# Geometry Seminar

The seminar meets Mondays at 3 o'clock in BLOC 220, and Fridays at 4 o'clock in BLOC 117. Talks are 50-60 minutes. Visitor Information. How to Give a Good Colloquium by John E. McCarthy.

Date Time |
Location | Speaker | Title – click for abstract | |
---|---|---|---|---|

01/173:00pm |
506A | Kevin Tucker Univ. of Illinois at Chicago |
Local Fundamental Groups of Strongly F-Regular VarietiesOne may study the non-manifold points of a complex algebraic
variety by analyzing the link of the singularity, i.e. the
intersection of a small nearby sphere with the variety. For varieties
of (complex) dimension two, it was shown by Mumford that the link is
simply connected if and only if the variety is actually smooth. A
similar statement fails in higher dimensions, but Kollár has
conjectured that fundamental group of the link should be finite for
certain mild and well-studied singularities called Kawamata Log
Terminal singularities. In this talk, I will give an overview of this
conjecture, and discuss ongoing recent work (with Carvajal-Schwede and
Bhatt-Carvajal-Graf-Schwede) on a positive characteristic weak variant
-- the finiteness of local fundamental groups for strongly F-regular
varieties. | |

01/303:00pm |
BLOC 220 | Taylor Brysiewicz TAMU |
The degree of SO(n)The degree of a variety is one of the most natural invariants to try to
compute. We give a formula for the degree of the special orthogonal
group SO(n) for the first time. This formula has a combinatorial
interpretation via non-intersecting lattice paths and also has
applications to low-rank semidefinite programming. We explain how to
verify this formula explicitly using a monodromy algorithm in numerical
algebraic geometry (for n<=7) and how such computations aid in further
study of the variety. | |

02/174:00pm |
BLOC 628 | Timo de Wolff TAMU |
Constrained Polynomial Optimization via SONCs and Relative Entropy ProgrammingDeciding nonnegativity of real polynomials is a fundamental problem in real algebraic geometry and polynomial optimization. Since this problem is NP-hard, one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. The standard certificates for nonnegativity are sums of squares (SOS). In practice, SOS based semidefinite programming (SDP) is the standard method to solve polynomial optimization problems.
In 2014, Iliman and I introduced an entirely new nonnegativity certificate based on sums of nonnegative circuit polynomials (SONC), which are independent of sums of squares. We successfully applied SONCs to global nonnegativity problems.
In Summer 2016, Dressler, Iliman, and I proved a Positivstellensatz for SONCs, which provides a converging hierarchy of lower bounds for constrained polynomial optimization problems. These bounds can be computed efficiently via relative entropy programming.
In this second of two talks on the topic I will give a brief overview about semidefinite, geometric, and relative entropy programming as well as Lasserre Relaxation. Afterwards, I will explain our converging hierarchy of lower bounds for constrained polynomial optimization and how they can be computed via relative entropy programming.
The first, corresponding talk will occur directly before in the algebra and combinatorics seminar. | |

02/244:00pm |
BLOC 628 | JM Landsberg TAMU |
Symmetry v. OptimalityThe talk will be a colloquium style talk - all are welcome.
I will discuss uses of algebraic geometry and representation theory in
complexity theory. I will explain how these geometric methods have been successful in proving lower complexity bounds: unblocking the problem of lower bounds for the complexity of matrix multiplication, which had been stalled for over thirty years, and providing the first exponential separation of the permanent from the determinant in any restricted model. (The permanent v. determinant problem is an algebraic cousin of the P v. NP problem.) I will also discuss exciting new work that indicates that these methods can also be used to provide complexity upper bounds, in fact construct explicit algorithms. This is joint work with numerous co-authors including G. Ballard, A. Conner, C. Ikenmeyer, M. Michalek, G. Ottaviani, and N. Ryder. | |

