Georgia Institute of Technology College of Sciences

Intersection bodies are a popular construction in convex geometry. I will give an introduction on these objects, convex algebraic geometry, and starshaped sets in general. Then, we will analyze some features of intersection bodies and focus on the polyotopal case. Intersection bodies of polytopes are always semialgebraic sets and they are naturally related to hyperplane arrangements, which reveal their boundary structure. Finally, we will investigate their convexity, in the two-dimensional case. The exposition will be enriched by examples and computations. This is based on joint works with Katalin Berlow, Marie-Charlotte Brandenburg and Isabelle Shankar.

Semi-supervised learning refers to the problem of recovering an input-output map using many unlabeled examples and a few labeled ones. In this talk I will survey several mathematical questions arising from the Bayesian formulation of graph-based semi-supervised learning. These questions include the modeling of prior distributions for functions on graphs, the derivation of continuum limits for the posterior, the design of scalable posterior sampling algorithms, and the contraction of the posterior in the large data limit.

The Kauffman bracket skein module S(M) of a 3-manifold M classifies polynomial invariants of links in M satisfying Kauffman bracket skein relations. Witten conjectured that the skein module (over a field, with generic A) is finite dimensional for any closed 3-manifold M. This conjecture was proved by Gunningham, Jordan, and Safronov, however their work does not lead to an explicit computation of S(M).
In fact, S(M) has been computed for a few specific families of closed 3-manifolds so far. We introduce a novel method of computing these skein modules for certain rational homology spheres. (This is joint work with R. Detcherry and E. Kalfagianni.)

In this talk, I will present a method to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. Starting from a global solution to the geometric reduced system satisfying several pointwise estimates, we find a matching exact global solution to the original quasilinear wave equations. As an application of this method, we will construct nontrivial global solutions to Fritz John's counterexample $\Box u=u_tu_{tt}$ and the 3D compressible Euler equations without vorticity for $t\geq 0$.

Ramsey proved that for every positive integer r, every sufficiently large graph contains as an induced subgraph either a complete graph on r vertices or an independent set with r vertices.  It is well known that every sufficiently large, connected graph contains an induced subgraph isomorphic to one of a large complete graph, a large star, and a long path.  We prove an analogous result for 2-connected graphs.  Similarly, for infinite graphs, every infinite connected graph contains an induced subgraph isomorphic to one of the following: an infinite complete graph, an infinite star, and a ray.  We also prove an analogous result for infinite 2-connected graphs.

Are you tired of having to read a bunch of words during a seminar talk? Well, you’re in luck! This talk will be a (nearly) word-free exploration of a construction called unicorn paths. These paths are incredibly useful and can be used to show that both the curve graph and the arc graph of a surface are hyperbolic. 

 In first-passage percolation (FPP), we let $\tau_v$ be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If $F$ is the distribution function of $\tau_v$, there are different regimes: if $F(0)$ is small, this weight typically grows like a linear function of the distance, and when $F(0)$ is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results that show that if the sum of $F^{-1}(1/2+1/2^k)$ diverges, then a.s. there are exceptional times at which the weight grows atypically, but if the sum of $k^{7/8} F^{-1}(1/2+1/2^k)$ converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. Then we will consider what the model looks like when the weight of a long path is unusually small by considering an analogous construction to Kesten's incipient infinite cluster in the FPP setting. This is joint work with M. Damron, J. Hanson, W.-K. Lam.

Finally, we discuss a result related to work of Damron-Lam-Wang ('16) that the growth of the passage time to distance $n$ ($\mathbb{E}T(0,\partial B(n))$, where $B(n) = [-n,n]^2$)  has the same order (up to a constant factor) as the sequence $\mathbb{E}T^{\text{inv}}(0,\partial B(n))$. This second passage time is the minimal total weight of any path from 0 to $\partial B(n)$ that resides in a certain embedded invasion percolation cluster. We discuss a result that claims this constant factor cannot be taken to be 1. This result implies that the time constant for the model is different than that for the related invasion model, and that geodesics in the two models have different structures. This was joint work with M. Damron. 

