Felipe Cucker (City University of Hong Kong) Computing the homology of real projective sets

14 / 09 / 2016 - 12:10

Contact: Martín Sombra | Location: Aula T1, Facultat de Matemàtiques, Universitat de Barcelona

Felipe Cucker
City University of Hong Kong
Computing the homology of real projective sets
Computational problems dealing with algebraic (or semialgebraic) sets are, with a few exceptions, difficult to solve. Feasibility problems are NP-complete and problems such as counting the number of solutions in a zero-dimensional set or computing the Euler characteristic are #P-complete. These results suggest that there is no hope of polynomial time algorithms. For the (clearly more complicated) problem of computing the homology groups of an algebraic set, the fastest algorithm remains the cylindrical algebraic decomposition, which has a doubly exponential cost. In the talk we will describe an algorithm doing the job for real projective sets with single exponential cost.

Antoine Chambert-Loir (Université Paris Diderot – Paris 7) 1, 2, 3; A, B, C

11 / 05 / 2016 - 12:10

Contact: Martín Sombra | Location: Aula T1, Facultat de Matemàtiques, Universitat de Barcelona

Antoine Chambert-Loir
Université Paris Diderot – Paris 7
1, 2, 3; A, B, C
Solving equations is one of the oldest tasks pursued by mathematicians, and the study of equations whose unknowns are integers goes back to the Ancient times. These are the Diophantine equations, so called in honor of the Greek mathematician Diophantus, the author of a book entitled Arithmetic, whose too narrow margin hosted the famous remark by Fermat that would change forever the stream of number theory.
During the 20th century, mathematicians understood that the answer to these questions does not so depend on the algebra of the equation, but rather on the form that this equation describes in space. The subject then became Diophantine geometry. Many questions are now understood, but the beauty of their solutions does not exhaust all the mysteries. I want to describe some chapters of this unfinished tale.

Mikhail Sodin (Tel Aviv University) Topology of zero sets of smooth random functions

10 / 02 / 2016 - 12:10

Contact: Joaquim Ortega | Location: Aula T1, Facultat de Matemàtiques, Universitat de Barcelona

Mikhail Sodin
Tel Aviv University
Topology of zero sets of smooth random functions
We describe recent progress in understanding the topology of zero sets of smooth random functions of several real variables and mention a number of basic questions which remain widely open.

Srikanth B. Iyengar (University of Utah) Group algebras and commutative rings

18 / 02 / 2015 - 12:10

Contact: Santiago Zarzuela | Location: Room T1, Facultat de Matemàtiques, Universitat de Barcelona

Srikanth B. Iyengar
University of Utah
Group algebras and commutative rings
The goal of this talk will be to describe a bridge between the modular representation theory of finite groups and modules over polynomial rings. This has provided new insights and results concerning modular representations, and has led to unexpected results in commutative algebra. The talk with be based on joint work with Avramov, Benson, Buchweitz, Carlson, Krause, and Miller.
This is a joint lecture with the IMUB workshop Homological Bonds between Commutative Algebra and Representation Theory.

Ian Morrison (Fordham University) Birational geometry of moduli spaces of stable curves

10 / 12 / 2014 - 12:10

Contact: Rosa M. Miró-Roig | Location: Room T1, Facultat de Matemàtiques, Universitat de Barcelona

Ian Morrison
Fordham University, USA
Birational geometry of moduli spaces of stable curves
Geometric questions about the moduli space of stable curves of a given genus are of interest in many cognate areas of algebraic geometry. For example, this moduli space has been extensively studied as a test case for general questions from the minimal model program in birational geometry, where its modular interpretation provides extra tools for answering these questions. I will introduce the notion of moduli, recall some basic properties of stable curves and their moduli spaces, and then touch on a few highlights of thirty years of progress in our understanding of the birational geometry of such moduli spaces.

