# 2020-2021 Colloquia

**Title: ** **Self-Referential Paradoxes**

**Speaker: ** Noson Yanofsky (Brooklyn College)

**Date: ** Wednesday, April 21 2021

**Time: ** 12:30pm-1:30pm

**Room: ** Please contact Jonathon Funk for the Zoom link

**Abstract**

Some of the most profound and famous theorems in mathematics and computer science of the past hundred and fifty years are instances of self-referential paradoxes. These theorems concern systems that have self-reference. We will remember

• Georg Cantor's theorem that shows there are different levels of infinity;

• Bertrand Russell's paradox which proves that simple set theory is inconsistent;

• Kurt Gödel's famous incompleteness theorems that reveals a limitation of the notion of proof;

• Alan Turing's realization that some problems can never be solved by a computer;

• and much more.

Amazingly, all these diverse theorems can be seen as instances of a single simple theorem of basic category theory. We will describe this theorem and show some of the instances. No category theory is needed for this talk.

**Title: ** **Nonlinear Splines - A Workable Model of the Draftsman's Spline **

**Speaker: ** Michael Johnson (Kuwait University)

**Date: ** Wednesday, April 14, 2021

**Time: ** 12:30pm-1:30pm

**Room: ** Please contact L. Boyadzhiev for the Zoom link

**Abstract**

A Draftsman’s Spline is a tool used by a draftsman to draw a smooth curve through given points \(P_1\); \(P_2\); : : : ; \(P_n\) in the \(xy\)-plane. It’s use dates back to the 1600’s where it was used in the design and construction of the hulls of sailing ships. The naive model asserts that the draftsman’s spline assumes the shape of an interpolating curve with minimal bending energy:

\(\frac{1}{2}\int_0^s k^2(s)ds \) where \(k\) denotes curvature and \(s\) arclength

Unfortunately, interpolating curves with minimal bending energy do not exist, except in the trivial case when the interpolation points are collinear. At General Motors Corporation (1964), it was proposed that the naive model be augmented by requiring that the bending energy of the interpolating curve be minimal when compared with ‘nearby’ curves that also (exactly) interpolate the given points.

In the literature, such curves are called stable nonlinear splines, and the techniques readily at hand (eg.,Calculus of Variations, Functional Analysis) seem insufficient for proving their existence. In this talk I’ll describe the above in more detail and also joint work with Albert Borbely that has produced the first broad existence proof for stable nonlinear splines.

**Title: ** **Mathematics of Neural Networks**

**Speaker: ** Azita Mayeli (QCC, Graduate Center CUNY)

**Date: ** Wednesday, March 24 2021

**Time: ** 12:30pm-1:30pm

**Room: ** Please contact D. Pham for the Zoom link

**Abstract**

Neural Networks (NN) are computational models inspired by human brain structure that have many applications in the field of AI such as voice, text, and image recognition. They are an important component of machine learning techniques. The key to understanding networks is the connectivity between network nodes, particularly whether or not two nodes are connected to each other. In this introductory talk, I will explain the mathematics underlying neural network algorithms, as well as how neurons interact within layers to produce accurate results.

**Title: ** **Fuglede Conjecture in \(\mathbb{Z}_p^2\times \mathbb{Z}_q^2\)**

**Speaker: ** Thomas Fallon (Graduate Center CUNY)

**Date: ** Wednesday, March 17 2021

**Time: ** 12:30pm-1:30pm

**Room: ** Please contact D. Pham for the Zoom link

**Abstract**

Fuglede's Conjecture states that sets that tile a space by translations and spectral sets are the same thing. The original problem was over \(\mathbb{R}^5\), and Terry Tao's counterexample in \(\mathbb{Z}_3^5\) that lifts to a counter example in \(\mathbb{R}^5\) motivated looking at the problem over other finite groups. I will discuss Fuglede's conjecture over \(\mathbb{Z}_p^2\times \mathbb{Z}_q^2\).

**Title: ** **Kinetic equation of coagulation and phase transition**

**Speaker: ** Pavel Dubovski (Stevens Institute of Technology)

**Date: ** Wednesday, March 10 2021

**Time: ** 12:30pm-1:30pm

**Room: ** Please contact L. Boyadzhiev for the Zoom link

**Abstract**

In 1962 McLeod found that Smoluchowski coagulation equation fails to conserve mass after certain critical time instant. This finding caused numerous research that allowed to figure out some classes of integral kernels, for which this nonlinear phenomenon holds. Recently, further results were obtained after a new coagulation model was derived by Safronov and Dubovski with surprisingly similar properties, even though this evolution nonlinear partial integrodifferential equation possesses certain hyperbolic properties unlike the classical parabolic-like Smoluchowski equation. Physically, the infringement of the mass conservation law implies the appearance of gel in colloid systems or raining in the science of clouds. In this talk the speaker shows mathematical background of the phase transition phenomenon and reviews the research results.

