Math Monday Archives

You can find more recent Math Mondays on the Math Monday page.

Spring 2024

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.


Fall 2023

Some Mathematics in Condensed Matter Physics

Mengxuan Yang
November 27

Abstract: Discovery of superconductivity and its physics theory has led to interesting mathematical studies of condensed matter physics. In this talk, using a model of twisted bilayer graphene recently developed by Becker-Embree-Wittsten-Zworski, I will introduce some mathematical ideas behind condensed matter physics, including representation theory and Bloch-Floquet theory. I will tell you some mathematical story behind "magic angles", at which you may find superconductivity.
Prerequisites: My talk only needs some linear algebra background. Some functional analysis would be helpful but that should not be necessary.


Differential Geometry of Surfaces

Alexander B. Givental
November 20

Abstract: In one hour, I'll try to formulate and prove the main result of Math 140, the celebrated Gauss--Bonnet theorem.
Prerequisites: The lecture will be very elementary; in particular, knowledge of calculus will not be assumed.


Ruler and compass constructions and abstract algebra

Emiliano Gomez
November 13

Abstract: This will be a pretty elementary and informal talk. We will see why some of the famous classical Greek ruler and compass constructions are impossible (doubling the square, squaring the circle, trisecting the angle). If time allows, we will also see which regular polygons are constructible and which ones are not. These problems were open for 2000 years, and their solution exemplifies the beauty and power of the interaction between geometry and algebra.
Prerequisites: There are very minimal requirements for the talk: I assume the audience knows a few Math 113 and Math 110 concepts: groups, rings, fields, rings of polynomials, vector spaces, and basic familiarity with complex numbers.


Algebraic Curves and the Riemann-Roch theorem

David Eisenbud
November 6

Abstract: The Riemann-Roch theorem is probably the most important single result in algebraic geometry. I'll try to explain what it says, and why it is important, without too many technical details.
Prerequisites: Math 113


The math behind a periodic table for 2-dimensional quantum matter

Colleen Delaney
October 30

Abstract: Ordinarily we think of particles like electrons as zipping around in 3 dimensions of space. But when these particles lose the freedom to move in 1 of these 3 dimensions, their collective behavior can form what is called a topological quantum phase of matter, the discovery of which was awarded the 2016 Nobel Prize in Physics. These exotic 2-dimensional phases (or states) of matter can support virtual or quasiparticles called anyons whose physics have a very mathematical flavor: (1) they depend only on the topology (global shape) and not the length or angles of the path the anyons carve out in spacetime and (2) they can be described using a branch of abstract algebra called category theory. The harmony between these topological and algebraic aspects of anyons can be expressed through a picture calculus obeyed by diagrams of anyon spacetime trajectories in which physics calculations are rigorously done by drawing intuitive pictures. Not only is this mathematical description of topological phases beautiful, it should also be useful! Topological phases are an active area of interdisciplinary research and development in both the public and private sectors motivated in part by the potential application of topological phases to quantum computing hardware and software, with national labs and companies like Microsoft and Google leading projects integrating theory and experiment.


Triple Symmetric Functions

Yegor Zenkevich
October 23

Abstract: Symmetric functions (e.g. Schur polynomials) play a major role in representation theory of general linear group and symmetric groups, a classical subjects. Recently it has been found that a mysterious generalization these functions which may be called triple symmetric functions seems to exist, but it is not clear what are the corresponding groups and representaitons. We will review the basics of Schur functions, then introduce the triple symmetric functions and play a bit with them.


Moduli spaces of curves

Hannah Larson
October 9

Abstract: I'll introduce the concept of moduli spaces in algebraic geometry with the example of the moduli space of circles. This is an example of a moduli space of "embedded curves." However, as I'll explain, the associated "abstract curves" are all the same. I'll finish by talking about moduli spaces of abstract curves and share some recent results, which are joint work with Samir Canning.
Prerequisites: The only requirements are complex numbers, polynomials, and a willingness to visualize!


Falconer's distance set conjecture and its many faces

Ruixiang Zhang
October 2

Abstract: In 1985, Falconer made the conjecture that whenever \(E \subset \mathbb{R}^2\) has dimension \(>1\), its distance set \(\Delta(E) = \{|x-y|: x, y \in E\}\) will have positive measure. Despite its innocent appearance, this conjecture remains unsolved as of today. We will introduce this conjecture and three interesting results towards it. The methods to prove these results are related to Fourier analysis, classical algebraic geometry/topology and geometric measure theory.
Prerequisites: Math 104 and 110


Large cardinal hypotheses and new axioms for set theory

Gabriel Goldberg
September 25

Abstract: In this talk I will introduce the ZFC axioms of set theory, which serve as an adequate foundation for almost all of mathematics but are inadequate to resolve many fundamental problems in set theory itself. I'll then turn to extensions of the ZFC axioms obtained by adding large cardinal hypotheses, also known as strong axioms of infinity, and finally I'll discuss axioms that are inspired by the ongoing search for canonical models of set theory.


The spectrum of an infinite product and the Stone-Čech compactification of \(\mathbb{N}\)

Owen Barrett
September 18

Abstract: Consider the spectrum of a countable product of a fixed field \(k\). Mumford wrote the following in his Red Book: ‘Those familiar with ultrafilters and similar far-out mysteries will have no problem proving that this topological space is the Stone-Čech compactification of \(\mathbb{N}\). Logicians assure us that we can prove more theorems if we use these outrageous spaces.’ This talk is about these outrageous spaces and their natural appearance in the study of commutative rings.


Crystals!

Nicole Gonzales
September 11

Abstract: A basic question in linear algebra is finding the coefficients of a vector in terms of a given basis. A fundamental goal in combinatorics is finding formulas for these coefficients when our vector spaces are certain polynomial rings. It turns out questions like these can be encoded and answered using certain directed graphs known as crystals. However, these crystals contain much more information than the original polynomials themselves. Algebraic properties like symmetries of the polynomials and multiplication rules are all encoded as symmetries and tensor product structures on these graphs. Hence, the polynomials can be seen as "shadows" of the crystals. In fact, the crystals themselves are also shadows of more complicated algebraic objects known as representations. In this talk we will introduce crystals from scratch and with lots of examples. We will show how the product of symmetric polynomials follows from tensoring certain symmetric crystal graphs. We will also discuss how certain bases in the full polynomial ring correspond to truncations of these symmetric crystals and how multiplying these truncated graphs turns out to be unexpectedly mysterious.


Monstrous Moonshine

Richard Borcherds
August 28

Abstract: This talk will be about the monster group of order 808,017,424,794,512,875,886,459,904, 961,710,757,005,754,368,000,000,000 and its relation to the elliptic modular function \(1/q + 744 + 196884q + 21493760q^2+ \ldots\)


Spring 2023

Differential Equations in Algebraic Geometry

Martin Olsson
April 17

Abstract: In math 1b we study ordinary differential equations in one variable, and these equations are normally viewed as in the domain of analysis. However, such differential equations also have a rich connection to algebraic geometry and arithmetic. In this talk I will discuss some of these algebraic and arithmetic aspects of differential equations, and the role they play in these more algebraic fields.


