Seminars Sorted by Series
Mathematical Conversations
Duality, Universality and Random Matrices
Provable Bounds in Machine Learning
Abstract: Machine learning is a vibrant field with many rich
techniques. However, most approaches in the field are heuristic: we
cannot prove good bounds on either their performance or their
running time, except in quite limited settings. This talk...
Quantum Theory and Topos Theory
We relate algebraic quantum mechanics (C*-algebras) to topos
theory, so as to capture the essence of quantum logic and quantum
spaces. Motivated by Bohr's idea that the empirical content of
quantum physics is accessible only through classical...
What is a Higher Order Object?
Nils Baas
I will discuss what we should mean by a higher order structure,
and relate it to examples in link theory,cobordism theory and
higher categories, motivating a general notion. Examples from
physics and chemistry will also appear.
These are a curious and only recently appreciated phenomenon in
celestial mechanics, in which (for example) a planet orbiting one
member of a binary star system undergoes large, slow oscillations
in its orbital eccentricity and inclination. I will...
Many times in theoretical computer science we meet codes that
have some local properties. For example Locally Decodable codes,
Locally Testable codes, codes with Low Density Parity Check Matrix,
Self Correctable codes and many others. In this talk...
Chaos Theory has been applied to find low energy spacecraft
trajectories beginning with the successful Japanese lunar mission
Hiten in 1991. The orbit design of this mission was based on the
empirical concept of a `weak stability boundary’, due to...
Constructing Invariants in Symplectic Geometry
Symmetry may be pretty, but asymmetry is subtle.
Zeros of Zeta Functions and the Riemann Hypothesis
Anders Sodergren
In this talk I will present some old and new results on zeros of
zeta functions. In particular, I will discuss the at first sight
shocking result that there are plenty of zeta functions for which
the Riemann hypothesis is false.
Stochastic integrable systems
Some parts of the random matrix/nonequilibrium workshop will be
concerned with stochastic models which are tagged as integrable. I
will briefly recall the notion of classical integrability and
quantum integrability, just to provide similarities and...
Category theory: what's it good for?
The most common question I get from people outside my field is
"what is category theory good for? does it actually help solve
problems?" The very very short answer is "yes"; in this talk, I
will give a slightly longer answer by explaining some of...
Etienne Ghys
In 1772 Euler characterized the surfaces that can be covered
with paper, allowing bending but not tretching, cutting or
wrinkling. For cloth in place of paper, it would be a different
question, as cloth is more flexible, and that was answered
by...
The cosmic Galois group, a tale of geometry, number theory and physics
Grothendieck has proposed, under the name of "motives" a kind of
Galois theory for non algebraic numbers. A mystery of the so-called
"standard model" in high energy physics is the occurrence of about
twenty numerical constants, independent of the...
Multiplying Integer Matrices
We will state a number of problems with completely different
origins, and reformulate them in terms of questions about what
happens when you multiply integer matrices. (In fancy words, these
are called the "Affine Sieve" or a "Local-Global Principle...
Christopher Brav
6:00pm|West Bldg. Lect. Hall
We describe a method of Selberg for constructing fundamental
domains for discrete groups of \(\mathrm{SL}(n,\mathbb R)\) in
terms of trace inequalities. We show how these Selberg domains can
be used to play ping-pong, establishing freeness of some...
A new viewpoint on analytic geometry
Oren Ben-Bassat
What is the difference between algebraic and analytic geometry?
Is there some way to construct moduli "spaces" in analytic geometry
(in the Archimedean or non-Archimedean contexts)? Is there a common
language for expressing the foundations of...
Boltzmann's Entropy and the Time Evolution of Macroscopic Systems
Boltzmann defined the entropy, \(S(M)\), of a macroscopic system
in a macrostate \(M\) as the "log of the volume of phase space"
corresponding to the system being in \(M\). This definition was
extended by von Neumann to quantum systems as "the log...
Games, strategies, and computational complexity
The following questions are quite intimately related. Please
consider them before the talk. Some have surprising answers which
are highly nontrivial theorems in computational complexity.
- Do you find Tic-Tac-Toe an interesting game? Why?
- Do you...
From the quantum Hall effect to integral lattices and braided tensor categories
Starting with a few remarks on hurricanes, I will sketch some
basic facts about the physics of the Quantum Hall Effect. Assuming
that the longitudinal conductance of a two-dimensional electron gas
in a uniform magnetic field vanishes, I will explain...
Non-associative division algebras and projective geometry
Projective planes satisfying certain symmetry conditions
correspond to non-associative division algebras, by the work of
Hilbert, Dickson, Albert, Wedderburn and Veblen. Knuth began the
attempt to classify them over finite fields, and very few...
From Matrix Multiplication to Digital Sculpting
As a prelude to an enjoyable mathematical conversation, I will
present both a visual depiction and a combinatorial interpretation
of matrix and hypermatrix multiplication. Finally I will discuss
how hypermatrix multiplication relates to Sculpting...
Correlation of magic sequences and some ideas from outer space
Magic sequences were introduced for the first radar measurements
of the distance to Venus. They are now used in GPS, satellite and
cell phone communications. In many cases their correlation
properties can be determined using some crazy ideas of...
The math and magic of Jorge Luis Borges
One needs no advanced mathematics to understand Borges' stories,
but with some mathematical insight is able to see unexpected and
nontrivial connections to rigorous math(s). I shall discuss two of
those, one combinatorial, the other analytic, and...
