Information theory

More broadly, I’m interested in mathematical aspects of information theory, particularly in connection with category theory, algebra and geometry (metric geometry, geometric measure theory, complex algebraic geometry…).

Gaussian completely monotone conjecture

In the past decade, J. Huh solved several long-standing open problems on log-concave sequences in combinatorics. The ground-breaking techniques developed in those work are from algebraic geometry: “We believe that behind any log-concave sequence that appears in nature there is such a Hodge structure responsible for the log-concavity”.

A function is called completely monotone if its derivatives alternate in signs; e.g., $e^{-t}$. A fundamental conjecture in mathematical physics and Shannon information theory is on the complete monotonicity of Gaussian distribution (GCM conjecture), which states that $I(X+Z_t)$ is completely monotone in $t$, where $I$ is Fisher information, random variables $X$ and $Z_t$ are independent and $Z_t \sim \mathcal{N}(0,t)$ is Gaussian. Inspired by the algebraic geometry method introduced by J. Huh, GCMC is reformulated in the form of a log-convex sequence. In general, a completely monotone function can admit a log-convex sequence and a log-convex sequence can further induce a log-concave sequence. The new formulation may guide GCM conjecture to the marvelous temple of algebraic geometry.

In another formulation, Gaussian completely monotone conjecture is referred to as: the $m$-th time-derivative of the entropy along the heat flow on $\mathbb{R}^d$ is positive for $m$ even and negative for $m$ odd. Recent progress on this topic is due to Guillaume Wang. In his article, A Higher-Order Otto Calculus Approach to the Gaussian Completely Monotone Conjecture, he have proven the GCM conjecture for orders up to $m=5$, assuming that the initial measure is log-concave, in any dimension. His work based on Otto calculus and on the interpretation of the heat flow as the Wasserstein gradient flow of the entropy.

• Notes for GCM conjecture and Hodge theory

Algebraic approach to information theory

In “simple” terms, information topology regards a statistical system (a collection of interrelated observables) as a generalized topological space (a topos) and identifies Shannon entropy, along other important “measures of information” used in information theory, as a possible invariant associated to this space.

Toposes or topoi are an abstraction of topological spaces in the language of category theory and sheaves introduced by Grothendieck and his collaborators (Artin, Verdier,…). Toposes allow richer cohomology theories than set-theoretic topological spaces, and some of these theories (e.g. étale cohomology) play a key role in modern algebraic geometry. Moreover, these Grothendieck toposes are particular cases of elementary toposes, which are “nice” categories with properties analogous to those of the category of sets that play an important role in logic.

Baudot and Bennequin first identified Shannon’s discrete entropy as a toposic invariant of certain categories of discrete observables. Juan Pablo Vigneaux and a series of articles extended their results in several directions. Namely, the general homological constructions were abstracted from the concrete setting of discrete variables via information structures (categories that encode the relations of refinement between observables), allowing seamless extensions and generalizations to other settings such as continuous vector-valued observables.

Coding theory

I’m also interested in algebraic approach to coding theory and error-correcting codes, particularly in connection with group theory, field theory and representation theory.

Maximun distance separable codes conjecture

If an \([n, k, d_{\min}]_q\) code \(\mathcal{C}\) satisfies the singleton bound with equality \(d_{\min} = n-k+1\), then we call \(\mathcal{C}\) a maximun distance separable codes (MDS code).

The main content of MDS conjecture is that, except for the trivial MDS codes or those with parameters \([2^m+2,\,3,\,2^m]_{q=2^m}\) or \([2^m+2,\,2^m-1,\,4]_{q=2^m}\), all MDS codes satisfy \(n \le q + 1\).

In algebraic formulation, MDS conjecture is that: a set $S$ of vectors of the vector space $F_q^k$ such that every subset of $S$ of size $k \leq q$ is a basis, has size at most $q + 1$, unless $q$ is even and $k = 3$ or $k = q − 1$, in which case it has size at most $q + 2$.