A few ways to lose your money

F. Alberto Grünbaum
April 10

Abstract: Be careful before you trust your intuition. A few cautionary tales.


Quantum Advantage

Lin Lin
April 2

Abstract: Quantum computers offer the potential to compress \(2^n\) bits of classical information using only \(n\) qubits. While this may suggest exponential quantum advantage across various applications, it is important to recognize the limitations of this perspective. In this talk, I will delve into these limitations and explore different approaches towards realizing quantum advantages in scientific computation. The discussion will be relatively informal, and no previous knowledge of quantum computation is required to understand most of the content.


Curvature and the Dirac Operator

McFeely Jackson Goodman
March 20

Abstract: The Dirac operator is a differential operator on a Riemannian manifold which can be used to relate the curvature - a local notion of shape - of a manifold to its topology - a global notion of shape. I will illustrate that story with concrete examples and computations. I will then discuss the implications to the following questions: Can a given manifold be deformed to have some notion of "positive curvature" everywhere? When can one manifold with positive curvature be deformed into another, maintaining positive curvature along the deformation?


Hyperbolicity From Two Perspectives

Jackson Morrow
March 13

Abstract: I will begin with a gentle survey about the arithmetic of algebraic curves describing when algebraic curves have infinitely many rational solutions and when they only have finitely many solutions. Next, I will transition to talking about how the arithmetic of curves is related to the space of maps between them, and to conclude, I will state some of my own results in this area.


Three Short Stories about Numbers and Logic: How imperfection can lead to beauty

Ian Sprung
March 6

Abstract: The first story: In the year 1900, people dreamed up machines that could do calculations. They asked if those machines would one day learn how to think and feel. An easier question they asked was whether they could solve polynomial equations. (The answer, found in 1970, was 'no!' So machines won't replace us.) The second story: The simplest equations ("elliptic curves") humanity does not understand look deceivingly simple, but have been a riddle for more than a century now. One major piece of light has been shed on this riddle in 1922 using a really simple idea. The third story: Some equations don't have formulas for finding \(x\), but this imperfection leads to something new -- we can still 'find' \(x\) by using \(x\) to make up new numbers instead. If time permits, I will mention a result from about a decade and a half ago that weaves these three stories together.


Diffusion Curves: Images via Differential Equations

Christopher Ryba
February 27

Abstract: Properties of image formats (e.g. raster vs vector, lossy vs lossless, compression, ...) determine their best use cases. We will discuss an unusual format: Diffusion Curves. Colour discontinuities in the image are stored, and the rest of the image is interpolated using the Laplace equation or biharmonic equation. There will be some discussion of the underlying mathematics, practical considerations, and examples.


The Mathematics of Cake-Cutting

Connor Halleck-Dubé
February 13

Abstract: "Fair division problems," that is, algorithmic questions of equitably dividing valuable resources, have a history stretching into antiquity and widespread applications to the real world. Focusing on the well-studied problem of cake-cutting, we'll discuss a variety of mathematical frameworks for handling the (surprisingly subtle!) question of what makes a division "fair" or "equitable," and some cases of solutions to these problems.


An Invitation to Enumerative Geometric Combinatorics

Andrés R. Vindas Meléndez
February 6

Abstract: Enumerative geometric combinatorics is an area of mathematics concerned with counting properties of geometric objects described by a finite set of building blocks. Lattice polytopes are geometric objects that can be formed by taking the convex hull of finitely many integral points. In this talk I will present background on polytopes, lattice-point enumeration, and share some results on a special family of polytopes that can be further studied. Throughout the talk I will present questions and open problems. (No prior knowledge will be assumed, and I will attempt to explain all concepts.


Numerical Computations in Science and Engineering

Per-Olof Persson
January 30

Abstract: Find out how mathematics and computers can be used to predict physical phenomena, and why this is important for analysis, design, and optimization in the applied sciences.


Fall 2022

Category theory in mathematics and beyond

David Spivak
November 28

Abstract: Since the 1940s, category theory has become an increasingly useful language for describing common structures seen across mathematics and in the wider world of science, engineering, art, and technology. In that sense the subject is very broad, but it's centered around a relatively compact core of definitions and theorems. In this talk, I'll begin by giving a general overview and survey of category theory and its applications. Then I'll work with the audience to choose a more specific topic, either giving a flavor of some area of active research, such as interconnected dynamical systems, or explaining some of the aforementioned mathematical definitions and theorems that sit at the core of the subject.


Solving the divergence equation

Sung-Jin Oh
November 21

Abstract: The problem of finding a vector field with prescribed divergence arises from many corners of mathematics and physics. It is an example of an underdetermined PDE -- one equation for multiple components. Accordingly, solutions are highly non-unique, and the right question is often not about the existence of just some solutions, but rather solutions with extra desirable properties. In this expository talk, I will discuss a way to solve this problem using ideas motivated by electrostatics (or mathematically, distribution theory). Then I will discuss its applications, ranging from differential topology to fluid mechanics to general relativity.


Quantum computing algorithms for quantum dynamics simulation

Di Fang
November 14

Abstract: Recent years have witnessed tremendous progress in developing and analyzing quantum computing algorithms for quantum dynamics simulation of bounded operators (Hamiltonian simulation). However, many scientific and engineering problems require the efficient treatment of unbounded operators, which frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure theory, quantum control and quantum machine learning. In this talk, we will start with a brief high-level introduction of quantum computing, and then discuss some recent advances in quantum algorithms for efficient unbounded Hamiltonian simulation, including Trotter type splitting and the quantum highly oscillatory protocol (qHOP) in the interaction picture. (The talk does not assume a priori knowledge on quantum computing; but preliminaries on linear algebra and baby differential equations are helpful.)


Partial Differential Equations in fluid dynamics

Krutika Tawri
November 7

Abstract: In this talk we will discuss mathematical problems arising from fluid dynamics, geophysics and material science. We will discuss the models, open problems and challenges in the field and recent progress.


On Hilbert's Irreducibility Theorem

Lea Beneish
October 31

Abstract: A polynomial with integer coefficients is said to be irreducible if it cannot be factored as a product of two non-constant polynomials. Hilbert's irreducibility theorem states that asymptotically, 100% of integer polynomials of degree \(n\) are irreducible. Since Hilbert's original version, "Hilbert irreducibility" refers to a broader class of theorems that apply in a variety of other settings. In this talk, we will state the theorem, do some examples, and discuss some of the ideas in the proof.


