Geometry Seminar
The seminar meets Mondays at 3 o'clock and Fridays at 4 o'clock,
in BLOC 628.
Talks are 50-60 minutes.
Visitor Information.
How to Give a Good Colloquium by John E. McCarthy.
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Date Time |
Location | Speaker |
Title – click for abstract |
 |
01/20 4:00pm |
BLOC 302 |
Tim Seynnaeve |
Matrix product states, geometry, and invariant theory
Matrix product states play an important role in quantum
information theory to represent states of many-body systems. They can be
seen as low-dimensional subvarieties of a high-dimensional tensor space.
In this talk, I will introduce two variants: homogeneous matrix product
states and uniform matrix product states. Studying the linear spans of
these varieties leads to a natural connection with invariant theory of
matrices. For homogeneous matrix product states, a classical result on
polynomial identities of matrices leads to a formula for the dimension
of the linear span, in the case of 2x2 matrices. I will explain this
connection, and the difficulties that arise when generalizing it to the
case of uniform matrix product states.
This talk is partially based on joint work with Claudia De Lazzari and
Harshit Motwani.
|
 |
01/23 3:00pm |
BLOC 302 |
Julia Lindberg University of Texas |
Estimating Gaussian mixtures using sparse polynomial moment systems
The method of moments is a statistical technique for density estimation that solves a system of moment equations to estimate the parameters of an unknown distribution. A fundamental question critical to understanding identifiability asks how many moment equations are needed to get finitely many solutions and how many solutions there are. We answer this question for classes of Gaussian mixture models using the tools of polyhedral geometry. Using these results, we present a homotopy method to perform parameter recovery, and therefore density estimation, for high dimensional Gaussian mixture models. The number of paths tracked in our method scales linearly in the dimension. |
 |
01/30 3:00pm |
BLOC 302 |
Suhan Zhong Texas A&M University |
Solving Nash equilibrium problems of polynomials
In this talk, we introduce a new approach for solving Nash equilibrium problems of polynomials. It is based on a hierarchy of semidefinite relaxations, which are constructed with Lagrange multiplier expressions and feasible extensions. Under some general assumptions, we show this approach either returns a Nash equilibrium or detects its nonexistence. |
 |
02/06 3:00pm |
BLOC 302 |
JM Landsberg TAMU |
An old problem in linear algebra relevant for quantum information theory and complexity theory, and what modern algebraic geometry can tell us about it.
A linear subspace of the space of bxc matrices is of
bounded rank r if no matrix in the space has rank greater than r. Such
spaces have been studied for a long time, but
little is known about them. I'll explain classical and modern results
about them, and why people in complexity theory and quantum information theory care about them.
|
 |
02/13 3:00pm |
BLOC 302 |
Liena Colarte Gómez IPAM (Warsaw) |
aCM projections of Veronese varieties
Gröbner's problem is a longstanding open question, posed by
Gröbner in 1967, about determining when a monomial projection of a
Veronese variety is an arithmetically Cohen-Macaulay variety. In this
talk, we review the state of the art of Gröbner's problem and we present
new contributions. Our approach takes advantage of the invariants of
finite abelian groups and their combinatorics. |
 |
02/20 3:00pm |
BLOC 302 |
Mounir NIsse Xiamen University Malaysia |
|
 |
03/03 4:00pm |
BLOC 302 |
Yen-Kheng Lim Xiamen University Malaysia |
Solving physics problems from the perspective of (tropical) algebraic geometry
In the first part of the talk, it will be shown how the partition function in statistical mechanics can be interpreted as an algebraic variety. In accordance to earlier literature, the zero-temperature limit is equivalent to taking the tropical limit of the algebraic variety. Previous literature have also generalised the temperature parameter to an n-vector. Here, we show that in the case of n=2, the two components of this generalised quantity are the inverse temperature and inverse temperature times chemical potential, respectively. Other values of n can also be similarly interpreted as various intensive thermodynamic parameters. The second part of the talk concerns null geodesics in four dimensional spacetimes. In particular, we observe that the condition for null circular orbits defines an A-discriminantal variety. A theorem by Rojas and Rusek for A-discriminants leads to the interpretation that there are two branches of null circular orbits for certain classes of spacetimes. A physical consequence of this theorem is that light rings around generic black holes with non-degenerate horizons are unstable. [Joint work with Mounir Nisse] |
 |
03/06 3:00pm |
BLOC 302 |
H. Huang Auburn |
TBA |
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03/20 3:00pm |
BLOC 628 |
Arthur Bik IAS and MPI Leipzig |
TBA |
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03/24 4:00pm |
BLOC 302 |
H, Keneshlou U. Konstanz |
TBA |