Seminars Sorted by Series
Mathematical Conversations
On the cap-set problem and the slice rank polynomial method
5:30pm|Remote Access Only
In 2016, Ellenberg and Gijswijt made a breakthrough on the
famous cap-set problem, which asks about the maximum size of a
subset of \mathbb{F}_3^n not containing a three-term arithmetic
progression. Ellenberg and Gijswijt proved that any such set...
Singularities of solutions of the Hamilton-Jacobi equation. A toy model: distance to a closed subset.
5:30pm|Remote Access Only
This is a joint work with Piermarco Cannarsa and Wei Cheng. Most
of the lecture is about the distance function to a closed subset in
Euclidean subset, at the level of a beginning graduate student. If
$A$ is a closed subset of the Euclidean space $...
Robustness, Verifiability and Privacy in ML
Shafi Goldwasser
Cryptography and Machine Learning have shared a curious history:
a scientific success for one often provided an example of an
impossible task for the other. Today, the goals of the two fields
are aligned. Cryptographic models and tools can and...
Joshua Greene
I will discuss a little about the context and solution of the
rectangular peg problem: for every smooth Jordan curve and
rectangle in the Euclidean plane, one can place four points on the
curve at the vertices of a rectangle similar to the one
given...
The Mumford-Shah conjecture
Silvia Ghinassi
The Mumford-Shah functional has been introduced by Mumford and
Shah in 1989 as a variational model for image reconstruction. Since
then, it has been widely studied both from a theoretical and an
applied point of view. In this talk we will focus on...
In 1979, Kaufman constructed a remarkable surjective Lipschitz
map from a cube to a square whose derivative has rank $1$ almost
everywhere. In this talk, we will present some higher-dimensional
generalizations of Kaufman's construction that lead to...
Three-term arithmetic progressions in sets of integers
Olof Sisask
It turns out that certain additive patterns in the integers are
very hard to get rid of. An instance of this is captured in a
conjecture of Erdős, which states that as long as a set of natural
numbers is 'somewhat dense' -- namely the sum of the...
Deep learning for the working mathematician
Artificial intelligence or "deep learning" is becoming
ubiquitous in new fields of mathematical applications stemming from
the internet economy. This has led to the creation of powerful new
tools. We would like to explore how these techniques can be...
Higher order Fourier analysis and generalizations of Szemerédi's theorem
Several of the most important problems in combinatorial number
theory ask for the size of the largest subset of some abelian group
or interval of integers lacking points in a fixed arithmetic
configuration. One example of such a question is, "What...
Isolated points on curves
Bianca Viray
Let $C$ be an algebraic curve over the rational numbers, that
is, a 1-dimensional complex manifold that is defined by polynomial
equations with rational coefficients. A celebrated result of
Faltings implies that all algebraic points on $C$ come in...
Determinants, hyperbolicity, and interlacing
Hyperbolic polynomials are a multivariate generalization of
real-rooted polynomials that originated in the study of partial
differential equations and have since found applications in many
other fields, including operator theory, optimization, and...
Nike Sun
In high dimensions, what does it look like when we take the
intersection of a set of random half-spaces with either the sphere
or the Hamming cube? This is one phrasing of the so-called
perceptron problem, whose study originated with a toy model
of...
The Möbius function $\mu(n)$ measures the parity of number of
prime factors of $n$ (if $n$ is square free). Understanding the
randomness in this function (often referred to as the Möbius
randomness principle) is a fundamental and very difficult...
How hard is it to tell two knots apart?
Many problems in classical topology can be formulated as
decision problems, with yes/no answer and an algorithm as a
solution. While such problems often appear to be intuitively hard,
we still know little about lower bounds on their algorithmic...
Quantum Integer Valued Polynomials
In algebraic combinatorics there well known objects called
q-integers, q-binomial coefficients, and q-factorials which for
lack of a better word "q-ify" the usual integers, binomial
coefficients, and factorials. I will explain these notions (and
say...
Why is $N_{\Gamma_0(12)}^{\mathrm{new}}(\lambda)$ of cocompact type?
I will speak, broadly, on, the arithmetic and geometry of
quaternion algebras and aspects of the spectral theory of
automorphic laplacians. I will describe spectral correspondences
between spaces of Maass waveforms on the unit group of a
quaternion...
Space vectors forming rational angles
Bjorn Poonen
In 1895, Hill discovered a $1$-parameter family of tetrahedra
whose dihedral angles are all rational multiples of $\pi$. In 1976,
Conway and Jones related the problem of finding all such tetrahedra
to solving a polynomial equation in roots of unity...
Newton, Euler, Navier, and Green
We touch lightly on the background of four mathematicians over
four centuries whose names are famous in mathematics with my
personal emphasis on fluid dynamics.
Many interacting quantum particles: open problems, and a new point of view on an old problem
The main object of interest in this talk will be a system of
many particles, modeled using the prescriptions of quantum
mechanics. A significant challenge to studying such systems is that
particles interact with each other, via weak or strong...
Embedded Contact Homology of Prequantization Bundles
Embedded Contact Homology (ECH) is a Floer type invariant due to
Hutchings. This talk provides a gentle overview of ECH (in part
through a video of the Hopf fibration) and sketches why ECH of a
prequantization bundle over a Riemann surface is...
Surfaces and Point Processes
Jayadev Athreya
We'll give several concrete examples of how to go from the
geometry of surfaces to the study of point processes, following
work of Siegel, Veech, Masur, Eskin, Mirzakhani, Wright, and
others. We'll discuss how this "probabilistic" perspective
helps...
