Seminars and Colloquia by Series

On the hardness of finding balanced independent sets in random bipartite graphs

Series
Graph Theory Seminar
Time
Tuesday, October 24, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Clough Commons room 102
Speaker
Yuzhou Wang – Georgia Tech –

We consider the algorithmic problem of finding large balanced independent sets in sparse random bipartite graphs, and more generally the problem of finding independent sets with specified proportions of vertices on each side of the bipartition. In a bipartite graph it is trivial to find an independent set of density at least half (take one of the partition classes). In contrast, in a random bipartite graph of average degree d, the largest balanced independent sets (containing equal number of vertices from each class) are typically of density (2 + od(1)) log d/d . Can we find such large balanced independent sets in these graphs efficiently? By utilizing the overlap gap property and the low-degree algorithmic framework, we prove that local and low-degree algorithms (even those that know the bipartition) cannot find balanced independent sets of density greater than (1 + ε) log d/d for any ε > 0 fixed and d large but constant.

Flexible Krylov methods for advanced regularization

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 23, 2023 - 14:00 for
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Malena Landman Sabate – Emory University –

Inverse problems involve the reconstruction of hidden objects from possibly noisy indirect measurements and are ubiquitous in a variety of scientific and engineering applications. This kind of problems have two main features that make them interesting yet challenging to solve. First, they tend to be ill-posed: the reconstruction is very sensitive to perturbations in the measurements. Second, real-world applications are often large-scale: resulting in computationally demanding tasks. In this talk I will focus on discrete linear problems: giving a general overview of the well-established class of solvers called Krylov subspace methods and its regularizing properties; as well as flexible variants that make them suitable to solve more challenging optimization tasks. I will show results and examples in different imaging applications.

Standard monomials and Gröbner bases for positroid varieties

Series
Algebra Seminar
Time
Monday, October 23, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ayah Almousa – University of Minnesota - Twin Cities –

Please Note: There will be a pre-seminar (aimed toward grad students and postdocs) from 11 am to 11:30 am in Skiles 006.

Positroid varieties are subvarieties of the Grassmannian that arise in the study of total positivity. Knutson, Lam, and Speyer described a certain type of Gröbner degeneration called the Hodge degeneration as projections of order complexes of intervals in the Bruhat order, but their description does not give an explicit Gröbner basis nor initial ideal. We give an explicit, combinatorial description of the Gröbner basis and initial ideal corresponding to the Hodge degeneration for an arbitrary positroid variety. As an application, we show that promotion on rectangular-shaped semistandard tableaux gives a bijection between standard monomials of a positroid variety and its cyclic shifts. This is joint work with Shiliang Gao (UIUC) and Daoji Huang (Minnesota).

Computation of high-order normal forms in diffeomorphisms

Series
CDSNS Colloquium
Time
Friday, October 20, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Joan Gimeno – Georgia Tech –

This talk will delve into a method specifically designed for
constructing high-order normal forms in Poincaré maps with high-order
precision and without any major assumption or structure of the
dynamical system itself. We will use the result to generate explicit
twist maps, calculating invariant tori, and determining the flying
time expansions around an elliptic fixed point of a Poincaré map. In
particular, this approach is able to check some non-degenerate
conditions in perturbation theory.

Electromagnetism and Falling Cats II

Series
Geometry Topology Working Seminar
Time
Friday, October 20, 2023 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Daniel Irvine – Georgia Institute of Technology –

In this talk I will continue to develop a parallel between the classical field theory of electromagnetism and geometric mechanics of animal locomotion. The focus of the previous talk was on electromagnetism, and the focus of this talk will be on the geometric mechanics of animal locomotion. We will investigate the aphorism that a cat dropped (from a safe height) upside-down always lands on her feet. I will explain how non-trivial topology of the configuration space of the cat can act as a "source" of locomotion.

No prior knowledge of classical field theory will be assumed. I will rely on some results from part 1, but I will review the relevant definitions.

An algorithm for comparing Legendrian links

Series
Geometry Topology Seminar
Time
Wednesday, October 18, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ivan Dynnikov – Steklov Mathematical Institute –

The talk is based on my joint works with Maxim Prasolov and Vladimir Shastin, where we studied the relation between rectangular diagrams of links and Legendrian links. This relation allows for a complete classification of exchange classes of rectangular diagrams in terms of equivalence classes of Legendrian links and their symmetry groups. Since all rectangular diagrams of given complexity can be searched, this yields a method to algorithmically compare Legendrian links. Of course, the general algorithm has too high complexity for a practical implementation, but in some situations, the most time consuming parts can be bypassed, which allows us to confirm the non-equivalence of Legendrian knots in several previously unresolved cases.

On the spectral synthesis for the unit circle in ${\mathcal F} L_s^q ({\mathbf R}^2)$

Series
Analysis Seminar
Time
Wednesday, October 18, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
TBA
Speaker
Masaharu Kobayashi – Hokkaido University –

Let ${\mathcal F}L^q_s ({\mathbf R}^2)$ denote the set of all tempered distributions $f \in {\mathcal S}^\prime ({\mathbf R}^2)$ such that the norm $ \| f \|_{{\mathcal F}L^q_s} = (\int_{{\mathbf R}^2}\, ( |{\mathcal F}[f](\xi)| \,( 1+ |\xi| )^s )^q\, d \xi )^{ \frac{1}{q} }$ is finite, where ${\mathcal F}[f]$ denotes the Fourier transform of $f$. We investigate the spectral synthesis for the unit circle $S^1 \subset {\mathbf R}^2$ in ${\mathcal F}L^q_s ({\mathbf R}^2)$ with $1\frac{2}{q^\prime}$, where $q^\prime$ denotes the conjugate exponent of $q$. This is joint work with Prof. Sato (Yamagata University).

The Acyclic Edge Coloring Conjecture holds asymptotically

Series
Graph Theory Seminar
Time
Tuesday, October 17, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lina Li – Iowa State University –

The Acyclic Edge Coloring Conjecture, posed independently by Fiam\v{c}ik in 1978 and Alon, Sudakov and Zaks in 2001, asserts that every graph can be properly edge colored with $\Delta+2$ colors such that there is no bicolored cycle. Over the years, this conjecture has attracted much attention. We prove that the conjecture holds asymptotically, that is $(1+o(1))\Delta$ colors suffice. This is joint work with Michelle Delcourt and Luke Postle.

The convergence problem in mean field control

Series
PDE Seminar
Time
Tuesday, October 17, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joe Jackson – University of Chicago –

This talk will be about the convergence problem in mean field control (MFC), i.e. the challenge of rigorously justifying the convergence of certain "symmetric" N-particle control problems towards their mean field counterparts. On the one hand, this convergence problem is already well-understood from a qualitative perspective, thanks to powerful probabilistic techniques based on compactness. On the other hand, quantitative results (i.e. rates of convergence) are more difficult to obtain, in large part because the value function of the mean field problem (which is also the solution to a certain Hamilton-Jacobi equation on the Wasserstein space) may fail to be C^1, even if all the data is smooth. After giving an overview of the convergence problem, I will discuss the results of two recent joint works with Cardaliaguet, Daudin, Delarue, and Souganidis, in which we use some ideas from the theory of viscosity solutions to overcome this lack of regularity and obtain rates of convergence of the N-particle value functions towards the value function of the corresponding MFC problem.

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