Tudor Dimofte
I'll give an introduction to some of the new relations between
geometry and physics that have arisen in recent years by
considering compactifications of "The 6-dimensional (2,0) theory"
-- with ties to (and among) instanton counting, Hitchin
moduli...
On characters and words in groups
In 1896, Frobenius obtained a remarkable character-theoretic
formula for the number of solutions to the equation
\(xyx^{-1}y^{-1}=g\), for any finite group \(G\) and element \(g
\in G\). While more than a century has since passed, our
understanding...
Randomness in the Mobius function and dynamics
I'll explain a nice way of visualizing the topology of a smooth
complex hypersurface in \((\mathbb{C}^*)^n\), by decomposing it
into `generalized pairs of pants'. Then I'll explain some useful
symplectic constructions arising from this picture, by...
Differential forms and homotopy groups
Richard Hain
This talk will be an introduction to K.-T. Chen's iterated
integrals and to the de Rham theory of homotopy groups. I will give
an historical introduction, starting with works of Frank Adams
(1956) and John Stallings (1975) that have been lost in the...
The Surprise Examination Paradox and the Second Incompleteness Theorem
Few theorems in the history of mathematics have inspired
mathematicians and philosophers as much as Godel’s first and second
incompleteness theorems. I will present a new proof for Godel's
second incompleteness theorem, based on Kolmogorov...
Trivializing the trivial group
I will discuss a 1965 conjecture of J. Andrews and M. Curtis---a
beguilingly straightforward statement about presentations of the
trivial group, which has striking significance in low-dimensional
topology: e.g. in relation to the 3-dimensional...
Can one decide on being free or thin?
We discuss some problems in group theory, some of which are
undecidable. For example, we would like to know whether or not
being thin is a decidable property of a group.
Galois groups and hyperbolic 3-manifolds
I will describe a result of Peter Scholze (reproved a bit later
by George Boxer) establishing a remarkable connection between
Galois theory and the homology of certain arithmetic hyperbolic
3-manifolds.
I will try to explain why one expects there to be a Galois
theory of a certain class of transcendental numbers, called
periods, and illustrate with some simple examples.
What is common to the Szemeredi Regularity lemma in graph
theory, the Green-Tao result on arithmetic progressions in the
primes, the Schapire Boosting algorithm in machine learning and
Impagliazzo Hard-Core set theorem in computational
complexity...
The study of free groups via Stallings core graphs
Introduced by Stallings in '83, core graphs provide a simple and
natural combinatorial-geometric approach to the study of free
groups. This approach yields simple elementary proofs to classical
results and solutions to various algorithmic problems...
One of the central topics in Complex Analysis deals with the
issue of identifying natural conditions that uniquely determine a
holomorphic function. A prominent role in this regard is played by
the one-sided and transmission Riemann-Hilbert problems...
The algebraic fundamental group of a topologically simply-connected algebraic variety
6:00pm|West Bldg. Lect. Hall
This will be an elementary/intuitive introduction to the $\pi_1$
of smooth algebraic varieties in $\mathbf A^1$-homotopy theory
(over algebraically closed fields of char 0). We will give some
flavor on the $\pi_0$ and $\pi_1$. Most of the...
Symmetries and deformation invariants in quantum mechanics
I begin with a geometric discussion of Wigner's theorem
concerning the linearization of quantum mechanical symmetries; it
first appeared in a joint paper with von Neumann. This is the
starting point for joint work with Gregory Moore in which we...
Spectral curves appear in many integrable systems. Their quantum
cousins are equally ubiquitous and describe among others random
matrices, the topology of Riemann's moduli space, and hyperbolic
knots. Quantum curves are the simplest example of what...
The ABC conjecture, Belyi's theorem and applications
We will present the ABC conjecture, Belyi's mapping theorem and
explain how they combine into a powerful tool for diophantine
problems, following the ideas of Elkies, Bombieri, Granville,
Langevin. Finally we will speculate a bit about the function...
Quasi-crystals and subdivision tilings
The Penrose tiling (Roger Penrose(1974)) and the "quasi-crystal"
made by Ron Schactman (1985) are beginning landmarks here. Our
objects today are tilings $T$, of $\mathbb R^d$, [$d = 1, 2$
mostly] which like Penrose's is aperiodic and can be a...
The classification of finite simple groups is a singular event
in the history of mathematics. It has one of the longest and most
complicated proofs any theorem (indeed just to define the terms in
the statement of theorem requires a lot). It has many...
Volumes of hyperbolic link complements
Thurston realized that certain link complements admit a complete
hyperbolic metric, which is a complete invariant of the manifold.
We'll discuss the volumes of hyperbolic link complements and what
is known about them and open questions.
Effective hyperbolic geometry
Powerful theorems of Thurston, Perelman, and Mostow tell us that
almost every 3-dimensional manifold admits a hyperbolic metric, and
that this metric is unique. Thus, in principle, there is a 1-to-1
correspondence between a combinatorial description...
An introduction to chromatic homotopy theory
Chromatic homotopy theory is the philosophy that homotopical
phenomena should be understood via the periodicities they exhibit.
Equivalently, it's the viewpoint that every prime number p hides an
infinite hierarchy of "chromatic primes" of...
Limitations for Hilbert's tenth problem over the rationals
Héctor Pastén Vásquez
In 1900 Hilbert asked for a decision procedure to determine
solvability of polynomial equations over the integers. Seventy
years later, Y. Matiyasevich showed that this problem is
unsolvable, building on earlier work of M. Davis, H. Putnam and
J...
Combinatorics to geometry to arithmetic of circle packings