A partition with no hook lengths divisible by a is called an a-core partition. For two coprime numbers a and b, a partition is called an (a,b)-core partition if it is both a-core and b-core partition. It is well-known that the number of a-core partitions is infinite, and Anderson proved the number of (a,b)-core partitions is a rational Catalan number. Inspired by work of Johnson, we give an expression for the number of (a,b,c)-core partitions. This is ongoing work with Jineon Baek and Myungjun Yu.

My talk will cover two key areas of voting rights in the area of redistricting: partisan gerrymandering and racial gerrymandering.  In the 30 years since Davis v. Bandemer 106 S. Ct. 2797 (1986) declared that partisan gerrymandering was indeed justiciable, there has been only one federal court that has struck down a single seat plurality election-based plan as an unconstitutional partisan gerrymander. The finding of unconstitutionality in that legislative case from Wisconsin, Whitford v. Gill, No. 15-cv-421-bbc, United States District Court, W.D. Wisconsin, decided in November 2016, is now on appeal to the U.S. Supreme Court. Earlier courts confronting partisan gerrymandering challenges had insisted that there were no clear and manageable standards to separate out "ordinary" politics from unconstitutional gerrymandering.  Beginning with a definition of gerrymandering, I offer a multi-componented standard that has as necessary conditions (a) a clear and severe injury involving a disparate impact on a political party that serves as the vehicle for the expression of particular ideas and values, (b) effects that can be expected to be durable, (c) effects that can be shown NOT to be explicable by chance nor explicable by specific features of the electoral geography that impact all plans, (d) compelling direct or indirect evidence of partisan intent  and (e) evidence that there exist one or more remedial plans that address the constitutional violation while also satisfying, on balance, all relevant constitutional and statutory criteria at least as well as the challenged plan.  Here "neutral" treatment as the baseline for comparison. Implementing this test requires the application of a number of different statistical ideas.  In particular, the concept of disparate impact is related to the mathematical/statistical ideal of asymmetry in seats-votes distributions.  I will also talk about racial gerrymandering and the problem of avoiding, on the one hand, situations where racial concerns play too great a role in districting and avoiding on the other hand, situations where race has not adequately been taken into account, leading to the unconstitutional dilution of minority voting strength.  Here I will talk about the application of statistical  tests for racially polarized voting and the implementation, in the districting context, of mathematical ideas such a compactness and contiguity, as well as the definition of terms such as "packing" and "cracking."

Deep learning, which exploits multiple levels of data representations that give rise to hierarchies of concept abstraction, has been the driving force in the recent resurgence of Artificial Intelligence (AI). In this talk, I will summarize rapid advances in cognitive AI, particularly including comprehension, reasoning, and generation across vision and natural language, and applications in vision-to-text captioning, text-to-image synthesis, and reasoning grounded on images for question answering and dialog. I will also discuss future AI breakthrough that will benefit from multi-modal intelligence, which empowers the communication between humans and the real world and enables enormous scenarios such as universal chat-bot and intelligent augmented reality.

We will discuss recent results of M. Hochman and B. Solomyak.