## Arf Lectures

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Arf Lectures have been supported by Fame-Crypt, İstanbul Center for Mathematical Sciences, Mathematics Foundation of Turkey, TÜBİTAK and Middle East Technical University.

### Advisory Board Members

### Organising Committee Members

### David Harvey (May 2, 2024, 15:40, Arf Auditorium) Poster

**Title: **Integer multiplication and its applications

**Abstract:** Integer multiplication is an ancient problem, with countless applications across all fields of human endeavour. Until the late 1950s, the fastest known multiplication algorithm, asymptotically speaking, was the long multiplication method that we learn at primary school. This method requires O(n²) basic operations to calculate the product of two n-digit numbers. The current fastest algorithm, discovered by Joris van der Hoeven and myself a few years ago, calculates the same product in only O(n log n) operations.

In this talk, I will briefly discuss the history of fast multiplication algorithms, and then turn to some applications, drawn from fields such as number theory and cryptography. In particular, I will discuss a few examples of problems whose resolution depends crucially on the ability to quickly multiply enormous integers, with millions or even billions of digits.

### Andrew Sutherland (October 13, 2022, 15:40, Arf Auditorium)

**Title:** Diophantine computations

**Abstract: **Diophantine problems, in which one seeks integer solutions to a polynomial equation, include some of the oldest problems in mathematics. There is a notable disparity between the difficulty of stating a Diophantine problem and that of solving it. This feature was formalized in the 20th century by the negative answer to Hilbert’s tenth problem: It is impossible to determine whether some Diophantine equations have solutions or not. One need not look very far to find examples whose status is unknown. A striking example noted by Mordell in 1953 involves the Diophantine equation x^3 + y^3 + z^3 = 3. In this talk I will discuss the resolution of Mordell's question and some related problems. I will also offer some thoughts on how we may approach questions that might be undecidable, and the role that computation can and cannot play.

### Geordie Williamson (2019)

**Title: **Representation theory and geometry

**Abstract: **One of the most fundamental questions in representation theory asks for a description of the simple representations. I will give an introduction to this problem with an emphasis on the representation theory of Lie groups and algebras. A recurring theme throughout the history of the subject is the appearance of geometric techniques in seemingly algebraic problems.

### Fernando Rodriguez Villegas (2018)

**Title:** Combinatorics and Geometry

**Abstract:** Can you tell a ball from a doughnut by counting? A. Weil discovered in the 1940's that the geometry of a complex algebraic variety is reflected in a precise way on its versions over finite fields. In this talk we will discuss the interaction between complex geometry and finite field geometry. This interaction between the continuous and the discrete has been a very fruitful two-way street that allows the transfer of important results from one context to the other.

### Vladimir Voevodsky (2015)

**Title:** Univalent Foundations of Mathematics

**Abstract:** Today's mathematics is using foundations which have been developed in the late 19th - early 20th century. Since these foundations had been completed mathematics has grown from a field advanced by a few outstanding minds to a large enterprise involving tens of thousands of mathematicians. On this new scale of existence it is becoming impossible for mathematics to rely entirely on the old peer-review model of ensuring the correctness of the results. Univalent Foundations are new foundations of mathematics invented with the intention of being usable for the development of computer proof assistants that will facilitate the crafting and verification of complex mathematical constructions.

### Persi Diaconis (2013-2014)

**Title:** The Mathematics of Coincidences

**Abstract:** Amazing coincidences are all around us, where we work and live and what we do often seem determined by seeming coincidences. I will review the history of ways of thinking about such things (work of Jung and Freud) and then show how a bit of mathematics can suggest that things are not so surprising after all. The math involves graph theory and probability in surprising new ways.

### David E. Nadler (2012)

**Title:** Traces and loops

**Abstract:** This talk will focus on the interplay between two basic notions originating in algebra and topology. In algebra, there is the trace of a matrix, important for its equality with the sum of the eigenvalues of the matrix. In topology, there are the loops on a space, which play a central role in the computation of homotopy groups and in the structure theory of spaces. There is a well-developed understanding of the intimate relation between traces and loops coming from non-commutative geometry and mathematical physics. We will explain how modern formulations elucidate fundamental identities in geometry and representation theory.