Subgroups of \(SL_2(\mathbb{Z})\) and modular forms

Yunqing Tang
October 24

Abstract: There is a natural action of \(SL_2(\mathbb{Z})\) on the set of complex numbers with positive imaginary part (such set is called the upper half plane). One way to study a finite index subgroup of \(SL_2(\mathbb{Z})\) is to study the holomorphic functions on the upper half plane invariant under this subgroup. I will use concrete examples to show that the arithmetic of the Fourier coefficients of these holomorphic functions is closely related to the so-called congruence property of the subgroup (including the recent joint work of Calegari, Dimitrov and myself). (I will explain all the terminologies mentioned here in the talk.)


Fundamental Groups

Owen Barrett
October 17

Abstract: What does Galois theory have to do with covering spaces? As it turns out, quite a lot. I will recount this beautiful classical story, which goes back to Grothendieck, underpins étale cohomology, and is ubiquitous in algebraic geometry. Time permitting, I will allude to recent developments.


Restricted Patterns of the Past, Present and Future

Zvezdelina Stankova
October 10

Abstract: Whether designing the new tile pattern in your family's kitchen backsplash, trying to avoid bad investment sequences, or simply counting all possible paths from your home to school that do not cross over the local river, inescapably you are venturing into the realm of restricted patterns. In this talk, we shall discuss several paths of pattern-exploration, and think about whether or not there is a "true" way of approaching pattern-avoidance equivalence and ordering among the array of generated ideas and methods. No matter what your math background is, you will find your own path between realistic visualization and abstract thinking, and perhaps, you will fall in love with one of the open problems. Of course, the more math you know, the more adventurous you may feel about attacking these open problems.


L-functions and reciprocity

Sug Woo Shin
October 3

Abstract: I will review how the quadratic reciprocity and the Riemann zeta function have evolved to a more general reciprocity law and more general zeta/L-functions, culminating with class field theory in the early 20th century. If time permits, I will hint at later developments.


Representation theory beyond Schur-Weyl duality.

Vera Serganova
September 26

Abstract: I will start with beautiful classical story of duality between representations of symmetric and general linear groups. Then I explain how this story extends to infinite-dimensional groups, supersymmetry and tensor categories.


The unreasonable effectiveness of elliptic curves

Ken Ribet
September 19

Abstract: If a and b are integers that satisfy a simple nonvanishing condition, the cubic equation \(y^2 = x^3 + ax + b\) defines an elliptic curve over the field of rational numbers. Elliptic curves have been studied for millennia and seem to occur all over the place in mathematics, physics and other sciences. In my talk, I'll explain how a specific elliptic curve provides the solution to a surprisingly hard "brain teaser" that had a big run on social media a few years ago.


Foundations of Mathematics

Richard Borcherds
September 12

Abstract: This talk will be about the early attempts to put mathematics on a firm footing and some of the problems that turned up, such as Russell's paradox, Godel's incompleteness theorem, the failure of the law of the excluded middle, and so on.


Mysteries of Linear Algebra

David Eisenbud
August 28

Abstract: How do you tell when you have found all the solutions to a system of linear equations? There are at least 3 levels of mystery in the answers to this question: 1. When the coefficients in the equations are real or complex numbers, it is a matter of computing the ranks of some matrices. 2. When the coefficients are polynomials, there is still a pretty good answer. 3. When the coefficients are still more complicated, it is a problem that can sometimes be solved computationally, but still remains completely mysterious in many cases. I will review level 1, explain some things about level 2, and touch on some of the mysteries of level 3.


Spring 2022

Searching for Rigidity in Algebraic Starscapes

Gabriel Dorfsman-Hopkins
25 April

Abstract: The creation and study of plots of algebraic integers has a rich and collaborative history, bringing together pure and computational mathematics with digital art. These images exhibit deep relationships between geometry and arithmetic, and serve as invitations to explore the mysterious patterns lying within the integers. We will cover some history of the visual geometry of algebraic numbers, and then explore the effect of emphasizing algebraic integers according to arithmetic invariants arising in Galois theory, exhibiting previously hidden geometries in these number starscapes. Finally, we will explain how the resulting imagery informs and inspires research questions in algebraic number theory. This work is joint with Shuchang Shu.


Understanding the Riemann hypothesis

Jesse Elliot
18 April

Abstract: The Riemann hypothesis, first conjectured by Riemann in 1859, is widely considered to be one of the most important, if not the most important, unsolved problems in all of mathematics. We provide several ways, including some that are new, of understanding what the Riemann hypothesis means and why it is important to the study of the prime numbers. We also discuss some ways in which the hypothesis can be extended further.


An invitation to hyperbolic geometry

Nick Miller
11 April

Abstract: In this talk I will give an informal introduction to hyperbolic geometry, focusing on dimensions 2 and 3, and discuss how hyperbolic manifolds arise as natural objects to study in mathematics. I will keep the talk as non-technical as possible, focusing on intuitively understanding geometric concepts, so no prerequisites will be required.


The cobordism hypothesis in low dimensions

Peter Haine
28 March

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 proven by Hopkins and Lurie. We'll give a hint at how the cobordism hypothesis is related to many interesting areas of math.


Long-time behaviour of nonlinear hyperbolic partial differential equations (PDEs)

Shi-Zhuo Looi
14 March

Abstract: Part of the historical arc of understanding the propagation of light, sound and gravity, the vibration of membranes, and other physical phenomena, involves a rigorous understanding of nonlinear hyperbolic partial differential equations (PDEs). I will introduce some themes in the investigation of the long-time asymptotics of such PDEs.


Traces and Loops

David Nadler
7 March

Abstract: We'll discuss the amazing relationship between traces of matrices and loops in topological spaces. No prior knowledge will be assumed other than some linear algebra and a little topology.


Long Time Dynamics in Nonlinear PDE's

Daniel Tataru
28 February

Abstract: One very interesting question in the study of evolution problems in nonlinear partial differential equations is to understand their long time dynamics. In this talk I will try to give some idea both of what one can expect, and what are some cool ideas that can be used in order to prove some things.This will include notions such as dispersion, scattering and normal forms.


Magic Angles in Twisted Bilayer Graphene

Maciej Zworski
14 February

Abstract: This may seem like a physics title but the maths involve very "pure" topics such as behaviour of non-self-adjoint operators, representation of finite groups and theta function, all presented via colourful figures and movies.


Moduli Spaces: study the forest, not the trees

Rachel Webb
7 February

Abstract: We spend a lot of time studying mathematical objects (e.g. vector spaces) as discrete phenomena, "the trees." Moduli spaces are a way to assemble related mathematical objects into a "forest" (e.g., the moduli space of 1-dimensional subspaces of \(\mathbb{R}^3\)). Moduli spaces are often interesting mathematical objects in their own right, and can sometimes provide insight into the original objects you were trying to study.In this talk we will try out many examples of moduli spaces, favoring those that use only basic set theory. Come prepared to get your hands---er, papers dirty!


