Scientific

Deep learning-based approaches have succeeded surprisingly in various fields of sciences. In particular, one of the first and most successful achievements of them is image classification. Now, deep learning-based algorithms perform even better than humans in classification problems. However, people in machine learning community observed that it is possible to deteriorate the performance of neural networks seriously by adding a well-designed small noise, which is called ‘adversarial attack’. Although humans still classify this new image correctly. the machine completely fails to classify this new image. Since this issue is serious in practice, for example security or self-driving car, practioners want to develop more robust machines against such adversarial attack, which motivates ‘adversarial training problem’. However, until very recently there has been no theoretical understanding of it. In this talk, I will present the recent progress of understanding adversarial training problem. The key idea of connecting the two areas originates from (Wasserstein) barycenter problem, one of the famous implications of optimal transport theory. I will introduce ‘generalized barycenter problem’, the extension of classical barycenter problem, and its multimarginal optimal transport formulations. Through the lens of those tools, one can understand the geometric structure of adversarial training problems. One crucial advantage of this result is that it allows to utilize many computational optimal transport tools. Lastly, if time is permitted, I will give the result of the existence of optimal robust classifiers which not only extends the binary setting case to the multiclass one but also provides a clean interpretation by duality.

For an integer k≥3; Δk (x) :=∑n≤xdk(n)-Ress=1 (ζk(s)xs/s), where dk(n) is the k-fold divisor function, and ζ(s) is the Riemann zeta-function. In the 1950's, Tong showed for all large enough X; Δk(x) changes sign at least once in the interval [X, X + CkX1-1/k] for some positive constant Ck. For a large parameter X, we show that if the Lindelöf hypothesis is true, then there exist many disjoint subintervals of [X, 2X], each of length X1-1/k-ε such that Δk (x) does not change sign in any of these subintervals. If the Riemann hypothesis is true, then we can improve the length of the subintervals to << X1-1/k (logX)-k^2-2. These results may be viewed as higher-degree analogues of a theorem of Heath-Brown and Tsang, who studied the case k = 2. This is joint work with Siegfred Baluyot.

Euler’s famous formula tells us that (with appropriate caveats), a map on the sphere with f countries (faces), e borders (edges), and v border-ends (vertices) will satisfy v-e+f=2. And more generally, for a map on a surface with g holes, v-e+f=2-2g. Thus we can figure out the genus of a surface by cutting it into pieces (faces, edges, vertices), and just counting the pieces appropriately. This is an example of the topological maxim “think globally, act locally”. A starting point for modern algebraic geometry can be understood as the realization that when geometric objects are actually algebraic, then cutting and pasting tells you far more than it does in “usual” geometry. I will describe some easy-to-understand statements (with hard-to-understand proofs), as well as easy-to-understand conjectures (some with very clever counterexamples, by M. Larsen, V. Lunts, L. Borisov, and others). I may also discuss some joint work with Melanie Matchett Wood.

Speaker biography:

Ravi Vakil is a Professor of Mathematics and the Robert K. Packard University Fellow at Stanford University, and was the David Huntington Faculty Scholar. He received the Dean's Award for Distinguished Teaching, an American Mathematical Society Centennial Fellowship, a Frederick E. Terman fellowship, an Alfred P. Sloan Research Fellowship, a National Science Foundation CAREER grant, the presidential award PECASE, and the Brown Faculty Fellowship. Vakil also received the Coxeter-James Prize from the Canadian Mathematical Society, and the André-Aisenstadt Prize from the CRM in Montréal. He was the 2009 Earle Raymond Hedrick Lecturer at Mathfest, and a Mathematical Association of America's Pólya Lecturer 2012-2014. The article based on this lecture has won the Lester R. Ford Award in 2012 and the Chauvenet Prize in 2014. In 2013, he was a Simons Fellow in Mathematics.

Turing patterns have been suggested as an explanation for morphogenesis in a variety of organisms. Despite the fact that morphogenesis occurs during growth, most studies of Turing patterns have been conducted on static domains. We present experimental and computational studies of Turing patterns in a chemical reaction-diffusion system on growing two-dimensional domains. We also investigate the effect of inert obstacles on pattern evolution. We find that the rate of domain growth significantly affects both how the patterns are laid down and their ultimate morphology.