03/104:00pm |
BLOC 220 - NOTE | Scott Aaronson UT Austin |
Boson Sampling and the Permanents of Gaussian MatricesI'll discuss BosonSampling, a proposal by myself and Alex
Arkhipov to demonstrate "quantum supremacy" (that is, an exponential
computational speedup over classical computers), using a
linear-optical setup that falls far short of being a universal quantum
computer. The goal, in BosonSampling, is to sample from a certain
kind of probability distribution, in which the probabilities are given
by the absolute squares of permanents of complex matrices (n-by-n
matrices, if there are n photons involved). Of particular interest to
mathematicians is that the BosonSampling program leads naturally to
rich mathematical questions---some of which we've answered, but many
of which remain open---about the permanent itself. (For example: are
permanents of i.i.d. Gaussian matrices close to lognormally
distributed? Is there an efficient algorithm to estimate them?) I'll
focus mainly on those questions. No quantum computation background is
needed for this talk. | |

03/244:00pm |
BLOC 628 | Robert Williams TAMU |
An introduction to convex neural codesThe brain encodes spatial structure via special neurons called
place cells which are associated with convex regions of space. We seek
to answer the decoding problem that arises from this situation: knowing
only the firing pattern of neurons, how can we tell if it corresponds to
convex regions? We will introduce tools from algebra and geometry and
show how they can be used to determine if a given neural code can arise
from place cells.
This talk is a practice talk for a job talk. | |

03/314:00pm |
BLOC 628 | Ata Firat Pir TAMU |
Irrational Toric VarietiesClassical toric varieties come in two flavors: Normal toric varieties are given by rational fans in R^n. A (not necessarily normal) affine toric variety is given by finite subset A of Z^n. Toric varieties are well understood and they can be approached in a combinatorial way, making it possible to compute examples of abstract concepts. Applications of mathematics have long studied the positive real part of a toric variety as the main object, where the points in A may be arbitrary points in R^n. In 1963, Birch showed the such an irrational toric variety is homeomorphic to the convex hull of the set A.
Recent work showing that all Hausdorff limits of translates of irrational toric varieties are toric degenerations suggested the need for a theory of irrational toric varieties associated to arbitrary fans in R^n. These are R^n_>-equivariant cell complexes dual to the fan. Among the pleasing parallels with the classical theory is that the space of Hausdorff limits of the irrational projective toric variety of a finite set A in R^n is homeomorphic to the secondary polytope of A.
This talk will sketch this story of irrational toric varieties. It represents work in progress with Sottile. | |

04/033:00pm |
BLOC 220 | Igor Zelenko TAMU |
On absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongationCR (Cauhy-Riemann or Complex-Real) geometry studies geometry of real submanifolds of complex spaces. The basic characteristic of a CR structure of hypersurface type is a special Hermitian form, called the Levi form. The geometry of CR structures with nondegenerate Levi form is well understood via construction of the canonical absolute parallelism (E. Cartan, Tanaka, Chern-Moser). When the Levi form is uniformly degenerate, one can define a more subtle nondegenericity condition, called k-nondegenericity, k>1, which is an obstruction for a real hypersurface to be a direct product of a real submanifold of a smaller dimension with a complex line. The absolute parallelism for 2-nondegenerate CR structures was constructed recently only in the minimal possible dimension 5 (Isaev-Zaitsev, Medori-Spiro, Pocciola) and in dimension 7 for some special cases (Porter). We extend these results to natural classes of 2-nondegenerate CR structures in arbitrary odd dimensions greater or equal to 5 by developing a bigraded analog of Tanaka prolongation procedure. The talk is based on the joint work with Curtis Porter. | |

04/103:00pm |
BLOC 220 | Fulvio Gesmundo TAMU |
Polynomials with maximal catalecticant rank and consequences in complexity theoryThe rank of catalecticant maps associated to polynomials has
been the main tool to prove lower bounds on the Waring rank of
polynomials since the nineteenth century. Despite a sufficiently
generic polynomial has maximal catalecticant rank, very few examples
of explicit polynomials with this property are known. In the first
part of this seminar, I present an extremely easy example of such
polynomials. In particular, this example leads to a no-go result for
the method of shifted partials that was introduced by
Gupta-Kamath-Kayal-Saptharishi as a promising approach to Valiant's
VPvsVNP conjecture. In the second part of the seminar, I show that
another example of polynomial with maximal catalecticant rank is the
complete symmetric function. This leads to a lower bound for its
Waring rank. This is joint work with JM Landsberg. | |