What is... Zonotopal Algebra?

Olga Holtz
30 January

Abstract: What do the following seven things have in common: splines, matroids, zonotopes, hyperplane arrangements, polynomial interpolation, \(D\)-invariant ideals, and integer point counting? Many surprising connections among these seemingly disparate phenomena can be explained using the so-called Zonotopal Algebra. I will give a quick overview of the subject along with basic examples. No particularly advanced background will be assumed.


Fall 2021

Arithmetic and hyperoblic geometry: pattern producing numbers

David Fisher
6 December

Abstract: I will attempt to give a lay persons description of some recent work on hyperbolic manifolds, totally geodesic submanifolds and arithmeticity. To do this, I will focus on the geometry of tesselations or tiling patterns, how they arise, what we know about them, and how they are surprisingly closely related to number theory. If time permits at the end, I will talk a bit about further open problems I've been thinking about lately. The talk derives from one I gave to a group of scientists at the Miller Institute. I will use slides from that talk and provide more details on the mathematics as the audience desires on the board. So the talk should highly accessible.


From points and lines to algebraic varieties and schemes

Martin Olsson
29 November

Abstract: The fundamental theorem of projective geometry concerns recovering a projective space from the collection of points and lines in that space. Ideas behind this theorem can be traced all the way back to Menelaus of Alexandria. Modern algebraic geometry moved in quite a different direction with the introduction of schemes, sheaves, and other notions. In recent work with Koll©r, Lieblich, and Sawin we proved a theorem showing that in many cases one can recover a given scheme from classically defined data. I will explain some of the ideas behind this result. Most of the talk will be accessible to students who have completed our lower division classes.


Moduli Spaces

Juliette Bruce
22 November

Abstract: An overarching theme, appearing in many different parts of math, is that if one wishes to study some type of geometric objects it is often useful to package all of your geometric objects together into one space (often called a moduli space). We will explore this idea by working together to investigate some down to Earth examples of moduli spaces and why they might be useful.


What happens inside a black hole?

Sung-Jin Oh
8 November

Abstract: Two important problems in the study of partial differential equations are (1) understanding the long-term behavior of solutions and (2) understanding the possible singularities. In this talk, I will introduce these two problems in the context of nonlinear wave equations, explain how they come together in the mathematical study of singularities of the Einstein gravitational field equation inside rotating black holes, and survey some recent progress.


An Introduction to the Finite Element Method

Per-Olof Persson
1 November

Abstract: I will give an informal introduction to the widely popular Finite Element Method for numerical solution of PDEs on arbitrarily shaped domains. Topics include unstructured mesh generation, definition of the relevant linear vector spaces and the variational formulation, proofs of convergence and optimality, and the practical implementation in the Julia programming language. If time permits, I will demonstrate some advanced applications of the method for solving the Navier-Stokes equations (so-called Computational Fluid Dynamics)


Curvature of a triangulated surface

John Lott
25 October

Abstract: A general goal is to give a good notion of the curvature of a singular space. I'll talk about the case of a triangulated surface


A pictorial introduction to p-adic geometry

Jackson Morrow
4 October

Abstract: Most people on earth know about the real numbers. That being said, most people do not know how to construct the real numbers i.e., how one can construct the real numbers from the rational numbers. For those of you who have taken a course in real analysis, you know that this process is called the completion of Q with respect to the usual Archimedean distance. In the first part of the talk, I will first define the p-adic distance on Q which roughly measures how divisible the rational number is by some fixed prime p. By imitating the same process that one goes from the rationals to the reals, I will define the p-adic numbers, denoted by Q_p, and we will spend some time exploring this new number system and observe strange phenomena (e.g., every triangle in the p-adics is isosceles). I will be providing lots of pictures (really cartoons) to help give a geometric intuition for what the p-adics looks like. In the second part, I will talk about geometric objects over the p-adics (e.g., curves over the p-adics) and how they relates (and are different) to geometric objects over the complex numbers. While the definition is quite technical, I will provide a lot of cartoons of these objects so one can attempt to pictures them. Time permitting, I will discuss some of my research in this area, which revolves around using geometry over the p-adics to study solutions to systems of polynomial equations.


What is "Arithmetic Statistics"

Lea Beneish
27 September

Abstract: The field of Arithmetic Geometry is in large part motivated by the study of rational points on varieties. For example, for an elliptic curve over the rationals, the set of its rational points forms a finitely generated abelian group. An example of a statistic is a quantity (computed from values in a sample) that gives us a numerical description of how likely an event is to occur. The field of "Arithmetic Statistics" is concerned with the frequency and distribution of arithmetic quantities. In this talk, I will discuss several examples of results and questions in arithmetic statistics.


The Riemann Hypothesis

Richard Borcherds
20 September

Abstract: The Riemann hypothesis about the zeros of the Riemann zeta function is one of the most notorious open problems in mathematics. I will try to explain what it is and why it is important, and will discuss some analogs of it that have been proved.


Spring 2021

The Lucas-Lehmer test

Ken Ribet
12 April

Abstract: The largest known prime numbers have typically been of the form 2^p-1, where p is prime. For example, the current record-holder corresponds to p=82589933; it was identified in 2018. If p is a prime, the corresponding "Mersenne number" 2^p-1 may or may not be prime; there is a relatively rapid test to decide whether or not 2^p-1 is prime. This test is named for François édouard Anatole Lucas, who lived in the middle of the 19th century, and D. H. Lehmer, a Cal math major who graduated in 1927 and joined the Berkeley faculty in 1940. Lehmer was our colleague in Evans Hall until his death in 1991. His wife and close mathematical collaborator, Emma Lehmer, lived in Berkeley beyond her 100th birthday; she died in 2007. My talk will attempt to give some insight into the Lucas-Lehmer test. I'll speak at the beginning about prime numbers and about Lucas and the Lehmers.


Lights Out with linear algebra

Ethan Dlugie
8 March

Abstract: "Lights Out" was a popular electronic game in the 90's. The player is presented with a 5 x 5 grid of lighted buttons. Pressing any button toggles the state of its light and those of its adjacent numbers (there are at most 4 of these neighbors). The name of the game is...well it's the name of the game: turn out all of the lights! We'll talk about how to phrase the problem as a seasoned math undergraduate student might and why you took/should take Math 54. In the end, I'll float some tantalizing problems that could probably be turned into a nice undergraduate research project.


p-adic Numbers

Koji Shimizu
1 March

Abstract: In Math 104, we learn that the real numbers are obtained from the rationals by completion with respect to the standard absolute value. For each prime p, the completion of the rationals with respect to the p-adic absolute value yields p-adic numbers. We will discuss several topics about them.


