Seminars Sorted by Series
Mathematical Conversations
Heegaard splittings and the stabilization problem for 3-manifolds
All 3-manifolds can be decomposed into two simple objects, or
handlebodies. Some manifolds have many such decompositions, which
are distinct. All, however, are related by the operation of
stabilization and destabilization. Given two decompositions...
Computer algebra systems, formal proofs and interactive theorem proving
Computer algebra systems are large software systems and as such
they have bugs. A recent issue of the Notices of the AMS features
the article "The Misfortunes of a Trio of Mathematicians Using
Computer Algebra Systems. Can We Trust in Them?" in...
Hyperbolicity in dynamics
"Hyperbolic" ranks highly among the most-abused terms in
dynamics. I'll prolong this abuse, and argue for its value, by
illustrating a variety of dynamical systems with distinct forms of
hyperbolic behavior that have known or conjectured...
Quantum chaos and eigenvalue statistics
One of the outstanding insights obtained by physicists working
on "Quantum Chaos" is a conjectural description of local statistics
of the spectrum of the Laplacian on a Riemannian surface according
to crude properties of the dynamics of the geodesic...
Totally geodesic surfaces in hyperbolic 3-manifolds
Although the existence of a totally geodesic surface in a finite
volume hyperbolic 3-manifold is "rare", when they do exist, their
presence seems to have an impact on the geometry and topology of
the hyperbolic 3-manifold, as well as number...
Where a surface is determined by its boundary: the world of Lagrangian fillings of Legendrian knots
Given a smooth knot in the 3-sphere, there are many
topologically distinct smooth surfaces in the 4-ball that have this
knot as its boundary. However, if the knot is Legendrian, meaning
that it satisfies a geometrical condition imposed by a
contact...
p-adic numbers in cryptography and data compression
The $p$-adic numbers are finally entering the realm of
engineering. I will give several examples of how they arise in
applications.
Conversations on Schubert's Wandererfantasie
Schubert's Wandererfantasie is one of the most monumental and
revolutionary piano pieces ever composed. I will highlight some of
the structural novelties that were adapted later by Liszt, Wagner,
Franck and others, and I will play the piece.
What is the Fukaya category?
The (derived) Fukaya category is a symplectic invariant
developed out of work of Gromov, Floer, Fukaya, Kontsevich and
Seidel that encodes the rigidity properties of Lagrangian
intersections. The purpose of the talk is to discuss the
construction of...
Entrance path category of a stratified space
A covering space $C \\to M$ is classified by a subgroup of the
fundamental group of $M$. If we refuse to choose a basepoint, then
$C \\to M$ is equivalent to a functor from the fundamental groupoid
of $M$ to $\\mathsf{Set}$. Suppose $(M,S)$ is a...
Local-to-global approaches to homological mirror symmetry
I will try my best to make homological mirror symmetry into a
tautology. The tool to do so is Family Floer cohomology, which
produces a "mirror space" from a given Lagrangian torus fibration.
Mirror symmetry thus gets reduced to the geometric...
Phase transitions and symmetry breaking
In broad terms, a phase transition is a variation in the
qualitative behavior of a system under changes of some parameter.
For instance, as the temperature is changed, water goes through a
gaseous, a liquid, and several solid phases, each of which...
The Uncertainty Principle
Charles Fefferman
This talk recalls how Gromov's classic non-squeezing theorem
from symplectic geometry was first conjectured, based on a
connection between eigenvalue problems from PDE and the uncertainty
principle from elementary quantum mechanics.
Revisiting isoperimetric inequalities for Lagrangians
Isoperimetric problems are ubiquitous in mathematics. We shall
discuss some proved and some conjectural ones in symplectic
geometry, together with applications to other areas of
mathematics.
Negative correlation and Hodge-Riemann relations
All finite graphs satisfy the two properties mentioned in the
title. I will explain what I mean by this, and speculate on
generalizations and interconnections.
Voevodsky's Univalent Foundations for mathematics
Daniel Grayson
We'll take a glance at the world of mathematics as viewed
through the Univalent Foundations of Voevodsky. In it, "set" and
"proposition" are defined in terms of something more fundamental:
"type". The formal language fulfills the mathematicians'...
Lagrangian tori, mutations and toric degenerations
A basic open problem in symplectic topology is to classify
Lagrangian tori in a given symplectic manifold. In recent years,
ideas from mirror symmetry have led to the realization that even
the simplest symplectic manifolds (eg. vector spaces or...
The positive Grassmannian
I will give an informal introduction to the positive
Grassmannian, including its cell decomposition and its connection
to cluster algebras.
Random permutations and statistical mechanics
I will review some theorems and conjectures about the structure
of random permutations which arise in statistical mechanics.
Conjectures about the cycle structure are related Bose-Einstein
condensation and to universality of Wigner-Dyson statistics...
A "geometric group theory" for homeomorphisms groups?
Frédéric Le Roux
I propose to discuss classical geometric group theory, and its
potential extension to homeomorphisms groups suggested recently by
Kathryn Mann and Christian Rosendal.
Categories and filtrations
We will describe a new construction of filtration on categories.
Applications to classical questions in geometry and group theory
will be discussed.
Geometric realizations of algebraic objects
Dmitry Orlov
Considering some special examples as algebras of quivers I will
give an informal introduction to a field of geometric realizations
of noncommutative and derived varieties.
