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MATH BULLETIN FS 2018
Editorial. 2 Neues aus dem
FVM.
Der Herbst neigt sich zu Ende, das Semester
neigt sich zu Ende, bald beginnt ein neues Jahr,
ein neues Semester und die Chance unser
4 Beweis
methoden.
Wissen zu erweitern. Bis dahin sind noch viele
Prüfungen zu meistern, doch vor
Prüfungsstress geniesst die Weihnachtszeit
und erholt euch. In dieser Ausgabe erhält ihr
dem
6 Was passiert am
Institut?
einen kleinen Einblick in die
Forschungsgruppe
Ayoub.
von Professor
Wir wünschen euch viel Spass beim Lesen und
Joseph
9 Witz und
Rätselecke.
viel Erfolg bei den Prüfungen.
Liebe Grüsse,
Sarina Sutter und Stefan Kurz
11 Vorlesungs
verzeichnis
UZH.
27 Forschungs
seminare.
28 Vorlesungs
verzeichnis ETH.
MATH BULLETIN FS 2018 1
Neues aus dem FVM.
Grosse personelle Veränderungen im Vorstand, der Fachverein
erhält eine eigene Flagge und ein wichtiger Hinweis für euch.
Vorstandsänderungen Vinzenz Muser / Marc Egger: Unser
langjähriges Präsidialteam, das den FvM
An der letzten MV (lang ist’s her...) zu Ruhm und Ehre geführt hat!
durften wir feierlich eine ganze Reihe Nathan Brack: Unser ExKassier, unter
neuer Vorstandsmitglieder aufnehmen: dessen finanzieller Aufsicht sich unser
Stefan Kurz: Der Mann der frühen Geld beträchtlich vermehrt hat!
Morgenstunden, neu verantwortlich für Simon Grüning: Unser TopJournalist!
die Zensur. Euch allen ein riesiges Dankeschön für
Annina Cincera: Unsere gute Seele mit eure Arbeit, ihr werdet sie vermissen!
einem riesigen Herz für Erstsemestler
und Maturanden/innen. Vereinsflagge
Lukas Burch: Der Mann des schnellen
Aufstiegs, unser neuer Vizepräsident. Und natürlich das Wichtigste: Unser
Gian Deflorin: Berater des Präsidiums, Verein bekommt eine Flagge, die wir im
Präsident des Teegremiums und nächsten Semester feierlich einweihen
nebenbei Vertretung im werden!
Mathematikerrat.
David Berger: Unser neuer Mann fürs Kafferahm
Seriöse, er wird uns künftig beim Cüpli
trinken mit dem Dekan vertreten. Gelegentlich beschäftigt sich der
Kilian Werder: Unser neuer Finanzboss Vorstand ja auch mit seriösen Themen,
(ab FS 18). deshalb zu guter Letzt noch eine
Im Namen des ganzen Vorstandes wichtige Mitteilung: Bitte konsumiert
wünschen wir euch allen eine erfüllende keinen Kaffeerahm aus der
Vorstandsarbeit mit vielen Institutsküche, das gilt nach wie vor. Der
Glücksausbrüchen! gehört uns nicht. Das meinen wir ernst.
Mit diesen Neuwahlen waren natürlich Wirklich.
auch schmerzliche Abschiede
verbunden: ~ Alain Schmid
2 MATH BULLETIN FS 2018Beweismethoden.
Proof by vigorous handwaving: Works well in a classroom or seminar setting.
Proof by forward reference: Reference is usually to a forthcoming paper of the
author, which is often not as forthcoming as at first.
Proof by example: The author gives only the case n = 2 and suggests
that it contains most of the ideas of the general
proof.
Proof by omission: "The reader may easily supply the details" or "The
other 253 cases are analogous"
Proof by deferral: "We'll prove this later in the course".
Proof by picture: A more convincing form of proof by example.
Combines well with proof by omission.
Proof by intimidation: "Trivial."
Proof by adverb: "As is quite clear, the elementary aforementioned
statement is obviously valid."
Proof by seduction: "Convince yourself that this is true! "
Proof by exhaustion: An issue or two of a journal devoted to your proof is
useful.
Proof by mutual reference: In reference A, Theorem 5 is said to follow from
Theorem 3 in reference B, which is shown to follow
from Corollary 6.2 in reference C, which is an easy
consequence of Theorem 5 in reference A.
3 MATH BULLETIN FS 2018MATH BULLETIN FS 2018 4
Was passiert am Institut?
Vorstellung der Arbeit der Forschungsgruppe von Professor Dr.
Joseph Ayoub.
One of the biggest successes of numbers, or at least the real numbers.
