11-12 May, 2023

Eli Grigsby

Boston College

Professor Grigsby will present two lectures, the first intended for an undergraduate audience.

• Undergraduate lecture “Neural networks and boolean functions”
  4pm, Thursday, May 11 in 128 Chiles Hall
  Abstract: Deep neural networks can drive cars, produce images from text prompts, and write titles and abstracts for math lectures (J/K…or am I?). A major source of their power is their ability to approximate any function on a compact set. I’ll tell you just enough about neural networks to convince you that this is possible, by focusing on a toy result: any boolean function can be modeled by a neural network of sufficiently high complexity.

  Recording

• Colloquium lecture : “The topology, geometry, and combinatorics of feedforward neural networks”
  4pm, Friday, May 12 in 128 Chiles Hall
  Abstract: Deep neural networks are a class of parameterized functions that have proven remarkably successful at making predictions about unseen data from finite l abeled data sets. They do so even in settings when classical intuition suggests that they ought to be overfitting (aka memorizing) the data.

  I will begin by describing the structure of neural networks and how they learn. I will then advertise one of the theoretical questions animating the field: how does the relationship between the number of parameters and the size of the data set impact the dynamics of how they learn? Along the way I will emphasize the many ways in which topology, geometry, and combinatorics play a role in the field.

  Recording

1-5 October, 2020

Mark Goresky

Institute for Advanced Study

Professor Goresky will present two lectures, the first intended for an undergraduate audience.

• Undergraduate lecture : “A glamorous Hollywood star, a renegade composer, and the mathematical development of spread spectrum communications”
  4pm, Thursday, October 1 on Zoom.
  Zoom meeting Link
  Abstract: Virtually all modern digital communications involve spread spectrum techniques. One of the earliest was invented by an unlikely pair: a Hollywood actress and a composer. This talk will survey their lives, their work and some of the later developments in spread spectrum.
• Colloquium lecture : Pseudo-random numbers and sequences
  4pm, Monday, October 5 on Zoom
  Zoom meeting Link
  Abstract: I will discuss three or four striking historical applications of pseudo-random numbers and some of the mathematics behind their synthesis and analysis.

6-7 May, 2019

Alison Etheridge

University of Oxford

Professor Etheridge will present two lectures, the second intended for an undergraduate audience.

• Colloquium lecture : “Modelling evolution in a spatial continuum.”
  4pm, Monday, May 6, 2019 in Deady Hall 208
  Abstract: Since the pioneering work of Fisher, Haldane and Wright at the beginning of the 20th Century, mathematics has played a central role in theoretical population genetics. In turn, population genetics has provided the motivation both for important classes of probabilistic models, such as coalescent processes, and for deterministic models, such as the celebrated Fisher-KPP equation. Whereas coalescent models capture ‘relatedness’ between genes, the Fisher KPP equation captures something of the interaction between natural selection and spatial structure. What has proved to be remarkably difficult is to combine the two, at least in the biologically relevant setting of a two-dimensional spatial continuum.

  In this talk we describe some of the challenges of modelling evolution in a spatial continuum and then, as time permits, turn to some results concerning the interplay between natural selection and spatial structure.

• Undergraduate lecture : “Modelling genes: the backwards and forwards of mathematical population genetics”
  4pm, Tuesday, May 7, 2019 in Deady Hall 208
  Abstract: When Mendelian genetics was rediscovered at the beginning of the 20th Century, it was widely believed to be incompatible with Darwin’s theory of natural selection. The mathematical sciences, in the hands of pioneers such as Fisher, Haldane and Wright, played a fundamental role in the reconciliation of the two theories, and the new field of theoretical population genetics was born.

  But fundamental questions remained (and remain) unresolved. The genetic composition of a population can be changed by natural selection, mutation, mating, and other genetic, ecological and evolutionary mechanisms. How do they interact with one another, and what was their relative importance in shaping the patterns that we see today?

  Whereas the pioneers of the field could only observe genetic variation indirectly, by looking at traits of individuals in a population, researchers today have direct access to DNA sequences, but making sense of this wealth of data presents a major scientific challenge and mathematical models play a decisive role.