Poincare duality in loop spaces
Geometers since Morse are interested in Morse Theory on the free
loop space $LM$ of a Riemannian manifold $M$, because the critical
points of the energy function on $LM$ are the closed geodesics on
$M$. I will discuss an observed symmetry of the...
String topology from the symplectic viewpoint
String topology, invented by Chas and Sullivan in their
eponymous 1999 paper, can be viewed as a systematic study of the
structure of spaces of free loops and strings on manifolds with
emphasis on two basic operations: concatenation and splitting.
I...
Almost commuting matrices: finite- and infinite-dimensional proofs
I will first give an outline of Lin-Friis-Rordam’s proof of the
fact that almost commuting matrices are close to commuting matrices
uniformly in the dimension. The proof is short and beautiful, but
it involves an infinite-dimensional argument which...
Equidistribution + Arakelov intersection theory = certain thin set of primes is infinite
In arithmetic geometry, there are lots of examples of natural
density zero sets of primes raised from the geometry of elliptic
curves or more generally abelian varieties. One may ask whether
such thin set is finite or not. For example, given any two...
Cohomology and cryptography
The Weil pairing is a bilinear form associated to an algebraic
curve. I will tell you about it and why it is interesting to
cryptographers. Then I'll talk about my (completely unsuccessful)
attempts to make an interesting trilinear analogue.
Spectral gaps without frustration
Marius Lemm
In spin systems, the existence of a spectral gap has
far-reaching consequences. "Frustration-free" spin systems form a
subclass that is special enough to make the spectral gap problem
amenable and, at the same time, broad enough to be
physically...
There is a very short proof that a graph is 3-colorable: you
simply give the coloring - it is linear in the size of the graph.
How long a proof is needed that a given graph is *not* 3-colorable?
The best we know is exponential in the size of the...
The three pillars of statistical machine learning: then and now
In this (short and informal) talk I will present the three
fundamental factors that determine the quality of a statistical
machine learning algorithm. I will then depict a classic strategy
for handling these factors, which is relatively well...
Approximate prime numbers
Unfortunately counting prime numbers is hard. Fortunately, we
can cheat by counting 'approximate prime numbers' which is much
easier. Moreover, this allows us to say something about the primes
themselves, and works in situations which seem well...
Proofs from algorithms, algorithms from proofs
Constructive vs Pure Existence proofs have been a topic of
intense debate in foundations of mathematics. Constructive proofs
are nice as they demonstrate the existence of a mathematical object
by describing an algorithm for building it. In computer...
Real zeros of random polynomials in several variables
The topology of the zero set and nesting properties of a random
homogeneous real polynomial of large degree has a universal
behavior depending only on the dimension. We discuss this and an
apparent relation to super-critical percolation in...
Connections between homotopy theory and number theory
For a formal group law G the group of automorphisms Aut(G) acts
on the space of deformations Def(G). The invariants of this action
miraculously recover an object of huge interest to algebraic
topologists, and this connection led to much progress in...
Zeroes of Laplace eigenfunctions
The classical Liouville theorem claims that any positive
harmonic function in $R^n$ is a constant function. Nadirashvili
conjectured that any non-constant harmonic function in $R^3$ has a
zero set of infinite area. The conjecture is true and we
will...
We will describe several situations in number theory and
geometry in which one recovers a sought-after structure by first
constructing a “random” approximation to it.
An Introduction to Univalent Foundations
Daniel Grayson
The Univalent Foundations of Voevodsky offer not only a formal
language for use in computer verification of proofs, but also a
foundation of mathematics alternative to set theory, in which
propositions and their proofs are mathematical objects, and...
The ubiquity of matrix tuples across mathematics
Our object of interest will be tuples matrices over a field.
I will explain how different views of this object by diverse fields
of mathematics give rise to important questions in these areas,
which turn out to be surprisingly tightly connected...
Dimension and support of the harmonic measure or What do Brownian travelers see?
Harmonic measure of a portion of the boundary is the probability
that a Brownian traveler starting inside the domain exits through
this portion of the boundary. It is also a simplest building block
of any harmonic function in a domain. Some...
Lillian Pierce
What do you do with a person who behaves in the worst possible
way at every point in time? Well, I don’t know. But if you ask
instead about an operator that picks out the worst possible
behavior of a function, we sometimes know how to control it.
We...
Synthetic homotopy theory: going beyond set-level mathematics
In addition to offering a formal system for doing ordinary (or
"set-level") mathematics, Vladimir Voevodsky’s Univalent
Foundations also suggest a new way of studying homotopy theory,
called "synthetic homotopy theory".
I will show how synthetic...
Hyperbolic geometry and quantum invariants in dimension 3
The end of the previous century saw radical changes to
three-dimensional topology, which arose from two completely
different approaches. One breakthrough came from Thurston's
introduction of hyperbolic geometry into the field. The second one
came...
Ordinary points mod $p$ of hyperbolic 3-manifolds
Hyperbolic 3-manifolds with arithmetic fundamental group exhibit
many remarkable number theoretic properties. Is it possible that
such manifolds live over finite fields (whatever that means)? In
this talk I will give some evidence for this...
I will consider a very simple open problem in the theory of ODEs
and give a brief overview on what is known about it. The problem is
also an excuse to talk about a widely open subject in modern
PDEs.
Mathematical Structures in the Jungles of Life
Misha Gromov