Mathematics during the XXth century is Equations over finite fields, for example,
the developing of algebraic topology to seemed to be out of the scope of
study Geometry. We call “algebraic algebraic topology, due to the “discrete”
topology” to the techniques attaching in nature of their solutions.
a natural way to every geometric space
an algebraic object—e.g.: a group, a In 1949, André Weil made four far
vector space—that encodes geometric reaching conjectures in Arithmetic and
properties of the space. Hence, we Algebraic Geometry over finite fields.
reduce the study of the geometry of a To be more precise, given some
space to simpler questions related to algebraic equations with rational
these algebraic objects. Probably the coefficients we can consider the
most successful of these algebraic algebraic variety they define modulo p.
topology techniques is the one called Then, Weil conjectured that the
“cohomology” which, in its simplest analogue of the Riemann hypothesis for
form, attaches a finite dimensional finite fields holds for the associated
vector space to a given geometric space. analogue of Riemann’s Zeta function.
Many theorems, such as the topological Even more, the Riemann’s Zeta function
classification of surfaces, GaussBonet was strongly related to the cohomology
formula, fixed point theorems, existence of the variety defined by its complex
of a relativistic structures, etc… can be solutions. Weil conjectures led
solved thanks to these techniques. Alexander Grothendieck to believe that
there should be unifying roots for
As we have said, since its beginnings Topology, Geometry (both Differential
algebraic topology had a strong and Algebraic) and Arithmetic: the
influence in Topology and Differential cohomology of a complex variety
and Complex Geometry. However, in (Complex Geometry, and Topology) was
principle algebraic geometers could not strongly determining its algebraic
profit from these techniques, unless solutions (Algebraic Geometry) to the
they were working over the complex extend that we can prove the analogue
MATH BULLETIN FS 2018 5Riemann’s hypothesis over finite fields introduced the notion of a motive in a
(Arithmetic). letter to Serre in 1964. Later he wrote
that, among the objects he had been
Between 1957 and 1970 Grothendieck privileged to discover, they were the
and his collaborators refounded most charged with mystery and formed
Algebraic Geometry and set up a perhaps the most powerful instrument
common unifying ground of work for of discovery.1 Motives should be the
algebraic topology. This set up could be object behind all cohomology theories,
applied in Algebraic Geometry and which are their avatars. Their name,
Arithmetic as well in Topology and “motif”, suggests it. As the motive
Differential Geometry, and it led melody in music, which we can make
Grothendieck to prove three of the Weil several variations or avatars in different
conjectures. After his retirement in 1970, styles, or of an image in painting2
one of his students, Pierre Deligne, motives in Mathematics would be the
finally proved in 1974 the last Weil reason and the leading intuition behind
conjecture. However, despite the many analogue phenomena in
enormous success of this program Arithmetic and Geometry, such as
Grothendieck was not satisfied. His cohomologies.
framework allowed to define a pleiad of
cohomology theories for algebraic Unfortunately, Grothendieck gave only
varieties: étale cohomology, crystalline the guidelines of the theory of motives.
cohomology, coherent sheaf Since then this subject has been one of
cohomology… However, the direct the major topics of research for
analogue of the classic cohomology Arithmetic and Algebraic Geometry.
with rational coefficients in Differential Professor Ayoub’s and its group
Geometry and Topology was not found. research lies precisely in the field of
Even more, JeanPierre Serre gave a motives. This area of Mathematics has
counterexample to the possibility that very clear objectives: Grothendieck
such cohomology could exist. before his retirement in 1970 set up the
Nevertheless all cohomologies found by so called “Standard conjectures” that
Grothendieck behaved as if they were would unlock the theory of motives.
avatars of this nonexisting analogue of Although these conjectures still remain
the Differential Geometry one. completely open until today, the world
of motives has seen an unprecedented
Before retiring in 1970, Grothendieck renewed progress in the last 20 years,
6 MATH BULLETIN FS 2018having Professor Ayoub as one of the achieving a proof of the Grothendieck
leading contributors. We will list only or KontsevichZagier conjecture in what
some of his main achievements in this is called the relative case.
topic.
Finally, a remarkable contribution of
Firstly, Professor Ayoub provided the Professor Ayoub is currently being
world of motives with a satisfactory achieved. One of the main roadblocks of
technical theory called “Grothendieck’s the desired standard conjectures is that
six functor formalism”. This technical they require to prove “existence of
theory is the basic tools that allowed cycles”. In other words, we require to
Grothendieck to give a common construct algebraic subvarieties with
framework for algebraic topology.
prescribed properties. Constructing
Hence, a motivic version of the six
subvarieties with prescribed properties
functor formalism was certainly to be
in the case of Differential Geometry,
placed in the realm of motives.
where there are partitions of the unity
and quite some flexibility to construct
One of the most interesting objects in
subvarieties, is often quite manageable.