  In this lecture we’ll discuss how to distill our understanding into workable models and then explore the remarkable power of simple mathematical caricatures in interrogating modern genetic data.

16-17 April, 2018

Melanie Matchett Wood

University of Wisconsin

Professor Matchett Wood will present two lectures, the first intended for an undergraduate audience.

• Undergraduate lecture : “The Chemistry of Primes”
  5pm, Monday, April 16, 2018 in Willamette Hall 100
  Abstract: We are familiar with the prime numbers as those integers which cannot be factored into smaller integers, but if we consider systems of numbers larger than the integers, the primes may indeed factor in those larger systems. We discuss various questions mathematicians ask about how primes may factor in larger systems, talk about both classical results and current research on the topic, and give a sense of the kind of tools needed to tackle these questions.
• Colloquium lecture : “Random groups from generators and relations”
  4pm, Tuesday, April 17, 2018 in Willamette Hall 100
  Abstract: We consider a model of random groups that starts with a free group on n generators and takes the quotient by n random relations. We discuss this model in the case of abelian groups (starting with a free abelian group), and its relationship to the Cohen-Lenstra heuristics, which predict the distribution of class groups of number fields. We will explain a universality theorem, an analog of the central limit theorem for random groups, that says the resulting distribution of random groups is largely insensitive to the distribution from which the relations are chosen. We will discuss joint work with Yuan Liu on the non-abelian random groups built in this way, including the existence of a limit of the random groups as n goes to infinity, as well as theorems about the distribution of non-abelian analogs of class groups of function fields that motivate this work.

1-2 May, 2017

Sergei Tabachnikov

Penn State

Professor Tabachnikov will present two lectures, the first intended for an undergraduate audience:

• Undergraduate lecture: “Proofs (not) from the Book”, 4pm, Monday, May 1, 2017 in Willamette Hall 100. There will be pizza after the lecture on the second floor of Fenton; undergraduates especially welcome.
  Abstract: The eminent mathematician of the 20th century, Paul Erdos, often mention “The Book” in which God keeps the most elegant proof of every mathematical theorem. So, attending a mathematical talk, he would say: “This is a proof from The Book!”, or “This is a correct proof, but not from The Book”. M. Aigner and G. Ziegler authored the highly successful “Proofs from THE BOOK” (translated into 13 languages). In this talk, I shall present several proofs that are not included in the Aigner-Ziegler book but that, in my opinion, could belong to “The Book”.

• Colloquium lecture: “Flavors of bicycle mathematics”, 4pm, Tuesday, May 2, 2017 in Willamette Hall 100
  Abstract: This talk concerns a naive model of bicycle motion: a bicycle is a segment of fixed length that can move so that the velocity of the rear end is always aligned with the segment. Surprisingly, this simple model is quite rich. Here is a sampler of the problems that I plan to discuss.
  1) The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal position arises. This monodromy mapping is a Moebius transformation, a remarkable fact that has various geometrical and dynamical consequences. Moebius transformations belong to one of the three types: elliptic, parabolic and hyperbolic. I shall mention a recent proof of a 100 years old conjecture: if the front wheel track of a unit length bicycle is an oval with area at least Pi then the respective monodromy is hyperbolic.
  2) The rear wheel track and a choice of the direction of motion uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. These two front tracks are related by the bicycle (Backlund, Darboux) correspondence which defines a discrete time dynamical system on the space of curves. What do pairs of curves in the bicycle correspondence have in common? It turns out, infinitely many quantities (the perimeter length, the total curvature squared,…) I shall explain that, in dimension three, the bicycle correspondence is closely related with another, well studied, completely integrable dynamical system, the filament (a.k.a binormal, smoke ring, local induction) equation.
  3) Given the rear and front tracks of a bicycle, can one tell which way the bicycle went? Usually, one can, but sometimes one cannot. The description of these ambiguous tire tracks is an open problem, intimately related with Ulam’s problem in flotation theory (in dimension two): is the round ball the only body that floats in equilibrium in all positions? The ambiguous tire tracks are solitons of the planar filament equation.

