Singularity formation in fluid dynamics

Federico Pasqualotto
November 15

Abstract: In physical models of fluids, a singularity occurs when a quantity of interest (velocity, pressure, vorticity…) becomes infinite at some finite time, starting from a "nice" initial configuration. In this talk, we will first introduce some classical models of compressible and incompressible fluids. We will then describe several mechanisms by which fluids can form singularities in finite time, allowing us to discuss the singularity formation problem for the incompressible three-dimensional Euler equations and the Navier—Stokes equations. We will finally touch upon computer-assisted methods and their applications to singularity formation in fluids. We will only assume familiarity with multi-variable calculus and some concepts of real analysis.
Prerequisites: Multi-variable calculus is enough. Intro to real analysis is preferable.

Interactive proofs

John Wright
Novermber 4

Abstract: What is a proof? In principle, we think of a proof as a sequence of axioms and inference rules that allow us to derive a mathematical statement. In practice, it's something that typically appears in a textbook or a math journal. In this talk, I'll discuss a different philosophy coming from theoretical computer science, in which a proof can be anything which convinces you. This opens the door to proofs having a variety of different attributes, including being interactive, zero-knowledge, and even quantum.

Symplectic embeddings and infinite staircases

Nicki Magill
October 28

Abstract: When mailing an object, a question arises: Can I fit this object into this particular box? The answer depends on which aspects of the object we want to preserve when placing it in the box. For example, different constraints apply when shipping a foam ball compared to a wooden one. For some objects, we might only care about preserving their volume, while others may be more rigid. In this talk, we will focus on preserving the symplectic structure, which can be quite flexible at times and rigid at others. Determining whether one object can fit into another while preserving the symplectic structure is often difficult to answer. Our focus will be on embedding four-dimensional ellipsoids. We will explore the intricate structure that determines when these symplectic ellipsoids can be embedded into other four dimensional objects.
Prerequisites: None!

L-functions and integrals

Danielle Wang
October 21

Abstract: We discuss how the Riemann zeta function, Dirichlet \(L\)-functions, and \(L\)-functions of modular forms are defined and analyzed by expressing them as Mellin transforms (a type of integral). This kind of relationship between \(L\)-functions and integrals has generalizations that are currently of interest in the relative Langlands program.
Prerequisites: I think really only things from calculus, like integrals on \(\mathbb R\) (except the functions are complex-valued), the idea of series/integrals converging, Taylor series. Some complex analysis feels like it would be nice, but shouldn't be necessary (at least in my mind). I will define holomorphic function, we won't do any contour integrals.

Finding and creating periodic orbits

Michael Hutchings
October 14

Abstract: A dynamical system can be described by a vector field. A periodic orbit is a trajectory of the vector field which starts and ends at the same point. This corresponds to repeating behavior in the dynamical system, such as a planet orbiting around a star. In general there are many open questions regarding when periodic orbits exist and what properties they have. This talk will introduce some recent results about existence of periodic orbits of certain vector fields in three dimensions.
Prerequisites: The beginning of the talk should be understandable to anyone who knows what a vector field is, e.g. from Math 53, while later in the talk we will touch on some more sophisticated notions, but without getting into technical details.

Kazhdan-Lusztig polynomials, combinatorial invariance, and machine learning

Christian Gaetz
October 7

Abstract: Kazhdan-Lusztig polynomials are an important family of polynomials with both algebrogeometric and representation-theoretic interpretations. I'll explain how to compute these polynomials and introduce the Combinatorial Invariance Conjecture (CIC), which asserts that they only depend on a certain directed graph. I'll then discuss work of Blundell, Buesing, Davies, Veličković, and Williamson, who used machine learning techniques to conjecture some recurrences which would imply the CIC. I'll end by mentioning some work of mine, joint with Grant Barkley, in which we use this approach to prove some new cases of the CIC.

The hardness of classification

Forte Shinko
September 30

Abstract: In mathematics, we usually talk about objects up to a notion of equivalence, such as groups up to isomorphism, manifolds up to homeomorphism, and so on. We typically use invariants to distinguish these objects; for instance, the groups \(\mathbb{Z}\) and \(\mathbb{Z}^2\) are not isomorphic, because "the size of the smallest generating set" is 1 in the former case, and 2 in the latter. In very nice situations, there are complete invariants which completely classify a given class of objects up to equivalence, such as for closed surfaces, which are homeomorphic iff they have the same genus. However, in many situations, there do not seem to exist such invariants, and we would like a formal way to say what it means for a nice classification to exist. We will use the language of descriptive set theory to formalize the notion of classification in terms of Borel equivalence relations, and then show that many natural classification problems have no complete invariants, or even much weaker versions of such.
Prerequisites: It will help to have some background in linear algebra and group theory.

Quantum advantage in scientific computation?

Lin Lin
September 23

Abstract: The advent of fully error-corrected quantum computers is anticipated to usher in a new era in computing, with Shor's algorithm poised to demonstrate practical quantum advantages in prime number factorization. However, cryptography problems are typically not categorized as scientific computing problems. This raises the question: which scientific computing challenges are likely to benefit from quantum computers? I will first explore some essential criteria and considerations to realize quantum advantages in these problems. I will then introduce some recent advancements in quantum algorithms related to scientific computing applications. At least the first half of the presentation is intended to be accessible to a broad audience.