An Introduction to Free Probability

Ian Charlesworth
22 February

Abstract: One way to understand the distribution of bounded random variables is to record the expectation of every polynomial, and if you can reconstruct the variables from this data. So what happens if you start with a list of expectations for non-commutative polynomials? It turns out many familiar things from Probability theory—like the Central Limit Theorem an don't Information Theory—have non-commutative versions with sometimes surprising twists; these are the focus of free probability.


Spin Chains

Ivan Danilenko
15 February

Abstract: AbsIntegrable systems are a rich subject that focuses on elegant systems with a lot of symmetries (and "conserved quantities"). We'll introduce and discuss some aspects of the integrable system called the Heisenberg spin chain. It can be defined using linear algebra, but the objects appearing there keep showing up in different parts of mathematics.


Diagonalizing matrices as fast as multiplying

Nikhil Srivastava
8 February

Abstract: In Math 54, we learn how to diagonalize a matrix by computing its eigenvalues and solving some linear equations. Unfortunately, this process cannot be implemented on a computer since there is no algorithm to exactly compute the roots of a polynomial. We will explain a new algorithm which gets around this issue, and provably diagonalizes matrices in just slightly more than the amount of time taken to multiply two matrices. The analysis of this algorithm uses ideas from complex analysis and random matrix theory.


Fall 2020

Development of new mathematics motivated by real-life problems: Micro-swimming soft robots, vascular stents, and bioartificial pancreas

Suncica Canic
Late October — Zoom

Abstract: Real-life problems are an important driving force in the development of new mathematics. With the recent developments of new technologies, biomedical engineering and medicine, the need for new mathematical and numerical methodologies to aid these developments has never been greater. Real-life problems are often times mathematically rich and very complex. In this talk I will focus on describing mathematical problems motivated by biomedical applications involving the interaction between fluids, such as blood, and various structures, such as cardiovascular tissue.


The Unreasonable Effectiveness of Infinity

Pierre Simon
19 October — Zoom

Abstract: I will discuss two (classical) examples of how the use of infinity can allow us to prove results about finite objects that we cannot prove otherwise. The first example is that of Hercules combatting the hydra, whose heads keep multiplying as he cuts them. We need an infinite set to prove that Hercules will eventually win. The second example is that of Laver tables: they are finite tables on natural numbers constructed using a simple rule. Proving that the period of the first line can be as large as one wants requires the use of very big infinite cardinals.


High-Order Discontinuous Galerkin Methods for Fluid and Solid Mechanics

Per-Olof Persson
12 October — Zoom

Abstract: It is widely believed that high-order accurate numerical methods, for example discontinuous Galerkin (DG) methods, will eventually replace the traditional low-order methods in the solution of many problems, including fluid flow, solid dynamics, and wave propagation. The talk will give an overview of this field, including the theoretical background of the numerical schemes, the efficient implementation of the methods, and examples of real-world applications. Topics include high-order compact and sparse numerical schemes, high-quality unstructured curved mesh generation, scalable preconditioners for parallel iterative solvers, fully discrete adjoint methods for PDE-constrained optimization, and implicit-explicit schemes for the partitioning of coupled fluid-structure interaction problems. The methods will be demonstrated on some important practical problems, including the inverse design of energetically optimal flapping wings and large eddy simulation (LES) of wind turbines.


NUMBERS BIG AND SMALL

Aidan Backus
5 October — Zoom

Abstract: One often says that "The derivative dy/dx is the quotient of two very small numbers dy and dx." This is all well and good, and is very useful for calculations, until you take real analysis and realize that this cannot be made precise... or can it? By replacing the real numbers with "hyperreal numbers", which contain the real numbers but also numbers which are "very small", we can play with infinitesimals to our hearts' content. I will prove that the hyperreal numbers exist, demonstrate how to use them to solve some calculus problems, and discuss how the same argument suggests the existence of "very large" numbers which can be used to tamper with the mathematical universe. The proof that hyperreal numbers exist will use some fancy machinery from modern logic and analysis, but I'll be gentle -- I'll only assume that you know calculus, some basic set theory, and proof-writing.


Bernoulli Randomness and Biased Normality

Andrew DeLapo
21 September — Zoom

Abstract: I will present some of the highlights of the senior honors thesis I completed last semester under the advisement of Professor Theodore Slaman. I will define what it means for real numbers to be Martin-Lof random, normal, and biased normal. I will present my algorithm for generating biased normal real numbers from normal reals. I will discuss some of the motivation for biased normality via an application to iterated function systems and fractals. Most of all, I am taking special care to make the talk accessible to everyone.


Solving Diophantine Equations with Geometry

Richard Borcherds
14 September — Zoom

Abstract: A Diophantine equation is an equation such as x^2+y^2=z^2 or x^3+y^3=9z^3, where one wishes to find integer solutions (such as 3^2+4^2=5^2). This talk will give some examples of using geometric ideas to find integer solutions.


Fall 2019

Just Guess! The Strength and Versatility of the Probabilistic Method

Rikhav Shah
7 October — 1015 Evans

Abstract: When showing constructions with particular properties exist, it is tempting to start trying to construct something with those properties. One alternative approach is to randomly construct something and show there's a nonzero probability it ends up with the desired properties. Sometimes, this perspective can lead to algorithms which are guaranteed to produce 'good' constructions.


Tropical Geometry of Curves

Madeline Brandt
30 September — 1015 Evans

Abstract: Tropical geometry is a new and exciting field of math which creates a link between algebraic geometry and combinatorics. As a result of this connection, surprising insights have developed in both areas. In this talk, I will show you how to carry out the tropicalization process for curves. This takes in an algebraic curve, and associates to it a metric graph. Even in this low-dimensional case, it is not easy to state a general algorithm. I will present a result for doing this in the special case of superelliptic curves.


Singularity is Nonlinear

Sung-Jin Oh
23 September — 1015 Evans

Abstract: Nonlinear hyperbolic/dispersive partial differential equations (PDEs) underlie the description of many wave phenomena in physics, such as the propagation of sound, light and gravity (i.e., gravitational waves). I will introduce a central theme in the analysis of such PDEs, namely, the tension between the stabilizing dispersive effect of the linear part and the (possibly) catastrophic effect of the nonlinear part. As an example of this theme, I will describe my recent work with D. Tataru on singularities of the energy-critical Yang-Mills equation.


Modular Forms

Richard Borcherds
16 September — 1015 Evans

Abstract: A (possibly apocryphal) quote from Martin Eichler says that "There are five elementary arithmetical operations: addition, subtraction, multiplication, division, and modular forms". This talk will explain what a modular form is and given some examples of how they can be used to do things such as prove Fermat's last theorem, find the best sphere packings, and understand the monster sporadic simple group.


What Did Sophie Do?

Ken Ribet
9 September — 1015 Evans

Abstract: I will explain the proof of Sophie Germain's theorem to the effect that the first case of Fermat's Last theorem is true for exponent p if p is a prime number such that 2p+1 is also a prime. Such primes are now known as Sophie Germain Primes.