Alexandre Turull (University of Florida) Representacions de grups finits

08 / 10 / 2014 - 12:10

Contact: Ignasi Mundet | Location: Aula T1, Facultat de Matemàtiques, Universitat de Barcelona

Alexandre Turull
University of Florida
Aquest col·loqui pretén portar l’audiència des de la prehistòria de la teoria de grups finits a problemes actualment oberts pausadament i remarcant alguns fets importants i curiosos.
La teoria dels grups finits ve marcada pel més gran teorema mai provat matemàticament: la classificació dels grups finits simples. Efectivament, tenim ara una col·lecció que creiem completa dels grups finits simples, que inclou grups tant interessants com el Monstre o el Nadó de Monstre. Aquesta llista infinita de grups dóna les peces per a formar els diferents grups. No hi ha cap esperança de classificar tots els grups finits.
Els grups són l’abstracció de la simetria. Els morfismes d’aquests grups en grups d’automorfismes d’altres estructures matemàtiques s’anomenen representacions. Un famós teorema de Jordan sobre representacions de grups finits com a automorfismes d’espais vectorials complexos té generalitzacions obertes. Parlarem una mica sobre unes reduccions recents d’aquestes generalitzacions.
Al mateix temps, els caràcters dels grups finits plantegen una sèrie de conjectures on l’objectiu és relacionar propietats locals amb propietats globals. Parlarem d’alguns resultats i conjectures recents en aquest camp.
Department of Mathematics, University of Florida, Gainesville, FL 32611, USA
E-mail address: turull@ufl.edu

Marc Noy (UPC) Random graphs, old and new

07 / 05 / 2014 - 12:10

Contact: Martín Sombra | Location: Aula T1, Facultat de Matemàtiques, Universitat de Barcelona

Marc Noy
Universitat Politècnica de Catalunya
The theory of random graphs was born in 1960 with the fundamental work of Erdős and Rényi. Since then, the field has grown enormously and has become one of the central topics in combinatorics. The model most analyzed is the G(n, p) model, consisting of n labelled vertices and in which the edges are chosen independently with probability p. The value p = p(n) is allowed to be a function of n, and one can show that many important properties have a threshold function. In particular, p = 1/n is the threshold for the emergence of the giant component, where a fundamental phase transition occurs (already discovered by Erdős and Rényi). In recent years, alternative models have been introduced in order to capture properties of real life networks.
In a different direction, we are also interested in analyzing random graphs from a constrained class of graphs defined by a global property, such as being regular, triangle-free, acyclic or planar, where the distribution is uniform among all graphs with the same number of vertices. In this context we cannot draw edges independently and we have to rely on counting. The first task is to estimate how many graphs are there with a given number of vertices, and this can be done using a variety of enumerative and probabilistic tools.
After reviewing classical models, we will discuss several key examples and different approaches. Then we will focus on random graphs with a topological flavour, where there has been much progress recently. We will also touch briefly on recent developments on logical limit laws. The talk is addressed to a wide audience including undergraduate and graduate students. The exposition will be mostly self-contained and will not assume familiarity with random graphs.

Jordan Stoyanov (Newcastle University) Intriguing problems from Combinatorics, Algebra, Analysis and Probability

09 / 04 / 2014 - 12:10

Contact: José Manuel Corcuera | Location: Aula B5, Facultat de Matemàtiques, Universitat de Barcelona

Newcastle University
This lecture is addressed to a wide audience: undergraduate and graduate students in Mathematics, PhD students, and professionals. It will be shown that diverse and intriguing mathematical problems from the above-mentioned areas can be solved quite elegantly by using ideas and techniques from Probability. Sometimes these are the only available solutions.
Some of the following topics will be discussed in detail:

• Combinatorial and algebraic identities.
• A problem involving two dice — fair or unfair?
• Toss 15 dice: if your sum equals your product, you win 1 million! Ready to play?
• Bernoulli’s Law of Large Numbers and Weierstrass’ theorem by Bernstein polynomials.
• Old Uspensky’s problem and its far-reaching extensions.
• The Buridan Donkey story (BD). Random walks in a random environment (JD).
• Many ways to interpret and solve the equation X + Y = XY.
• Values of the Riemann zeta function via Cauchy’s distribution.
• A few exercises and one unsolved problem.

