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.

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.

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.

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.

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.

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.

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.

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.