# On the Erdos-Szekeres convex polygon problem

## Speaker:

## Speaker Link:

## Institution:

## Time:

## Host:

## Location:

The classic 1935 paper of Erdos and Szekeres entitled "A combinatorial problem in geometry" was a starting point of a very rich discipline within combinatorics: Ramsey theory. In that paper, Erdos and Szekeres studied the following geometric problem. For every integer n ≥ 3, determine the smallest integer ES(n) such that any set of ES(n) points in the plane in general position contains n members in convex position, that is, n points that form the vertex set of a convex polygon. Their main result showed that ES(n) ≤ {2n - 4 \choose n-2} + 1 = 4^{n -o(n)}. In 1960, they showed that ES(n) ≥ 2^{n-2} + 1 and conjectured this to be optimal. In this talk, we will sketch a proof showing that ES(n) =2^{n +o(n)}.

# The space of metrics on an algebraic variety

## Speaker:

## Institution:

## Time:

## Host:

## Location:

The space of (Kahler) metrics on a projective algebraic variety can be given a natural infinite dimensional Riemannian structure. This leads to the notion of a geodesic in the space of metrics. I will discuss a recent result on the optimal regularity of these geodesics and how this relates to nonlinear PDEs and canonical metrics. This is joint work with J. Chu and V. Tosatti.

# Some problems on the distribution of prime numbers

## Speaker:

## Institution:

## Time:

## Host:

## Location:

We briefly describe some ideas and techniques that lead to solutions to certain problems in number theory, such as the bounded gaps between prime numbers, and others. This talk will be made understandable to general math audiences; technical details will be avoided.

# Deep learning in vision and language intelligence

## Speaker:

## Speaker Link:

## Institution:

## Time:

## Host:

## Location:

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.

# Harmonic measure in higher co-dimension

## Speaker:

## Institution:

## Time:

## Location:

Over the past century an effort to understand dimension and structure of the harmonic measure spanned many spectacular developments in Analysis and in Geometric Measure Theory. Uniform rectifiability emerged as a natural geometric condition, necessary and sufficient for classical estimates in harmonic analysis, boundedness of the harmonic Riesz transform in L^2, and, in the presence of some background topological assumptions, for suitable scale invariant estimates on harmonic functions. While many of geometric and analytic problems remain relevant in sets of higher co-dimension (e.g., a curve in $\RR^3$), the concept of the harmonic measure is notoriously missing. In this talk, we introduce a new notion of a "harmonic" measure, associated to a linear PDE, which serves the higher co-dimensional sets. We discuss its basic properties and give large strokes of the argument to prove that our measure is absolutely continuous with respect to the Hausdorff measure on Lipschitz graphs with small Lipschitz constant.

# Applications of Tauberian theorems to counting arithmetic objects

## Speaker:

## Speaker Link:

## Institution:

## Time:

## Host:

## Location:

The talk will start with some remarks on the role that zeta functions and Tuberian theorems have played in number theory in the last 180 years starting essentially with Dirichlet's proof of his Arithmetic Progression Theorem. The remainder of the talk will be devoted to giving a survey of recent applications of Tauberian theorems to counting arithmetic objects.

# On Geometric Quantization of Poisson Manifolds

## Speaker:

## Institution:

## Time:

## Host:

## Location:

Geometric Quantization is a program of assigning to

classical mechanical systems (symplectic manifolds and the associated

Poisson algebras of C-infinity functions) their quantizations ---

algebras of operators on Hilbert spaces. Geometric Quantization has

had many applications in Mathematics and Physics. Nevertheless the

main proposition at the heart of the theory, invariance of

polarization, though verified in many examples, is still not proved in

any generality. This causes numerous conceptual difficulties: For

example, it makes it very difficult to understand the functoriality of

theory.

Nevertheless, during the past 20 years, powerful topological and

geometric techniques have clarified at least some of the features of

the program.

In 1995 Kontsevich showed that formal deformation quantization can be

extended to Poisson manifolds. This naturally raises the question as

to what one can say about Geometric Quantization in this context. In

recent work with Victor Guillemin and Eva Miranda, we explored this

question in the context of Poisson manifolds which are "not too far"

from being symplectic - the so called b-symplectic or b-Poisson

manifolds - in the presence of an Abelian symmetry group.

In this talk we review Geometric Quantization in various contexts, and

discuss these developments, which end with a surprise.

# Three principles of data science: predictability, stability, and computability

## Speaker:

## Institution:

## Time:

## Host:

## Location:

In this talk, I'd like to discuss the intertwining importance and connections of three principles of data science in the title in data-driven decisions. The ultimate importance of prediction lies in the fact that future holds the unique and possibly the only purpose of all human activities, in business, education, research, and government alike.

Making prediction as its central task and embracing computation as its core, machine learning has enabled wide-ranging data-driven successes. Prediction is a useful way to check with reality. Good prediction implicitly assumes stability between past and future. Stability (relative to data and model perturbations) is also a minimum requirement for interpretability and reproducibility of data driven results. It is closely related to uncertainty assessment. Obviously, both prediction and stability principles can not be employed without feasible computational algorithms, hence the importance of computability. The three principles will be demonstrated through analytical connections, and in the context of two on-going neuroscience projects, for which "data wisdom" is also indispensable. Specifically, the first project interprets a predictive model used for reconstruction

of movies from fMRI brain signals; the second project employs deep learning networks (CNNs) to understand pattern selectivities of neurons in the difficult visual cortex V4.

# Blending Mathematical Models and Data

## Speaker:

## Institution:

## Time:

## Host:

## Location:

A central research challenge for the

mathematical sciences in the $21^{st}$ century is

the development of principled methodologies for the

seamless integration of (often vast) data sets

with (often sophisticated) mathematical models.

Such data sets are becoming routinely available

in almost all areas of engineering, science and technology,

whilst mathematical models describing phenomena of

interest are often built on decades, or even centuries, of

human knowledge creation. Ignoring either the data or the models

is clearly unwise and so the issue of combining them

is of paramount importance. In this talk we will give

a historical perspective on the subject, highlight some of

the current research directions that it leads to, and

describe some of the underlying mathematical frameworks

begin deployed and developed. The ideas will be illustrated

by problems arising in the geophysical, biomedical and

social sciences.