The rigorous study of spin systems such as the Ising model is
currently one of the most active research areas in probability
theory. In this talk, I will introduce one particular class of such
models, known as lattice gauge theories (LGTs), and go...
Extremal combinatorics is a central research area in discrete
mathematics. The field can be traced back to the work of Turán and
it was established by Erdős through his fundamental contributions
and his uncounted guiding questions. Since then it has...
Suppose you have a set S of integers from {1 , 2 , … , N} that
contains at least N / C elements. Then for large enough N , must S
contain three equally spaced numbers (i.e., a 3-term arithmetic
progression)?
Through the random matrix analogy, Fyodorov, Hiary and Keating
conjectured very precisely the typical values of the Riemann zeta
function in short intervals of the critical line, in particular
their maximum. Their prediction relied on techniques...
In Euclidean geometry, bisectors are perpendicular lines. In
random plane geometry, the situation is more complicated. I will
describe bisectors in the directed landscape, the universal
geometry in the KPZ class. These help answer some open...
For a compact subset K of a closed symplectic manifold,
Entov-Polterovich introduced the notion of (super)heaviness, which
reveals surprising symplectic rigidity. When K
is a Lagrangian submanifold, there is a well-established
criterion for its...
The Mackey-Zimmer representation theorem is a key structural
result from ergodic theory: Every compact extension between ergodic
measure-preserving systems can be written as a skew-product by a
homogeneous space of a compact group. This is used, e.g...
Let X be a smooth projective variety over the field of complex
numbers. The classical Riemann-Hilbert correspondence supplies a
fully faithful embedding from the category of perverse sheaves on X
to the category of algebraic D_X-modules. In this...
Define the Collatz map Col on the natural numbers by setting
Col(n) to equal 3n+1 when n is odd and n/2 when n is even. The
notorious Collatz conjecture asserts that all orbits of this map
eventually attain the value 1. This remains open, even if...
A central goal of physics is to understand the low-energy
solutions of quantum interactions between particles. This talk will
focus on the complexity of describing low-energy solutions; I will
show that we can construct quantum systems for which the...