Squaring the Circle

Andrew Marks
September 16

Abstract: In 1925 Tarski posed the problem of whether a solid two dimensional square can be partitioned into finitely many pieces which can be rearranged to form a disk of the same area. We will explain the history of this problem, and the ideas used in its 1990 solution by Laczkovich. We will emphasize tools for converting between solutions to combinatorial problems on finite vs infinite graphs, and also the study of effective bounds on the ergodic theorem. At the end of the talk, we will discuss some recent work joint with Spencer Unger giving better upper bounds on the number of pieces needed to square the circle. This uses results from diophantine approximation in number theory. Diophantine approximation studies how well irrational numbers can be approximated by rationals. A landmark result in this area is Roth's theorem on diophantine approximation of algebraic irrationals, for which Roth won the Fields medal in 1958. It is still an open problem if the constants in Roth's theorem can be computed effectively. We will explain how recent partial results on this problem (giving effective irrationality exponents for the cube root of 2) can be used to improve the upper bound on the number of pieces used to square the circle.
Prerequisites: It may be helpful (but is not required) to know the definition of a group action for the talk.

Sequences and Hidden Structure

Alexander Paulin
September 9

Abstract: A very simple way to generate a sequence of numbers is to fix a function and repeatedly apply it to zero. What do sequences constructed in this manner look like? What happens when we vary the function in a continuous way? In this lecture we will discover that such questions hold vast hidden depth, opening the door to some of the most beautiful mathematics ever discovered.

The cobordism hypothesis in low dimensions

Peter Haine
April 15

Abstract: In this talk, we’ll introduce cobordisms and talk about topological quantum field theories (TQFTs) in low dimensions. Though these words might sound intimidating, we’ll keep things concrete by sticking to dimensions 1 and 2 where we can draw pictures of everything. Only a good understanding of linear algebra should be necessary to enjoy the talk. The main result we’ll explain is how a 2 dimensional TQFT is determined by a single piece of algebraic data. This result led Baez and Dolan to conjecture that the same is true in all dimensions. This is referred to as the cobordism hypothesis; it was recently proven by a number of mathematicians. We’ll give a hint at how the cobordism hypothesis is related to many interesting areas of math.
Prerequisites: Linear algebra (linear transformations, tensor products, pairings, dual vector spaces, etc.). Knowing what a category is would be helpful, but is not necessary.

A Model theorist looks at familiar mathematical systems

Carol Wood
April 15

Abstract: Model theory is a relatively young branch of logic, not much older than I am (but I am old). I would like to give a taste of its flavor by describing a consequence of the compactness theorem, a basic result in logic. Compactness allows nonstandard analogs e.g. of integers and real numbers. In these nonstandard structures we find infinitely large integers and both infinitely large and infinitesimally small real numbers.
Prerequisites: No prerequisites are required. It could be helpful if you ever wondered why the concept of limit in calculus is hard to understand, or maybe it was easy for you!

Congruent numbers and elliptic curves

Ken Ribet
April 8

Abstract: Congruent numbers are positive square-free integers that occur as areas of right triangles whose sides all have rational length. The numbers 1–4 are not congruent numbers, but 5, 6 and 7 all are congruent numbers. These numbers appear in Fibonacci's book "Liber Quadratorum (book of squares)" and were studied by Lagrange, Fermat and others. Determining the set of congruent numbers amounts to identifying elliptic curves of a certain type that have positive rank. The conjecture of Birch and Swinnerton-Dyer translates the identification problem into an analytic problem involving L-series of elliptic curves. I will explain how the main themes in this story fit together and point to some ongoing research about elliptic curves and their L-series. Although Fibonacci numbers appear prominently in Math 55, I had little idea about Fibonacci himself until I encountered his Book of Squares. Spoiler: he was born around 1170 and published his book in 1225. He acquired the name "Fibonacci" only in 1838!

Dispersion and global solutions in nonlinear dispersive flows

Daniel Tataru
April 1

Abstract: Dispersive flows are partial differential equations where linear waves travel in different directions, depending on their frequency. Nonlinear effects may potentially disrupt this pattern over long time scale. The aim of this talk will be to provide some insights into these competing phenomena, leading up to some very recent conjectures.
Prerequisites: Real analysis as well as some differential equations for the second part of the talk (e.g. math 123 or 126)

Representations of groups and tensor categories

Aleksandra Utiralova
March 18

Abstract: To any group \(G\) one can associate the category \(\operatorname{Rep}(G)\) of its finite-dimensional representations. It turns out that it is possible to reconstruct \(G\) from just this categorical data but to do this one should take into account the tensor product structure on representations of \(G\), which turns \(\operatorname{Rep}(G)\) into the so-called tensor category. I will give a brief description of this construction and then proceed to discuss tensor categories without the underlying group. The examples I'm interested in include the category of super vector spaces and Deligne categories \(\operatorname{Rep}(S_t)\), which give interpolations of the category of representations of the symmetric group \(S_n\) to complex values of the parameter \(n\).