**Title: ** **Calabi-Yau Theorem: Calabi's Conjecture and Yau's Proof**

**Speaker: ** Caner Koca (New York City College of Technology)

**Date: ** Wednesday, March 3 2021

**Time: ** 12:30pm-1:30pm

**Room: ** Please contact D. Pham for the Zoom link

**Abstract**

Calabi-Yau Theorem is without doubt one of the most celebrated theorems in modern geometry. The statement of the theorem was initially conjectured by Eugenio Calabi in 1954. Shing-Tung Yau finally provided a proof in 1977, for which he was awarded a Fields Medal in 1982. The goal of this talk is to introduce this theorem, to explain its significance, and to sketch the steps of Yau's proof with as minimal background knowledge as possible.

**Title: Random Knotting**

**Speaker: **Moshe Cohen (SUNY New Paltz)

**Date: ** Wednesday, February 17, 2021

**Time: ** 12:30pm-1:30pm

**Room: ** Please contact Fei Ye for the Zoom link

**Abstract**

Take the ends of a piece of rope, tie a knot into it, and fuse the two ends together. Mathematicians ask: Is your knot the same as my knot? Knot Theorists like to find computations that can be performed on knots whose answers can help distinguish them.

**Title: ** **Some topics in stochastic heat and wave equations**

**Speaker: ** Jian Song (Shandong University)

**Date: ** Wednesday, February 10, 2021

**Time: ** 12:30pm-1:30pm

**Room: ** Please contact Fei Ye for the Zoom link

**Abstract**

The purpose of this talk is to present some results of my research on SPDEs. For solutions of a class of stochastic heat equations with multiplicative Gaussian noise, I will explain the Feynman-Kac formula, the Holder-continuity, the regularity of the density, and the long-term asymptotic behavior. I will also briefly explain the relationship between SPDEs and BDSDEs, and our recent work on directed polymer in colored environment. As a comparison, some of the topics for stochastic wave equations will also be discussed.

**Title: Subordination Principle for the Fractional Diffusion-Wave Equation**

**Speaker: Yuri Luchko**

**Date: Wednesday, November 11, 2020**

**Time: 12:30pm—1:30pm**

**Room: See Zoom web address below**

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**Abstract**

**Title: Theory and numerics of some types of fractional differential equations **

**Speaker: Jeffrey Slepoi**

**Date: Wednesday, October 28, 2020**

**Time: 1:00pm—2:00pm**

**Room: See Zoom web address below**

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**Abstract**

**Title: Polyalgorithms, Number Fields, and Motives**

**Speaker: Ivan Horozov**

**Date: Wednesday, October 7, 2020**

**Time: 12:00pm—1:00pm**

**Room: See Zoom web address below**

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**Abstract**

**Title: Almost complex and complex structures on manifolds**

**Speaker: Luis Fernandez**

**Date: Wednesday, September 23, 2020**

**Time: 12:30pm—1:30pm**

**Room: See Zoom web address below**

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**Abstract**

An *almost complex* structure on a manifold is a way to define an operation similar to multiplication by *i* (the imaginary unit) on the tangent space of the manifold; a manifold with an almost complex structure is called an *almost complex *manifold.

On the other hand, a *complex manifold* is a manifold modeled over complex vector spaces.

I will explain these concepts carefully and compare them, showing several examples, especially those based on algebra structures like the quaternions and the octonions. Then I will show examples of almost complex manifolds that cannot be complex manifolds, but can be as close as we want to complex manifolds in a certain sense. This work is in collaboration with Scott Wilson, from Queen's College and the GC.

If there is enough time, I will also talk about almost complex curves on the 6-sphere.

**Title: Left Invariant Complex Structures on Double Lie Groups**

**Speaker: D. N. Pham**

**Date: Wednesday, September 23, 2020**

**Time: 1:10pm—2pm**

**Room: See Zoom web address below**

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**Abstract**

Roughly speaking, a Lie bialgebra is a Lie algebra g such that its dual space g* is equipped with its own Lie algebra structure. The Lie algebras g and g* are compatible in the sense that they define a Lie algebra on g⨁g* which contain g and g* as subalgebras and for which the natural scalar product on g⨁g* is invariant. This Lie algebra structure on g⨁g* is denoted as D(g) and is called a double Lie algebra. A Lie group whose Lie algebra is D(g) is called a double Lie group. In this talk, I will discuss the problem of constructing left invariant complex structures on double Lie groups (which will in effect turn these objects into complex manifolds). The only prerequisites for this talk are basic differential geometry and familiarity with Lie groups and Lie algebras. No familiarity with complex manifolds is assumed. We will review the relevant background on complex manifolds as part of the talk. This project is joint work with Fei Ye.