TBA by Speaker at Joint UGA/GT Topology seminar
- Series
- Geometry Topology Seminar
- Time
- Monday, February 26, 2024 - 16:30 for 1 hour (actually 50 minutes)
- Location
- University of Georgia
- Speaker
Designing effective positional encodings for graphs is key to building powerful graph transformers and enhancing message-passing graph neural networks’ expressive power. However, since there lacks a canonical order of nodes in the graph-structure data, the choice of positional encodings for graphs is often tricky. For example, Laplacian eigenmap is used as positional encodings in many works. However, it faces two fundamental challenges: (1) Non-uniqueness: there are many different eigen-decompositions of the same Laplacian, and (2) Instability: small perturbations to the Laplacian could result in completely different eigenvectors, leading to unpredictable changes in positional encoding. This is governed by the Davis-Kahan theorem, which further negatively impacts the model generalization. In this talk, we are to introduce some ideas on building stable positional encoding and show their benefits in model out-of-distribution generalization. The idea can be extended to some other types of node positional encodings. Finally, we evaluate the effectiveness of our method on molecular property prediction, link prediction, and out-of-distribution generalization tasks, finding improved generalization compared to existing positional encoding methods.
I will mainly talk about three papers:
1. Distance Encoding: Design Provably More Powerful Neural Networks for Graph Representation Learning,NeurIPS20, Pan Li, Yanbang Wang, Hongwei Wang, Jure Leskovec
2. Equivariant and Stable Positional Encoding for More Powerful Graph Neural Networks, ICLR22 Haorui Wang, Haoteng Yin, Muhan Zhang, Pan Li
3. On the Stability of Expressive Positional Encodings for Graphs, ICLR24 Yinan Huang, William Lu, Joshua Robinson, Yu Yang, Muhan Zhang, Stefanie Jegelka, Pan Li
Please Note: Note unusual date and length for the seminar!
It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties (e.g. tightness, fillability, vanishing or non-vanishing of various Floer or symplectic homology classes) of contact structures are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. The case of positive contact surgeries much more subtle. In this talk, extending an earlier work of the speaker with Conway and Etnyre, I will discuss some new results about symplectic fillability of positive contact surgeries, and in particular we will provide a necessary and sufficient condition for contact (positive) integer surgery along a Legendrian knot to yield a fillable contact manifold. When specialized to knots in the three sphere with its standard tight structure, this result can be rather efficient to find many examples of fillable surgeries along with various obstructions and surprising topological applications. This will report on joint work with T. Mark.
In this talk, I will present some joint works with Tom Courtade on characterizing probability measures that optimize the constant in a given functional inequalitiy via integration by parts formulas, and how Stein's method can be used to prove quantitative bounds on how close almost-optimal measures are to true optimizers. I will mostly discuss Poincaré inequalities and Gaussian optimizers, but also some other examples if time allows it.
When studying symplectic 4-manifolds, it is useful to consider Lefschetz fibrations over the 2-sphere due to their one-to-one correspondence uncovered by Freedman and Gompf. Lefschetz fibrations of genera 0 and 1 are well understood, but for genera greater than or equal to 2, much less is known. However, some Lefschetz fibrations with monodromies that respect the hyperelliptic involution of a genus-g surface have stronger properties which make their invariants easier to compute. In this talk, we will explore Terry Fuller's results from the late 90's which explore two-fold branched covers of hyperelliptic genus-g Lefschetz fibrations. We will look at his proof of why a Lefschetz fibration with only nonseparating vanishing cycles is a two-fold cover of $S^2 \times S^2$ branched over an embedded surface. The talk will include definitions, constructions, and Kirby pictures of branched covers in 4 dimensions. If time, we will discuss his results on hyperelliptic genus-g Lefschetz fibration which contain at least one separating vanishing cycles.
The study of Ramsey properties of the binomial random graph G_{n,p} was initiated in the 80s by Frankl & Rödl and Łuczak, Ruciński & Voigt. In this area we are often interested in establishing what function f(n) governs G_{n,p} having a particular Ramsey-like property P or not, i.e. when p is sufficiently larger than f(n) then G_{n,p} a.a.s. has P and when p is sufficiently smaller than f(n) then G_{n,p} a.a.s. does not have P (the former we call a 1-statement, the latter a 0-statement). I will present recent results on this topic from two different papers.
In the first, we almost completely resolve an outstanding conjecture of Kohayakawa and Kreuter on asymmetric Ramsey properties. In particular, we reduce the 0-statement to a necessary colouring problem which we solve for almost all pairs of graphs. Joint work with Candy Bowtell and Robert Hancock.
In the second, we prove similar results concerning so-called anti- and constrained-Ramsey properties. In particular, we (essentially) completely resolve the outstanding parts of the problem of determining the threshold function for the constrained-Ramsey property, and we reduce the anti-Ramsey problem to a necessary colouring problem which we prove for a specific collection of graphs. Joint work with Natalie Behague, Robert Hancock, Shoham Letzter and Natasha Morrison.
It is known that the symplectic property is preserved by the mean curvature flow in a K\"ahler-Einstein surface which is called "symplectic mean curvature flow". It was proved that there is no finite time Type I singularities for the symplectic mean curvature flow. We will talk about recent progress on an important Type II singularity of symplectic mean curvature flow-symplectic translating soliton. We will show that a symplectic translating soliton must be a plane under some natural assumptions which are necessary by investigating some examples.
In dimension 4, there exist simply connected manifolds which are homeomorphic but not diffeomorphic; the difference between the distinct smooth structures can be localized using corks. Similarly, there exist diffeomorphisms of simply connected 4-manifolds which are topologically but not smoothly isotopic to the identity. In this talk, I will discuss some preliminary results towards an analogous localization of this phenomena using corks for diffeomorphisms. This project is joint work with Slava Krushkal, Anubhav Mukherjee, and Mark Powell.
Please Note: There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am to 11:30 am in Skiles 005.
Given a matroid and a group of its matroid automorphisms, we study the induced group action on the Chow ring of the matroid. This turns out to always be a permutation action. Work of Adiprasito, Huh and Katz showed that the Chow ring satisfies Poincar\'e duality andthe Hard Lefschetz theorem. We lift these to statements about this permutation action, and suggest further conjectures in this vein.