Spring 2019

Learning Algebraic Varieties from Samples

Prof. Bernd Sturmfels
4 February — 1015 Evans

Abstract: We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package.


Fall 2018

What is reverse mathematics?

Java Darleen Villano

Abstract: To do mathematics is to prove or disprove an array of statements. Occasionally, such statements involve the explicit construction of an object (e.g. a maximal ring ideal, a vector space basis, etc.). This talk will serve as a rough introduction to the research program known as "reverse mathematics", where we examine which axioms and principles are necessary to permit such constructions.


The Kakeya problem

Marina Iliopoulou

Abstract: A Kakeya set in Euclidean space is a set containing a unit line segment in each direction. The problem of determining that, in some form or other, Kakeya sets are large appeared about a century ago, and lies at the heart of harmonic analysis today, with various applications in many mathematical areas. In this talk we will discuss this problem and see how its study has affected modern harmonic analysis and incidence geometry.


Arrow's theorem and infinite electorates

Wesley Holliday

Abstract: Given a group of individuals each with a ranking over some set of options, how can the individuals' rankings be combined to form a group ranking of the options? Is there a method of combination that satisfies reasonable axioms? I'll give a crash course on these questions from the mathematical theory of social choice. In particular, I'll give a quick proof of the most famous result in the field, Arrow's Impossibility Theorem, which shows that several apparently desirable axioms for combining individual rankings are jointly inconsistent for a finite group of individuals. Finally, we'll see that these axioms become consistent if we allow the group of individuals to be infinite.


Coloring maps on surfaces

Ian Agol

Abstract: We'll discuss various questions about coloring maps on surfaces. It is well-known that coloring a map on a surface can require more than four colors (with appropriate assumptions, e.g. connectivity of the countries). We'll consider whether a finite map on a surface can admit a 4-coloring in a finite-sheeted cover.


Mirror symmetry in six dimensions


Catherine Cannizzo

Abstract: I will describe my thesis work in understanding an example of a six dimensional manifold, a symplectic manifold, through a concept arising from string theory known as mirror symmetry.


Within-host and between-host viral dynamics

Aidan Backus

Abstract: Since the discovery of HIV as the etiological agent of AIDS, mathematical biologists have studied HIV and other viruses through the new branch of applied mathematics known as viral dynamics. We'll present a candidate model for the spread of HIV in a community, which incorporates the probability of transmission as a function of infection age. Knowledge of differential equations, chemical kinetics, and probability would be nice, but is certainly not necessary.


Irrational rotations on infinite measure

James Leng

Abstract: In dynamical systems, one of the first examples one considers are rotations by an irrational multiple of pi. We give a construction of a transformation on the real line that has many of the same measure theoretic properties as the irrational rotations on the circle.


The Combinatorics of Electrical Networks

Brian Burks

Abstract: A electrical network is a weighted graph, with some vertices "internal" and the rest "boundary", and edge weights corresponding to conductances.


Planar Graphs

Clark Lyons

Abstract: A planar graph is a set of vertices and edges between them that you can draw in the plane without crossing edges. Kuratowski's theorem gives a surprisingly simple condition for a graph to be planar. In this talk, we will see how to tell if a graph is planar and why every map can be colored with four colors without adjacent regions being colored the same. No advanced background is required.


Spring 2018

A European application of random matrix theory

Brent Nelson
25 April, 2018—939 Evans

Abstract: Random matrix theory is a field of mathematics that blends probability theory and linear algebra. It was originally developed by Wigner to model heavy nuclei of atoms, but today offers applications to many other areas of math and science. In this talk, I will provide an introduction to random matrix theory and discuss a surprising application to genetics and cartography. Note the special time of 5:00-6:00pm on Wednesday; you might call this Math Monday + 2.


Differential game theory and how to use PDE to run down pedestrians optimally

Craig Evans
23 April, 2018—939 Evans

Abstract: I will discuss how to use certain nonlinear PDE to design optimal strategies for game problems involving differential equations and will in particular explain applications to the em homicidal chauffeur problem.


Chaos in dynamical systems

Semyon Dyatlov
16 April, 2018—939 Evans

Abstract: We will study several mathematical ways to describe when a given dynamical system exhibits chaotic, or unpredictable, behavior, such as the notions of ergodicity and mixing. These concepts will be illustrated on several examples, both basic (where I will attempt to give a rigorous proof of ergodicity) and more interesting ones, such as chaotic billiards (which will be demonstrated by numerical simulations).


An introduction to the Langlands program

Sug Woo Shin
9 April, 2018—939 Evans

Abstract: This is a gentle introduction to the Langlands program based on the developments of reciprocity law. Some attention will be drawn to the birth of the Langlands program in a letter of Langlands to Weil in 1967. Time permitting a snapshot of some current developments may be given.


Existence and regularity of soap film solutions to Plateau's problem

Jenny Harrison
2 April, 2018—939 Evans

Abstract: Plateau's problem asks if there is an area minimizing surface spanning a given boundary curve. Roughly speaking, the competing surfaces must contain the curve and not have any holes. Is there an area minimizer amongst the class of all such surfaces? Jesse Douglas was awarded one of the first Fields medals in 1936 for his solution of Plateau's problem. Federer and Fleming were awarded the Steele Prize for their 1960 solution. Reifenberg produced a completely different solution, also in 1960. We will briefly discuss how these papers were independent of each other and solved special distinct cases. Most mathematicians believe this was the end of the story. But was there anything significant left to do? We will answer this question in the affirmative and describe some recent results arising from a flurry of interest in the last four years. The talk should be accessible to students who have studied real analysis, although some simple algebraic topology will be invoked to say what it means for a surface to span a given boundary.


The integers and the 3-sphere

Eric Chen
19 March, 2018—939 Evans

Abstract: Studying the properties of the integers is interesting, but also very hard. Sometimes, instead of staring at lots of equations and identities, it helps to draw a geometric picture of what's going on. In this talk, I will introduce enough fancy words (not that many) to explain how one could think of quadratic reciprocity geometrically. If there's extra time, I'll also say something about more general rings of integers.


What is a quantum metric space?

Marc Rieffel
12 March, 2018—939 Evans

Abstract: I will indicate some issues in high-energy quantum physics that suggest the need for a definition of a "quantum metric space", and then I will proceed to show how to obtain a useful definition, concentrating on "compact" (and "finite") ones. I will indicate many examples. If time allows, I will also indicate how to define the distance between compact quantum metric spaces, with examples. This is a relatively new topic, with many aspects remaining to be explored. Useful background for my talk is Math 104, 110, and a bit of 113, but this is not necessary. No physics background is needed (though it would be a bit helpful).


The maths behind fruit ninja

Srivatsav Kunnawalkam
5 March, 2018—1015 Evans (Note that this Math Monday will occur at 4 PM!

