Georgia Institute of Technology College of Sciences

Stability and bifurcation conditions for a vibroimpact motion in an inclined energy harvester with T-periodic forcing are determined analytically and numerically. This investigation provides a better understanding of impact velocity and its influence on energy harvesting efficiency and can be used to optimally design the device. The numerical and analytical results of periodic motions are in excellent agreement. The stability conditions are developed in non-dimensional parameter space through two basic nonlinear maps based on switching manifolds that correspond to impacts with the top and bottom membranes of the energy harvesting device. The range for stable simple T-periodic behavior is reduced with increasing angle of incline β, since the influence of gravity increases the asymmetry of dynamics following impacts at the bottom and top. These asymmetric T-periodic solutions lose stability to period doubling solutions for β ≥ 0, which appear through increased asymmetry. The period doubling, symmetric and asymmetric periodic motion are illustrated by bifurcation diagrams, phase portraits and velocity time series.

I will give a brief survey of concordance in high-dimensional knot theory and how slice results have classically been obtained in this setting with the aid of surgery theory. Time permitting, I will then discuss an example of how some non-abelian slice obstructions come into the picture for 1-knots, as intuition for the seminar talk about L^2 invariants.

Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue. In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the  properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences to conclude that the variety of a prime (tropical) ideal is either empty or consists of a single point. This is joint work with D. Joó.

The question of which high-dimensional knots are slice was entirely solved by Kervaire and Levine. Compared to this, the question of which knots are doubly slice in high-dimensions is still a largely open problem. Ruberman proved that in every dimension, some version of the Casson-Gordon invariants can be applied to obtain algebraically doubly slice knots that are not doubly slice. I will show how to use L^2 signatures to recover the result of Ruberman for (4k-3)-dimensional knots, and discuss how the derived series of the knot group might be used to organise the high-dimensional doubly slice problem.

I this talk I will summerize some of our contributions in the analysis of parabolic elliptic Keller-Segel system, a typical model in chemotaxis. For the case of linear diffusion, after introducing the critical mass in two dimension, I will show our result for blow-up conditions for higher dimension. The second part of the talk is concentrated in the critical exponent for Keller-Segel system with porus media type diffusion. In the end, motivated from the result on nonlocal Fisher-KPP equation, we show that the nonlocal reaction will also help in preventing the blow-up of the solutions.  

Optimal transport is a thoroughly studied field in mathematics and introduces the concept of Wasserstein distance, which has been widely used in various applications in computational mathematics, machine learning as well as many areas in engineering. Meanwhile, control theory and path planning is an active branch in mathematics and robotics, focusing on algorithms that calculates feasible or optimal paths for robotic systems. In this defense, we use the properties of the gradient flows in Wasserstein metric to design algorithms to handle different types of path planning and control problems as well as the K-means problems defined on graphs.

We will see some instances of swindles in mathematics, primarily focusing on some in geometric topology due to Barry Mazur.

We discuss the asymptotic value of the maximal perimeter of a convex set in an n-dimensional space with respect to certain classes of measures. Firstly, we derive a lower bound for this quantity for a large class of probability distributions; the lower bound depends on the moments only. This lower bound is sharp in the case of the Gaussian measure (as was shown by Nazarov in 2001), and, more generally, in the case of rotation invariant log-concave measures (as was shown by myself in 2014). We discuss another class of measures for which this bound is sharp. For isotropic log-concave measures, the value of the lower bound is at least n^{1/8}.

In addition, we show a uniform upper bound of Cn||f||^{1/n}_{\infty} for all log-concave measures in a special position, which is attained for the uniform distribution on the cube. We further estimate the maximal perimeter of isotropic log-concave measures by n^2. 

The well known Erdos-Hajnal Conjecture states that every graph has the Erdos-Hajnal (EH) property. That is, for every $H$, there exists a $c=c(H)>0$ such that every graph $G$ with no induced copy of $H$ has the property $hom(G):=max\{\alpha(G),\omega(G)\}\geq |V(G)|^{c}$. Let $H,J$ be subdivisions of caterpillar graphs. Liebenau, Pilipczuk, Seymour and Spirkl proved that the EH property holds if we forbid both $H$ and $\overline{J}.$ We will discuss the proof of this result.

I will talk about a conjecture that in Gibbs states of one-dimensional spin chains with short-ranged gapped Hamiltonians the quantum conditional mutual information (QCMI) between the parts of the chain decays exponentially with the length of separation between said parts. The smallness of QCMI enables efficient representation of these states as tensor networks, which allows their efficient construction and fast computation of global quantities, such as entropy. I will present the known partial results on the way of proving of the conjecture and discuss the probable approaches to the proof and the obstacles that are encountered.