Seminars and Colloquia by Series

On the well-posedness of the Mortensen observer for a defocusing cubic wave equation

Series
PDE Seminar
Time
Tuesday, February 27, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Jesper Schröder – Technische Universität Berlin –

In this presentation the analytical background of nonlinear observers based on minimal energy estimation is discussed. Originally, such strategies were proposed for the reconstruction of the state of finite dimensional dynamical systems by means of a measured output where both the dynamics and the output are subject to white noise. Our work aims at lifting this concept to a class of partial differential equations featuring deterministic perturbations using the example of a wave equation with a cubic defocusing term in three space dimensions. In particular, we discuss local regularity of the corresponding value function and consider operator Riccati equations to characterize its second spatial derivative.

Splitting of vector bundles on toric varieties

Series
Algebra Seminar
Time
Tuesday, February 27, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mahrud Sayrafi – University of Minnesota

In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Generalizations of this criterion, under additional hypotheses, have been proven for other toric varieties, for instance by Eisenbud-Erman-Schreyer for products of projective spaces, by Schreyer for Segre-Veronese varieties, and Ottaviani for Grassmannians and quadrics. This talk is about a splitting criterion for arbitrary smooth projective toric varieties, as well as an algorithm for finding indecomposable summands of sheaves and modules in the more general setting of Mori dream spaces.

On Expressivity and Stability of Positional Encoding for Graph Neural Networks and Graph Transformers

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 26, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Pan Li – Georgia Institute of Technology –

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

 

 

Contact surgeries and symplectic fillability

Series
Geometry Topology Seminar
Time
Friday, February 23, 2024 - 10:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 249
Speaker
Bülent Tosun – IAS and U. Alabama

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.

Stein's mathod and stability for sharp constants in functional inequalities

Series
Analysis Seminar
Time
Wednesday, February 21, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Max Fathi – Etablissement public experimental, Paris, France –

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.

Two-fold branched covers of hyperelliptic Lefschetz fibrations

Series
Geometry Topology Student Seminar
Time
Wednesday, February 21, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sierra Knavel – Georgia Tech

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. 

Thresholds for random Ramsey problems (Joseph Hyde (UVic))

Series
Graph Theory Seminar
Time
Tuesday, February 20, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Joseph Hyde – University of Victoria –

 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.

On symplectic mean curvature flows

Series
PDE Seminar
Time
Tuesday, February 20, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jiayu Li – University of Science and Technology of China –

  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.

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