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

 

Abstract

In this talk, a subordination principle for the solution operators to a family of the linear multi-dimensional space-time-fractional diffusion-wave equations is addressed. These equations are obtained from the diffusion equation by replacing the first order time-derivative by the Caputo fractional derivative of order β, 0 < β ≤ 2 and the Laplace operator by the fractional Laplacian −(−∆) α 2 with 0 < α ≤ 2. First, a special representation of the fundamental solution to these equations is obtained in form of a Mellin-Barnes type integral. This representation is then employed for derivation of a subordination formula that connects the solutions to the space-time-fractional diffusion-wave equations with different orders α and β of the fractional derivatives. The talk is mainly based on the results published in [1].
[1] Yu. Luchko, Subordination principles for the multi-dimensional space-time-fractional diffusionwave equation. Theory of Probability and Mathematical Statistics 98, 1, 2018, 121-141.

 

 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

 

Abstract

A number of methods were developed to numerically solve fractional differential equations. This work contains two methods for numerically solving non-linear fractional differential equations, necessity of multiple approaches is demonstrated to assure their validity. The new substitution method is analyzed and used in this work as a base for all calculations. The convergence theorem assures precision of the method, discretization schemes present implementation. Sufficient condition for a well-condition problem for the method is developed and proved. Solutions for Bessel equation in fractional derivatives were attempted before and the topic interests many scientists. This work presents the theory for solving the generalized fractional Bessel equation, conditions for existence of a solution and its uniqueness for equations with Caputo derivatives. A step further expands the Bessel equation into a more general quasi-linear fractional Bessel equation, where the matching of powers and the order of the derivatives is not required for all terms but one. The discovered methodology for quasi-linear Bessel equation is further developed in its application to the homogeneous equations with constant coefficients and fractional equations with power functions as coefficients. Simple equations in this domain were considered before and analytical solutions were identified. We expand the class of equations and cross check the results with simpler equations. Some cases of fractional Cauchy Euler equations were addressed in the past. Generalized fractional Cauchy Euler equations for both Riemann-Liouville and Caputo fractional derivatives are analyzed next in this work. The solutions similar to the classical Cauchy Euler equation are identified. Independence of solutions is proved.

 

 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

 

Abstract

The first half of the talk will be an overview of relations between poylogarithms, Riemann zeta values and multiple zeta values, and their relations to algebraic geometry. Some people who have worked on this topic are Deligne, Zagier, Kontsevich, Manin, Goncharov and many others. The next half of the course will be on a generalization of multiple zeta values to number fields, which I call multiple Dedkind zeta values. At the end I will mention a current work with Pavel Sokolov on analogues of polylogarithms to number fields, which we call Dedekind polylogarithms and their applications to values L-functions.

 

 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

 

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

 

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.

Campus Cultural Centers

Kupferberg Holocaust Center (KHC)Opens in a new window
Kupferberg Holocaust Center Opens in a new window

The KHC uses the lessons of the Holocaust to educate current and future generations about the ramifications of unbridled prejudice, racism and stereotyping.

Queensborough Performing Arts CenterOpens in a new window
QPAC: Performing Arts CenterOpens in a new window

QPAC is an invaluable entertainment company in this region with a growing national reputation. The arts at QPAC continues to play a vital role in transforming lives and building stronger communities.

Queensborough Art GalleryOpens in a new window
QCC Art GalleryOpens in a new window

The QCC Art Gallery of the City University of New York is a vital educational and cultural resource for Queensborough Community College, the Borough of Queens and the surrounding communities.