This is a colloquium-style seminar.  It runs on Wednesdays. The current organizer is Han-Bom Moon. 

The seminar is broadcasted via zoom. Check the announcement emails for the link.

Spring 2024

Date: February 14, 3:00 PM

Speaker: Han-Bom Moon (Fordham University)

Place: JMH 405 (RH)

Title: On roots of complex polynomials

Abstract: This talk is prepared for students. I want to explore the following question -- as a student or a (yet) non-professional mathematician, how can we find an interesting question? I will give some suggestions with a concrete example from the study of the roots of complex polynomials. Only Calculus is required to follow the talk.

Slide

Date: February 28, 3:45 PM (Speical Time!)

Speaker: Francesco Iafrate (Sapienza University of Rome)

Place: LL 508 (LC)

Title: Regularized statistical problems for diffusion processes

Abstract: We address the estimation of stochastic differential equations parameters in sparse, high-dimensional settings. In order to recover the true reduced model, we apply adaptive penalized estimation techniques, such as the bridge and the Lasso, under a high frequency sampling scheme. These methods are adaptive in the sense that they build on initial, unpenalized, estimators, such as those based on quasi-likelihood. We present the asymptotic behaviour of these estimators, in particular we argue that they possess the so-called oracle properties. This class of problems has applications in determining causal relations in high-dimensional systems of stochastic differential equations, for example in the context of financial time series. We introduce generalizations and future extensions, including problems with multiple penalties in the mixed-rates asymptotic regime.

Date: April 3, 3:00 PM

Speaker: Melkana Brakalova (Fordham University)

Place: JMH 132 (RH)

Title: On one-quasiconformality at a point

Abstract: The theory of plane qusiconformal (q.c.) mappings (originally known as {\bf almost conformal}) was developed in the first half of the 20th  century, most notably in the works of  Gr\"otzsch, Teichm\"uller, Lavrentieff, Ahlfors, and Bers. However, properties of such maps were used  in the early/mid 19th century e.g. by Gauss in his study of isothermal coordinates and by Tissaut in his work on cartography. As of now the theory of quasiconformal mappings is a major area of geometric function theory and an ubiquitous tool used in Nevanlinna theory, Teichm\"uller theory, hyperbolic geometry,  complex dynamics,  fluid mechanics,  computer vision, to mention a few.

Quasiconformal mappings are useful in applications as they are less rigid then conformal maps while distorting shapes and their measurements such as area, angle, length in a bounded fashion. 

In this presentation we  introduce the notion of the conformal module and geometric and analytic definitions of the q.c. mapping, one--quasiconformality of a q.c. mapping at a point, and present some known, some of our own results and open questions for the case of conformality at a point. 

Further results on one--quasiconformality and applications to the study of the integrable Teichm\"uller subspaces of the universal Teichm\"uller space will be discussed in the future.

Date: April 10

Speaker: Tristan Ozuch (MIT)

Place: LL 508 (LC)

Title: Einstein metrics and Ricci flow in dimension 4

Abstract: Einstein manifolds and Ricci flow were instrumental tools in answering long standing topological questions in dimension 3. It is now in dimension 4 that most open questions remain, and geometers hope that those same tools will provide some answers. However, Einstein 4-manifolds are still far from being fully understood and many 4-dimensional specific techniques have yet to be applied to Ricci flow.

I will motivate Einstein metrics and the Ricci flow approach to topology and discuss what are some of the main difficulties and specificities of dimension 4. 

Slide

Date: April 17

Speaker: Jennifer Li (Princeton)

Place: JMH 132 (RH)

Title: On the cone conjecture for log Calabi-Yau mirrors of Fano 3-folds

Abstract: Generally speaking, the three building blocks in the birational classification of algebraic varieties are Fano, Calabi-Yau, and varieties of general type. Convex geometry is a tool that can be used to understand these varieties. More specifically, we may study the cone of curves of a variety (or, dually, the nef cone of the variety). For Fano varieties, the cone of curves (and therefore the nef cone) is always rational polyhedral, meaning the cone has finitely many rational extremal rays. Varieties of general type lie at the other extreme - they form a class that is impossible to classify, and there are examples of the cone of curves of such variety being round. In the Calabi-Yau case, we do not have complete control, but Morrison's cone conjecture (1993) provides hope: given a Calabi-Yau variety Y, there exists a rational polyhedral cone which is a fundamental domain for the action of the automorphism group of Y on the nef cone. In this talk, I will explain a version of Morrison's cone conjecture that holds for certain types of 3-folds, and it turns out that there are many examples of such 3-folds.

Date: April 24

Speaker: Ishan Datt (J.P. Morgan)

Place: JLL 508 (LC)

Title: Statistical Methods for trading commodity markets

Abstract: In commodity markets, there are often dislocations within the market caused by discrepancies between supply and demand.  Here we explore a few methodologies to identify and capitalize on these dislocations across the market.  The statistical methods presented provide a baseline model for quickly identifying and ranking dislocations based on a variety of factors.

Date: May 1

Speaker: Paul Jung (Fordham)

Place: JMH 132 (RH)

Title: Infinite-width limits of Neural Nets and the Neural Tangent Kernel

Abstract: Artificial Neural Networks (ANNs) have played a pivotal role in the AI revolution, contributing to the development of large language models like ChatGPT. In this talk, we explore the concept of taking the infinite-width limit of ANNs using a scaling technique known as neural tangent scaling. Surprisingly, this limit reveals an intriguing link to machine learning Kernel Methods of yore (harking all the way back to the early parts of this millenium!).

Date: May 8

Speaker: Scott Chapman (Sam Houston State University)

Place: LC

Title: The Mathematics of Chicken McNuggets

Abstract: Every day, 34 million Chicken McNuggets are sold worldwide. At most McDonalds locations in the United States today, Chicken McNuggets are sold in packs of 4, 6, 10, 20, 40, and 50 pieces. However, shortly after their introduction in 1979 they were sold in packs of 6, 9, and 20. The following problem spawned from the use of these latter three numbers.

The Chicken McNugget Problem. What numbers of Chicken McNuggets can be ordered using only packs with 6, 9, or 20 pieces? In particular, what is the largest number of McNuggets that cannot be ordered?

While the answers to this question are relatively easy, it opens a new line of exploration into the world of numerical monoids. We consider the basic properties of numerical monoids and discuss how elements in such a monoid can be “factored” (much like a positive integer can be factored into a product of prime numbers). The talk is accessible to students with a fundamental background in number theory and abstract algebra.