the theory of motives is the so called
In Algebraic Geometry this is not the
“motivic Galois group”. This group
case. Indeed, even the simplest linear
encodes properties of the
algebraic differential equation, δyxδx,
cohomological avatars of motives and,
among them, it was conjectured to has the exponential as solution and, as a
govern a very important class of consequence, it has no algebraic
numbers called “multiple zeta values”. solutions. Hence, there is very little
These numbers, a generalization of the freedom. In the case of curves, where
values of Riemann’s Zeta function at the subvarieties are simply points, this
integers, appear not only in the theory problem leads to the RiemannRoch
of transcendence and algebraic theorem, which is one of the major
elements, but they also play an theorems of algebraic curves. However,
important role in quantum mechanics. in higher dimensions the problem
In this area the main conjecture is becomes more difficult. To put it in few
known as the Grothendieck or words, very little is known on existence
KontsevichZagier conjecture. Professor of cycles, and too much is required for
Ayoub made important contributions to the standard conjectures. However, the
the study of the motivic Galois group,
MATH BULLETIN FS 2018 7so called “conservativity conjecture” of
motives states that the cohomological
avatars of motives preserve (or conserve)
a lot of information from the motive, and
a very important one from algebraic
subvarieties. In particular this conjecture
directly implies the existence of cycles in
a great general setting and it would
certainly be a major advance for the
theory of motives. Professor Ayoub has a
tentative proof of the conservativity
conjecture which is currently being
checked. A seminar will be run next
semester on our University about the
validity of this proof.
~ Alberto Navarro GarmendiaWitz und Rätselecke.
Zusammengestellt von Sarina Sutter und Stefan Kurz.
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9 MATH BULLETIN FS 2018Square Factorials Multiply together the first 100 factorials 1!x...x100!. Find a natural number n, such that dividing this product by n! produces a square number. Odd squares How many square numbers are these, whose digits are all odd? Blackboard Suppose that numbers 1 to 20 are written on a blackboard. At each turn, you may erase two numbers a and b and write the sum a+b in their place. After 19 such turns, there will be one number left at the blackboard. What will this number be? Odd sums What is (1+3)/(5+7)? What is (1+3+5)/(7+9+11)? What is (1+3+5+7)/(9+11+13+15)? What is (The sum of first n odd numbers)/(The sum of the next n odd numbers)? Great Riddles reprinted with permission from Matthew Scroggs, Find the solutions and more at mscroggs.co.uk 10 MATH BULLETIN FS 2018
Vorlesungsverzeichnis UZH. MAT 112 Lineare Algebra II (9 ECTS) Euklidische und normierte Vektorräume Bilinearformen und Skalarprodukte Orthonormalbasen und GramSchmidtOrthogonalisierung Spektralsatz fuer selbsadjungierte Endomorphismen Normalform symmetrischer Bilinearformen Quadratische Formen Unitäre Vektorräume Spektralsatz für normale Endomorphismen Tensor Produkte und deren universelle Eigenschaften Symmetrische und alternierende Tensoren Dozierende: Anna Beliakova MAT 122 Analysis II (12 ECTS) Topologie des Euklidischen Raums Differentialrechnung mehrerer Variablen Taylorentwicklung, lokales Verhalten einer Abbildung, Konvexität Satz ueber die Umkehrabbildung, Satz über implizite Funktionen Untermannigfaltigkeiten des Euklidischen Raums. Lokale Extrema mit Nebenbedingungen. Vektoranalysis: Vektorfelder, Rotation, Divergenz, Gradient. Elementare Lösungsmethoden für gewöhnliche Differentialgleichunten. Lineare Systeme von gewöhnlichen Differentialgleichungen, Exponential einer Matrix. Existenz und Eindeutigkeitssatz für gewöhnliche Differentialgleichungen. Differenzierbare Abhängigkeit, Fluss eines Vektorfeldes. Folgen und Reihen von Funktionen. Fourierreihe. Dozierende: Viktor Schroeder MATH BULLETIN FS 2018 11
MAT 801 Numerik I (9 ECTS) Grundlegende Konzepte werden eingeführt. Effiziente Berechnungsmethoden für relativ einfache mathematische Probleme werden vorgestellt. Inhaltsübersicht: Zahlendarstellung und Rundungsfehler, numerische Verfahren für lineare und nichtlineare Gleichungen, Approximation und Interpolation, numerische Integration, numerische Methoden für Differentialgleichungen. Dozierende: Alexander Veit MAT 901 Stochastik (9 ECTS) Die Fähigkeit, die grundlegenden Ideen der Wahrscheinlichkeitsrechnung sowohl theoretisch als auch in Anwendungen einzusetzen. Voraussetzungen: MAT 111 Lineare Algebra, MAT 121 Analysis und MAT 221 Analysis III Dozierende: Jean Bertoin MAT 506 Kommutative Algebra (9 ECTS) Es wird eine Einführung in die kommutative Algebra gegeben, die sich an Studierende ab dem 4. Semester richtet. Die kommutative Algebra befasst sich mit den kommutativen Ringen. Einfache Beispiele solcher Ringe sind der Ring der ganzen Zahlen, Polynomringe über Körpern und deren Restklassen und Bruchringe. Die kommutative Algebra ist ein wirksames Werkzeug für die algebraische und die analytische Geometrie, für die Zahlentheorie aber auch für die Kombinatorik und Algorithmik. Das Gebiet ist sehr reichhaltig und vermag deshalb sicher auch für sich alleine genommen algebraisch interessierte Hörerinnen und Hörer anzusprechen. Im Zentrum unserer Vorlesung steht die Theorie der noetherschen lokalen Ringe. Höhepunkt ist die homologische Charakterisierung der regulären Ringe gemäss 12 MATH BULLETIN FS 2018