### Jonathan Pila (2011)

**Title:** Diophantine Geometry via O-minimality

**Abstract:** The talk will be about an application of mathematical logic to certain Diophantine problems. Diophantine geometry traditionally considers rational points on algebraic varieties. I will describe some results about the distribution of rational points on certain non-algebraic sets in real space which were developed in rough analogy with ideas in Diophantine geometry. The starting point was a result, obtained jointly with Bombieri, that rational points on the graph of a real-analytic but non-algebraic functions are "sparse" in a suitable sense. This may be generalised to a result, obtained with Wilkie, applicable to sets that are "definable in an o-minimal structure over the real field". I will define this notion, which comes from model theory, and explain why this is the natural setting. The result may be applied to certain Diophantine problems, using a strategy proposed by Zannier in the context of the Manin-Mumford conjecture. I will describe this and some further applications of this strategy to cases of the Andre-Oort and Zilber-Pink conjectures.

### John W. Morgan (2010)

**Title: ** The Topology of 3-Dimensional Manifolds

**Abstract:** Poincaré launched the subject of 3-dimensional topology in 1904. At the end of a long treatise on 3-manifolds he asked what became known as the Poincaré Conjecture: Is every simply connected 3-manifold homeomorphic to the 3-sphere. This problem sparked a century of work on manifolds of dimensions 3 and higher, work that made topology one of the most dynamic and exciting areas of mathematics during the 20th century. But in spite of all this work, at the end of the 20th century the Poincaré Conjecture still stood unresolved. Then in 2002 and 2003, Grigory Perelman put a series of 3 preprints on the archive that completely resolved this conjecture and the more general conjecture, due to Thurston, about the structure of all 3-manifolds. His approach was to use work of Richard Hamilton concerning what is called the Ricci flow. This is a parabolic evolution equation for a Riemannian metric on a manifold. In this talk we will review the motivating questions and the Ricci flow. After giving this background we will then sketch Perelman’s method of solution.

### Ben Joseph Green (2009)

**Title: ** Patterns of Primes

### Günter Harder (2008)

**Title:** Cohomology of Arithmetic Groups and Applications to Arithmetic

**Abstract:** This is an exposition of the basic notions and concepts which are needed to build up the cohomology theory of arithmetic groups. We are dealing with objects which have a certain degree of complexity and in this text I give some explanation of the background needed to understand the the definitions and the general structural elements of these objects.

### Hendrik Lenstra (2007)

**Title:** Escher and Droste effect

**Abstract:** In 1956, the Dutch graphic artist M.C. Escher made an unusual lithograph with the title "Print Gallery". It shows a young man viewing a print in an exhibition gallery. Amongst the buildings depicted on the print, he sees a paradoxically the very same gallery that he is standing in. A lot is known about the way in which Escher made his lithograph. It is nor nearly as well known that it contains a hidden "Droste effect", or infinite repetition; but this is brought to light by a mathematical analysis of the studies used by Escher. On the basis of this discovery, a team of mathematicians at Leiden produced a series of hallucinating computer animations. These show, among others, what happens inside the mysterious spot in the middle of the lithograph that Escher left blank.

### Jean-Pierre Serre (2006)

**Title:** Variation with p of the number of solutions mod p of a family of polynomial equations

### Peter Sarnak (2005)

**Title:** Hilbert's Eleventh Problem

**Abstract:** Hilbert's 11-th problem asks about the representation of integers (respectively rationals) in a number field by integral (respectively rational) quadratic form. The rational case was resolved a long time ago, being the well known Hasse Minkowski theorem. The integral case is apparently much more difficult and was only resolved recently. Its solution depends on works of many people. We will describe the problem and outline its solution.

### Robert P. Langlands (2004)

**Title: **Conformal Field Theory and the Mathematician & Descartes and Fermat

### David Mumford (2003)

**Title: **Variational Problems Arising from Computer Vision for Objects and Their Shape

### Don B. Zagier (2002)

**Title:** Taylor Coefficients of Modular Forms

**Abstract:** Modular forms are a special class of holomorphic functions with an infinite group of symmetries and with many important arithmetic properties. They play a crucial role in much of modern number Theory, the most spectacular example being in the proof of Fermat's Last Theorem by Andrew Wiles a few years ago. Usually one derives arithmetic information from modular forms by looking at their Fourier expansions or "Taylor coefficients at infinity", but it turns out that their Taylor expansions at suitably chosen finite points also have beautiful arithmetic properties and many applications (in particular to Diophantine equations such as the question: "which prime numbers can be written as the sum of two perfect cubes?"). Moreover, these Taylor coefficients, unlike the more familiar Fourier coefficients, can be computed by means of a quite elementary algorithm. The lecture will treat several aspects of the theory, examples, and applications.

### Gerhard Frey (2001)

**Title:** Brauer Groups and Data Security

**Abstract:** We will deal with public key cryptosystems based on discrete logarithms in elliptic curves and multiplicative groups of finite fields. It will be shown that these problems lead to 'explicit' class field theory as well as local and global computations in Brauer groups of number fields.