Math Calendar
An important invariant of a symmetric monoidal ∞-category is its Picard group. The Picard group of K(n)-local spectra was first studied by Hopkins-Mahowald-Sadofsky, and despite the great efforts of many others since then, this group is still unknown at heights above 2. Recently, Mor has described an approach to computing the K(n)-local Picard group using methods of descent. Descent is a well-established technique for computing Picard groups, but to apply it to the K(n)-local setting, one must carefully take its profinite nature into account. Mor does this using the language of condensed mathematics. In this talk, I will give an introduction to descent methods for Picard groups, and outline how condensed mathematics can be used to set up a spectral sequence converging to the K(n)-local Picard group.
Algebraic Geometry codes (AG codes) are a family of error-correcting codes introduced by Goppa in the '80s and constructed using algebraic curves defined over a finite field F_q.
AG codes, which can be seen as a generalization of Reed-Solomon codes, provide many excellent examples of error-correcting codes. Furthermore, a celebrated result by Tsfasman, Vladut and Zink (1982) showed that, if q is the square of a prime and is larger than or equal to 49, then there exist sequences of AG codes beating the Gilbert-Varshamov bound, for infinitely many values of q. This result is particularly striking, as it means that there are some families of AG codes that are better than random codes.
A central role in the study of the parameters of AG codes is played by the properties of the underlying curve, and especially by Weierstrass semigroups at the points of the curve. Given a point P on an algebraic curve X, the Weierstrass semigroup H(P) is defined as the set of natural numbers n for which there exists a function f on X having pole divisor (f)_\infty = nP. The structure of H(P) in general varies as the point varies, however, it is known that generically the semigroup is the same, and there can exist only finitely many points of X, called Weierstrass points, with a different semigroup. The reasons for interest in Weierstrass semigroups are multifold: on one hand, they hold an intrinsic theoretical interest which arises from Stöhr-Voloch theory, where they are used to obtain characterizing properties of the curve. On the other hand, together with their generalizations to the case of multi-point Weierstrass semigroups, they represent a key ingredient for computing excellent bounds for the minimum distance of AG codes from the curve.
In this talk, I will give an introduction to AG codes, their properties and their connections with Weierstrass semigroups. With this regard, I will present some results concerning the determination of one-point and two-point Weierstrass semigroups at the points of certain maximal curves. In the two-point case, our results lead to the study of new families of two-point AG codes with good parameters from two well-known maximal curves. In the one-point case, our results deal instead with the computation of the Weierstrass semigroup at every point of a maximal curve with the third largest genus. One surprising result is that, unlike what happens for all the other maximal curves where the Weierstrass points are known, the set of Weierstrass points of this curve is much richer than the set of its F_{q^2}-rational points.
Joint work with Peter Beelen, Leonardo Landi, Maria Montanucci and Marco Timpanella.
In this joint work in progress with Helge Ruddat and Bernd Siebert, we employ a particular type of Log Smooth Degeneration (LSD) to study the Geometry of Enumerative Mirror Symmetry (GEMS).
Mirror Symmetry is a broad conjecture that predicts that symplectic invariants of a Kähler manifold correspond to algebro-geometric invariants of a mirror-dual complex algebraic variety. This is generally proven by computing both sides.
In this work, we take the first steps towards a full enumerative correspondence that canonically identifies the invariants of both sides. To do so, we employ the Intrinsic Mirror Construction of Gross-Siebert. Then the enumerative correspondence passes through an intermediary tropical manifold and tropical invariants thereof.
UGC seminar webpage https://utrechtgeometrycentre.nl/ugc-seminar/
In this talk, I will report on current developments which go beyond the classical uniform dimension growth bounds, focusing on an affine variant (which implies the projective one). This is based on recent work in the case of affine hypersurfaces, joint with Cluckers, Dèbes, Nguyen and Vermeulen.