For a given finite group G, which spaces can be the fixed point set of a G-action on a given compact space?
Which spaces can be the fixed point sets of actions of $G$ on
finite cell-complexes of a given homotopy type? The general answers
to such questions, for $G$ not a group of prime-power order, will
be expressed, even for non-simply-connected spaces...
Embedding Symplectic Ellipsoids and Diophantine equations
This talk will explain work stemming from a group project that
investigated the ellipsoidal embedding capacity function for the
family of Hirzebruch surfaces. This problem turns out to have
unexpected arithmetic structure, leading to an intricate...
Floer's Jungle: 35 years of Floer Theory
An exceptionally gifted mathematician and an extremely complex
person, Floer exhibited, as one friend put it, a “radical
individuality.” He viewed the world around him with a singularly
critical way of thinking and a quintessential disregard for...
Math & Computation: some principles, anecdotes and questions
I planned to give a different talk, about recent work I am
excited about. But then Helmut asked me to instead talk of "Dreams
of mathematics and computer science". And his wish is my
command...
I'll describe, mainly through works of some great...
Amol Aggarwal
In this talk we discuss two models of a discrete random surface.
The first is a Markov process, like a simple random walk, under
which the surface is grown according to random updates. The second
chooses the surface uniformly at random, after...
Symmetries in symbolic dynamics
Bryna Kra
Originating in the work of Hadamard in the 1890’s on the coding
of geodesic flow, symbolic dynamics has become a key tool for
studying topological, smooth, and measurable dynamical systems. The
automorphism group of a symbolic system capture its...
Tangent cones and their uniqueness, maybe a meeting ground for hard analysis and algebraic geometry
6:00pm|Birch Garden, Simons Hall
In several diverse settings (variational problems and geometric
flows, elliptic, parabolic, but also some dispersive PDEs)
monotonicity formulas allow to get a first coarse description of
singularities, which are commonly called tangent cone. Their...
The Unreasonable Effectiveness of Convexity in Symplectic Geometry
6:00pm|Birch Garden, Simons Hall
Convexity plays a central role in several geometric and
dynamical problems in symplectic geometry. However, convexity is
not preserved under structure preserving isomorphisms and it is
unknown whether there exists an intrinsic property
responsible...
Gaussian Elimination with Complete Pivoting: Searching for a Needle in a Haystack
6:00pm|Birch Garden, Simons Hall
Gaussian elimination is one of the oldest and most popular
techniques for factoring a matrix. The growth of entries in
Gaussian elimination is an important practical problem. Modern
results as well as practice show that entry growth is not a...
6:00pm|Birch Garden, Simons Hall
Stochastic Characteristics: ellipticity and hypoellipticity from finite to infinite dimensions
6:00pm|Birch Garden, Simons Hall
I will give an elementary introduction to the connections
between diffusions and stochastic characteristics in $\mathbb R^n$.
I will then explain how one might think about what it means to be
elliptic or hypoellipticity in an infinite dimensional...
Noether's Theorem in the Calculus of Variations and Hyperbolic Manifolds
6:00pm|Birch Garden, Simons Hall
I will remind the audience of Noether’s theorem in the calculus
of variations and give a little of the history. An elementary
application to integrals of Lagrangians defined on functions with
domain a hyperbolic surface will be given, ending with a...
6:00pm|Birch Garden, Simons Hall
A magnetic interpretation of the nodal count on graphs
6:00pm|Birch Garden, Simons Hall
The study of nodal sets, i.e. zero sets of eigenfunctions, on
geometric objects can be traced back to De Vinci, Galileo, Hook,
and Chladni. Today it is a central subject of spectral geometry.
Sturm (1836) showed that in 1D, the $n$-th eigenfunction...
Can one hear the winding number?
6:00pm|Birch Garden, Simons Hall
6:00pm|Birch Garden, Simons Hall
From Stein to Weinstein and Back
6:00pm|Birch Garden, Simons Hall
I will discuss some open questions about the relation between
Stein and Weinstein structures.
The vision of the sets according to Brownian travelers
6:00pm|Birch Garden, Simons Hall
The harmonic measure is an important tool, which allows one to
reconstruct a harmonic function from its values on the boundary.
But it also admits a very simple and beautiful probabilistic
interpretation: it is the probability that the path of the...
Random hyperbolic surfaces
6:00pm|Birch Garden, Simons Hall
I will introduce an extremely natural model for random
hyperbolic surfaces and discuss how little we know about it in
large genus.
The Strong Cosmic Censorship conjecture in general relativity
6:00pm|Birch Garden, Simons Hall
The statement that general relativity is a deterministic theory
finds its mathematical formulation in the Strong Cosmic Censorship
conjecture due to Roger Penrose. I will introduce the
conjecture and report on some recent progress.
Lego in finite groups, Hurwitz spaces, and Markoff triples
6:00pm|Birch Garden, Simons Hall
Let G be a group, and let (g,h) be a pair in G x G. Consider the
group of symmetries of G x G generated by the "moves" sending (g,h)
to (g,gh), (g,g^{-1}h), (g,hg), (g,hg^{-1}), (gh,h),...etc. An old
question from the 50's, motivated by the study of...
Crooked geometry: Crystallography in the geometry of (2+1)-special relativity
6:00pm|Birch Garden, Simons Hall
Bieberbach's 1912 theory of Euclidean crystallographic groups
provides a satisfying qualitative classification of flat Riemannian
manifolds. In 1977 Milnor asked whether a similar picture could
extend to flat affine manifolds, that is, when the...
6:00pm|Birch Garden, Simons Hall