Serre, Auslander und Buchsbaum und deren Anwendung. Die benötigten Begriffe aus der homologischen Algebra werden in der Vorlesung mit entwickelt. Im einzelnen werden folgende Themen behandelt: 1. Noethersche Ringe und Moduln 2. Primideale 3. KrullDimension 4. Assoziierte Primideale 5. Tiefe 6. CohenMacaulayRinge 7. Projektive Dimension und BettiZahlen 8. Globale Dimension 9. Reguläre lokale Ringe 10.Faktorialität 11.Normalität 12.Ganze Erweiterungen und Primideale 13.Algebren über Körpern Voraussetzungen: MAT211 Algebra I Dozierende: Andrew Kresch MAT 604 Funktionentheorie (9 ECTS) This lecture is an introduction to the theory of holomorphic function in one variable. Vorraussetzung: Analysis I & Lineare Algebra I Dozierende: Valentin Féray MATH BULLETIN FS 2018 13
MAT 721 Differentiable Manifolds (9 ECTS) Differenzierbare Mannigfaltigkeiten sind Mengen (wie z.B. die Sphäre), die lokal wie ein Rn aussehen, so dass man die übliche Differential und Integralrechnung verwenden kann: Auf differenzierbaren Mannigfaltigkeiten kann man z.B. Differentialgleichungen studieren und Funktionen oder Differentialformen integrieren. Die Vorlesung ist eine Voraussetzung für Vorlesungen in den Bereichen Differentialgeometrie, Differentialtopologie, globale Analysis und mathematische Physik. Die hier eingeführten Begriffe liegen auch zugrunde, u.a., der allgemeinen Relativitätstheorie und der Eichtheorien. Themen der Vorlesung: Differenzierbare Mannigfaltigkeiten und Untermannigfaltigkeiten. Vektorfelder, Flüsse und der Satz von Frobenius. Lie Gruppen, Prinzipal und assoziierte Bündel, Zusammenhänge. Differentialformen, Integration und der Satz von Stokes. Vorraussetzung: Grundbegriffe der Analysis I+II, Lineare Algebra I+II, Mehrfachintegrale, Differentialgleichungen Dozierende: Alberto S. Cattaneo MAT 753 Topology of surfaces (3 ECTS) The course is considered as a continuation of „Topology and Geometry". The main objects discussed are surfaces, two dimensional spaces that look locally like a plane. The material will cover classification results, certain topological invariants such as the Euler characteristic and homology, and will end with a number of applications. Voraussetzungen: Linear algebra, Topology & Geometry Dozierende: Krzysztof Putyra 14 MATH BULLETIN FS 2018
MAT 754 An Introduction to Geometric Quantization (2 ECTS) A classical mechanical system can be described by a symplectic manifold called the state space and a distinguished observable, which is a function on the state space, called the Hamiltonian. More generally, in classical mechanics we have the state space a symplectic manifold and observables which are functions on the sympletic manifold. In quantum mechanics, on the other hand, the state space is a Hilbert space and the observables are operators on the Hilbert space. Roughly speaking, Quantization of a classical theory is a process which associates a quantum theory to the classical theory. Ideally, one would like to associate each classical observable a quantum observable. This is impossible because there are no go theorems. In practice, one has to lower one’s expectation so that there is a "reasonable quantization" ofa classical system. In this course, we will study one particular method of quantization called geometric quantization. Position space quantization, moment space quantization and holomorphic quantization are particular instances of geometric quantization. Voraussetzungen: Some background on Analysis and manifolds will be helpful. Dozierende: Santosh Kandel MAT 832 Mathematische Modellierung (9 ECTS) In der Vorlesung wird die mathematische und numerische Modellierung physikalischer Probleme behandelt. Die Herleitung der Gleichungen aus physikalischen Prinzipien, deren Vereinfachung und effiziente numerische Lösung stehen im Mittelpunkt dieser Vorlesung. Voraussetzung: Numerik I Dozierende: Stefan A. Sauter MATH BULLETIN FS 2018 15
MAT 512 Elliptische Kurven (9 ECTS) Introductory course on the theory of elliptic curves, with a point of view towards their applications in cryptography. Elliptic curves and their equation, the group law on the points of an elliptic curve, endomorphisms, jinvariant, elliptic curves defined over finite fields, Krational points, torsion points, Frobenius endomorphisms, Hasse's Theorem, Schoof's algorithm, the Discrete Logarithm Problem in the group of points of an elliptic curve, pairings, cryptographic protocols which exploit elliptic curves. Voraussetzungen: Knowledge of the material covered in a basic algebra course (e.g. MAT211 Algebra I) will be assumed. This includes, but is not limited to, groups (finite groups, Chinese Reminder Theorem), rings (in particular polynomial rings), fields, and their homomorphisms. Very basic knowledge of algebraic geometry and/or cryptography will be helpful, but not required. Dozierende: Joachim Rosenthal MAT 525 Hopf algebras (9 ECTS) This lecture is an introduction to the theory of Hopf algebras. We introduce and discuss basic algebraic concepts (coalgebras, bialgebras, Hopf algebras, Hopf modules and comodules, universal enveloping algebras). The highlight of the lecture will be a proof of the CartierKostant theorem for pointed cocommutative Hopf algebras, that describes how a large class of Hopf algebras may described as a smash product algebra composed out of the primitive and grouplike elements. Voraussetzungen: Linear Algebra I, II and Algebra Dozierende: Benedikt Stufler 16 MATH BULLETIN FS 2018