Abstract: Hilbert, way back in 1900, knew about fruit ninja. Indeed he posed the problem, also known as Hilbert's 3rd problem: "Can you fruit ninja a fruit in the shape of a regular tetrahedron, into a fruit in the shape of a cube?". In this talk, we will attempt to solve this problem using a powerful tool known as linear algebra. We will then discuss about other such questions which involve finite decompositions and reassembling of subsets of R^n.


An introduction to ordinal analysis

James Walsh
26 February, 2018—939 Evans

Abstract: Hilbert's Program was an early twentieth century research program with two goals: (i) axiomatize mathematics and (ii) prove the consistency of the axioms by indubitable means. Interest in Hilbert's Program waned after Gödel's discovery that, roughly, no interesting axiomatic theory can be proved consistent on the basis of indubitable means. This discovery sowed the seeds for a refined version of Hilbert's Program known as ordinal analysis. In ordinal analysis, the strength of axiomatic theories is measured and compared by determining what principles are necessary and sufficient for proving their consistency. I will provide a non-technical introduction to the subject with no background knowledge assumed.


To Infinity and Beyond

Clark Lyons
11 February, 2018—939 Evans

Abstract: In the 19th century, George Cantor first showed that some infinite sets are larger than other infinite sets. He showed that the size of the set of real numbers is larger than the size of the set of natural numbers, but he left open the question of how much larger. It turns out that this question lies at the foundation of mathematics, and determining exactly how large the continuum can be motivated much of the set theory research done in the 20th century. In this talk we will see what the current state of "continuum problem" is. No prerequisite knowledge assumed.


How to count like a logician

Sung Hyup Lee
5 February, 2018—939 Evans

Abstract: Ramsey's theorem provides a generalisation of the classic pigeonhole principle. This aspect of infinitary combinatorics, which tells us when certain homogeneity properties hold given certain conditions, has lent itself naturally to a slew of results in logic and the foundations of mathematics. In this talk, I plan to give a brief and friendly survey of this interaction.


Infinite games

John R. Steel
29 January, 2018—1015 Evans

Abstract: We consider games of the following sort. There are two players, I and II, and a payoff set $A$. The players alternate making moves, and at the end of the game have produced an infinite sequence s of moves. Player I wins if and only if $s$ is in $A$. This game is called $G_A$, and it is said to be determined if one of the two players has a winning strategy. For which $A$ is $G_A$ determined? The question turns out to be a basic one in the foundations of mathematics. Its answer is intimately connected to the existence of infinities much larger than the countable infinity involved in the description of the game. We shall explain this connection further, in a non-technical talk aimed at a general audience.


Fall 2017

Between characteristics

Silvain Rideau
20 November, 2017—740 Evans

Abstract: In many ways, a field of characteristic zero behaves like a field of large positive characteristic. For example, if the characteristic of a field K is zero or prime to m, then the polynomial X^m - 1 only has simple roots. In this talk, we will explain how, using the tools of logic, this observation can be formalized and how one can actually prove that characteristic zero behavior is the limit for large p of characteristic p behavior. We will then use this to prove a result of Ax on polynomial maps over the complex numbers.


Linear Algebra, Optimization, and Financial Risk

Robert M. Anderson
13 November, 2017—740 Evans

Abstract: The commercially dominant models for estimating risk in stocks require an army of analysts reading accounting data to classify individual stocks' exposures to predefined risk factors. We discuss a simple machine learning method, based on linear algebra and optimization, that may have the potential to disrupt these models.


Elo Ratings and the Sports Model — A Neglected Topic in Applied Probability

David Aldous
6 November, 2017—740 Evans

Abstract: Teaching Probability as mathematics without any reference to data is (arguably) foreign to the spirit of the 21st century. One interface between Probability and readily available data involves Elo-type rating algorithms for sports teams. Combined with a model for win/lose with given strengths, and a model for time-varying strengths, one gets a model within which one can compare observed algorithmic ratings to unseen strengths. In particular, the sentence ratings tend to converge on a team's true strength relative to its competitors after about 30 matches" has been widely copied online. Is there any theory or data to support this assertion?


Riemannian geometry and Ricci flow

Richard Bamler
30 October, 2017—740 Evans

Abstract: This talk will be about the fruitful interplay of three different mathematical fields — topology, geometry and analysis. The talk will consist of two parts. In the first, more classical part, I will introduce the concept of a Riemannian manifold. I will then explain how the topology and the geometry of a Riemannian manifold are related to one another. In dimensions 2 and 3, this relationship will even give us a way of classifying all possible topologies. This classification in dimension 3 is called the "Geometrization Conjecture" and it was just recently proven by Perelman. In the second part of the talk, analysis will enter the picture. I will introduce the Ricci flow, which is a method of "spreading out" curvature similar to a heat equation. I will then sketch how Perelman used the Ricci flow to prove the Geometrization Conjecture. If time permits, I will also discuss some more recent results and open questions.


The geometry of groups

Carolyn Abbott
23 October, 2017—740 Evans

Abstract: A group is an algebraic object, but it also possible (and useful!) to study the geometry of a group. To do this, we need to understand what a group looks like. In this talk, I will introduce a way to "draw a picture" of a group, and we will discuss what can be said about the group depending on the geometry of the picture.


Homology theories and their applications

Kat Christianson
16 October, 2017 at 6 PM—740 Evans

Abstract: Homology is a powerful algebraic tool that arises naturally in a wide range of different areas of modern mathematics. The main goal of this talk is to provide an introduction to how homology theories are defined and how they are used in practice. We will focus on two standard homology theories in algebraic topology as examples. After laying some theoretical foundations for these two homology theories, we will consider a couple applications of those foundations to questions in geometry and commutative algebra.


The Fibonacci Sequence modulo p — Around and around we go

Alexander Youcis
9 October, 2017—740 Evans

Abstract: The Fibonacci sequence has, in the past decade or so, taken on an almost meme like quality. That being said, it is an interesting sequence of integers, the structure of which can get surprisingly complex. We will be interested in this talk in what sort of periodicity properties the Fibonacci sequence has. Of course, this can't be the normal notion of periodicity (since the Fibonacci sequence is increasing). Instead, we mean the periodicity of the sequence modulo a prime p. It turns out that the study of such a problem naturally leads one to consider slightly more sophisticated problems in number theory and algebra.


Partition identities

Jeremy Lovejoy
1 October, 2017—740 Evans

Abstract: A partition of n is a non-increasing sequence of natural numbers whose sum is n. The study of partitions goes back to Euler, who proved the original partition identityu. For any n, the number of partitions of n into distinct parts is equal to the number of partitions of n into odd parts. Another famous partition identity is the first Rogers-Ramanujan identity, which says that for all n, the number of partitions of n into parts which differ by at least two is equal to the number of partitions of n into parts congruent to 1 or 4 modulo 5. (Try it for some small values of n to see that it works!) While these two partition identities have similar statements, one is much deeper than the other. In this talk I will discuss the proofs of these identities, the search for a generalization, and some unsolved problems.