Physics of flows for most efficient heat transfer between two walls

Anuj Kumar
March 11

Abstract: We are interested in the design of forcing in the Navier–Stokes equation such that the resultant flow maximizes the heat transfer between two differentially heated walls for a given power supply budget. Previous work established that heat transport cannot scale faster than 1/3-power of the power supply. Recently, Tobasco & Doering (PRL'17) constructed self-similar two-dimensional steady branching flows, saturating this upper bound up to a logarithmic correction to scaling. We present a construction of three-dimensional "branching pipe flows" that eliminates the possibility of this logarithmic correction and for which the corresponding heat transport scales as a clean 1/3-power law in power supply. Our flows resemble previous numerical studies of the three-dimensional wall-to-wall problem by Motoki, Kawahara & Shimizu (JFM'18). However, using an unsteady branching flow construction, it appears that the 1/3 scaling is also optimal in two dimensions. After carefully examining these designs, we extract the underlying physical mechanism that makes the branching flows "efficient," based on which we present a design of mechanical apparatus that, in principle, can achieve the best possible case scenario of heat transfer. We will further discuss some interesting implications of branching flows, for example, anomalous dissipation in turbulent flows and Rayleigh--B'enard convection.
Prerequisites: The talk will use a few concepts from PDEs and calculus of variations but I will try to keep it very simple.

Nash's \(C^1\) isometric embedding theorem

Sung-Jin Oh
March 4

Abstract: The goal of this talk is to present a proof of the remarkable Nash \(C^1\) embedding theorem, which states, for instance, that the unit sphere \(S^2\) can be "crumpled" in a \(C^1\) fashion into an arbitrarily small ball in \(\mathbb R^3\). Note that such a statement is obviously false if one replaces "\(C^1\)" by "smooth", by consideration of curvature. This theorem turned out to be more than a mere curiosity; its proof foreshadowed an important technique called "convex integration", which found remarkable applications in a wide array of fields, such as symplectic topology, calculus of variations, and fluid dynamics.

An overview of microlocal geometry

Kendric Schefers
February 26

Abstract: Notions of "smoothness" and "singularity" exist for a wide variety of objects in geometry. The first example one encounters in life are the notions for real-valued functions, where a smooth function is one which is differentiable, and a singular one is one which is not. There are also such notions for distributions, D-modules, coherent sheaves, etc. Working microlocally in each of these setting means trying to get a handle on the singularities in question by looking not only at where singularities occur, but also in what direction they arise. In this talk, I will try to give an impression of how the microlocal philosophy is implemented in modern geometry.
Prerequisites: should probably be linear algebra and multivariable calculus, but truthfully, the talk is going to be very impressionistic. Most of what I talk about would require a lot of advanced material to fully understand, but I'll be sort of hand-waving the whole time. No one who shows up will fully understand the talk, but conversely, anyone who shows up will understand at least the gist of it.

Generalized Parking Function Polytopes

Andrés R. Vindas Meléndez
February 12

Abstract: A classical parking function is a list of positive integers whose nondecreasing rearrangement (b1, ..., bn) satisfies b <= i. The convex hull of {parking functions of length n} is an n-dimensional polytope (think high dimensional polygon), called the classical parking function polytope. We can loosen our notion of "parking function" to create generalized parking function polytopes. These new polytopes are relatives of the Pitman-Stanley polytope and some partial permutohedra. We leverage these connections to compute the volume of special parking polytopes.

Can we use math to improve traffic flow?

Franziska Weber
February 5

Abstract: Braess’s paradox is a proposed explanation for when the modification of a road system by for example adding a new road, leads to a worse traffic situation instead of an improvement. It was proposed in 1968 by the mathematician Dietrich Braess and can mathematically be formulated as a Nash Equilibrium that is a worse situation than the best overall flow through the road network. In this talk, we review Braess’s paradox and its emergence in real life situations and then explore whether it can be observed in traffic flow models involving partial differential equations.

A mirror into the higher dimensional world

Catherine Cannizzo
January 29

Abstract: We live in a three dimensional world. If we consider time as a fourth “coordinate”, we have four dimensions. These four dimensions are known as “space-time.” In physics, string theory conjectures that, at scales much smaller than an electron (of a similar order of magnitude of the universe to the atom), there are small strings vibrating. For the theory to work, the strings must vibrate in 6 compact extra dimensions, for a total of 10 dimensions! It turns out that two geometric models for the strings give the same physics. These pairs opened up mathematicians to the notion of “mirror symmetry” which gives us a lot of interesting information about geometric spaces. In this talk, I will define a “geometric space”, give some examples, and illustrate how to visualize higher dimensions as well as touch on the connection to my research in homological mirror symmetry.

Moduli

Martin Olsson
January 22

Abstract: One of the key notions in modern mathematics is the notion of moduli. It arises in algebraic geometry from the fact that classification of various objects of interest often leads to the exploration of other spaces (called moduli spaces). I will discuss some elementary examples of moduli in this talk.