A discussion of Khovanov-Lee homology, how to extract some invariants of braid closures from the homology theory, and motivation for studying both the homology theory and the invariants.
While the existence and properties of the SRB measure for the billiard map associated with a periodic Lorentz gas are well understood, there are few results regarding other types of measures for dispersing billiards. We begin by proposing a naive definition of topological entropy for the billiard map, and show that it is equivalent to several classical definitions. We then prove a variational principle for the topological entropy and proceed to construct a unique probability measure which achieves the maximum. This measure is Bernoulli and positive on open sets. An essential ingredient is a proof of the absolute continuity of the unstable foliation with respect to the measure of maximal entropy. This is joint work with Viviane Baladi.
It is a classical theorem of Roth that every dense subset of $\left\{1,\ldots,N\right\}$ contains a nontrivial three-term arithmetic progression. Quantitatively, results of Sanders, Bloom, and Bloom-Sisask tell us that subsets of relative density at least $1/(\log N)^{1-\epsilon}$ already have this property. In this talk, we will discuss some sets of $N$ integers which unlike $\left\{1,\ldots,N\right\}$ do not contain nontrivial four-term arithmetic progressions, but which still have the property that all of their subsets of density at least $1/(\log N)^{1-\epsilon}$ must contain a three-term arithmetic progression. Perhaps a bit surprisingly, these sets turn out not to have as many three-term progressions as one might be inclined to guess, so we will also address the question of how many three-term progressions can a four-term progression free set may have. Finally, we will also discuss about some related results over $\mathbb{F}_{q}^n$. Based on joint works with Jacob Fox and Oliver Roche-Newton.
Tutte proved that every 4-connected planar graph contains a Hamilton cycle, but
there are 3-connected $n$-vertex graphs whose longest cycles have length
$\Theta(n^{\log_32})$. On the other hand, Jackson and Wormald proved that an
essentially 4-connected $n$-vertex planar graph contains a cycle of
length at least $(2n+4)/5$, which was improved to $5(n+2)/8$ by Fabrici {\it et al}. We improve this bound to $\lceil (2n+6)/3\rceil$ for $n\ge 6$ by proving a quantitative version of a result of Thomassen,
and the bound is best possible.
The classical isoperimetric problem consists in finding among all sets with the same volume (measure) the one that minimizes the surface area (perimeter measure). In the Euclidean case, balls are known to solve this problem. To formulate the isoperimetric problem, or an isoperimetric inequality, in more general settings, requires in particular a good notion of perimeter measure.
The starting point of this talk will be a characterization of sets of finite perimeter original to Ledoux that involves the heat semigroup associated to a given stochastic process in the space. This approach put in connection isoperimetric problems and functions of bounded variation (BV) via heat semigroups, and we will extend these ideas to develop a natural definition of BV functions and sets of finite perimeter on metric measure spaces. In particular, we will obtain corresponding isoperimetric inequalies in this setting.
The main assumption on the underlying space will be a non-negative curvature type condition that we call weak Bakry-Émery and is satisfied in many examples of interest, also in fractals such as (infinite) Sierpinski gaskets and carpets. The results are part of joint work with F. Baudoin, L. Chen, L. Rogers, N. Shanmugalingam and A. Teplyaev.
Let $K$ be a n dimensional convex body with of volume $1$. and barycenter of $K$ is the origin. It is known that $|K \cap -K|>2^{-n}$. Via thin shell estimate by Lee-Vempala (earlier versions were done by Guedon-Milman, Fleury, Klartag), we improve the bound by a sub-exponential factor. Furthermore, we can improve the Hadwiger’s Conjecture in the non-symmetric case by a sub-exponential factor. This is a joint work with Boaz A. Slomka, Tomasz Tkocz, and Beatrice-Helen Vritsiou.
In the world of 4 manifolds, finding exotic structures on 4 manifolds is considered one of most interesting and difficult problems. I will give a brief history of this and explain a very interesting tool "knot surgery" defined by Fintushel and Stern. In this talk I will mostly focused on drawing pictures. If time permits, I will talk various interesting applications.
We are interested in arithmetic progressions in positive measure subsets of [0,1]^d. After a counterexample by Bourgain, it seemed as if nothing could be said about the longest interval formed by sizes of their gaps. However, Cook, Magyar, and Pramanik gave a positive result for 3-term progressions if their gaps are measured in the l^p-norm for p other than 1, 2, and infinity, and the dimension d is large enough. We establish an appropriate generalization of their result to longer progressions. The main difficulty lies in handling a class of multilinear singular integrals associated with arithmetic progressions that includes the well-known multilinear Hilbert transforms, bounds for which still constitute an open problem. As a substitute, we use the previous work with Durcik and Thiele on power-type cancellation of those transforms, which was, in turn, motivated by a desire to quantify the results of Tao and Zorin-Kranich. This is joint work with Polona Durcik (Caltech).
Pollen grain surface morphologies are famously diverse, and each species displays a unique, replicable pattern. The function of these microstructures, however, has not been elucidated. We show electron microscopy evidence that the templating of these patterns is formed by a phase separation of a polysaccharide mixture on the cell membrane surface. Here we present a Landau theory of phase transitions to ordered states describing all extant pollen morphologies. We show that 10% of all morphologies can be characterized as equilibrium states with a well-defined wavelength of the pattern. The rest of the patterns have a range of wavelengths on the surface that can be recapitulated by exploring the evolution of a conserved dynamics model. We then perform an evolutionary trait reconstruction. Surprisingly, we find that although the equilibrium states have evolved multiple times, evolution has not favored these ordered-polyhedral like shapes and perhaps their patterning is simply a natural consequence of a phase separation process without cross-linkers.
