Algebra, Geometry, and Number Theory Seminar

Abstracts

Sean Sather-Wagstaff - Semidualizing modules arise independently in various algebraic contexts, e.g., commutative algebra and representation theory. In spite of this, our understanding of these modules is quite limited. For instance, until recently, it was not know that a given local ring admits only finitely many semidualizing modules (up to isomorphism). We will discuss recent progress on our understanding of these modules, over arbitrary local rings and over special classes of local rings. We will also discuss some motivation for studying these modules, including Avramov and Foxby's composition question for ring homomorphisms of finite G-dimension and growth rates for Bass numbers of local rings. This is joint work with Avramov, Iyengar, and Singh.

Sergey Galkin - I will give a uniform construction of a polynomial that is mirror dual to minuscule homogeneous varieties (e.g. Grassmannians), that we developed with Alexey Bondal. The construction involves a symmetry breaking mechanism of minuscule descent, that I will also explain. Homological mirror symmetry in this case predicts that derived categories of coherent sheaves on these varieties admit full exceptional collections, and it is likely that a uniform construction similar to our mirror construction could produce them, but it have not been found yet. In the pre-talk I will give some background on interplay between projective geometry of homogeneous varieties, representation theory and combinatorics. In particular I'll define (co)minuscule varieties and representations, recall geometric classification of Lie groups (after Landsberg and Manivel), describe Schubert cells, and define Bruhat and Hasse partially ordered sets, and explain Gonciulea-Lakshmibai's toric degenerations to moduli spaces of quiver representations. Pre-talk will start at 2:00 in LC 317R.

Marco Aldi - Lev Borisov gave an elegant proof of Berglund Hübsch duality, a special form of mirror symmetry relating explicit pairs of Calabi-Yau hypersurfaces. In this talk we present an alternate approach to Borisov's proof based on D-module theoretic considerations. An advantage of our method is that when specialized to overconvergent p-adic D-modules it yields non-trivial Frobenius chain-maps which are intertwined by the duality. We conclude by describing the arithmetic information encoded by the eigenvalues of these Frobenius map. This is joint work with Andrija Perunicic. Pre-talk will start at 2:00 in LC 317R.

Justin Sawon -In this talk we will describe some results about Betti, Hodge, and characteristic numbers of compact hyperkahler manifolds. In (complex) dimension four one can find universal bounds for all of these invariants (Beauville, Guan); in higher dimensions it is still possible to find some bounds. We also describe how these bounds are related to the question: are there finitely many hyperkahler manifolds in each dimension, up to deformation?

Sarah Trebat-Leder - (Pretalk: Ranks of Elliptic Curves and a Theorem of Stewart and Top) The first part of the talk will be an overview of elliptic curves, geared towards BSD and the distribution of ranks, and the second part will explain a result of Stewart and Top about the number of cubic twists of x^3 + y^3 = 1 with rank at least 2 or 3.

(Research Talk:) We revisit the mathematics that Ramanujan developed in connection with the famous ``taxi-cab" number 1729. A study of his writings reveals that he had been studying Euler's diophantine equation a^3+b^3=c^3+d^3. It turns out that Ramanujan's work anticipated deep structures and phenomena which have become fundamental objects in arithmetic geometry and number theory. We find that he discovered a K3 surface with Picard number 18, one which can be used to obtain infinitely many cubic twists over Q with rank \geq 2.

Hailong Dao - Let R be a commutative Noetherian ring of finite Krull dimension. It is a classical result that R is regular if and only if it has finite global dimension. In recent years, certain non-commutative rings which are modules-finite over R and have finite global dimension have become objects of intense interests. They can serve as "non-commutative desingularizations" of Spec(R) and have come up in the three-dimensional solution of the Bondal-Orlov conjecture, higher Auslander-Reiten theory and non-commutative minimal model program. Despite all that attention, these objects remain rather mysterious, for example we do not know fully when they exist, or what global dimensions can occur. In this talk I will describe some very recent work on these questions. Some of the work are joined with E. Faber, C. Ingalls, O. Iyama, R. Takahashi, I. Shipman and C. Vial.

Yuecheng Zhu - Thanks to the progress of minimal model program, there is a canonical way of compactifying the moduli space of KSBA stable pairs. A stable pair is roughly a variety and a divisor with some stability conditions. For the moduli space of abelian varieties, the compactification problem thus boils down to how to choose ample divisors. Luckily, mirror symmetry provides a natural choice of ample divisors. Strictly speaking, the divisors provided by mirror symmetry is not unique. Mirror symmetry provides a set of divisors near each large complex limit. However, choose any divisor from the set, we always get the same compactification of the moduli space. I will introduce both KSBA compactification and mirror symmetry, and explain how these two things fit together and provide a canonical compactificaiton of moduli of polarized abelian varieties.

Abbey Bourdon - Let E be an elliptic curve defined over a number field F. It is a classical theorem of Mordell and Weil that the collection of points of E with coordinates in F form a finitely generated abelian group. We seek to understand the subgroup of points with finite order. In particular, given a positive integer d, we would like to know precisely which abelian groups arise as the torsion subgroup of an elliptic curve defined over a number field of degree d. I will discuss recent progress on this problem for the special class of elliptic curves with complex multiplication (CM). In particular, if d is odd, we now have a complete classification of the groups that arise as the torsion subgroup of a CM elliptic curve defined over a number field of degree d. This is joint work with Paul Pollack. Pretalk in LC 310 at 1:10-2:00 pm.

Paul Pollack - For each positive integer d, let T(d) denote the supremum of all orders of groups E(F)[tors] appearing for an elliptic curve E defined over a degree d number field F. A celebrated theorem of Merel asserts that T(d) < infinity for all d. However, the known quantitative results in this direction are far from the conjectured truth. Let T_{CM}(d) be defined the same way as T(d), but with the restriction to CM elliptic curves. I will discuss some recent statistical results concerning T_{CM}(d) and related functions. Perhaps surprisingly, the "anatomy of integers" (as pioneered by Paul Erdos) plays a key role in the proofs. Joint work with Abbey Bourdon and Pete L. Clark.

Jerry Wang - The problem of the density of squarefree discriminant polynomials is an old one, being considered by many people, and the density being conjectured by Lenstra. A proof has been out of question for a long time. The reason it was desired is that a squarefree discriminant polynomial f immediately gives the ring of integers of Q[x]/f(x) and its Galois group. In recent joint work with Manjul Bhargava and Arul Shankar, we counted the number of odd degree polynomials with squarefree discriminant and proved the conjecture of Lenstra. In this talk, I will explain the general strategy of the squarefree sieve and the specific strategy to deal with discriminants which in turn leads to counting integral orbits for a representation of a non-reductive group.

Pretalk at 2:00 in 317R.

Frank Moore - The notion of a matrix factorization was introduced by Eisenbud in the commutative case in his study of bounded (periodic) free resolutions over complete intersections. Since then, matrix factorizations have appeared in a number of applications. We extend the notion of (homogeneous) matrix factorizations to regular normal elements of connected graded algebras over a field. If time permits, examples using the NCAlgebra package for Macaulay2 will be given.

Yusuf Mustopa - The CM (Castelnuovo-Mumford) regularity of a coherent sheaf F on a polarized projective variety X gives a measure of the algebraic complexity of F. A sheaf with CM regularity 0 is globally generated, while a sheaf with CM regularity 1 may not have any global sections at all. In this talk I will discuss results showing that on many well-known polarized irregular surfaces, torsion-free sheaves with CM regularity 1 are generic vanishing sheaves.

Last semester's seminar.