We will define Newton polygons for polynomials over a valued field and prove a couple theorems using them. For example, relating the valuations of the roots of the polynomial to the slopes of the Newton polygon and proving the algebraic closure of the Puiseux series in characteristic 0.
This is a general audience Geometry-Topology talk where I will give a broad overview of my research interests and techniques I use in my work. My research concerns the study of link concordance using groups, both extracting concordance data from group theoretic invariants and determining the properties of group structures on links modulo concordance. Milnor's invariants are one of the more fundamental link concordance invariants; they are thought of as higher order linking numbers and can be computed using both Massey products (due to Turaev and Porter) and higher order intersections (due to Cochran). In my work, I have generalized Milnor's invariants to knots inside a closed, oriented 3-manifold M. I call this the Dwyer number of a knot and show methods to compute it for null-homologous knots inside a family of 3-manifolds with free fundamental group. I further show Dwyer number provides the weight of the first non-vanishing Massey product in the knot complement in the ambient manifold. Additionally, I proved the Dwyer number detects knots K in M bounding smoothly embedded disks in specific 4-manifolds with boundary M which are not concordant to the unknot in M x I. This result further motivates my definition of a new link concordance group in joint work with Matthew Hedden using the knotification construction of Ozsv'ath and Szab'o. Finally, I will briefly discuss my recent result that the string link concordance group modulo its pure braid subgroup is non-abelian.
Starting from Total Variation, this talk will overview some mathematical approaches for image processing, such as removing noise. We will also consider numerical application to data understanding. A few more application maybe presented.
I will show some operations that preserve Lorentzian property following Section 6 of https://arxiv.org/pdf/1902.03719.pdf
Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. As data objects, they are characterized by the challenges associated with "big data," as well as the complication that their discrete geometric structure results in a non-Euclidean phylogenetic tree space, which poses computational and statistical limitations.
In this talk, I will compare the geometric and statistical properties between a well-studied framework - the BHV space, and an alternative framework that we propose, which is based on tropical geometry. Our framework exhibits analytic, geometric, and topological properties that are desirable for theoretical studies in probability and statistics, as well as increased computational efficiency. I also demonstrate our approach on an example of seasonal influenza data.
Let $f$ be defined on $\mathbb{Z}$. Let $A_N f$ be the average of $f$ along the square integers.
Cutting a polyhedron along some spanning tree of its edges will yield an isometric immersion of the polyhedron into the plane. If this immersion is also injective, we call it an unfolding. In this talk I will give some general results about unfoldings of polyhedra. There is also a notion of pseudo-edge unfolding, which involves cutting on a pseudo edge graph, as opposed to an edge graph. A pseudo edge graph is a 3-connected graph on the surface of the polyhedron, whose vertices coincide with the vertices of the polyhedron, and whose edges are geodesics. I will explain part of the paper "Pseudo-Edge Unfoldings of Convex Polyhedra," a joint work of mine with Professor Ghomi, which proves the existence of a convex polyhedron with a pseudo edge graph along which it is not unfoldable. Finally, I will discuss some connections between pseudo edge graphs and edge graphs.
Using ideas from information theory, we establish lower bounds on the volume and the surface area of a geometric body using the size of its slices along different directions. In the first part of the talk, we derive volume bounds for convex bodies using generalized subadditivity properties of entropy combined with entropy bounds for log-concave random variables. In the second part, we investigate a new notion of Fisher information which we call the L1-Fisher information and show that certain superadditivity properties of the L1-Fisher information lead to lower bounds for the surface areas of polyconvex sets in terms of its slices.
A combinatorial structure is said to be quasirandom if it resembles a random structure in a certain robust sense. For example, it is well-known that a graph G with edge-density p is quasirandom if and only if the density of C_4 in G is p^4+o(p^4); this property is known to equivalent to several other properties that hold for truly random graphs. A similar phenomenon was established for permutations: a permutation is quasirandom if and only if the density of every 4-point pattern (subpermutation) is 1/4!+o(1). We strengthen this result by showing that a permutation is quasirandom if and only if the sum of the densities of eight specific 4-point patterns is 1/3+o(1). More generally, we classify all sets of 4-point patterns having such property.
The talk is based on joint work with Timothy F. N. Chan, Jonathan A. Noel, Yanitsa Pehova, Maryam Sharifzadeh and Jan Volec.
Graphs, which in their simplest forms are vertices connected by edges,
are widely used in high performance computing, machine learning and
network science. This talk will use recent progresses on two
well-studied algorithmic problems in static and dynamic graph,
max-flows and dynamic matchings, to discuss a methodology for
designing faster algorithm for large graphs. This approach is
motivated by a fundamental phenomenon in data structures: the
advantages of offline data structures over online ones.
I will start by describing how work on max-flows led to a focus on
finding short paths in residual graphs, and how investigating more
global notions of progress in residual graphs led to a more
sophisticated and general understanding of iterative methods and
preconditioning. I will then discuss a similar phenomenon in dynamic
graphs, where maintaining a large matching seems to require the online
detection of short augmenting paths, but can once again be
circumvented through the offline construction of smaller equivalent
graphs.
