A self-avoiding walk (SAW) on a graph is a path which does not visit any vertex twice. In this talk, we study an enumeration problem consisting in counting such walks of given lengths. More precisely, we will present the proof (obtained jointly with S. Smirnov) of a conjecture of Nienhuis stating that the number of SAWs of length $n$ on the hexagonal lattice grows like $\sqrt{2+\sqrt 2}^{n+o(n)}$. The proof will also shed new light on a very instructive and beautiful phase transition in the geometric properties of long SAWs.

We review how historically the problem of string phenomenology lead theoretical physics first to algebraic/diffenretial geometry, and then to computational geometry, and now to data science and AI.

With the concrete playground of the Calabi-Yau landscape, accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in machine-learning can help explore problems of physical and mathematical interest, from geometry, to group theory, to combinatorics and number theory.

We review approaches to understanding the competition between energy production from different sources, that are based on game theory. The players in such games, ranging from fossil fuel producers, such as the OPEC nations, to renewables, are vastly heterogeneous, with differing costs, sustainability and emission levels. The past several years have seen new discoveries of fossil fuels facilitated by new technologies such as fracking, cheaper solar panels, as well as uncertain demand from major markets like China. The games studied are in continuous time and may be deterministic, stochastic with controlled jumps (random discoveries), finite player or of continuum mean field type. Some results can be derived analytically, and quantitative analysis comes from small parameter approximations and numerical PDE solutions. We also discuss a related mean field game model to study the question of how centralization of reward and computational power occur in Bitcoin-like cryptocurrencies.

The internet and much of the modern infrastructure relies on public key cryptosystems to remain secure. These, in turn, rely on hard mathematical problems. Classical systems rely on the difficulty of factoring large integers (RSA) or solving various discrete logarithm problems (Diffie-Hellman, ECC). In the mid-1990's Peter Shor explained how a quantum computer would be able to break all of these systems, which has led to the construction and study of so-called quantum-resistant cryptosystems, that is, systems for which there is no (known) fast quantum algorithm. In this talk I will give a brief history of pre- quantum cryptography, after which I will describe a hard lattice problems that lies at the heart of many quantum-resistant algorithms and give the details of one such cryptosystem called NTRU, which was constructed by myself, along with Jill Pipher and Joe Silverman.

The heat equation and the wave equation are
two classical and important evolution equations,
which have been widely studied in various settings.
In this talk, I will consider them on non-compact symmetric spaces,
which include in particular hyperbolic spaces.
I will try to give the state of the art in this subject
and I will emphasize some features showing that
the large scale behavior in negative curvature may be different
and actually better compared to the Euclidean case.

The characters of the general linear group are symmetric functions. In fact, the irreducible characters are found by evaluating Schur polynomials at eigenvalues of matrices. In this talk I will discuss bases of ring of symmetric functions that evaluate to characters of the symmetric group when evaluated at roots of unity. I will discuss how these basis relate to two major outstanding open problems in algebraic combinatorics: the restriction problem and the Kronecker problem. The main combinatorial objects in this talk are tableaux filled with multisets.

This is joint work with Mike Zabrocki.