Questions and suggestions from the audience during the lecture will be very welcome.

Alain Yger (Université de Bordeaux 1) Currents versus effectivity questions in complex geometry

12 / 02 / 2014 - 12:10

Contact: Martín Sombra | Location: Aula T1, Facultat de Matemàtiques, Universitat de Barcelona

Alain Yger
Université de Bordeaux 1
It is well known in multivariate complex geometry that currents (i.e., differential forms with distribution coefficients) provide quite flexible tools that happen to be helpful (even sometimes necessary) in order to make effective concepts or methods of algebraic nature. They basically allow to profit, besides the n degrees of freedom, of the “ghost” ones, which on one side have vocation to be treated as “constants” through algebraic computations and on the other side allow to keep track of positivity. Such “complex” currential ideas even inspired recently methods in a “real” (instead of complex) setting (for example in tropical or analytic geometry in the ultrametric frame), where the idea of involving “ghost” objects has also been carried through. Introducing currents in order to materialize algebraic or geometric objects allow to conciliate local requirements with semi-local or even global ones. But there is of course a price to pay: appealing to current theory (in order that it could be such helpful) involves simultaneously some kind of “averaging process”: the construction of Bochner-Martinelli type currents (which, as we know today, provide nice currential realizations for duality) essentially by combining the averaging of the (algebraic) Cauchy kernel with the geometric frame of holomorphic hermitian bundles, that of explicit currential representatives in improper intersection theory which could fit with local as well as global requirements via Crofton’s formula, explicit realisations of the logarithmic height in arithmetic geometry (think about the Mahler measure of a projective hypersurface via the explicitation of normalized Green currents through the Bost-Gillet-Soulé product, provide indeed a wide range of illustrations of such a fact. I will present some of these examples, and discuss positive as well as negative points.
The talk intends to be an elementary invitation to such “currential» methods, focusing on their recent developments, their interest, but at the same time on their limits (concerning precisely effectivity).

W. Hugh Woodin (University of California, Berkeley) The Continuum Hypothesis and the mathematical conception of infinity

11 / 12 / 2013 - 15:30

Contact: Joan Bagaria | Location: Room T1, Facultat de Matemàtiques, Universitat de Barcelona

W. Hugh Woodin
University of California, Berkeley
By the middle of the 20th century the problem of the Continuum Hypothesis was widely regarded as one of the prominent problems in all of Mathematics. Remarkably, this question cannot be solved on the basis of the basic principles (these are the ZFC axioms) on which the entire subject is based. This discovery of Cohen, announced 50 years ago, is arguably one of the major discoveries in the history of modern Mathematics.
But does this mean that the question of the Continuum Hypothesis has no answer? Any «solution» must involve the adoption of new principles, but which principles should one adopt? Alternatively, perhaps the correct assessment of Cohen’s discovery is that the entire enterprise of the mathematical study of Infinity is ultimately doomed because the entire subject is simply a human fiction without any true foundation. In this case, any «solution» to the Continuum Hypothesis is just an arbitrary (human) choice.
Over the last few years a scenario has emerged by which with the addition of a single new principle not only can the problem of the Continuum Hypothesis be resolved, but so can all of the other problems which Cohen’s method has been used to show that are also unsolvable (and there have been many such problems). Moreover the extension of the basic (ZFC) principles by this new principle would be seen as an absolutely compelling option based on the fundamental intuitions on which the entire mathematical conception of Infinity is founded.
However, this scenario critically depends upon the outcome of a single conjecture. If this conjecture is false then the entire approach, which is the culmination of nearly 50 years of research, fails or at the very least has to be significantly revised. Thus the mathematical study of Infinity has arguably reached a tipping point. But which way will it tip?