Over the moduli space of stable curves, integrals of tautological classes have been an important subject. For example, Witten-Kontsevich theorem says that the generating series of integrals of \psi classes satiesfies the KdV hierarchy. What can we say about relative compactified Jacobian? Using the quasi-stable model, there is a natural choice of tautological classes on compactified Jacobian. In this talk, we will see that the pushforward along the forgetful morphism from the compactified Jacobian to the moduli space of stable curves preserves tautological classes. Our main ingredient is the universal double ramification cycle formula. Using the Witten-Kontsevich theorem, one can compute integrals of tautological classes on compactified Jacobian. Along the way, I will explain how this idea can be adapted to study the logarithmic Picard group constructed by Molcho-Wise. This is a joint work in progress with S. Molcho.
Seminar webpage https://utrechtgeometrycentre.nl/ugc-seminar/
In many control problems there is only limited information about the actions that will be available at future stages. We introduce a framework where the Controller sequentially chooses actions $a_{0}, a_{1}, \ldots$, one at a time. Her goal is maximize the probability that the infinite sequence $(a_{0}, a_{1}, \ldots)$ is an element of a given subset $G$ of $\N^\N$. The set $G$ is assumed to be a Borel tail set. The Controller's choices are restricted: having taken a sequence $h_{t} = (a_{0}, \ldots, a_{t-1})$ of actions prior to stage $t \in \N$, she must choose an action $a_{t}$ at stage $t$ from a non-empty, finite subset $A(h_{t})$ of $\N$. The set $A(h_{t})$ is chosen from a distribution $p_{t}$, independently over all $t \in \N$ and all $h_{t} \in \N^{t}$. We consider several information structures defined by how far ahead into the future the Controller knows what actions will be available.
In the special case where all the action sets are singletons (and thus the Controller is a dummy), Kolmogorov’s 0-1 law says that the probability for the goal to be reached is 0 or 1. We construct a number of counterexamples to show that in general the value of the control problem can be strictly between 0 and 1, and derive several sufficient conditions for the 0-1 ``law" to hold.
JOINT WORK WITH: J\'{a}nos Flesch, William Sudderth, Xavier Venel
A CRASH COURSE ON MEAN FIELD MODELS: DERRIDA’S GREM AND APPLICATIONS
Part 5 - On the GREM Approximation of TAP Free Energies. By Giulia Sebastiani.
The free energy of TAP-solutions for the SK-model of mean field spin glasses can be expressed as a nonlinear functional of local terms:
we exploit this feature in order to contrive abstract GREM-like models which we then solve by a classical large deviations treatment.
This allows to identify the origin of the physically unsettling quadratic (in the inverse of temperature) correction to the Parisi free energy
for the SK-model, and formalizes thetrue cavity dynamics which acts on TAP-space, i.e. on the space of TAP-solutions.
Joint works with Nicola Kistler, and Marius A. Schmidt.
Part 6 - From log-correlated models to (un)directed polymers in the mean field limit. By Adrien Schertzer.
As seen in the previous lectures, Derrida's Random Energy Models have played a key role in the understanding of certain issues in spin glasses.
The mathematical analysis of these models - in particular the multi-scale refinement of the second moment method as devised by Kistler,
is also particularly efficient to analyse the so-called log-correlated class; the latter consists of Gaussian fields with - as the name suggests,
logarithmically decaying correlations. I will introduce/recall some models falling into this class, and the main steps in their analysis
through the paradigmatic Branching Brownian motion / Branching Random Walk. Finally, I will conclude with recent results on models
which are not even Gaussian, but for which the multiscale treatment still goes through swiftly:
the directed and undirected first passage percolation in the limit of large dimensions, a.k.a. the (un)directed polymers in random environment.
Joint works with Nicola Kistler, and Marius A. Schmidt.
Talk 2: 14.15-16.00
Title: Fluctuations and mixing of Internal DLA on cylinders
Abstract: Internal DLA models the growth of a random discrete set by subsequent aggregation of particles. At each step, a new particle starts inside the current aggregate, and it performs a simple random walk until reaching an unoccupied site, where it settles. The large scale properties of IDLA clusters are by now well understood. In these two talks I will instead focus on Internal DLA on cylinder graphs, seen as a Markov chain on the space of particle configurations. I will present several techniques for bounding the maximal fluctuations of IDLA clusters, which allow one to show that the stationary distribution concentrates on a small subset of the infinite state space. I will then discuss the mixing time of the chain, and its dependence on the choice of the cylinder's base graph.
Seminar webpage https://utrechtgeometrycentre.nl/ugc-seminar/