A surface of genus $g$ has many symmetries. These form the surface’s mapping class group $Mod(S_g)$, which is finitely generated. The most commonly used generating sets for $Mod(S_g)$ are comprised of infinite order elements called Dehn twists; however, a number of authors have shown that torsion generating sets are also possible. For example, Brendle and Farb showed that $Mod(S_g)$ is generated by six involutions for $g \geq 3$. We will discuss our extension of these results to elements of arbitrary order: for $k > 5$ and $g$ sufficiently large, $Mod(S_g)$ is generated by three elements of order $k$. Generalizing this idea, in joint work with Margalit we showed that for $g \geq 3$ every nontrivial periodic element that is not a hyperelliptic involution normally generates $Mod(S_g)$. This result raises a question: does there exist an $N$, independent of $g$, so that if $f$ is a periodic normal generator of $Mod(S_g)$, then $Mod(S_g)$ is generated by $N$ conjugates of $f$? We show that in general there does not exist such an $N$, but that there does exist such a universal bound for the class of non-involution normal generators.
We study self-similar and fractal networks from the combinatorial perspective. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to known graph-theoretical characteristics: rank dimension and product (or Prague or Nešetřil-Rödl) dimension. Our approach reveals how self-similarity and fractality of a network are defined by a pattern of overlaps between densely connected network communities. It allows us to identify fractal graphs, explore the relations between graph fractality, graph colorings and graph Kolmogorov complexity, and analyze the fractality of several classes of graphs and network models, as well as of a number of real-life networks. We demonstrate the application of our framework to evolutionary studies by revealing the growth of self-organization of heterogeneous viral populations over the course of their intra-host evolution, thus suggesting mechanisms of their gradual adaptation to the host's environment. As far as the authors know, the proposed approach is the first theoretical framework for study of network fractality within the combinatorial paradigm. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.
Based on joint work with Leonid Bunimovich
Motivated by the Dikin walk, we develop aspects of an interior-point
theory for sampling in high dimensions. Specifically, we introduce symmetric
and strong self-concordance. These properties imply that the corresponding
Dikin walk mixes in $\tilde{O}(n\bar{\nu})$ steps from a warm start
in a convex body in $\mathbb{R}^{n}$ using a strongly self-concordant barrier
with symmetric self-concordance parameter $\bar{\nu}$. For many natural
barriers, $\bar{\nu}$ is roughly bounded by $\nu$, the standard
self-concordance parameter. We show that this property and strong
self-concordance hold for the Lee-Sidford barrier. As a consequence,
we obtain the first walk to mix in $\tilde{O}(n^{2})$ steps for an
arbitrary polytope in $\mathbb{R}^{n}$. Strong self-concordance for other
barriers leads to an interesting (and unexpected) connection ---
for the universal and entropic barriers, it is implied by the KLS
conjecture.
Consider a system of rotators subject to a small quasi-periodic forcing which (1) is analytic, (2) satisfies a time-reversibility property, and (3) has a Bryuno frequency vector. Without imposing any non-degeneracy condition, we prove that there exists at least one quasi-periodic solution with the same frequency vector as the forcing. The result can be interpreted as a theorem of persistence of lower-dimensional tori of arbitrary codimension in degenerate cases. This is a joint work with Livia Corsi.
We provide a martingale proof of the fact that the number of descents in random permutations is asymptotically normal with an error bound of order n^{-1/2}. The same techniques are shown to be applicable to other descent and descent-related statistics as they satisfy certain recurrence relation conditions. These statistics include inversions, descents in signed permutations, descents in Stirling permutations, the length of the longest alternating subsequences, descents in matchings and two-sided Eulerian numbers.
Please Note: Melvin Leok is a professor in the Department of Mathematics at the University of California, San Diego. His research interests are in computational geometric mechanics, computational geometric control theory, discrete geometry, and structure-preserving numerical schemes, and particularly how these subjects relate to systems with symmetry. He received his Ph.D. in 2004 from the California Institute of Technology in Control and Dynamical Systems under the direction of Jerrold Marsden. He is a three-time NAS Kavli Frontiers of Science Fellow, and has received the NSF Faculty Early Career Development (CAREER) award, the SciCADE New Talent Prize, the SIAM Student Paper Prize, and the Leslie Fox Prize (second prize) in Numerical Analysis. He has given plenary talks at the Society for Natural Philosophy, Foundations of Computational Mathematics, NUMDIFF, and the IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control. He serves on the editorial boards of the Journal of Nonlinear Science, the Journal of Geometric Mechanics, and the Journal of Computational Dynamics, and has served on the editorial boards of the SIAM Journal on Control and Optimization, and the LMS Journal of Computation and Mathematics.
Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry. A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-$h$ flow map of a Hamiltonian system on the space of probability distributions.
In his book Convex Polyhedra, ch. 7 (end of subsection 2) A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of convex polytopes with given geometric data. As an example of a geometric problem in which variational approach was successfully applied, Aleksandrov quotes the Minkowski problem. He also mentions the Weyl problem of isometric embedding for which a variational approach was proposed (but not fully developed and not completed) by W. Blashke and G. Herglotz. The first goal of this talk is to give a variational formulation and solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gaussian curvature (also posed by Aleksandrov who solved it using topological methods). The second goal of this talk is to show that in variational form the Aleksandrov problem is closely connected to the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations.
The problem of phase retrieval for a set of functions $H$ can be thought of as being able to identify a function $f\in H$ or $-f\in H$ from the absolute value $|f|$. Phase retrieval for a set of functions is called stable if when $|f|$ and $|g|$ are close then $f$ is proportionally close to $g$ or $-g$. That is, we say that a set $H\subseteq L_2({\mathbb R})$ does stable phase retrieval if there exists a constant $C>0$ so that
$$\min\big(\big\|f-g\big\|_{L_2({\mathbb R})},\big\|f+g\big\|_{L_2({\mathbb R})}\big)\leq C \big\| |f|-|g| \big\|_{L_2({\mathbb R})} \qquad\textrm{ for all }f,g\in H.
$$
It is known that phase retrieval for finite dimensional spaces is always stable. On the other hand, phase retrieval for infinite dimensional spaces using a frame or a continuous frame is always unstable. We prove that there exist infinite dimensional subspaces of $L_2({\mathbb R})$ which do stable phase retrieval. This is joint work with Robert Calderbank, Ingrid Daubechies, and Nikki Freeman.
