Department of Mathematics Colloquium 2021-2022 Fall

August 31

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September 7

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September 14

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September 21

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September 28

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October 5

Dan Ashlock (University of Guelph)

Title: Problem Factories: almost endless problem generation

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Math educators often have a few cool problems that can be used to dispel the idea that math is awful, a view too often inculcated by pre-university math education.  Some of these wonderful problems fit into larger frameworks which we call problem factories. The idea for problem factories arose at a math education conference, probably because of the density of these wonderful problems.  The title of the talk is also the title of a book that Andrew McEachern and I wrote that consists of a brief discussion of the idea of problem factories and several examples.

A problem factory is a body of mathematical knowledge that defines a collection of similar problems together with effective techniques for generating instances of the problems.  The initial book sticks with problems that use classical theorem-and-proof mathematics with a computer serving as a convenience.  A second book is in progress which treats problem factories that seem to require artificial intelligence to locate problem instances.  The transition from classical to AI based problem factories is gradual, not crisp, and the talk will contain examples.

Link: Virtual (the link is above)

October 12

Bernard Lidicky (Iowa State University)

Title: Flag Algebras and Its Applications

Abstract: Flag algebras is a method, developed by Razborov, to attack problems in extremal combinatorics. Razborov formulated the method in a very general way which made it applicable to various settings. The method was introduced in 2007 and since then its applications have led to solutions or significant improvements of best bounds on many long-standing open problems, including several problems of Erdős. The main contribution of the method was transferring problems from finite settings to limits settings. This allows for clean calculations ignoring lower order terms. The method can utilize semidefinite programming and computers to produce asymptotic results. This is often followed by stability type arguments with the goal of obtaining exact results. In this talk we will give a brief introduction of the basic notions used in flag algebras and demonstrate how the method works. Then we will discuss applications of flag algebras in different settings.

Link: In-person with a zoom stream (see the Zoom link above).

October 19

Jue Yan (Iowa State University)

Title: Recent development of direct discontinuous Galerkin method for compressible Navier-Stokes equations and cell-average based neural network method for time dependent problems 

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Compressible Navier-Stokes (N-S) equation is one of the most important partial differential equations arising in physics and engineering. It can accurately capture the air density and pressure distributions around fast flying object (i.e. airplane wings). In the first part of my talk, I will discuss a new direct discontinuous Galerkin method solving two-dimensional compressible N-S equations. The new method is based on the observation that the complicated nonlinear diffusion process can be written out as a sum of multiple individual diffusion processes corresponding to density, momentum and total energy conserved variables.The new method is shown to be able to accurately calculate physical quantities such as lift, drag and friction coefficients. In the second part of my talk, I will introduce the newly developed cell-average based neural network method. It is found the neural network method can be relieved from the small time step size restriction and can adapt almost any large time step size for solution evolution, which makes the method extremely fast and efficient. Potential of the neural network method will be addressed at the end of the talk.

Link: In-person with a zoom stream (see the Zoom link above). 

October 21

Tim Pennings (Davenport University)

Title: Battle of the Titans: Mathematics vs Truth

Abstract: What is Truth?  This question asked by Pontius Pilate at Jesus's trial has haunted reflective people for centuries. Supreme Court Justice Potter Stewart said of pornography (roughly): I can't define it, but I know it when I see it. Similarly, philosophers (and others who study truth and our ability to know truth) have used mathematics as a proof for the existence of truth. Does truth exist? Yes, consider mathematics. What is truth? Consider mathematics. But now the soft underbelly of mathematics has been exposed, showing that mathematics - and the world of Truth - is much richer and more uncertain than ever imagined. Come explore the world of alternative facts. Not for the faint-of-heart.

Link: In person

October 26

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November 2

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November 9

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November 16

Isaac Goldbring (University of California-Irvine)

Title: The Connes Embedding Problem, MIP*=RE, and the Completeness Theorem

Abstract: The Connes Embedding Problem (CEP) is arguably one of the most famous open problems in operator algebras.  Roughly, it asks if every tracial von Neumann algebra can be approximated by matrix algebras.  Earlier this year, a group of computer scientists proved a landmark result in complexity theory called MIP*=RE, and, as a corollary, gave a negative solution to the CEP.  However, the derivation of the negative solution of the CEP from MIP*=RE involves several very complicated detours through C*-algebra theory and quantum information theory.  In this talk, I will present joint work with Bradd Hart where we show how some relatively simple model-theoretic arguments can yield a direct proof of the failure of the CEP from MIP*=RE while simultaneously yielding a stronger, Gödelian-style refutation of CEP as well as the existence of “many” counterexamples to CEP.  No prior background in any of these areas will be assumed.

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November 23

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November 30

Gunhee Cho (University of California, Santa Barbara)

Title: Introduction to a Hopf fibration and an anti-de Sitter space in sub-Riemannian geometry

Abstract: Sub-Riemannian geometry is a generalization of Riemannian geometry and we introduce Hopf fibration and anti-de Sitter space as natural objects in sub-Riemannian geometry that appears very naturally in various contexts of Physics and Mathematics. On the other hand, Theorem of Hurwitz says that every normed division finite-dimensional algebra has four types: real numbers, complex numbers C, quaternions Q, and octonions O; we introduce heat kernel analysis and its sub-Laplacian, and horizontal Brownian motion on Hopf-fibrations and anti-de Sitter spaces over C, Q, and O, and introduce recent progress including the octonionic setting. These works are the joint work with Fabrice Baudoin.

Link: Virtual

December 6

Maria Chudnovsky (Princeton University) 

Title: Induced subgraphs and tree decompositions

Abstract: Tree decompositions are a powerful tool in both structural graph theory and graph algorithms. Many hard problems become tractable if the input graph is known to have a tree decomposition of bounded “width”. Exhibiting a particular kind of a tree decomposition is also a useful way to describe the structure of a graph. Tree decompositions have traditionally been used in the context of
forbidden graph minors; bringing them into the realm of forbidden induced subgraphs has until recently remained out of reach. Over the last couple of years we have made significant progress in this direction, exploring both the classical notion of bounded tree-width, and concepts of more structural flavor. This talk will survey some of these ideas and results.

Link: In person

Time: 3.10pm - 4.00pm

Note: This talk will be as a part of the Joint AWM-MOCA Speaker Series (JAMSS) and will be followed by an hour (4.00pm-5.00 pm) of discussion with graduate/undergraduate students at the same room.