Existing unconditional progress on the abc conjecture and Szpiro's
conjecture is rather limited and coming from essentially only two
approaches: The theory of linear forms in $p$-adic logarithms, and
bounds for the degree of modular parametrizations...
In the lectures I will formulate a conjecture asserting that
there is a hidden action of certain motivic cohomology groups on
the cohomology of arithmetic groups. One can construct this action,
tensored with $\mathbb C$, using differential forms...
Geometric Complexity Theory (GCT) was developed by Mulmuley and
Sohoni as an approach towards the algebraic version of the P vs NP
problem, VP vs VNP, and, more generally, proving lower bounds on
arithmetic circuits. Exploiting symmetries, it...
We will explain a definition of open Gromov-Witten invariants on
the rational elliptic surfaces and explain the connection of the
invariants with tropical geometry. For certain rational elliptic
surfaces coming from meromorphic Hitchin system, we...
We show that there exist binary locally testable codes (for all
rates) and locally correctable codes (for low rates) with rate and
distance approaching the Gilbert-Varshamov bound (which is the best
rate-distance tradeoff known for general binary...
A celebrated theorem of Duke states that Picard/Galois orbits of
CM points on a complex modular curve equidistribute in the limit
when the absolute value of the discriminant goes to infinity. The
equidistribution of Picard and Galois orbits of...
In the lectures I will formulate a conjecture asserting that
there is a hidden action of certain motivic cohomology groups on
the cohomology of arithmetic groups. One can construct this action,
tensored with $\mathbb C$, using differential forms...
Neural networks have been around for many decades. An important
question is what has led to their recent surge in performance and
popularity. I will start with an introduction to deep neural
networks, covering the terminology and standard approaches...