We discuss flows (and group-connectivity) in signed graphs, and prove a new result about group-connectivity in 3-edge-connected signed graphs. This is joint work with Alejandra Brewer Castano and Kathryn Nurse.
We will discuss the one-phase Muskat problem concerning the free boundary of Darcy fluids in porous media. It is known that there exists a class of non-graph initial boundary leading to self-intersection at a single point in finite time (splash singularity). On the other hand, we prove that the problem has a unique global-in-time solution if the initial boundary is a periodic Lipschitz graph of arbitrary size. This is based on joint work with H. Dong and F. Gancedo.
This talk will have two parts. The first half will describe how to construct symplectic structures on trisected 4-manifolds. This construction is inspired by projective complex geometry and completely characterizes symplectic 4-manifolds among all smooth 4-manifolds. The second half will address a curious phenomenon: symplectic 4-manifolds appear to not admit any interesting connected sum decompositions. One potential explanation is that every embedded 3-sphere can be made contact-type. I will outline some strategies to prove this from a trisections perspective, describe some of the obstructions, and give evidence that these obstructions may be overcome.
Please Note: The speaker will present in person.
The group SL(n) x SL(n) acts on m-tuples of n x n matrices by simultaneous left-right multiplication. Visu Makam and the presenter showed the ring of invariants is generated by invariants of degree at most mn^4. We will also discuss geometric aspects of this action and connections to algebraic complexity and the notion of noncommutative rank.
Please Note: Zoom: https://gatech.zoom.us/j/98256586748?pwd=SkJLZ3ZKcjZsM0JkbGdyZ1Y3Tk9udz09 Meeting ID: 982 5658 6748 Password: 929165
A class of graphs is said to be $\chi$-bounded with binding function $f$ if for every such graph $G$, it satisfies $\chi(G) \leq f(\omega(G)$, and polynomially $\chi$-bounded if $f$ is a polynomial. It was conjectured that chair-free graphs are perfectly divisible, and hence admit a quadratic $\chi$-binding function. In addition to confirming that chair-free graphs admit a quadratic $\chi$-binding function, we will extend the result by demonstrating that $t$-broom free graphs are polynomially $\chi$-bounded for any $t$ with binding function $f(\omega) = O(\omega^{t+1})$. A class of graphs is said to satisfy the Vizing bound if it admits the $\chi$-binding function $f(\omega) = \omega + 1$. It was conjectured that (fork, $K_3$)-free graphs would be 3-colorable, where fork is the graph obtained from $K_{1, 4}$ by subdividing two edges. This would also imply that (paw, fork)-free graphs satisfy the Vizing bound. We will prove this conjecture through a series of lemmas that constrain the structure of any minimal counterexample.
Several experts from the M2internals group will give tutorials and lead the discussion. This is a part of Meeting on Applied Algebraic Geometry.
(Macaulay2 is a software system devoted to supporting research in algebraic geometry and commutative algebra.)
The Meeting on Applied Algebraic Geometry (MAAG 2023) is a regional gathering which attracts participants primarily from the South-East of the United States. Previous meetings took place at Georgia Tech in 2015, 2018, and 2019, and at Clemson in 2016.
For more information and to register, please visit https://sites.google.com/view/maag-2023. The registration is free until February 28th, 2023, and the registration fee will become $50 after that.
MAAG will be followed by a Macaulay2 Day on April 16.
Organizers: Abeer Al Ahmadieh, Greg Blekherman, Anton Leykin, and Josephine Yu.
Zoom Link: Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09
Abstract: Linear response refers to the smooth change in the statistics of an observable in a dynamical system in response to a smooth parameter change in the dynamics. The computation of linear response has been a challenge, despite work pioneered by Ruelle giving a rigorous formula in Anosov systems. This is because typical linear perturbation-based methods are not applicable due to their instability in chaotic systems. Here, we give a new differentiable splitting of the parameter perturbation vector field, which leaves the resulting split Ruelle's formula amenable to efficient computation. A key ingredient of the overall algorithm, called space-split sensitivity, is a new recursive method to differentiate quantities along the unstable manifold.
In the second part, we discuss a new KAM method-inspired construction of transport maps. Transport maps are transformations between the sample space of a source (which is generally easy to sample) and a target (typically non-Gaussian) probability distribution. The new construction arises from an infinite-dimensional generalization of a Newton method to find the zero of a "score operator". We define such a score operator that gives the difference of the score -- gradient of logarithm of density -- of a transported distribution from the target score. The new construction is iterative, enjoys fast convergence under smoothness assumptions, and does not make a parametric ansatz on the transport map.
