In this talk, we'll sketch how one might hope to construct spaces (or spectra) from Floer theories, including framed flow categories and finite-dimensional approximation. If time allows, we'll talk about some questions Floer spaces (or spectra) can be useful for.
There will be a pre-seminar from 11 am to 11:30 am in Skiles 005.
We describe an algorithm to compute Whitney stratifications of real algebraic varieties, and of polynomial maps between them, by exploiting the algebraic structure of certain conormal spaces. One of the map stratification algorithms described here yields a new method for solving the real root classification problem. We also explore applications of this new map stratification algorithm to the study of the singularities of Feynman integrals; understanding and evaluating these integrals is a fundamental component in a wide variety of problems arising in quantum field theory.
Speaker will present in person.
Diffusion models have become ubiquitous for image generation and are increasingly being used for scientific applications. To date, many flavors of diffusion models have been developed by varying the stochastic process that noises data, but also the domain on which these processes act. Typically, generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact analysis. We relate the theory to the existing continuous-state Gaussian diffusion in discrete and continuous time. This approach is validated using a simple stochastic decay process, in which the reverse process generates images from a single all-black image, rather than a noisy prior distribution.
We'll give a short description of what exactly monopole Floer spectra are, and then explain how to calculate them for AR plumbings, a class of 3-manifolds including Seifert spaces. This is joint work with Irving Dai and Hirofumi Sasahira.
An optimal control problem in the space of probability measures, and the viscosity solutions of the corresponding dynamic programming equations defined using the intrinsic linear derivative are studied. The value function is shown to be Lipschitz continuous with respect to a novel smooth Fourier Wasserstein metric. A comparison result between the Lipschitz viscosity sub and super solutions of the dynamic programming equation is proved using this metric, characterizing the value function as the unique Lipschitz viscosity solution. This is joint work with Prof. H. Mete Soner.
For a graph $H$ and an $n$-vertex graph $G$, the $H$-bootstrap percolation process on $G$ is the process which starts with $G$ and, at every time step, adds any missing edges on the vertices of $G$ that complete a copy of $H$. This process eventually stabilises and we are interested in the extremal question raised by Bollob\'as, of determining the maximum \emph{running time} (number of time steps before stabilising) of this process, over all possible choices of $n$-vertex graph $G$. We initiate a systematic study of this parameter, denoted $M_H(n)$, and its dependence on properties of the graph $H$. In a series of works we determine the precise running time for cycles and asymptotic running time for several other important classes. In general, we study necessary and sufficient conditions on $H$ for fast, i.e. sublinear or linear $H$-bootstrap percolation, and in particular explore the relationship between running time and minimum vertex degree and connectivity. Furthermore we also obtain the running time of the process for typical $H$ and discover several graphs exhibiting surprising behavior. The talk represents joint work with David Fabian and Patrick Morris.
In recent years, the few classical results in large deviations for random matrices have been complemented by a variety of new ones, in both the math and physics literatures, whose proofs leverage connections with Harish-Chandra/Itzykson/Zuber integrals. We present one such result, focusing on extreme eigenvalues of deformed sample-covariance and Wigner random matrices. This confirms recent formulas of Maillard (2020) in the physics literature, precisely locating a transition point whose analogue in non-deformed models is not yet fully understood. Joint work with Jonathan Husson.
This talk has two goals. The first is to talk through Keynes-Newton’s construction of minimal non-uniquely ergodic interval exchange transformations. The second is to explain why I’m talking about this in the student topology seminar.
The game of HEX has deep mathematical underpinnings despite its simple rules. What could this game possibly have to do with coffee?! And how does that connection, once identified, lead to consideration of ferromagnetism and even to the melting polar ice caps? Join Hugo Duminil-Copin, Professor of Mathematics at IHES and the University of Geneva, for an exploration of the way in which mathematical thinking can help us make some truly surprising connections.
The Ising model is one of the most classical lattice models of statistical physics undergoing a phase transition. Initially imagined as a model for ferromagnetism, it revealed itself as a very rich mathematical object and a powerful theoretical tool to understand cooperative phenomena. Over one hundred years of its history, a profound understanding of its critical phase has been obtained. While integrability and mean-field behavior led to extraordinary breakthroughs in the two-dimensional and high-dimensional cases respectively, the model in three and four dimensions remained mysterious for years. In this talk, we will present recent progress in these dimensions based on a probabilistic interpretation of the Ising model relating it to percolation models.
This series of talks will discuss connections between Riemannian geometry and contact topology. Both structures have deep connections to the topology of 3-manifolds, but there has been little study of the interactions between them (at least the implications in contact topology). We will see that there are interesting connections between curvature and properties of contact structures. The talks will give a brief review of both Riemannian geometry and contact topology and then discuss various was one might try to connect them. There will be many open problems discussed (probably later in the series).
Corrine Yap: The Ising model is a classical model originating in statistical physics; combinatorially it can be viewed as a probability distribution over 2-vertex-colorings of a graph. We will discuss a fixed-magnetization version—akin to fixing the number of, say, blue vertices in every coloring—and a natural Markov chain sampling algorithm called the Kawasaki dynamics. We show some surprising results regarding the existence and location of a fast/slow mixing threshold for these dynamics. (joint work with Aiya Kuchukova, Marcus Pappik, and Will Perkins)
Changxin Ding: For trees on a fixed number of vertices, the path and the star are two extreme cases. Many graph parameters attain its maximum at the star and its minimum at the path among these trees. A trivial example is the number of leaves. I will introduce more interesting examples in the mini talk.
Jing Yu: We show that all simple outerplanar graphs G with minimum degree at least 2 and positive Lin-Lu-Yau Ricci curvature on every edge have maximum degree at most 9. Furthermore, if G is maximally outerplanar, then G has at most 10 vertices. Both upper bounds are sharp.