MAT 579 Lie groups and Lie algebras (9 ECTS) This class is devoted to the analysis of Lie algebras and Lie groups, their structure theory and their representations. Lie groups are groups which are also a differentiable manifolds equipped then with group operations compatible with the smooth structure. They represent the best developed theory of continuous symmetry of mathematical objects, which makes them indispensable tools for many parts of contemporary mathematics and modern theoretical physics; they provide, thus, a natural framework for analysing differential equations and they play an enormous role in modern geometry, on several different levels. In this course, after a short recollection on basic facts on differentiable manifolds, we introduce and discuss a relationship between Lie groups and Lie algebras. We will then study representation theory of the simplest Lie algebras and finish the course with the structure theory of general complex semisimple Lie algebras. Voraussetzungen: MAT111 Lineare Algebra I & II, MAT121 Analysis I & II Dozierende: Pavel Safronov MAT 516 Analytic number theory (4 ECTS) This course will introduce some of the fundamental theorems and results of classical Analytic Number Theory. The course will start with basic notions, including the fundamental theorem of arithmetic, Euclid's theorem for the infinitude of primes, rational/irrational numbers and it will continue with the study of arithmetic functions, perfect numbers and Fermat numbers, congruences, quadratic residues, Dirichlet series and also aspects of the prime counting function and the Riemann zeta function. During the class, some special topics such as the proof of Chebyshev’s inequality will be presented as well. Voraussetzungen: Students who attend this course should have already followed a course in Differential and Integral Calculus. Dozierende: Michail Rassias MATH BULLETIN FS 2018 17
A First Course in Homotopy Theory (3ECTS) Homotopy theory is one of the main branches of algebraic topology, which uses various algebraic gadgets to decide when topological spaces have the same "shape". The central idea of homotopy theory is to regard two topological spaces as "the same" when they can be continuously deformed into one another, instead of asking if they are homeomorphic to one another (the latter being a vastly stronger condition). With some caveats, this weaker notion of equivalence is detected by the homotopy groups of a space and so homotopy theory is largely about studying the behaviour of these groups. In this course, we will discuss a variety of core concepts of the theory including, but not limited to; Homotopy groups and long exact sequences CW complexes and simplicial sets Fibrations and cofibrations The Whitehead and Hurewicz Theorems EilenbergMac Lane spaces Postnikov and Whitehead towers The Homotopy Excision Theorem Rational Homotopy Theory Voraussetzung: Geometrie & Topologie, Lineare Algebra Dozierende: Vincent Schlegel MAT927 Probabilistic models and methods in arithmetic (9 ECTS) Voraussetzungen: MAT901 Stochastik I and MAT604 Funktionentheorie Dozierende: Ashkan Nikeghbali 18 MATH BULLETIN FS 2018
MAT 631 Mathematical aspects of quantum mechanics (6 ECTS) In this course, we will consider selected tools in functional analysis playing an important role in the mathematical study of quantum systems. This will include the spectral theorem for unbounded operators, Stone’s theorem for the unitary group generated by selfadjoint operators, perturbation theory. We will discuss how these mathematical results can be applied to understand energetic and dynamical properties of quantum systems. The course is addressed primarily to advanced bachelor students and master students in mathematics, with some experience in analysis. No prior knowledge of physics will be assumed. Vorraussetzung: Analysis IIII Dozierende: Benjamin Schlein MAT 752 Advanced Topics in Field Theory (2 ECTS) Dozierende: Alberto S. Cattaneo MAT 827 Numerical Methods for Hyperbolic Partial Differential Equations (10 ECTS) Introduction to hyperbolic problems: Conservation, flux modeling, examples and significance in physics and engineering. Linear Advection equations in one dimension: Characteristics, energy estimates, upwind schemes. Scalar conservation laws: shocks, rarefactions, solutions of the Riemann problem, weak and entropy solutions, some existence and uniqueness results, finite volume schemes of the Godunov, EngquistOsher and LaxFriedrichs type. Convergence for monotone methods and Eschemes. Secondorder schemes: LaxWendroff, TVD schemes, limiters, strong stability preserving RungeKutta methods. Linear systems: explicit solutions, energy estimates, first and high order finite MATH BULLETIN FS 2018 19