6-7 April, 2016

Cameron Gordon

University of Texas at Austin

Professor Gordon will present two lectures, the first intended for an undergraduate audience:

• Undergraduate lecture: “Knots”, 4pm, Wednesday, April 6, 2016 in Fenton Hall 110. There will be pizza after the lecture on the second floor of Fenton; undergraduates especially welcome.
  Abstract: A knot is just a closed loop in 3-dimensional euclidean space, embedded in some possibly highly tangled fashion. The mathematical study of knots began in the second half of the nineteenth century, motivated by problems in physics, and continues to be an active area of research. Probably the most basic question in knot theory is: is it possible to tell if a knot is really knotted (and if so, how)? It was also recognized early on that any knot becomes unknotted in four dimensions. This leads to the question of how many “jumps” into the 4th dimension are needed to untie a given knot, a question that is still very mysterious. We will describe some of the history of the subject and discuss these questions.

• Colloquium lecture: “Left-orderability of 3-manifold groups”, 4pm, Thursday, April 7 2016 in Straub Hall 145
  Abstract: The fundamental group is a more or less complete invariant of a 3-dimensional manifold. We will discuss how the purely algebraic property of this group being left-orderable is related to two other aspects of 3-dimensional topology, one geometric and the other essentially analytic.

27-28 January 2015

Pavel Etingof

MIT

Professor Etingof will present two lectures, the first of which is intended for undergraduates:

• Undergraduate lecture: “Representations of quivers and Gabriel’s theorem”, 4pm, Tuesday, 27 January 2015 in Walnut room, EMU. There will be a reception in Fenton 219 immediately after the talk.
• Colloquium lecture: “Double affine Hecke algebras”, 2pm, Wednesday, 28 January 2015 in Maple room, EMU. After the lecture there will be tea in Fenton 219.

Here is a poster giving more information about the two lectures (PDF).

3-4 April 2014

Mikhail Kapranov

Yale University

Professor Kapranov will present two lectures, the first of which is intended for undergraduates:

• Undergraduate lecture: “How simple is the simplex?”, 4pm, Thursday, 3 April 2014 in 166 Lawrence Hall. There will be a reception in Fenton 219 immediately after the talk.
• Colloquium lecture: “2-dimensional symmetry of homological algebra”, 4pm, Friday, 4 April 2014 in 208 Deady Hall. Before the lecture there will be tea at 3:15 in Fenton 219.

Here is a poster giving more information about the two lectures (PDF).

13-14 May 2013

Sergey Fomin

University of Michigan

Professor Fomin will present two lectures, the first of which is intended for undergraduates:

• Undergraduate lecture: “Periodicity of Cluster Transformations”, 4pm, Monday, 13 May 2013 in 208 Deady Hall. There will be a reception in Fenton 219 immediately after the talk.
• Colloquium lecture: “Cluster Algebras and Invariant Theory”, 4pm, Tuesday, 14 May 2013 in 112 Lillis Hall. Before the lecture there will be tea from 3:15 in Fenton 219.

Here is a poster giving more information about the two lectures (PDF).

14-15 May 2012

Ravi Vakil

Stanford University

Professor Vakil will present two lectures, the first of which is intended for undergraduates:

• Undergraduate lecture: “The Mathematics of Doodling”, Monday, 14 May 2012, 4:00 pm in 208 Fenton Hall.
• Colloquium lecture: “Cutting and Pasting in Algebraic Geometry”, Tuesday, 15 May 2012, 4:00 pm in 208 Fenton Hall.

24-25 May 2011

Denis Auroux

University of California, Berkeley

Professor Auroux will present two lectures, the first of which is intended for undergraduates:

• Undergraduate lecture: “Seeing into the fourth dimension”, Tuesday, 24 May 2011, 4:00 pm in 100 Willamette Hall. Reception to follow in Willamette Hall atrium.
• Colloquium lecture: “Building 3-manifold invariants by composing correspondences”, Wednesday, 25 May 2011, 4:00 pm in 100 Willamette Hall. Tea from 3:00 pm to 4:00 pm in Willamette Hall atrium.

