RTG Workshop
U.C. Berkeley
September 26-29, 2012

Invited lecturers:

The U.C. Berkeley NSF RTG in representation theory, geometry, and combinatorics will host a 4-day workshop September 26-29, 2012, featuring three minicourses (abstracts), as well as talks by some participants.

There are no registration fees, but individuals interested in attending are kindly requested to register (here ).  There is funding available for graduate students and postdocs, persons from under-represented groups are especially encouraged to apply. The deadline to apply for funding is August 13, 2012.

If you have any questions, do not hesitate to contact the organizers: Luke Oeding, Noah Giansiracusa.

Schedule
Wednesday, September 26, 2012
9:00AM - 9:30AM Alcove Coffee/Tea
9:30AM - 9:45AMSibley Auditorium Welcome
9:45AM - 11:00AMSibley Auditorium Giorgio OttavianiTensor decomposition and tensor rank. (slides)
11:00AM - 12:30PMSibley Auditorium Jan Draisma
12:30PM - 2:00PM Northside Lunch
2:00PM - 3:30PMSibley Auditorium Andrew SnowdenTwisted Commutative Algebras: (slides)
3:30PM - 4:00PM Alcove Coffee/Tea
4:00PM - 4:30PMSibley Auditorium Andrew CritchAlgebraic constraints on MPS-entangled qubits (slides)
4:30PM - 5:00PMSibley Auditorium Carmeliza Navasca New algorithms based on the reduced least-squares functional of the canonical polyadic decomposition (slides)
5:00PM - 5:30PMSibley Auditorium Shenglong Hu
6:30PM - 8:30PMCha-Am Conference Dinnerat Cha-Am, 1543 Shattuck Avenue, Berkekey (website)
Thursday, September 27, 2012
9:00AM - 9:30AMAlcove Coffee/Tea
9:30PM - 11:00AMSibley Auditorium Giorgio OttavianiNon abelian apolarity and applications (slides)
11:00AM - 11:30PMSibley Auditorium Galya Dobrovolska
11:30AM - 12:00PMSibley Auditorium Robert Krone
12:00PM - 1:30PMNorthside Lunch
1:30PM - 2:00PMSibley Auditorium Daniel Litt
2:00PM - 2:30PMSibley Auditorium Gus Schrader
2:30PM - 3:00PMSibley Auditorium Qingchun Ren
3:00PM - 4:00PMEvans 1015 Coffee/Tea
4:10PM - 5:00PMEvans 60 Jan Draisma
Friday, September 28, 2012
9:00AM - 9:30AMEvans 1015 Coffee/Tea
9:30AM - 11:00AMEvans 1015 Giorgio Ottaviani The complexity of the Matrix Multiplication Algorithm. (slides)
11:00AM - 12:30PMEvans 1015 Andrew SnowdenDelta Modules. (slides)
12:30PM - 2:00PM Northside Lunch
2:00PM - 2:30PMEvans 1015 Shamil ShakirovUndulation invariant of plane curves. (slides)
2:30 - 3:00PMEvans 1015 Claudiu Raicu Representation stability for syzygies of line bundles on Segre--Veronese varieties. (slides)
3:00PM - 3:30PMEvans 1015 Alexandru ConstantinescuKalai's conjecture and beyond. (slides)
3:30PM - 4:00PMEvans 1015 Coffee/Tea
4:00PM - 4:30PMEvans 1015 Daniel Halpern-Leistner The derived category of a GIT quotient (slides)
4:30PM - 5:00PMEvans 1015 Will Traves From Pascal's Mystic Hexagon Theorem to secant varieties. (slides)
Saturday, September 29, 2012
9:00AM - 9:30AMEvans 1015 Coffee/Tea
9:30AM - 11:00AMEvans 1015 Andrew SnowdenSyzygies of Segre embeddings and related varieties. (slides)
11:00AM - 12:30PMEvans 1015 Jan Draisma Tensors of bounded rank. (slides)
12:30PM - 2:00PM Northside Lunch
2:00PM - 2:30PMEvans 1015 Pablo SolisA wonderful embedding of the loop group. (slides)
2:30PM - 3:00PMEvans 1015 Greg BlekhermanReal Waring Rank. (slides)
3:00PM - 3:30PMEvans 1015 Jason Morton
3:30PM - 4:00PMEvans 1015 tea/coffee
4:00PM - 5:00PMEvans 1015 Ravi Vakil Stabilization of discriminants in the Grothendieck ring (slides)