This talk is centered around a symplectic approach to eigenvalue problems for systems of ordinary differential operators (e.g., Sturm-Liouville operators, canonical systems, and quantum graphs), multidimensional elliptic operators on bounded domains, and abstract self-adjoint extensions of symmetric operators in Hilbert spaces. The symplectic view naturally relates spectral counts for self-adjoint problems to the topological invariant called the Maslov index. In this talk, the notion of the Malsov index will be introduced in analytic terms and an overview of recent results on its role in spectral theory will be given. 

Whitney showed that every planar triangulation without separating triangles is Hamiltonian. This result was extended to all $4$-connected planar graphs by Tutte. Hakimi, Schmeichel, and Thomassen showed the first lower bound $n/ \log _2 n$ for the number of Hamiltonian cycles in every $n$-vertex $4$-connected planar triangulation and in the same paper, they conjectured that this number is at least $2(n-2)(n-4)$, with equality if and only if $G$ is a double wheel. We show that every $4$-connected planar triangulation on $n$ vertices has $\Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{\Omega(n^{1/4})}$ Hamiltonian cycles.

The Laplacian of a graph is a real symmetric matrix given by $L=D-A$, where $D$ is the degree matrix of the graph and $A$ is the adjacency matrix. The Laplacian is a central object in spectral graph theory, and the spectrum of $L$ contains information on the graph. In the case of a random graph the Laplacian will be a random real symmetric matrix with dependent entries. These random Laplacian matrices can be generalized by taking $A$ to be a random real symmetric matrix and $D$ to be a diagonal matrix with entries equal to the row sums of $A$. We will consider the eigenvalues of general random Laplacian matrices, and the challenges raised by the dependence between $D$ and $A$. After discussing the bulk global eigenvalue behavior of general random Laplacian matrices, we will focus in detail on fluctuations of the largest eigenvalue of $L$ when $A$ is a matrix of independent Gaussian random variables. The asymptotic behavior of these Gaussian Laplacian matrices has a particularly nice free probabilistic interpretation, which can be exploited in the study of their eigenvalues. We will see how this interpretation can locate the largest eigenvalue of $L$ with respect to the largest entry of $D$. This talk is based on joint work with Kyle Luh and Sean O'Rourke.

We will start sketching the proof of the quantitative unique continuation principle used in Ding-Smart from their key lemma. We will discuss the proof of a growth lemma from our key lemma, which (roughly) says that with high probability, eigenfunctions which are small on a high proportion of sites do not grow too rapidly.

Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world. More specifically, we will talk about the history of  exotic 4-manifolds and we will define the nice tools used to construct exotic 4-manifolds, like symplectic normal connect sum, Rational Blow-Down, Luttinger Surgery, Branch Covers, and Knot Surgery. Finally, we will provide various constructions of exotic 4-manifolds.

In this talk, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph $K_{m,n}$, extending a method from de Klerk et al. [SIAM J. Discrete Math. 20 (2006), 189--202] and the subsequent reduction by De Klerk, Pasechnik and Schrijver [Math. Prog. Ser. A and B, 109 (2007) 613--624].
 
We exploit the full symmetry of the problem by developing a block-diagonalization of the underlying matrix algebra and use it to improve bounds on several concrete instances. Our results imply that $\mathrm{cr}(K_{10,n}) \geq  4.87057 n^2 - 10n$, $\mathrm{cr}(K_{11,n}) \geq 5.99939 n^2-12.5n$, $\mathrm{cr}(K_{12,n}) \geq 7.25579 n^2 - 15n$, $\mathrm{cr}(K_{13,n}) \geq 8.65675 n^2-18n$ for all~$n$. The latter three bounds are computed using a new relaxation of the original semidefinite programming bound, by only requiring one small matrix block to be positive semidefinite. Our lower bound on $K_{13,n}$ implies that for each fixed $m \geq 13$, $\lim_{n \to \infty} \text{cr}(K_{m,n})/Z(m,n) \geq 0.8878 m/(m-1)$. Here $Z(m,n)$ is the Zarankiewicz number: the conjectured crossing number of $K_{m,n}$.
 
This talk is based on joint work with Sven Polak.

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.

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.

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.)