Here are the abstracts.

26-28 April 2010

Alexei Borodin

California Institute of Technology

Professor Borodin will present two lectures, the first of which is intended for undergraduates:

• Undergraduate lecture, “Around Longest Increasing Subsequences”, Wednesday, 26 May 2010, 4:00 pm in 166 Lawrence Hall.
• Colloquium lecture, “Interlacing Particle Systems”, Friday, 28 May 2010, 4:00 pm in 208 Deady Hall.

Here is the poster, including the abstracts.

21-22 April 2008

Allen Knutson

UC San Diego

Professor Knutson will present two lectures, the second of which is intended for undergraduates:

• Colloquium lecture, “The Space of Commuting Matrices, and Statistical Mechanics”, 4pm, 21 April 2008, 208 Deady.
• Undergraduate lecture (likely to be interesting to graduate students too), “Juggling Patterns and Gaussian Elimination”, noon, 22 April 2008, 301 Deady.

Here are the abstracts.

11-12 April 2007

Dan Freed

University of Texas at Austin

Professor Freed will present two lectures, the first of which is intended for undergraduates:

• Lecture 1: “The geometry and topology of Maxwell’s equations”.
  12:00 p.m., Wednesday, 11 April 2007
• Lecture 2: “Secondary differential-geometric invariants, generalized cohomology, and QCD”
  4:00 p.m., Thursday, 12 April 2007

28 March – 1 April 2005

Alexandre Kirillov

University of Pennsylvania

Professor Kirillov will present three lectures, the second two of which are undergraduate lectures:

• Lecture 1: “A tale on two fractals”
  4:00 p.m., Monday, 28 March 2005, 106 Deady Hall
• Lecture 2: “Self-similar fractal sets and generalized numerical systems”
  12:00 p.m., Wednesday, 30 March 2005, 229 McKenzie Hall
• Lecture 3: “Descartes theorem and its generalization”
  4:00 p.m., Friday, 1 April 2005, 106 Deady Hall

Here are the abstracts for all three talks (pdf).

24-25 May 2004

Peter Sarnak

Princeton University and Courant Institute

Professor Sarnak will present two lectures, the second of which is intended for undergraduates:

• Lecture 1: “The spectrum of the modular surface”
  4:00 p.m., Monday, 24 May 2004, 208 Deady Hall
• Lecture 2: “Sums of squares”
  12:00 p.m., Tuesday, 25 May 2004, 229 McKenzie Hall

14-15 October 2002

John Conway

Princeton University

Professor Conway will present two lectures, the second of which is intended for undergraduates:

• Lecture 1: “The 219 space groups”
  4:00 p.m., Monday, 14 October 2002, 110 Fenton Hall
• Lecture 2: “What’s new about polyhedra and polytopes?”
  4:00 p.m., Tuesday, 15 October 2002, 110 Fenton Hall

8-10 May 2000

George Andrews

Pennsylvania State University

Professor Andrews will present two lectures, the second of which is intended for undergraduates:

• Lecture 1: “Some things Ramanujan may have had up his sleeve”
  4:00 p.m., Monday, 8 May 2000, 110 Fenton Hall
• Lecture 2: “Ramanujan, continued fractions and teaching mathematics”
  4:00 p.m., Wednesday,10 May 2000, 210 Deady Hall

26-28 April 1999

David Eisenbud

Director of MSRI and University of California, Berkeley

Professor Eisenbud will present three lectures, the last of which is intended for undergraduates:

• Lecture 1: “What free resolutions (might) teach us about geometry”
  4:00 p.m., Monday, 26 April 1999, 110 Fenton Hall
• Lecture 2: “Free resolutions over exterior algebras”
  4:00 p.m., Tuesday, 27 April 1999, 210 Deady Hall
• Lecture 3: “How to take a walk using a commutative algebra”
  12:00 p.m., Wednesday, 28 April 1999, 210 Deady Hall

May 1997

Michael Artin

Massachusetts Institute of Technology

June 1996

Persi Diaconis

Stanford University

October 1994

Hugh Montgomery

University of Michigan