Sharks in the Shallows

Sri Kunnawalkam Elayvalli
25 September, 2017—740 Evans

Abstract: I will talk about problems in mathematics that are so easy to state that even your grandmother can understand them, but are so difficult to solve that no one ever in history has solved them. The interesting thing about the problems I will present is that they are not very famous in the public eye (unlike the twin prime conjecure, Goldbach's conjecture, the abc conjecture, and so on). The platter will include problems of various different flavors, like numbers, sets, triangles, points, sums, etc.


To Infinity and Beyond — The strange world of ordinal arithmetic

Professor Richard E. Borcherds
11 September, 2017—740 Evans

Abstract: Beyond the usual numbers 0, 1, 2, 3, ..., lie Cantor's countable ordinal numbers \(\omega\), \(\omega+1\), ..., \(\omega+\omega\);, ..., \(\epsilon_0\), .... In some way their arithmetic is similar to the natural numbers; one can add, multiply and raise them to powers and define primes and so on. However in other ways their arithmetic looks a little strange; for example, \(1+\omega\leq\omega+1\).


Older

Eigenvectors of Tensors

Bernd Sturmfels
12 September, 2016

Abstract: Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalization. The spectral theory of tensors was pioneered about a decade ago, and has since found numerous applications, but it is also intimately connected to classical questions in optimization and dynamics. We present an introduction to this theory.


Cutout Instructions and Geometric Invariants

Julian Chaidez
19 September, 2016

Abstract: Suppose that you are given paper cutout instructions for a very large looking polyhedron, with many sides. How could you distinguish between a real set of instructions, which describes a polyhedron that can actually be constructed, and fake set of instructions, which appears valid at a glance but actually described an unbuildable shape?


The Global Attractor Conjecture

Christopher Eur
26 September, 2016

Abstract: The Global Attractor Conjecture (GAC) is a fundamental conjecture in chemical reaction network theory (CRNT) that has eluded proofs for decades despite appearing intuitively obvious. In this talk, we introduce CRNT following works of Jackson, Horn, and Feinberg, and discuss its ties to algebraic geometry and convex geometry. We conclude with a sketch of the recently proposed proof of GAC.


Where to find Galois representations, and why

Sander Mack-Crane
3 October, 2016

Abstract: Starting from prime numbers, we'll discuss one of the motivating problems in number theory, and how it connects to Galois theory and then to Galois representations. This sets us up for a gentle overview of what the Langlands program is all about. As an example we'll discuss the simplest case, known as class field theory. Knowledge of abstract algebra will be helpful.


The p-Group Fixed Point Theorem

Qiaochu Yuan
10 October, 2016

Abstract: In this talk, we'll describe a surprisingly useful result from group theory that can be applied to prove many of the basic results in elementary number theory. We'll start with Fermat's little theorem and Wilson's theorem and work our way up from there. Time permitting we may make it all the way up to quadratic reciprocity.


Differential Equations in Algebraic Geometry

Martin Olsson
17 October, 2016

Abstract: I will discuss some of the algebraic aspects of ordinary linear differential equations. While the theory of such equations, as presented for example in math 1B, is analytic in nature involving things like the derivative and limits, one can, in fact, study such equations more algebraically over the rationals, or even finite fields.


From Linear Maps to Multilinear Maps Through Cryptography

Mahrud Sayrafi
24 October, 2016

Abstract: In this talk we will see how almost all aspects of cryptography essentially boil down to studying functions that are easy to compute but hard to invert without some extra information. While analyzing these functions often involves tools from statistics or computer science, the problems at the core are mathematical in nature, and in fact, you have known all of them since precalc.


Classical and Quantum Chaos

Maciej Zworski
31 October, 2016

Abstract: Classical chaos was famously described by Edward Lorenz (of butterfly effect fame) as the situation when the present determines the future, but the approximate present does not approximately determine the future. I will illustrate that by some example and also, for contrast, show completely integrable systems (that is, predictable systems).


Invariants of nonunique factorization

Christopher O'Neill
7 November, 2016

Abstract: The fundamental theorem of arithmetic states that every integer factors uniquely as a product of primes. Non-unique factorization theory examines settings where uniqueness of factorization fails to hold (that is, settings in which every element can be factored, but possibly in more than one way). This relatively new field has many algebraic and combinatorial aspects.


The calculus of variations and weak KAM theory

Craig Evans
14 November, 2016

Abstract: I will first discuss the Newtonian, the Lagrangian and the Hamiltonian interpretations for the dynamics of a particle moving in a conservative field of forces, and explain how to extract interesting information such as conservation laws and the Virial Theorem. I will then generalize to weak KAM (= Kolmogorov, Arnold, Moser) theory.


Supersymmetry in algebra and geometry

Vera Serganova
28 November, 2016

Abstract: I will give a brief introduction to superalgebras and illustrate on examples how they appear in different areas of mathematics. My examples will include the proof of Amitzur-Levitski identity, the duality between orthogonal and symplectic groups and de Rham complex. We will also discuss the notion of supertrace.


An Introduction to Analytic Number Theory

Cailan Li
30 January, 2017

Abstract: In this talk we will first give a crash course in complex analysis and then talk about the beautiful Riemann zeta function and its generalization. We will then talk about the shiny objects known as modular forms, and some of their applications. In particular, we will discuss the role modular forms played in Andrew Wiles' proof of Fermat's Last Theorem.


Local fields and the geometry of the p-adics

Alex Carney
6 February, 2017

Abstract: Local fields are a formalization of the idea that often in number theory, problems are easier to solve mod p than over the integers or the rationals. We'll start by defining the p-adics, the most common example of a local field, and describe some of their strange and interesting properties. At the end, I will briefly point to some exciting modern directions in the study of local fields.


Reflection groups, wallpaper groups, and orbifolds

Richard Borcherds
13 February, 2017

Abstract: This talk will describe how to classify things like the 18 archimedean solids and the 17 wallpaper groups using orbifolds.


Helpful ways to visualize spaces

Jeff Hicks
27 February, 2017

Abstract: A lot of my intuition for mathematics comes from drawing pictures of the problem I want to solve. As a topologist, this means trying to come up with clean visual representations of various topological spaces. A good drawing for a space should ideally be mathematically motivated and intuitive.


4-manifolds revisited

Peter Teichner
20 March, 2017

Abstract: I'll give a classification of manifolds up to dimension 4.


Quantum sets

Mariusz Wodzicki
10 April, 2017

Abstract: A set is generally considered to be the simplest structure of Mathematics. Every set comes disguised under a number of other structures, however. While exploring some of them, we shall encounter what I propose to call a quantum set, in analogy to quantum groups.