04/144:00pm |
BLOC 628 | Kaitlyn Phillipson St. Edwards Univ. |
Groebner Bases of Neural IdealsThe neural ideal was introduced recently as an algebraic object that can be used to better understand the combinatorial structure of neural codes. Every neural ideal has a particular generating set, called the canonical form, that directly encodes a minimal description of the receptive field structure intrinsic to the neural code. On the other hand, for a given monomial order, any polynomial ideal is also generated by its unique (reduced) Groebner basis with respect to that monomial order. How are these two types of generating sets - canonical forms and Groebner bases - related? In this talk, we will demonstrate that when the canonical form of the neural ideal is a Groebner basis, it is the union of all reduced Groebner bases for the ideal (i.e. the universal Groebner basis). A natural question to pursue, then, is under what conditions will the canonical form be a Groebner basis? We will give some partial answers to this question. In addition, we will discuss what the Groebner basis elements can tell us about the structure of the receptive field. This is joint work with numerous co-authors, including Rebecca Garcia, Luis David Garcia-Puente, and Anne Shiu. | |

04/174:00pm |
BLOC 628 | Frank Sottile TAMU |
The trace test in numerical algebraic geometryNumerical algebraic geometry uses tools from numerical analysis
to study algebraic varieties on a computer. In numerical algebraic
geometry, a variety X is represented by a witness set, which is
a linear section of X in a projective or affine space.
A fundamental step is numerical irreducible decomposition that
decomposes a witness set into subsets corresponding to the
irreducible components of X using monodromy and the trace test.
In this talk I will introduce numerical algebraic geometry and
witness sets, and describe numerical irreducible decomposition,
including a new and elementary proof of the trace test. I will
then explain versions of witness sets, the trace test, and numerical
irreducible decomposition for multihomogeneous varieties X
that take advantage of this structure.
This is joint work with Anton Leykin and Jose Rodriguez. | |

04/2110:00pm |
TAGS Conference | | ||

04/243:00pm |
BLOC 220 | Corey Harris Florida State |
Chern-Mather class of the multiview varietyThe multiview variety associated to a collection of N cameras
records which sequences of image points in P^2N can be obtained by taking
pictures of a given world point x∈P3 with the cameras. In order to
reconstruct a scene from its picture under the different cameras it is
important to be able to find the critical points of the function which
measures the distance between a general point u∈P^2N and the multiview
variety. We calculate a specific degree 3 polynomial that computes the
number of critical points as a function of N. In order to do this, we
construct a resolution of the multiview variety, and use it to compute its
Chern-Mather class. | |

04/284:00pm |
BLOC 628 | Elham Izadi UC San Diego |
The primal cohomology of theta divisorsThe celebrated Hodge conjecture predicts that Hodge substructures of the cohomology of an algebraic variety ``come from’’ its subvarieties. Part of the difficulty of the conjecture is due to the fact that examples for which the Hodge conjecture would be nontrivially true are few and far in between. The primal cohomology of the theta divisor of a principally polarized abelian variety is an interesting test case for the general Hodge conjecture. I will talk about some results and interesting open problems concerning the primal cohomology.
| |

05/013:00pm |
BLOC 220 | Jeff Sommars Univ. of Illinois at Chicago |
Algorithms for Computing Tropical PrevarietiesThe computation of the tropical prevariety is the first step in the application of polyhedral methods to compute positive dimensional solution sets of polynomial systems. In particular, pretropisms are candidate leading exponents for the power series developments of the solutions. The computation of the power series may start as soon as one pretropism is available, so the parallel computation of the tropical prevariety has an application in a pipelined solver.
I’ll present a parallel work-stealing implementation of a new algorithm to compute tropical prevarieties. This new software has made the first computation of the tropical prevariety of the cyclic 16-roots problem. I’ll also report on computational experiments of the n-body and n-vortex problems, as well as a tropical problem. | |

05/054:00pm |
BLOC 628 | Jens Forsgaard TAMU |
TBATBA | |

05/124:00pm |
BLOC 628 | Luca DiCerbo ICTP |
TBA | |

05/154:00pm |
BLOC 628 | G. Ballard Wake Forest |
TBA |

### Archives

Spring: | 2008 • 2007 • 2005 |

Fall: | 2008 • 2007 • 2006 |

Please contact Timo de Wolff for more information.