Algebraic geometry codes

Algebraic–geometry (AG) codes use the language of smooth projective curves over a finite field $\mathbb F_q$. The key ingredients are the function field $\mathbb F_q(X)$ of a curve $X$, divisors $D$ and $G$ on $X$, and the Riemann–Roch space $L(G)$ of rational functions whose poles are bounded by $G$. Evaluating functions in $L(G)$ at a set of $\mathbb F_q$-rational points produces codewords; dually, using spaces of differentials $\Omega(G-D)$ and their residues gives parity–check information. This viewpoint can be phrased either in divisor/line–bundle language (via $L(G)=H^0(X,\mathcal O_X(G))$) or in function–field language, and yields a clean duality $C_L^\perp=C_\Omega$.

The standard construction proceeds as follows. Pick $n$ distinct rational points $P_1,\dots,P_n\in X(\mathbb F_q)$ and set $D=\sum_i P_i$. Choose a divisor $G$ whose support is disjoint from $D$. The evaluation code

has length $n$, and—when $2g-2<\deg G<n$—dimension and minimum distance obey the Riemann–Roch bounds

The differential code

satisfies $C_L^\perp=C_\Omega$ and $d_\Omega\ge \deg G-2g+2$. Classical Reed–Solomon and BCH codes appear as the genus-$0$ special case ($X=\mathbb P^1$), while “Goppa codes” arise from particular choices of $G$ on $\mathbb P^1$. In practice, a basis of $L(G)$ gives a generator matrix; a basis of $\Omega(G-D)$ gives a parity–check matrix.

Performance-wise, AG codes leverage curves with many rational points to obtain long lengths over fixed alphabets with strong distance. Asymptotically, for square $q$ there are explicit towers of curves (e.g., the Garcia–Stichtenoth towers) that meet the Tsfasman–Vladut–Zink (TVZ) bound, surpassing the Gilbert–Varshamov bound for large $q$. Concrete families such as Hermitian and Suzuki curves already yield excellent finite-length parameters. On the algorithmic side, there are unique-decoding methods (Feng–Rao, Berlekamp–Massey–Sakata/Gröbner-basis) and list-decoding (Guruswami–Sudan for one-point AG codes) that go beyond the designed distance. The rich duality structure also makes AG codes useful in cryptography (Goppa-based McEliece variants), secret sharing, and constructing quantum CSS codes.

Probability theory

Integrable probability

Integrable probability is an area of research at the interface between probability theory, mathematical physics, combinatorics and representation theory. It refers to the study of probabilistic models that are exactly solvable.

The notion of exact solvability or integrability is somewhat vague. A model is called exactly solvable when observables of interest can be computed by a formula involving well-known functions (rational functions, $\exp$, $\sin$, $\Gamma$), so that the complexity of the formula does not increase as parameters go to $\infty$.

• Introduction to Integrable Probability
• Topics in Representation Theory

Statistical mechanics

The field of applications of probabilistic ideas was considerably expanded in the XIX century. Most notably, Ludwig Boltzmann (Austria, 1844-1906) proposed that thermodynamic macroscopic observables, such as the temperature and the entropy, could be explained from microscopic considerations, using a probabilistic description of the possible configurations of the molecules that compose a given substance. For this, it was necessary to take limits as the number of particles goes to infinity and/or to consider continuous models (particles distributed in Euclidean space).

Boltzmann ideas were further developed by Josiah Williard Gibbs (USA, 1839-1903), who also coined the term statistical mechanics. For instance, Boltzmann proposed that the entropy $S$ of a system is given by \(S=k_B \log(\#\text{ microscopic configurations of the system}),\) when all the configurations $X$ are equiprobable and the argument of the logarithm is finite. Here $k_B$ denotes a universal constant (Boltzmann’s constant). More generally, Gibbs gave the formula \(S= - k_B \sum_{x\in X} p(x) \log p(x),\) where $p:X\to \mathbb R_{\geq 0}$ satisfies $\sum_{x\in X} p(x) = 1$.