# Abstracts

(we're using mathjax)

Let $$V_1, \dots, V_n$$ be complex vector spaces.  Inside the tensor product $$V_1 \otimes \dots \otimes V_n$$, there are many interesting subvarieties:  the Segre variety of pure tensors; higher subspace varieties; secant and tangent varieties to these varities; etc.  As one varies the $$V_i$$ and $$n$$, the coordinate rings, defining ideals and syzygy modules of these varieties each form an algebraic structure called a Delta-module.  Delta-modules are reasonable objects --- for instance, finitely generated Delta-modules are noetherian and have rational Hilbert series, in a suitable sense --- and thus provide a tool to study these varieties.  I plan to discuss the theory of Delta-modules, other closely related algebraic structures and how they can be used to study varieties like those mentioned above.

Systems of polynomial equations in infinitely many variables arise naturally in many areas of applied algebraic geometry. For instance, they may be limits of systems in finitely many variables that describe statistical models where some of the parameters tend to infinity. Typically, these infinite-dimensional systems have a lot of symmetry. In these lectures I will explain how to exploit this symmetry to obtain finiteness results, both theoretical and computational. In particular, I will give a detailed exposition of joint work with Kuttler showing that tensors of bounded (border) rank are defined in bounded degree. Beautiful combinatorics of well-quasi-ordered sets plays a fundamental role.

Every matrix of rank $$r$$ can be decomposed as the sum of exactly $$r$$ matrices of rank one. This elementary fact is the starting point of a broad theory that generalizes this decomposition and the concept of rank to tensors (multidimensional matrices) and to polynomials (symmetric tensors). The roots of this theory were developed in the 19th century by Sylvester and others, who studied the notion of apolarity and the Waring decomposition of a polynomial. In recent years tensor decomposition has found many striking applications in several fields like signal processing and phylogenetics. We review classical apolarity, Sylvester's algorithm and its modern generalizations in the setting of vector bundles (non abelian apolarity). We discuss the open problem to compute the rank of a general tensor and uniqueness of tensor decomposition. We apply this machinery to the complexity of Matrix Multiplication with the aid of representation theory.

Matrix product states (MPS) are tensors which, as models of 1-dimensional quantum spin systems, approximate the ground states of gapped local Hamiltonians. To classify such states of matter, we are interested in the projective variety of entangled spin systems which can arise as MPS. Algebraically, MPS models bear a similarity to hidden Markov models (HMM), statistical models used in fields as diverse as natural language processing, genomics, and aeronautics. In this talk, I will explain how reparametrization techniques from [C,2012] for binary HMM, including various classical results on trace algebras, aid in computing algebraic constraints on MPS states, and exhibit two interesting hypersurfaces of entangled qubit systems found by this method. (This work is joint with Jason Morton.)

We study the reduced least-squares functional of the canonical polyadic decomposition through elimination of one factor matrix. An analysis of the reduced functional gives several equivalent optimization problems, like a Rayleigh quotient or a projection. These formulations are the basis of several new algorithms: the centroid projection method for efficient computation of suboptimal solutions and two fixed point iterations for approximating the best rank-one and best rank-R decompositions under certain non-degeneracy conditions.
(This is joint work with S. Kindermann.)

E-eigenvalues of tensors was introduced in 2005. For a regular tensor, its E-eigenvalues are exactly the roots of its E-characteristic polynomial. The degree of the E-characteristic polynomials of generic tensors can be computed through Chern classes. The coefficients of the E-characteristic polynomial are invariants under the action of the orthogonal linear group. The constant term of the E-characteristic polynomial has a resultant formula. For tensors of dimension two, explicit formulae for the coefficients of the E-characteristic polynomials are given. Especially, the leading coefficient is a power of a sum of squares.

