This series of talks will introduce the immersed curve perspective of knot Floer homology and give applications to to 3-manifold topology.
We consider Klein-Gordon equations with an external potential and cubic nonlinearity in three spatial dimensions. It is assumed that the linear operator has internal modes, and hence the unperturbed linear equation has multiple time-periodic solutions known as bound states. In 1999, Soffer and Weinstein treated the case when the linear operator has one large eigenvalue and proved the decay of the solution. In 2022, we solved the general one eigenvalue case. In our recent work, we solved the multiple internal modes case: the operator can has multiple and possibly degenerate eigenvalues. Indeed, we determined the sharp decay rate of the overall solution, as well as distinct decay rates for different modes of the solution. This is a joint work with Prof. Zhen Lei and Dr. Jie Liu.
This series of talks will introduce the immerved curve perspective of knot Floer homology and give applications to to 3-manifold topology.
Statistical mechanics explains that systems in thermal equilibrium spend a greater fraction of their time in states with apparent order because these states have lower energy. This explanation is remarkable, and powerful, because energy is a "local" property of states. While non-equilibrium steady states can similarly exhibit order, there can be no local property analogous to energy that explains why, as Landauer argued 50 years ago. However, recent experiments suggest that a local property called “rattling” predicts which states are favored, at least for a broad class of non-equilibrium systems.
I will present a Markov chain theory that explains when and why rattling predicts non-equilibrium order. In brief, it "works" when the correlation between a Markov chain's effective potential and the logarithm of its exit rates is high. It is therefore important to estimate this correlation in different classes of Markov chains. As an example, I will discuss estimates of the correlation exhibited by reaction kinetics on disordered energy landscapes, including dynamics of the random energy model and the Sherrington–Kirkpatrick spin glass. (Joint work with Dana Randall.)
Mathematics is at the core of artificial intelligence, from the linear algebra that powers deep learning to the probability and optimization driving new algorithms. We will explore how mathematical ideas can open new directions for AI innovation and how recent U.S. AI policy trends are shaping research priorities. Together, these perspectives reveal opportunities for mathematicians to influence the design and future of AI technologies.
The `middle third integer Cantor set' consists of those integers which do not have a 2 in their base 3 representation. We will review and extend some results about such sets. For a general integer Cantor set K, with 0 as an allowed digit, it is known that K is intersective, a result of Furstenberg-Katznelson. That is, for a dense set of integers A, A-A must intersect K. Writing K={k_1, k_2, ...}, we show that the set of n such that k_n\in A-A has positive density. The set p(K), where p is an integer polynomial with zero constant term, is also intersective due to Bergelson-McCutcheon. We show the same density result for p(K). We also show an L^2 Ergodic Theorem along K. The pointwise Ergodic Theorem lies beyond current techniques. Joint work with A Burgin, A Fragkos, D. Mena, M Reguera.
Let $r_2(k)$ denote the smallest integer $n$ such that every $2$-edge-colored complete graph $K_n$ has a monochromatic $k$-connected subgraph. In 1983, Matula established the bound $4(k-1)+1 \leq r_2(k) < (3+\sqrt{11/3})(k-1)+1$. Furthermore, In 2008, Bollobás and Gyárfás conjectured that for any $k, n \in \mathbb{Z}^+$ with $n > 4(k-1)$, every 2-edge-coloring of the complete graph on $n$ vertices
leads to a $k$-connected monochromatic subgraph with at least $n-2k+2$ vertices. We find a counterexample with $n = \lfloor 5k-2.5-\sqrt{8k-\frac{31}{4}} \rfloor$ for $k\ge 6$, thus disproving the conjecture,
and we show the conclusion holds for $n > 5k-2.5-\sqrt{8k-\frac{31}{4}}$ when $k \ge 16$. Additionally, we improve the upper bound of $r_2(k)$ to $\lceil (3+\frac{\sqrt{497}-1}{16})(k-1) \rceil$ for all $k \geq 4$.
A knot is slice if it bounds a smoothly embedded disk in the four-ball and a knot is ribbon if it bounds such a disk with no local maxima. The slice-ribbon conjecture posits that every slice knot is ribbon. We prove that a linear combination of iterated cables of tight fibered knots is ribbon if and only if it is of the form K # -K, generalizing work of Miyazaki and Baker. Consequently, either iterated cables of tight fibered knots are linearly independent in the smooth concordance group, or the slice–ribbon conjecture fails.
Please Note: There will be a pre-seminar 10:55-11:15 in Skiles 005.
Traditionally, chip-firing is a discrete dynamical system where poker chips move around the vertices of a graph. One fascinating result is that number of configurations of a fixed number of chips, modulo a firing equivalence relation, is the number of spanning trees of the graph. This relationship gives the set of spanning trees group-like properties.
In this talk, I will discuss how chip-firing ideas can be generalized from graphs to regular matroids, where bases play the role of spanning trees. This will lead to an overview of joint work with Ding, Tóthmérész, and Yuen on the consistency of the Backman-Baker-Yuen Sandpile Torsor.
============(Below is the information on the pre-talk.)============
Title (pre-talk): Transforming Spanning Trees Using Mathematicians and Coffee Cups
Abstract (pre-talk): There is a fascinating structure to the set of spanning trees of a plane graph, which allows this set to behave much like a group. Perhaps most incredibly, there is a sense in which this structure is canonical.
In this talk, I will show you how spanning trees can be transformed after introducing mathematicians and coffee cups on some of the vertices. This is a variant of the rotor-routing process which takes advantage of a special property of plane graphs.
Random matrix products perhaps among some of the most extensively studied examples of random dynamical systems, and moreover are central to the study of one-dimensional disordered systems. We discuss recent results by the author (joint with S. Raman) obtaining estimates on heavy-tailed random matrix products which are robust under perturbations in an appropriate sense, and hence "uniform" on compact sets of measures in an appropriate topology. We also discuss recent localization results for the Anderson model with heavy tails which make use of these uniform bounds.
