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1 September 2023
Speaker: Sohail Farhangi, Adam Mickiewicz University (Poland)
Title: On the partition regularity of $ax+by=cw^mz^n$
Click to read the Abstract.
The study of Ramsey theory is about structures that are partition regular. A diophantine equation is partition regular if for any finite partition of $\mathbb{N}$, one of the cells of the partition contains a solution to the given equation. Schur proved in 1916 using elementary arguments that the equation $x+y = z$ is partition regular, which is one of the earliest results of this type. It was only in 2010-2011 that Bergelson and Hindman independently proved that the equation $x+y = wz$ is partition regular, and their arguments were much more sophisticated than those of Schur since they used very special types of ultrafilters. In joint work with Richard Magner, we gave an almost complete classification of those $a,b,c,m,n \in \mathbb{N}$ for which the equation $ax+by = cw^mz^n$ is partition regular. We also study the partition regular of $ax+by = cw^mz^n$ over rings of integers of algebraic number fields, which necessitated new results about ultrafilters which are of independent interest.
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8 September 2023
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No Colloquium - Welcome Party
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22 September 2023
Speaker: Asamoah Nkwanta , Morgan State U.
Title: Linear Trees, Lattice Walks, and RNA Arrays
Click to read the Abstract.
The leftmost column entries of RNA arrays I and II count the RNA numbers that are related to RNA secondary structures from molecular biology. RNA secondary structures sometimes have mutations and wobble pairs. Mutations are random changes that occur in a structure and wobble pairs are known as non-Watson-Crick base pairs. We provide a combinatorial interpretation of RNA array II as RNA secondary structures with n bases and k base-point mutations where $\omega$ of the structures contain wobble base pairs. This is research based on Dr. Jasmine Evans dissertation "RNA Combinatorics and Secondary Structure Prediction with Applications to MicroRNA Structure Analysis" Ph.D., Morgan State University, Baltimore, MD, USA, 2020.
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26 September 2023
Speaker: William Velez, U. of Arizona
Title: TBA
Click to read the Abstract.
TBA
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29 September 2023
Speaker: Alessandra Celletti, University of Rome, Tor Vergata (Italy)
Title: The Mathematics of Space science: the weak boundary between theories and applications
Click to read the Abstract.
Mathematical theories have been often motivated by problems of Celestial Mechanics and Astrodynamics; among the others, perturbative methods, KAM theory, Nekhoroshev theorem.
On the other hand, the investigation of the dynamics of celestial bodies often requires a delicate balance between theoretical and numerical methods, nowadays including also machine learning techniques.
In this context, I will illustrate some examples that pertain to satellite and space debris dynamics, whose studies make use of advanced perturbative and computational methods.
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6 October 2023
Speaker: Alberto Bressan, PennState
Title: Hyperbolic conservation laws in one space dimension: old and new results.
Click to read the Abstract.
Hyperbolic conservation laws provide the basic mathematical models for continuum physics. A typical example is provided by the Euler
equations of gas dynamics, accounting for the conservation of mass, momentum and energy.
The first part of the talk will review the basic concepts in
the theory of hyperbolic conservation laws, including: weak solutions admissibility conditions, existence and continuous dependence of solutions to the initial value problem.
The second part will cover some very recent advances concerning the uniqueness of entropy admissible solutions. Implications of these results toward a posteriori error estimates for numerical approximations will be discussed, together with some
open questions.
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13 October 2023
Speaker: Jerome Vetois, Mc Gill University
Title: Compactness and Stability Issues for Second Order Elliptic Equations on Closed Manifolds
Click to read the Abstract.
I will discuss questions of compactness and stability under perturbation of the data for solutions to nonlinear second order elliptic PDE in the curved setting of a closed Riemannian manifold.
Equations of this type arise both in the fields of Conformal Geometry (Yamabe Equation) and Mathematical Physics (Nonlinear Schrodinger Equations, Einstein-Lichnerowicz Constraint Equations, etc.).
Several results have been obtained which now give quite an accurate picture of the conditions under which the sets of solutions to such equations are compact/stable or non-compact/unstable.
I will give an overview of some of these results as well as some elements of the methods we can use to either detect or rule out the possibility of non-compactness/instability phenomena.
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20 October 2023
Speaker: Lia Vas, Saint Joseph's University
Title: Algebras of graphs
Click to read the Abstract.
The focus of the talk are some algebras defined for a directed graph. We present an overview of the reasons for their introduction, the motivation for their consideration, as well as history of their development. We expand on the meaning of the axioms defining these algebras and summarize current research trends. No background other than a standard (undergraduate) linear algebra course is necessary.
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27 October 2023
Speaker: Thomas Judson, Stephen F. Austin State University
Title: UTMOST: A Journey through Open Source Textbooks (talk's slides)
Click to read the Abstract.
The quantity and quality of open-source mathematics textbooks has grown considerably in the past decade. Not only are open-source textbooks less expensive than commercially published textbooks, but many open-source textbooks can be of equal or even better quality than their commercial counterparts. The focus of the UTMOST Project, funded by the National Science Foundation, has been to develop resources for writing open-source textbooks and conduct educational research on how students and faculty use highly interactive textbooks. Products of the UTMOST Project include PreTeXt, CoCalc, and the Sage Cell Server. PreTeXt is a markup language that allows an author to produce textbooks and all scholarly work in any discipline and in any format, including PDF, HTML, ePub, Jupyter Notebook, and braille. Currently, there are 80+ books and projects written in PreTeXt. Moreover, books written in PreTeXt can be hosted on Runestone, a platform for hosting interactive textbooks on the Internet. This presentation will focus on what we have learned about open-source textbooks and tools from the beginning of UTMOST to the present day as well as a brief look in to the future.
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3 November 2023
Speaker: Dimiter Vassiliev, University of New Mexico
Title: The fractional Yamabe equation on homogeneous groups.
Click to read the Abstract.
The general themes of the talk are Dirichlet forms, fractional (non-local) operators and associated Sobolev type spaces on groups of homogeneous type.
I will recall some general motivating examples for considering non-local operators and particular equations before focusing on the respective questions in the setting of homogenous groups.
The considered groups are not assumed to be Carnot groups or to satisfy a Hormander-type conditions.
I will describe a result on sharp asymptotic decay of solutions to non-linear equations modeled on the fractional Yamabe equation.
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10 November 2023
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Veteran's Day, University Closed.
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17 November 2023
Speaker: Ali Maalaoui, Clark University
Title: Singular Solutions for Critical Equations involving the Dirac Operator
Click to read the Abstract.
In this talk I will address the construction of Delaunay type solutions for the critical Dirac equation and the conformal Dirac-Einstein equation on the sphere. After a reduction to a Hamiltonian system, we prove the existence of a family of periodic solutions indexed by their period corresponding to a solution on the sphere with two points singularities. This family converges to the spherical solution when the period diverges to infinity. This is a joint work with V. Martino, Y. Sire and T. Xu.
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24 November 2023
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Thanksgiving Recess, University Closed