In actual systems, the number of configurations is a priori infinite. The formulas above only make sense on finite portions of the system, and one is forced to consider a limiting procedure. The simplest setting on which one can perform this kind of limiting operation is the Ising model: Consider a finite and discrete set $\Lambda \subset \mathbb Z^2$ of sites organized in a square array. Let $s_{x,y}\in \lbrace +1,-1 \rbrace$ be a “spin” associated to the site $(x,y)\in \Lambda$. Then $\vec s = \lbrace s_{x,y} \rbrace_{(x,y)\in \Lambda}$ is a possible configuration of the system. The energy of this configuration is given by the Hamiltonian: \(\mathcal{H}(\vec s, BC) = - \sum\limits_{(x,y)} s_{x,y} \left(s_{x+1,y} + s_{x,y+1} \right) - h \sum\limits_{(x,y)} s_{x,y}\) Because of the interactions between neighbors in the sum, the hamiltonian depends on some given boundary conditions $BC$. Remark that two neighboring spins that are equal decrease the energy. The Boltzmann-Gibb’s theory postulates that, in equilibrium, the configurations of the system are distributed according to the probability law \(\mathbb P(\vec s) \propto \exp \left(- \frac{\mathcal{H}(\vec s)}{k_B T} \right),\) where $T$ is the temperature of the system. This is called today a Gibb’s state.

It turns out that the ``typical’’ or more probable configurations are very different, depending on the value of $T$. If $T$ is much smaller that a certain critical temperature $T_c$, then the alignment tendency of the spins predominates and one sees big clusters with the same spin. Whereas if $T\gg T_c$, disorder predominates. A phase transition happens at $T_c$. In fact, this $T_c$ is only well defined in the limit where the diameter of $\Lambda$ goes to infinity and the boundary conditions become irrelevant.

Algebra and Geometry

Tropical algebraic geometry

Tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition:

• \[x \oplus y = \min\{x, y\},\]
• \[x \otimes y = x + y.\]

Tropical geometry is a variant of algebraic geometry in which polynomial graphs resemble piecewise linear meshes, and in which numbers belong to the tropical semiring instead of a field. Because classical and tropical geometry are closely related, results and methods can be converted between them. Algebraic varieties can be mapped to a tropical counterpart and, since this process still retains some geometric information about the original variety, it can be used to help prove and generalize classical results from algebraic geometry.

Fruitful interactions are emerging between combinatorics and Hodge theory, both in the applications of Hodge theory to address problems in combinatorics. My interest is the tropical analog of the classical Hodge theory for Kähler manifolds, containing Poincaré Duality, Hard Lefschetz theorem and Hodge-Riemann relations, together with their applications in information and channel coding theory.

Braid group and Knizhnik–Zamolodchikov equations

For $n\ge 2$, the braid group $B_n$ has the Artin presentation

this is called the Artin presentation, and the $\sigma_i$ are Artin’s generators.

In mathematical physics the Knizhnik–Zamolodchikov equations, or KZ equations, are linear differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex partial differential equations with regular singular points satisfied by the N-point functions of affine primary fields and can be derived using either the formalism of Lie algebras or that of vertex algebras.

Let \(\hat{\mathfrak{g}}_{k}\) denote the affine Lie algebra with level $k$ and dual Coxeter number $h$. Let $v$ be a vector from a zero mode representation of \(\hat{\mathfrak{g}}_{k}\) and $\Phi(v, z)$ the primary field associated with it. Let $t^a$ be a basis of the underlying Lie algebra \(\mathfrak{g}\), $t^{a}_{i}$ their representation on the primary field \(\Phi(v_i, z)\) and \(\eta\) the Killing form. Then for $i, j = 1, 2, \ldots , N$ the Knizhnik–Zamolodchikov equations read