| Date: | January 20, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Tim Seynnaeve |
| Title: | Matrix product states, geometry, and invariant theory |
| Abstract: | 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.
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| Date: | January 23, 2023 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Julia Lindberg, University of Texas |
| Title: | Estimating Gaussian mixtures using sparse polynomial moment systems |
| Abstract: | 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. |
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| Date: | January 30, 2023 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Suhan Zhong, Texas A&M University |
| Title: | Solving Nash equilibrium problems of polynomials |
| Abstract: | 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. |
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| Date: | February 6, 2023 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | JM Landsberg, TAMU |
| Title: | An old problem in linear algebra relevant for quantum information theory and complexity theory, and what modern algebraic geometry can tell us about it. |
| Abstract: | 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.
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| Date: | February 13, 2023 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Liena Colarte Gómez, IPAM (Warsaw) |
| Title: | aCM projections of Veronese varieties |
| Abstract: | 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. |
|
| Date: | February 20, 2023 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Mounir NIsse, Xiamen University Malaysia |
| Title: | Moment map, knots, and coamoebas |
| Abstract: | We give a topological description of the inverse image of the moment map. In fact, considering a closed mechanical linkage, there is a connection between the fibers of the moment map and polygon linkages. Also, we realize any torus knot (more generally, a torus link) as the critical values of the argument map restricted to a complex plane curve. I will give some applications in physics and tropical geometry. Moreover, I will ask some interesting questions in both complex and tropical algebraic geometry. This work is based on a joint work in progress with Yen Kheng Lim. |
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| Date: | March 3, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Yen-Kheng Lim, Xiamen University Malaysia |
| Title: | Solving physics problems from the perspective of (tropical) algebraic geometry |
| Abstract: | I will show 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. [Joint work with Mounir Nisse] |
|
| Date: | March 6, 2023 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | H. Huang, Auburn |
| Title: | New examples of basic vector spaces of matrices of low rank |
| Abstract: | 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. Classifications of such spaces is an interesting and important problem. The only cases known are when r \leq 3 with two proofs given by Artkinson via the study of Atkinson normal form and Eisenbud-Harris via the study of the first Chern class of a sheaf associated to the space. We will discuss a connection between these two approaches. Then we will discuss recent developments of this problem when r = 4 by linking it with the study of space of complexes of length 3. |
|
| Date: | March 20, 2023 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Arthur Bik, IAS and MPI Leipzig |
| Title: | Strength of infinite polynomials |
| Abstract: | The topic of this talk is the strength of polynomials (previously known as Schmidt rank) defined by Ananyan and Hochster in their paper proving Stillman's conjecture. It is the minimal number of reducibles that sum up to the polynomial. In many settings, one can divide polynomials into two classes, those of high strength and those of low strength, and investigate these classes separately. By definition, polynomials of low strength have structure in the sense that they have a description in terms of a small number of lower degree polynomials. For high strength polynomials, we search for other kinds of structure. During the talk, I will discuss ways in which polynomials of high strength are similar to infinite polynomials of infinite strength, which are often easier to understand. |
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| Date: | March 24, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 506A |
| Speaker: | H, Keneshlou, U. Konstanz |
| Title: | The construction of regular maps to the Grassmannian |
| Abstract: | A continuous map f : C^n −→ C^N is called k-regular, if the image of any
k distinct points in C^N are linearly independent. The study of existence of
regular map was initiated by Borusk 1957, and later attracted attention due
to its connection with the existence of interpolation spaces in approximation
theory, and certain inverse vector bundles in algebraic topology.
In this talk, based on a joint work with Joachim Jelisiejew, we consider the
general problem of the existence of regular maps
to Grassmannians C^n −→Gr(τ, C^N ). We will discuss the tools and methods of algebra and algebraic
geometry to provide an upper bound on N, for which a regular map exists.
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| Date: | March 31, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Máté L. Telek, University of Copenhagen |
| Title: | Reaction networks and a generalization of Descartes’ rule of signs to hypersurfaces |
| Abstract: | The classical Descartes’ rule of signs provides an easily computable upper bound for the number of positive real roots of a univariate polynomial with real coefficients. Descartes' rule of signs is of special importance in applications where positive solutions to polynomial systems are the object of study. This is the case in reaction network theory where variables are concentrations or abundances. Motivated by this setting, we give conditions based on the geometrical configuration of the exponents and the sign of the coefficients of a polynomial that guarantee that the number of connected components of the complement of the hypersurface where the defining polynomial attains a negative value is at most one or two. Furthermore, we discuss how these results can be applied to show that the parameter region of multistationarity of a reaction network is connected. |
|
| Date: | April 28, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Derek Wu, TAMU |
| Title: | Border rank bounds for GL_n-invariant tensors arising from spaces of matrices of constant rank |
| Abstract: | One measure of the complexity of a tensor is its border rank.
Finding the border rank of a tensor, or even bounding it, is a difficult problem that is currently an area of active research, as several problems in theoretical computer science come down to determining the border ranks of certain tensors.
For a class of $GL(V)$-invariant tensors lying in a $GL(V)$-invariant space $V\otimes U\otimes W$, where $U$ and $W$ are $GL(V)$-modules, we can take advantage of $GL(V)$-invariance to find border rank bounds for these tensors.
I discuss a special case where these tensors correspond to spaces of matrices of constant rank. |
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