In this talk, I will present a complete coarse classification of strongly exceptional Legendrian realizations of the connected sum of two Hopf links in contact 3-spheres. This is joint work with Sinem Onaran.
Please Note: There will be a pre-seminar at 10:55 am in Skiles 006 (not 005).
The theory of stable polynomials features a key notion called proper position, which generalizes interlacing of real-rooted polynomials to higher dimensions. In a recent paper, I introduced a Lorentzian analog of proper position and used it to give a new characterization of elementary quotients of valuated matroids. This connects the local structure of spaces of Lorentzian polynomials with the incidence geometry of tropical linear spaces. A central object in this connection is the moduli space of codimension-1 tropical linear subspaces of a given tropical linear space. In this talk, I will show some new structural results on this moduli space and their implications for Lorentzian polynomials.
In this talk, I will present recent advancements in the study of smooth mapping class groups of 4-manifolds. Our work focuses on diffeomorphisms arising from Dehn twists along embedded 3-manifolds and their interaction with Seiberg-Witten theory. These investigations have led to intriguing applications across several areas, including symplectic geometry (related to Torelli symplectomorphisms), algebraic geometry (concerning the monodromy of singularities), and low-dimensional topology (involving exotic diffeomorphisms). This is collaborative work with Hokuto Konno, Jianfeng Lin, and Juan Munoz-Echaniz.
In this talk I will discuss our recent effort to develop structure-preserving machine learning (ML) for time series data, focusing on both dissipative PDEs and singularly perturbed ODEs. The first part presents a data-driven modeling method that accurately captures shocks and chaotic dynamics through a stabilized neural ODE framework. We learn the right-hand-side of an ODE by adding the outputs of two networks together, one learning a linear term and the other a nonlinear term. The architecture is inspired by the inertial manifold theorem. We apply this method to chaotic trajectories of the Kuramoto-Sivashinsky equation, where our model keeps long-term trajectories on the attractor and remains robust to noisy initial conditions. The second part explores structure-preserving ML for singularly perturbed dynamical systems. A powerful tool to address these systems is the Fenichel normal form, which significantly simplifies fast dynamics near slow manifolds. I will discuss a novel realization of this concept using ML. Specifically, a fast-slow neural network (FSNN) is proposed, enforcing the existence of a trainable, attractive invariant slow manifold as a hard constraint. To illustrate the power of FSNN, I will show a fusion-motivated example where traditional numerical integrators all fail.
In this talk, we will study random dynamical systems of smooth surface diffeomorphisms. Aaron Brown and Federico Rodriguez Hertz showed that, in this setting, hyperbolic stationary measures have the SRB property, except when certain obstructions occur. Here, the SRB property essentially means that the measure is absolutely continuous along certain “nice” curves (unstable manifolds). In this talk, we want to understand conditions that guarantee that SRB stationary measures are absolutely continuous with respect to the Lebesgue measure of the ambient space. Our approach is inspired on Tsujii’s “transversality” method, which he used to show Palis conjecture for partially hyperbolic endomorphisms. This is a joint work with Aaron Brown, Homin Lee and Yuping Ruan.
We consider the mean field Bose gas on the unit torus at temperatures proportional to the critical temperature of the Bose—Einstein condensation phase transition. We discuss trace norm convergence of the Gibbs state to a state given by a convex combination of quasi-free states. Two consequences of this relation are precise asymptotic formulas for the two-point function and the distribution of the number of particles in the condensate. A crucial ingredient of the proof is a novel abstract correlation inequality. This is joint work with Nam Panh Tanh and Marcin Napiorkowski.
A classical Fefferman-Stein inequality relates the distributional estimate for a square function for a harmonic function u to a non-tangential maximal function of u. We extend this ineuality to certain multiparameter settings, including the Shilov boundaries of tensor product domains, and the Heisenberg groups with flag structure.
Our technique bypasses the use of Fourier or the dependence of group structure. Direct applications include the the (global) weak type endpoint estimate for multi-parameter Calderon–Zygmund operators and maximal function characterisation of multi-parameter Hardy spaces.
This talk is based on the recent progress: Ji Li, ``Fefferman–Stein type inequality'', Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2024.
(Note the unusual location!)
We study an extension of the k-Disjoint Paths Problem where, in addition to finding k disjoint paths joining k given pairs of vertices in a graph, we ask that those paths satisfy certain constraints expressable by abelian groups. We give an O(n^8) time algorithm to solve this problem under the assumption that the constraint can be expressed as avoiding a bounded number of group elements; moreover, our O(n^8) algorithm allows any bounded number of such constraints to be combined. Group-expressable constraints include, but not limited to: (1) paths of length r modulo m for any fixed r and m, (2) paths passing through any bounded number of prescribed sets of edges and/or vertices, and (3) paths that are long detours (paths of length at least r more than the distance between their ends for fixed r). The k=1 case with the modularity constraint solves problems of Arkin, Papadimitriou and Yannakakis from 1991. Our work also implies a polynomial time algorithm for testing the existence of a subgraph isomorphic to a subdivision of a fixed graph, where each path of the subdivision between branch vertices satisfies any combination of a bounded number of group-expressable constraints. In addition, our work implies similar results addressing edge-disjointness. It is joint work with Youngho Yoo.
