Algebraic Geometry Seminar Fall 2016

The seminar meets on Fridays at 2:25 pm in Van Vleck B305.

Here is the schedule for the previous semester.

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Fall 2016 Schedule

date	speaker	title	host(s)
September 16	Alexander Pavlov (Wisconsin)	Betti Tables of MCM Modules over the Cones of Plane Cubics	local
September 23	PhilSang Yoo (Northwestern)	Classical Field Theories for Quantum Geometric Langlands	Dima
October 7	Botong Wang (Wisconsin)	Enumeration of points, lines, planes, etc.	local
October 14	Luke Oeding (Auburn)	Border ranks of monomials	Steven
October 28	Adam Boocher (Utah)	TBA	Daniel
November 4	Reserved	TBA	Daniel
November 11	Daniel Litt (Columbia)	TBA	Jordan
November 18	David Stapleton (Stony Brook)	TBA	Daniel
December 2	Rohini Ramadas (Michigan)	TBA	Daniel and Jordan
December 9	Robert Walker (Michigan)	TBA	Daniel

Abstracts

Alexander Pavlov

Betti Tables of MCM Modules over the Cones of Plane Cubics

Graded Betti numbers are classical invariants of finitely generated modules over graded rings describing the shape of a minimal free resolution. We show that for maximal Cohen-Macaulay (MCM) modules over a homogeneous coordinate rings of smooth Calabi-Yau varieties X computation of Betti numbers can be reduced to computations of dimensions of certain Hom groups in the bounded derived category D(X). In the simplest case of a smooth elliptic curve embedded into projective plane as a cubic we use our formula to get explicit answers for Betti numbers. In this case we show that there are only four possible shapes of the Betti tables up to a shifts in internal degree, and two possible shapes up to a shift in internal degree and taking syzygies.

PhilSang Yoo

Classical Field Theories for Quantum Geometric Langlands

One can study a class of classical field theories in a purely algebraic manner, thanks to the recent development of derived symplectic geometry. After reviewing the basics of derived symplectic geometry, I will discuss some interesting examples of classical field theories, including B-model, Chern-Simons theory, and Kapustin-Witten theory. Time permitting, I will make a proposal to understand quantum geometric Langlands and other related Langlands dualities in a unified way from the perspective of field theory.

Botong Wang

Enumeration of points, lines, planes, etc.

It is a theorem of de Brujin and Erdős that n points in the plane determines at least n lines, unless all the points lie on a line. This is one of the earliest results in enumerative combinatorial geometry. We will present a higher dimensional generalization to this theorem. Let E be a generating subset of a d-dimensional vector space. Let [math]\displaystyle{ W_k }[/math] be the number of k-dimensional subspaces that is generated by a subset of E. We show that [math]\displaystyle{ W_k\leq W_{d-k} }[/math], when [math]\displaystyle{ k\leq d/2 }[/math]. This confirms a "top-heavy" conjecture of Dowling and Wilson in 1974 for all matroids realizable over some field. The main ingredients of the proof are the hard Lefschetz theorem and the decomposition theorem. I will also talk about a proof of Welsh and Mason's log-concave conjecture on the number of k-element independent sets. These are joint works with June Huh.

Luke Oeding

Border ranks of monomials

What is the minimal number of terms needed to write a monomial as a sum of powers? What if you allow limits? Here are some minimal examples:

[math]\displaystyle{ 4xy = (x+y)^2 - (x-y)^2 }[/math]

[math]\displaystyle{ 24xyz = (x+y+z)^3 + (x-y-z)^3 + (-x-y+z)^3 + (-x+y-z)^3 }[/math]

[math]\displaystyle{ 192xyzw = (x+y+z+w)^4 - (-x+y+z+w)^4 - (x-y+z+w)^4 - (x+y-z+w)^4 - (x+y+z-w)^4 + (-x-y+z+w)^4 + (-x+y-z+w)^4 + (-x+y+z-w)^4 }[/math]

The monomial [math]\displaystyle{ x^2y }[/math] has a minimal expression as a sum of 3 cubes:

[math]\displaystyle{ 6x^2y = (x+y)^3 + (-x+y)^3 -2y^3 }[/math]

But you can use only 2 cubes if you allow a limit:

[math]\displaystyle{ 6x^2y = \lim_{\epsilon \to 0} \frac{(x^3 - (x-\epsilon y)^3)}{\epsilon} }[/math]

Can you do something similar with xyzw? Previously it wasn't known whether the minimal number of powers in a limiting expression for xyzw was 7 or 8. I will answer this and the analogous question for all monomials.

The polynomial Waring problem is to write a polynomial as linear combination of powers of linear forms in the minimal possible way. The minimal number of summands is called the rank of the polynomial. The solution in the case of monomials was given in 2012 by Carlini--Catalisano--Geramita, and independently shortly thereafter by Buczynska--Buczynski--Teitler. In this talk I will address the problem of finding the border rank of each monomial.

Upper bounds on border rank were known since Landsberg-Teitler, 2010 and earlier. We use symmetry-enhanced linear algebra to provide polynomial certificates of lower bounds (which agree with the upper bounds). This work builds on the idea of Young flattenings, which were introduced by Landsberg and Ottaviani, and give determinantal equations for secant varieties and provide lower bounds for border ranks of tensors. We find special monomial-optimal Young flattenings that provide the best possible lower bound for all monomials up to degree 6. For degree 7 and higher these flattenings no longer suffice for all monomials. To overcome this problem, we introduce partial Young flattenings and use them to give a lower bound on the border rank of monomials which agrees with Landsberg and Teitler's upper bound. I will also show how to implement Young flattenings and partial Young flattenings in Macaulay2 using Steven Sam's PieriMaps package.