volume schemes. Nonlinear Systems: Hugoniot Locus and integral curves, explicit Riemann solutions of shallowwater and Euler equations. Review of available theory. Dozierende: Rémi Abgrall MAT 946 (Random) Graphs with applications to risk modelling (9 ECTS) In this lecture, we introduce the most basic tools and facts about random graphs and focus on applications to (systemic) risk modelling and the propagation of stress in a financial network. Voraussetzungen: Basic probability theory (Stochastic I and II). Dozierende: Ashkan Nikeghbali STA 111 Stochastic Modeling (5 ECTS) We discuss relevant topics of aspects of stochastic modeling: including runs, ruins, queues, Markov processes, Brownian motion, POT/EVT, compartmental models, stochastic cellular automata. Vorraussetzung: STA 110 Introduction to Probability (or equivalent) Dozierende: Reinhard Furrer 20 MATH BULLETIN FS 2018
STA 390 Statistical Practice (4 ECTS) This module aims to offer a first glimpse into the practice of a statistician. To this end students will work under supervision on "real" statistical problems, e.g., consulting cases, statistical software implementations, methodology developments etc. Voraussetzungen: Completion of the 30 ECTS minor.Approval of the coordinator. Dozierende: Reinhard Furrer STA 408 Statistical Methods in Epidemiology (5 ECTS) Analysis of casecontrol and cohort studies. The most relevant measures of effect (odds and rate ratios) are introduced, and methods for adjusting for confounders (MantelHaenszel, regression) are thoroughly discussed. Advanced topics such as measurement error and propensity score adjustments are also covered. We will outline statistical methods for casecrossover and case series studies and finally describe methods to analyse registry data, including relative survival techniques. Voraussetzungen: Basic knowlegde of the programming language R.. Dozierende: Leonhard Held, Beate Sick STA 425 Survival Analysis (3 ECTS) The analysis of survival times, or in more general terms, the analysis of time to event variables is concerned with models for censored observations. Because we cannot always wait until the event of interest actually happens, the methods discussed here are required for an appropriate handling of incomplete observations where we only know that the event of interest did not happen within a certain period. Most prominently, survival analysis is used in randomised clinical trials in oncology, but also in epidemiology, and also outside biostatistics in economics (duration analysis) or sociology (event history analysis). During the course, we will study the most important methods and models for MATH BULLETIN FS 2018 21
censored data, including general concepts of censoring, simple summary statistics, estimation of survival curves, frequentist inference for two and more groups, and regression models for censored observations. Voraussetzungen: Basic knowlegde of the programming language R, Regression and likelihood methods. Dozierende: Torsten Hothorn STA 421 Bayesian Data Analysis (3 ECTS) Bayesian methodology for data analysis is well established and used in many domains of science. The unique ability to incorporate prior knowledge makes it especially attractive for empirical research. The growing number of practitioners appreciate the Bayesian paradigm as an intuituve approach to answer various relevant research questions. Moreover, in recent years, Bayesian modeling has become more accessible through generalpurpose software systems for Bayesian computation. In particular, JAGS and RINLA represent the current stateoftheart. However, Bayesian methodology is often used without a deeper understanding of its intricacies and therefore not tailored to optimal efficacy for a given setting. This lecture provides a widely accessible introduction to Bayesian methodology by contrasting the classical and Bayesian approaches. The underpinnings and mechanisms of the Bayesian paradigm will be made tangible by analyses in R, JAGS and RINLA. The presented applications in this lecture are from a biomedical research context but participants will be enabled to transfer their acquired skills in Bayesian methodology to other areas of empirical research. Voraussetzungen: A sound knowledge of classical statistics and programming in R. Dozierende: Malgorzata Roos 22 MATH BULLETIN FS 2018
MAT 820 Numerisches Praktikum (3 ECTS) Das numerische Praktikum kann sich formal aus einem Seminar und einer Seminararbeit oder auch nur aus einer Seminararbeit zusammensetzen. Inhaltlich geht es darum, eine erlernte numerische Methode für ein konkretes Problem anzuwenden und numerische Experimente durchzuführen, um deren Verhalten (Konvergenz, Robustheit, Anwendbarkeit, Effizienz) zu analysieren und mit der Theorie zu vergleichen. Ein numerisches Praktikum kann den Grundstock bilden für eine anschliessende Masterarbeit und steht typischerweise in Verbindung mit aktuellen Forschungsrichtungen der Forschungsgruppe Numerik am IMath in enger Zusammenarbeit mit den Mitgliedern der Arbeitsgruppe. Dozierende: Stefan A. Sauter MAT 557 Advanced topics in linear algebra (3 ECTS) The idea of the