A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

Kardar-Parisi-Zhang - interacting particle systems - random growth processes - directed polymers - Markov duality - quantum integrable systems - Bethe ansatz - asymmetric simple exclusion process - stochastic partial differential equations[-]

A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

Kardar-Parisi-Zhang - interacting particle systems - random growth processes - directed polymers - Markov duality - quantum integrable systems - Bethe ansatz - asymmetric simple exclusion process - stochastic partial differential equations[-]

In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach spaces), and give a conceptual proof of the Weierstrass division theorem.
Lecture two covers smooth, unramified and étale morphisms between noetherian rings. The relation of these notions with the concepts of submersion, immersion and diffeomorphism from differential geometry is given.
In the third lecture, we investigate ring extensions between the three power series rings and describe the respective flatness properties. This allows us to prove approximation in the linear case.
The last lecture is devoted to the geometry of solution sets in power series spaces. We construct in the case of one $x$-variable an isomorphism of an $m$-adic neighborhood of a solution with the cartesian product of a (singular) scheme of finite type with an (infinite dimensional) smooth space, thus extending the factorization theorem of Grinberg-Kazhdan-Drinfeld.
CIRM - Chaire Jean-Morlet 2015 - Aix-Marseille Université[-]

The $p$-adic Igusa zeta function, topological and motivic zeta function are (related) invariants of a polynomial $f$, reflecting the singularities of the hypersurface $f = 0$. The first one has a number theoretical flavor and is related to counting numbers of solutions of $f = 0$ over finite rings; the other two are more geometric in nature. The monodromy conjecture relates in a mysterious way these invariants to another singularity invariant of $f$, its local monodromy. We will discuss in this survey talk rationality issues for these zeta functions and the origins of the conjecture.[-]

One of the possible applications of Artin approximation is to prove that the local geometry of sets defined in affine space by real or complex analytic equations is not more complicated than the local geometry of sets defined by polynomial equations. A possible approach is to prove that a complex analytic (singular) germ, for example $(X,0) \subset (\mathbf{C} ^n,0)$, is the intersection, in some affine space $\mathbf{C}^N$, of an algebraic germ $(Z,0) \subset (\mathbf{C}^N,0)$ by a complex analytic non singular subspace $(W,0)$ of dimension $n$ which is "in general position" with respect to $Z$ at the origin. Approximating $Z$ by an algebraic subspace then yields the desired result, provided the "general position" condition is sufficiently precise. I will explain how one can attack this problem using a notion of "general position with respect to a singular space" which is based on the concept of minimal Whitney stratification, which will also be explained. Nested Artin approximation is essential in this approach.

nested Artin approximation - Whitney forms - singularities - stratifications - germ of subspace[-]

Let $A$ be the ring of formal power series in $n$ variables over a field $K$ of characteristic zero. Two power series $f$ and $g$ in $A$ are said to be equivalent if there exists a $K$-automorphism of $A$ transforming $f$ into $g$. In my talk I will review criteria for a power series to be equivalent to a power series which is a polynomial in at least some of the variables. For example, each power series in $A$ is equivalent to a polynomial in two variables whose coefficients are power series in $n - 2$ variables. In particular, each power series in two variables over $K$ is equivalent to a polynomial with coefficients in $K$. Similar results are valid for convergent power series, assuming that the field $K$ is endowed with an absolute value and is complete. In the special case of convergent power series over the field of real numbers some weaker notions of equivalence will be also considered. I will report on works of several mathematicians giving simple proofs. Some open problems will be included.

singularities - power series[-]

The Zariski problem concerns the analytical classification of germs of curves of the complex plane $\mathbb{C}^2$. In full generality, it is asked to understand as accurately as possible the quotient $\mathfrak{M}(f_0)$ of the topological class of the germ of curve $\lbrace f_0(x, y) = 0 \rbrace$ up to analytical equivalence relation. The aim of the talk is to review, as far as possible, the approach of Zariski as well as the recent developments. (Full abstract in attachment).

O. Zariski - analytic classification - foliation - germ - Puiseux expansion[-]

The space of formal arcs of an algebraic variety carries part of the information encoded in a resolution of singularities. This series of lectures addresses this fact from two perspectives. In the first two lectures, we focus on the topology of the space of arcs, proving Kolchin's irreducibility theorem and discussing the Nash problem on families of arcs through the singularities of the variety; recent results on this problem are proved in the second lecture. The last two lectures are devoted to some applications of arc spaces toward a conjecture on minimal log discrepancies known as inversion of adjunction. Minimal log discrepancies are invariants of singularities appearing in the minimal model program, a quick overview of which is given in the third lecture.[-]

The space of formal arcs of an algebraic variety carries part of the information encoded in a resolution of singularities. This series of lectures addresses this fact from two perspectives. In the first two lectures, we focus on the topology of the space of arcs, proving Kolchin's irreducibility theorem and discussing the Nash problem on families of arcs through the singularities of the variety; recent results on this problem are proved in the second lecture. The last two lectures are devoted to some applications of arc spaces toward a conjecture on minimal log discrepancies known as inversion of adjunction. Minimal log discrepancies are invariants of singularities appearing in the minimal model program, a quick overview of which is given in the third lecture.[-]

The space of formal arcs of an algebraic variety carries part of the information encoded in a resolution of singularities. This series of lectures addresses this fact from two perspectives. In the first two lectures, we focus on the topology of the space of arcs, proving Kolchin's irreducibility theorem and discussing the Nash problem on families of arcs through the singularities of the variety; recent results on this problem are proved in the second lecture. The last two lectures are devoted to some applications of arc spaces toward a conjecture on minimal log discrepancies known as inversion of adjunction. Minimal log discrepancies are invariants of singularities appearing in the minimal model program, a quick overview of which is given in the third lecture.[-]