Conference website: https://www.math.uci.edu/scgas
Please register at the conference website if planning to attend.
Conference website: https://www.math.uci.edu/scgas
Please register at the conference website if planning to attend.
Undergraduates are curious about research in mathematics: what kinds of questions do mathematicians ask, what does research entail, how do you begin to solve a new problem. In this talk, we will discuss integrating undergraduate research projects inside the classroom and how to expose students to new mathematical questions in both upper and lower division courses. We will then talk more generally about setting students up for success in the classroom.
In this talk, I will introduce our recent work on Gauss curvature flow with Xu-Jia Wang and Qi-Rui Li.
In this work we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\R^{n+1}$ with the speed $f r^{\alpha} K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. We prove that if $\alpha\ge n+1$, the flow exists for all time and converges smoothly after normalization to a hypersurface, which is a sphere if $f\equiv 1$. Our argument provides a new proof for the classical Aleksandrov problem ($\alpha = n+1$) and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang recently, for the case q<0 ($\alpha>n+1$). If $\alpha< n+1$, corresponding to the case q > 0, we also establish the same results for even function f and origin-symmetric initial condition, but for non-symmetric f, counterexample is given for the above smooth convergence.
A key challenge in Riemannian geometry is to find ``best" metrics on compact manifolds. To construct such metrics explicitly one is interested to know if approximation sequences contain subsequences that converge in some sense to a limit manifold.
In this talk we will present convergence results of sequences of closed Riemannian
4-manifolds with almost vanishing L2-norm of a curvature tensor and a non-collapsing bound on the volume of small balls. For instance we consider a sequence of closed Riemannian 4-manifolds,
whose L2-norm of the Riemannian curvature tensor is uniformly bounded from
above, and whose L2-norm of the traceless Ricci-tensor tends to zero. Here,
under the assumption of a uniform non-collapsing bound, which is very close
to the euclidean situation, and a uniform diameter bound, we show that there
exists a subsequence which converges in the Gromov-Hausdor sense to an
Einstein manifold.
To prove these results, we use Jeffrey Streets' L2-curvature
ow. In particular, we use his ``tubular averaging technique" in order to prove fine distance
estimates of this flow which only depend on significant geometric bounds.
For 3 years I served as the Director of Jump Labs, a new endeavor for cutting-edge research and recruiting launched by Jump Trading, a quantitative high frequency trading firm based in Chicago.
Jump Labs sponsors research in high performance computing and data science via gifts grants involving:
The crux is to create a long term and powerful pipeline for talent acquisition by challenging the faculty and students with real-world problems. The structure aligns relevant industrial research with the passions and expertise of the faculty member and students. Opportunities for publication are encouraged. In our first two years we sponsored over 60 undergraduate and graduate students and 20 professors spanning 25 projects. The structure seeks to advance relevant research and creates a powerful recruiting pipeline for talent that is long term and low risk.
We will discuss the successes and challenges encountered at Jump Labs in its first three years.
Abstract: Renormalization provides a powerful tool to approach universality and
rigidity phenomena in dynamical systems. In this talk, I will discuss
recent results on renormalization and rigidity theory of circle
diffeomorphisms (maps) with a break (a single point where the derivative
has a jump discontinuity) and their relation with generalized interval
exchange transformations introduced by Marmi, Moussa and Yoccoz. In a
joint work with K.Khanin, we proved that renormalizations of any two
sufficiently smooth circle maps with a break, with the same irrational
rotation number and the same size of the break, approach each other
exponentially fast. For almost all (but not all) irrational rotation
numbers, this statement implies rigidity of these maps: any two
sufficiently smooth such maps, with the same irrational rotation number
(in a set of full Lebesgue measure) and the same size of the break, are
$C^1$-smoothly conjugate to each other. These results can be viewed as
an extension of Herman's theory on the linearization of circle
diffeomorphisms.
I am going to talk about Carleman estimate with Carleman weight first. To prove Carleman estimate, I need to introduce some definitions of semiclassical analysis first. Then I am going to talk about Carleman estimate with limiting Carleman weight and some applications.
I will talk about the use of peers to enhance learning in three different contexts. The first context is a flipped integral calculus course. Students are expected to prepare for class ahead of time by watching video(s) and taking online quizzes. The instructor accesses the quiz data before class and uses student responses to tailor the classroom instruction. In-class time focuses on extending student understanding with a variety of active learning techniques, including peer-to-peer instruction. I will report the data we have collected about the impact of this experience on both student attitudes and learning.
The second context is a summer online bridge program for incoming students. We utilize undergraduate coach/mentors to meet online virtually with a team of 4-5 incoming students throughout the summer to help close some of their mathematical gaps. I will describe the design of this program, how it enhances Yale's desire to recruit and retain a diverse student body, and the impact it has on student attitudes and learning. I will also highlight data that describes the impact of peer coaches on both learning and the motivation to learn.
The third context is a systematic supervised reading/research program for ~1200 math majors at UC Irvine. I will provide some suggestions for how this program might be structured to leverage advanced undergraduates and graduate students to help motivated math majors.
In this talk, we report a recent joint work with Shuonan Wu that gives a universal construction of simplicial finite element methods for 2m-th order partial differential equations in ℝ^n, for any m≥1, n≥1. This family of finite element space consists of piecewise polynomials of degree not greater than m. It has some natural inclusion properties as in the corresponding Sobolev spaces in the continuous cases and it recovers the MWX element when n≥m. We establish quasi-optimal error estimates in an appropriate energy norm. The theoretical results are further validated by numerical tests.