seminar is to explore some topics which are „linear algebraic", but which lie outside the standard material from basic courses in linear algebra and algebra. The topics will be interrelated among themselves, and will also connect to various areas of mathematics. We will try to emphasize these connections as much as we can, while still having fun with each topic for its own sake. Here is a sample of some of the things we will learn about: some basic notions of general (universal) algebra; e.g. posets and lattices some basic methods of homological algebra; e.g. exact sequences, diagram chasing some theory of finitely generated modules over a principle ideal domain, e.g. for the classification of endomorphisms of a vector space over (nearly) any field some representation theory of quivers and posets, e.g. as a tool to study systems of linear spaces and maps. Voraussetzungen: Linear Algebra Iⅈ Algebra. Dozierende: Jonathan Michael Lorand MATH BULLETIN FS 2018 23
MAT 680 Seminar Analysis. Topics from the theory of Dynamical Systems (3 ECTS) The field of 'Dynamical Systems’ has its origin in celestial mechanics, which is concerned with the study of the movements of the planets, in particular the prediction of periodic orbits, possible collisions, or the existence of unbounded orbits (i.e., planets leaving the (solar) system). The subject of ‘Dynamical Systems’ is the (time) evolution of systems, described in terms of a law of evolution on a state space. Ordinary differential equations can be viewed as dynamical systems where the law of evolution is given by the flow (solution map). In this course, we will be mostly concerned with discrete dynamical systems. Their dynamics are described by the iteration of a map, acting on the state space, and the main object of study is their orbits, defined as sequences of states, obtained by applying to a starting point iterates of the given map. Fundamental questions are, among others, the following ones: (1) What is the long time asymptotics of a given orbit? Does it approach an equilibrium (attractor)? Does it return to starting point (periodic orbit)? Or does it return to a state nearby (recurrence)? Is the orbit dense on the state space (transitive)? (2) Does the system admit conserved quantities, such as the ‘energy’ or ‘momentum’, or does there exist an invariant measure? (3) How do the orbits depend on the starting point? Do they stay close to a given orbit if their starting points are slightly changed (Lyapunov / asymptotic / orbital stability)? Or is there a sensitive dependence on initial state (chaotic systems)? (4) Are there no qualitative changes of the orbits if the law of evolution is (slightly) perturbed (structural stability)? In particular, do periodic orbits persist under small perturbations? The aim of the course is to get to know basic examples and concepts of dynamical systems, pertinent to the questions raised above. It is based on selected chapters of the book ‘Lectures on Dynamical Systems. An Introduction’, EMS 2010, by E. Zehnder. If you are interested in attending the seminar, please let us know by email (thomas.kappeler@math.uzh.ch) as early as possible so that we can get organized ahead of the start of the spring semester. Dozierende: Thomas Kappeler 24 MATH BULLETIN FS 2018
MAT 592 Seminar on Galois theory (3 ECTS) Historically Galois theory arose as an attempt to systematically understand why a general polynomial of degree five or greater is not solvable by radicals. In more modern terms, it outlines an elegant connection between the study of field extensions and the study of groups. The aim of the seminar course is to understand this connection. The seminar course will start with a review of the basics of field theory and present main results of Galois theory with proofs. We will discuss applications of Galois theory to historically significant problems (solvability by radicals, Euclidean construction of regular polynomials, etc.) as well as present some connections with the more modern topics. Dozierende: Andrew Kresch MAT 960 Mathematical foundations of machine learning (3 ECTS) Machine learning is a collection of algorithms that allow computers to carry out complex tasks, such as determining whether an email is spam or not, by learning from data. It is an important field that spans engineering, computer science and statistics. The topic of this seminar is the mathematical theory of machine learning. We will study some basic machine learning algorithms, and introduce a mathematical framework that allows us to state and prove rigorous results about their performance. Students will present selected chapters from books on the subject. Voraussetzungen: Stochastics I, Linear Algebra I, Programming Dozierende: David Belius MATH BULLETIN FS 2018 25