I will derive a result of Brion on Kronecker coefficients from a computation of Fourier-Malgange transform of some local systems on projective space corresponding to representations of the symmetric group. I will also indicate an elementary proof of this result of Brion.

A symmetric ideal in the polynomial ring of a countable number of variables is an ideal that is invariant under any permutations of the variables. While such ideals are usually not finitely generated, Aschenbrenner and Hillar proved that such ideals are finitely generated if you are allowed to apply permutations to the generators, and in fact there is a notion of Gröbner bases of these ideals. With Chris Hillar and Anton Leykin, I am working on an algorithm for calculating such Gröbner bases, and implementing this algorithm in Macaulay2. We are also exploring how these methods can be generalized to other group actions, and possible applications of these algorithms.

I'll describe the geometry of the space of line bundles on plane curves satisfying various cohomological conditions. In particular, I'll describe a generalization of the 2000 result of Beauville that the relative Jacobian of the universal smooth degree d plane curve is unirational -- I'll show that all "cohomological loci" on the relative Jacobian are unirational as well. I'll also recover Beauville's result via different methods. We use these methods to describe, for example, the space of degree s line bundles on degree d curves globally generated by 2 sections, that is, maps from plane curves to $$\mathbb{P}^1$$. Time permitting, I'll discuss other "motivic" invariants of these cohomological loci, e.g. Poincare and Hodge polynomials.

Kummer varieties are quotients of abelian varieties by the map sending an element to its inverse. Over the complex numbers, the second order theta functions embed a Kummer variety in $$\mathbb{P}^{2^g-1}$$. In this talk we focus on the special case of 3-dimensional Kummers. It turns out that Kummer threefolds in $$\mathbb{P}^7$$ can be described as the singular locus of a certain quartic hypersurface called the Coble quartic. We obtain an explicit expression for this quartic polynomial, with its coefficients expressed in terms of second order theta constants. We'll also discuss the universal Kummer threefold, a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds, and present some results on the equations defining this variety.

This is the second part of our discussion on universal Kummer threefolds. The coefficients of Coble quartics lie in $$\mathbb{P}^{14}$$. The Zariski closure is the Gopel variety. It can be reembedded into $$\mathbb{P}^{134}$$, where elements of $$Sp_6(F_2)$$ act by signed permutations in the coordinates. In this way, the Gopel variety becomes the intersection of $$\mathbb{P}^{14}$$ and a toric variety inside $$\mathbb{P}^{134}$$. This makes it more convenient for studying the tropicalization of the Gopel variety. Finally, this leads to a version of universal Kummer threefold in $$\mathbb{P}^7\times \mathbb{P}^8$$, where the moduli part is parametrized by 7 points in $$\mathbb{P}^2$$.

A classical problem in algebraic geometry is to determine, whether a given plane curve has tangent lines of multiplicity four or higher. A.Cayley proved in 19th century that there exists an invariant of degree 6(r-3)(3r-2) that vanishes if and only if such lines exist. Because of extremely high degree, he did not give any explicit formula for these invariants. Using modern methods, we are now able to give an explicit formula for undulation invariants for quartic (r=4) and quintic (r=5) curves. .

I will discuss the multivariate version of Church and Farb's notion of representation stability and explain how it applies to the syzygies of line bundles on products of projective spaces. I will give bounds for when stabilization occurs and show that these bounds are sometimes sharp by describing the linear syzygies for a family of line bundles on Segre varieties. Part of the motivation for this work comes from the fact that Ein and Lazarsfeld's conjecture on the asymptotic vanishing of syzygies for arbitrary varieties reduces to the case of line bundles on a product of (at most three) projective spaces.

