves increases?
This is joint work with Brian Lehmann, Eric Riedl, and Sho Tanimoto.
 Spencer Bloch (University of Chicago)
Height pairings for algebraic cycles
There has been considerable work over the years on height pairings for algebraic cycles. Geometrically, height pairings for 0cycles and divisors on curves and on abelian varieties are now quite well understood. There has even been considerable progress arithmetically on the Birch and Swinnerton Dyer conjecture. The problem for cycles of dimension > 0 is more difficult, though it would seem to be important. There is, for example, a good deal of recent work on relations with the Langlands program.
I will explain a few ideas and examples.
I. The basic geometric and motivic setup. The Selmer group, Birch and Swinnerton Dyer, Tamagawa numbers.
II ''Stupid heights''; functions on Hilbert schemes.
III. padic Hodge theory and local heights.
IV. The double point example; Beilinson's theorem.
 Kirti Joshi (University of Arizona)
The absolute Grothendieck conjecture is false for FarguesFontaine curves
I will explain the theorem in the title, and its relevance and sketch a proof and some explicit examples.
 Madhav Nori (University of Chicago)
Quasiunipotency of local monodromy
We consider some variations and applications of Grothendieck's theorem on local monodromy to finite abelian coverings of varieties.
 Kapil Paranjape (IISER Mohali)
Algebraization of point set topology
In a collaboration with Alok Maharana, we studied the work of Stone, Hochster, Isbell, Joyal, Johnstone and others on locales and sober topological spaces. As a result we can exhibit a locale or a sober topological space as a dense sublocale of an algebraic scheme over a field. Such dense sublocales are understood in terms of nuclei. The embedding is functorial.
 Chris Peterson (Colorado State University)
Random thoughts and an old friend
Back in the previous century (19981999) I was a postdoc with Mohan Kumar (and, de facto, with Prabhakar Rao). It was an exciting time for me as these were two of my mathematical heroes. Inspired by Mohan's remarkable paper ''Construction of rank two vector bundles on
P^{4} in positive characteristic'', we worked on a project which used the tautological bundle on the Grassmannian Gr(2,4) to give a very concrete construction of low rank vector bundles on
P^{4}. This was over an algebraically closed field with positive characteristic. Working in positive characteristic was new for me as I had always worked over
C. The project definitely opened my mind to new ways of thinking. Today's talk revisits the tautological bundle on Gr(2,4) but over the reals using ''random algebraic geometry''. We consider the problem of determining the expected number of real lines lying on a random real cubic surface in
P^{3}.
 Bangere Purnaprajna (University of Kansas)
Deformation of canonical maps and its applications
A framework was developed in a joint work with F. J. Gallego and M. Gonzalez to systematically deal with the deformation
of finite morphisms, multiple scheme structures on algebraic varieties and their smoothing. There are several applications of this
framework. In this talk I will talk about some of them. First is the description of some components of the moduli space of varieties of
general type in all dimensions. In particular, we show the existence of components of the moduli space of general type in all dimensions,
that are analogue of the moduli space of curves of genus g>2. Secondly, we give a new method to construct smooth varieties in projective space
embedded by complete sub canonical linear series within the range of the Hartshorne conjecture and beyond. Are all of them complete intersections?
We also construct systematically, smooth non complete intersection subvarieties embedded by complete linear series outside the range of the Hartshorne
conjecture. As a byproduct, we construct simple canonical varieties of any dimension, expanding the original question posed by Enriques for algebraic surfaces.
This is joint work with F. J. Gallego, J Mukherjee and D. Raychaudhury.
 Prabhakar Rao (University of Missouri  St. Louis)
Vector bundles on varieties
I will revisit the important work of Professor Kumar on vector bundles on projective spaces, review some collaborations with Chris Peterson as well as with G. Ravindra on related topics and finish with some recent extensions of these ideas.
 Vasudevan Srinivas (TIFR)
On finite presentation for the tame fundamental group
This is a report on joint work with H. Esnault and M. Schusterman.
Recall that the etale fundamental group of a variety over an algebraically
closed field of characteristic 0 is known to be a finitely presented
profinite group; this is proved by first reducing to varieties over the
complex numbers, and then comparing with the topological fundamental
group. In positive characteristics, even if we restrict to smooth
varieties, finite generation fails in general for etale fundamental groups
of nonproper varieties (eg, for the affine line).
For a smooth variety with a smooth, projective compactification with a SNC
boundary divisor, we show that the tame fundamental group is a finitely
presented profinite group. In particular, this holds for the fundamental
groups of smooth projective varieties.
 David Wright (Washington University in St. Louis)
A survey of the illustrious Jacobian conjecture
The celebrated Jacobian Conjecture asserts:
Let F be a polynomial map from
C^{n} to
C^{n}. If the Jacobian determinant of F is everywhere nonvanishing, then F is a polynomial automorphism.
This conjecture, now 83 years old and still unsolved for n>1, can be viewed as a problem in differential geometry, algebraic geometry, commutative algebra, analysis, topology, and combinatorics. Over the years it has proved to be remarkably elusive and deceptive, with numerous false proofs appearing, some published by notable mathematicians. In this talk we will recount this sometimes amusing story, name the key players, discuss some of the problemâ€™s subtle obstructions, state known reductions of the problem, identify cases where it has been proved to be true, and state some important equivalent formulations.
Click on names to see titles and abstracts.
Schedule
Talks are nominally 50 minutes, with 10 minutes for questions.
Monday May 23 
Tuesday May 24 
Wednesday May 25 
9:3010:30 
Beheshti 
9:3010:30 
Bloch 
9:3010:30 
Peterson 
11:0012:00^{*} 
Joshi 
11:0012:00 
Nori 
11:0012:00 
Wright 
1:302:30 
Purna 
1:302:30 
Paranjape 


3:004:00 
Rao 
3:004:00 
Srinivas 








* There will be a group photo at 12:05 pm on the front steps of Brookings Hall, immediately after this talk.
We plan to have a party at David Wright's house in the Central West End (starting at 7:30pm) on Monday evening. [David's Address: 4548 Westminster Place, St. Louis]
On Tuesday, after the conference, there will be a banquet at Brasserie by Niche (6pm8:30pm; 4580 Laclede Ave., St. Louis), followed by a musical event at Mohan's house (''Geometry and Music'', performed by Purna, starting at 9pm). [Mohan's address: 7150 Waterman Ave., University City]
At the end of the conference on Wednesday, Mohan will provide lunch for participants at his house.
Accommodation
Clayton Plaza Hotel. Breakfast is included with a hotel reservation.
Clayton Plaza is 1.2 miles away from campus and 1.8 miles from the math department. The hotel used to have a shuttle service but this is temporarily suspended.
One option is to take the Metrolink from the Forsyth station to the Skinker station. It is also a reasonably nice walk down Forsyth Blvd if weather is good.
* = online participation