STA 380 Selected Topics in Statistics (3 ECTS) This is a seminar where students learn to present, discuss, and defend research results from a recent article in the statistical literature. We propose a set of papers from a certain topic, but may propose another recent paper motivated by personal interest. Voraussetzungen: STA402 Likelihood Inference or MAT924 Mathematical Statistics. Dozierende: Reinhard Furrer MAT 589 Seminar: Representation theory (3 ECTS) Representation theory is simple to define: it is the study of the ways in which a given group may act on vector spaces. It is almost certainly unique, however, among such clearly delineated subjects, in the breadth of its interest to mathematicians. This is not surprising: group actions are ubiquitous in 20th century mathematics, since it captures the concept of "symmetry", and where the object on which a group acts is not a vector space we have learned to replace it by one that is (e.g., a cohomology group, tangent space, etc.). As a consequence, many mathematicians other than specialists in the field—or even those who think they might want to be—come in contact with the subject in various ways. It is for such people that this course is designed. To put it another way, we intend this course for beginners and nonspecialists. The present courses program focuses exactly on the simplest cases: representations of finite groups and, if time permits, the basic examples of actions of Lie groups on finitedimensional real and complex vector spaces. Depending on the audience, the classification of modules over a PID, may also be covered. Vorraussetzungen: Linear Algebra I & II, Algebra Dozierende: Alberto Navarro Garmendia 26 MATH BULLETIN FS 2018
Forschungsseminare. Auf der Webseite unseres Instituts math.uzh.ch findet ihr unter Veranstaltungen Kolloquia/Seminare eine Liste von Forschungsseminaren. Ein Forschungsseminar besteht aus nicht zusammenhängenden Vorträgen in denen Doktoranden und Professoren unter anderem ihre aktuellen Forschungsprojekte vorstellen. Dies könnte vor allem für Studenten interessant sein, die sich überlegen, in welchem Bereich oder bei welchem Professor sie ihre Masterarbeit schreiben wollen. Wenn ihr in der Liste auf eines der Seminare klickt, könnt ihr für das jeweilige Seminar einen Newsletter abonnieren, der euch vor jedem Vortrag kurz über den Titel des Vortrags informiert. Achtung: Es findet nicht jede Woche ein Vortrag statt. Unten seht ihr die Liste von Forschungsseminaren für das Herbstsemester: Talks in mathematical physics: Do 15.0017.00 ETH HG G 43 Arbeitsgemeinschaft in Codierungstheorie und Kryptographie: Mi 15.0017.00 Y27 H28 Discrete mathematics: Di 11.1512.00 Y27 H46 Analysis Seminar: Di 15.1517.00 ETH PDE and Mathematical Physics: Do 17.0019.30 Y27 H35/36 Oberseminar: Algebraische Geometerie: Mo 13.0014.45 Y27 H25 GeometrieSeminar: Mi 15.4518.00 ETH Zurich Colloquium in Applied and Computational Mathematics: Mi 16.1517.15 Y27 H25 Stochastische Prozesse: Mi 17.1519.00 Y27 H25 Research Seminar on Statistics: Fr 15.1516.15 ETH Research Seminar in Applied Statistics: Do 16.1518.00 Y27 H46 Epidemiology and Biostatistics Methods Seminar: Mi 14.0016.00 HRS F05 27 MATH BULLETIN FS 2018
Vorlesungsverzeichnis ETH.
401391301L Mathematical Foundations for Finance 4 KP E. W. Farkas, M. Schweizer
401233300L Methoden der mathematischen Physik I 6 KP H. Knörrer
402220301L Allgemeine Mechanik 7 KP N. Beisert
252085100L Algorithmen und Komplexität Information 4 KP A. Steger
401200300L Algebra I 7 KP E. Kowalski
401353100L Differential Geometry I 10 KP Salamon
401346100L Functional Analysis I 10 KP A. Carlotto
401300161L Algebraic Topology I 8 KP W. Merry
401313200L Commutative Algebra Information 10 KP P. D. Nelson
401358167L Symplectic Geometry 8 KP A. Cannas da Silva
401360100L Probability Theory 10 KP A.S. Sznitman
401362100L Fundamentals of Mathematical Statistics 10 KP S. van de Geer
401390100L Mathematical Optimization Information 11 KP R. Weismantel
252020900L Algorithms, Probability, and Computing Information 8 KP E. Welzl,
M. Ghaffari, A. Steger, P. Widmayer
402020500L Quantum Mechanics I Information 10 KP C. Anastasiou
401305900L Kombinatorik II 4 KP N. Hungerbühler
401303400L Axiomatische Mengenlehre 8 KP L. Halbeisen
401311867L Classical Modular Forms 8 KP I. N. Petrow
401320367L Small Cancellation Theory 4 KP D. Gruber
401317767L Introduction to Vertex Operator Algebras 4 KP C. A. Keller
401062501L Applied Analysis of Variance and Experimental Design Information 5 KP
L. Meier
401064900L Applied Statistical Regression Information 5 KP M. Dettling
401362814L Bayesian Statistics 4 KP F. Sigrist
401463767L On Hypothesis Testing 4 KP F. Balabdaoui
401493567L Mean Field Games 4 KP M. Burzoni
401383365L Chaotically Singular Spacetimes 6 KP H. Knörrer, M. Reiterer,
E. Trubowitz
402083000L General Relativity Information 10 KP G. M. Graf
401305564L Algebraic Methods in Combinatorics 6 KP B. Sudakov
401390100L Mathematical Optimization Information 11 KP R. Weismantel
MATH BULLETIN FS 2018 28Impressum. Herausgeber Fachverein Mathematik der Universität Zürich Universität Irchel, Raum Y27K37 Winterthurerstrasse 190 8057 Zürich bulletin@math.uzh.ch http://fvm.math.uzh.ch/ Redaktion und Layout Sarina Sutter Stefan Kurz Druck Druckzentrum ETH Zentrum Rämistrasse 101 8092 Zürich Inserate ¼ Seite, ½ Seite, 1 Seite Preis auf Anfrage Bilder Cover: http://www.natuerlichonline.ch/magazin/artikel/derfibonaccicode/ Dank Wir danken folgenden Personen herzlich für ihre Beiträge im Bulletin FS 2018: Matthew Scroggs, Alberto Navarro Garmendia, Gian Deflorin. 29 MATH BULLETIN FS 2018
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