Starting from an unpublished conjecture of Kalai and from a conjecture of Eisenbud, Green and Harris, we study several problems relating h-vectors of Cohen-Macaulay, flag simplicial complexes and face vectors of simplicial complexes. We will first sketch a proof of Kalai's conjecture for vertex decomposable complexes. We then state two new conjectures which arise naturally from this proof, and provide some evidence in their support. These results are part of a joint work with Matteo Varbaro, cf. ArXiv:1004.0170

The derived category of an algebraic variety encodes all of the information about sheaf cohomology and lots of topological information. For a variety $$X$$ acted on by a reductive group, one can consider the derived category of equivariant coherent sheaves on $$X$$, or the derived category of a GIT quotient of $$X$$. I will describe a new method of studying the derived category of a GIT quotient by identifying it with a subcategory the equivariant category. This is analogous to a classical description of the cohomology of a GIT quotient due to F. Kirwan, L. Jeffrey, and others. I will apply this technique to produce examples of non-isomorphic varieties which have equivalent derived categories, and to produce examples of automorphisms of derived categories which don't come from automorphisms of the variety itself.

I describe the wonderful compacti cation of loop groups. These compacti cations are obtained by adding normal-crossing boundary divisors to the group LG of loops in a reductive group $$G$$ (or more accurately, to the semi-direct product $$\mathbb{C}^* \times LG$$ in a manner equivariant for the left and right $$\mathbb{C}^* \times LG$$-actions. The analogue for a torus group $$T$$ is the theory of toric varieties; for an adjoint group $$G$$, this is the wonderful compactications of De Concini and Procesi. The loop group analogue is suggested by work of Faltings in relation to the compacti cation of moduli of $$G$$-bundles over nodal curves. Using the loop analogue one can construct a 'wonderful' completion of the moduli stack of $$G$$-bundles over nodal curves.

I'll discuss an application of secants to Segre varieties in a constructive geometry problem that has roots in Pascal's Mystic Hexagon Theorem.

While symmetric tensor decomposition is usually studied over the complex numbers, the situation for real symmetric tensors is more complicated. I will discuss a proof of the conjecture of Comon and Ottaviani that typical real Waring ranks of bivariate forms of degree $$d$$ take all integer values between $$\lfloor \frac{d+2}{2}\rfloor$$ and $$d$$. That is for all $$d$$ and all $$\lfloor \frac{d+2}{2}\rfloor \leq m \leq d$$ there exists a bivariate form $$f$$ such that $$f$$ can be written as a linear combination of $$m$$ $$d$$-th powers of real linear forms and no fewer, and additionally all forms in an open neighborhood of $$f$$ also possess this property. Equivalently, for all $$d$$ and any $$\lfloor \frac{d+2}{2}\rfloor \leq m \leq d$$ there exists a symmetric real bivariate tensor $$t$$ of order $$d$$ such that $$t$$ can be written as a linear combination of $$m$$ symmetric real tensors of rank 1 and no fewer, and additionally all tensors in an open neighborhood of $$t$$ also possess this property. arXiv:1205.3257

I'll talk about some potential applications of high-dimensional and infinite-dimensional algebraic geometry arising in the study of quantum many-body systems. As with Matrix Product States, we can use what we've learned in algebraic statistics to say something about the models currently used to study such systems. The importance of the thermodynamic limit provides strong motivation for understanding the infinite-dimensional case.

We consider the limiting behavior'' of discriminants, by which we mean informally the closure of the locus in some parameter space of some type of object where the objects have certain singularities. We are looking for the kind of stabilization of algebraic structure as the "problem gets large", of the sort you will have seen in the lectures of Draisma and Snowden. We focus on the space of partially labeled points on a variety $$X$$, and linear systems on $$X$$. These are connected --- we use the first to understand the second. We describe their classes in the "ring of motives", as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization can be described in terms of the motivic zeta values. The results extend parallel results in both arithmetic and topology. I will also present a conjecture (on motivic stabilization of symmetric powers'') suggested by our work. Although it is true in important cases, Daniel Litt has shown that it contradicts other hoped-for statements. This is joint work with Melanie Wood. (This is less technical than it sounds, and I will define everything from scratch.)

Organized by: Luke Oeding, Noah Giansiracusa and the UC Berkeley RTG in Representation Theory, Geometry, and Combinatorics.
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