**18 January 2016**

16:00 (Room L3)

**Title:**4th Moment of quadratic Dirichlet $L$-functions in function fields.

**by:**Alexandra Florea (Stanford University)

**Abstract**

We discuss moments of $L$-functions in function fields, in the hyperelliptic ensemble, focusing on the fourth moment of quadratic Dirichlet $L$-functions at the critical point. We explain how to obtain an asymptotic formula with some of the secondary main terms.

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**03 December 2015**

16:00 (Room L5)

**Title:**Galois Theory of Periods and Applications.

**by:**Francis Brown (University of Oxford)

**Abstract**

A period is a certain type of number obtained by integrating algebraic differential forms over algebraic domains. Examples include $\pi$, algebraic numbers, values of the Riemann zeta function at integers, and other classical constants. Difficult transcendence conjectures due to Grothendieck suggest that there should be a Galois theory of periods. I will explain these notions in very introductory terms and show how to set up such a Galois theory in certain situations.

I will then discuss some applications, in particular to Kim's method for bounding $S$-integral solutions to the equation $u+v=1$, and possibly to high-energy physics.

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**26 November 2015**

16:00 (Room L5)

**Title:**On the Central Limit Theorem for the number of steps in the Euclidean algorithm.

**by:**Ian Morris (University of Surrey)

**Abstract**

The number of steps required by the Euclidean algorithm to find the greatest common divisor of a pair of integers $u,v$ with $1<u<v<n$ has been investigated since at least the 16th century, with an asymptotic for the mean number of steps being found independently by H. Heilbronn and J.D. Dixon in around 1970. It was subsequently shown by D. Hensley in 1994 that the number of steps asymptotically follows a normal distribution about this mean. Existing proofs of this fact rely on extensive effective estimates on the Gauss-Kuzman-Wirsing operator which run to many dozens of pages. I will describe how this central limit theorem can be obtained instead by a much shorter Tauberian argument. If time permits, I will discuss some related work on the number of steps for the binary Euclidean algorithm.

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**19 November 2015**

16:00 (Room L5)

**Title:**Prime number races with very many competitors.

**by:**Adam Harper (University of Cambridge)

**Abstract**

The prime number race is the competition between different coprime residue classes mod $q$ to contain the most primes, up to a point $x$ . Rubinstein and Sarnak showed, assuming two $L$-function conjectures, that as $x$ varies the problem is equivalent to a problem about orderings of certain random variables, having weak correlations coming from number theory. In particular, as $q\rightarrow\infty$ the number of primes in any fixed set of $r$ coprime classes will achieve any given ordering for $\sim1/r!$ values of $x$. In this talk I will try to explain what happens whenr is allowed to grow as a function of $q$. It turns out that one still sees uniformity of orderings in many situations, but not always. The proofs involve various probabilistic ideas, and also some harmonic analysis related to the circle method. This is joint work with Youness Lamzouri.

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**12 November 2015**

16:00 (Room L5)

**Title:**Iwasawa theory for the symmetric square of a modular form. (Cancelled)

**by:**Sarah Zerbes (University College London)

**Abstract**

I will discuss some new results on the Iwasawa theory for the $3$-dimensional symmetric square Galois representation of a modular form, using the Euler system of Beilinson-Flach elements I constructed in joint work with Kings, Lei and Loeffler.

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**5 November 2015**

16:00 (Room L5)

**Title:**Around the Möbius function.

**by:**Kaisa Matomäki (University of Turku)

**Abstract**

The Möbius function plays a central role in number theory; both the prime number theorem and the Riemann Hypothesis are naturally formulated in terms of the amount of cancellations one gets when summing the Möbius function. In a recent joint work with Maksym Radziwill we have shown that the sum of the Möbius function exhibits cancellation in "almost all intervals" of arbitrarily slowly increasing length. This goes beyond what was previously known conditionally on the Riemann Hypothesis. Our result holds in fact in much greater generality, and has several further applications, some of which I will discuss in the talk. For instance the general result implies that between a fixed number of consecutive squares there is always an integer composed of only "small" prime factors. This settles a conjecture on "smooth" or "friable" numbers and is related to the running time of Lenstra's factoring algorithm.

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**29 October 2015**

16:00 (Room L5)

**Title:**Arthur's multiplicity formula for automorphic representations of certain inner forms of special orthogonal and symplectic groups.

**by:**Olivier Taibi (Imperial College)

**Abstract**

I will explain the formulation and proof of Arthur's multiplicity formula for automorphic representations of special orthogonal groups and certain inner forms of symplectic groups $G$ over a number field $F$. I work under an assumption that substantially simplifies the use of the stabilisation of the trace formula, namely that there exists a non-empty set $S$ of real places of $F$ such that $G$ has discrete series at places in $S$ and is quasi-split at places outside $S$, and restricting to automorphic representations of $G(A_{F})$ which have algebraic regular infinitesimal character at the places in $S$. In particular, this proves the general multiplicity formula for groups $G$ such that $F$ is totally real, $G$ is compact at all real places of $F$ and quasi-split at all finite places of $F$. Crucially, the formulation of Arthur's multiplicity formula is made possible by Kaletha's recent work on local and global Galois gerbes and their application to the normalisation of Kottwitz-Langlands-Shelstad transfer factors.

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**22 October 2015**

16:00 (Room L5)

**Title:**Linear Algebra with Errors, Coding Theory, Cryptography and Fourier Analysis on Finite Groups.

**by:**Steven Galbraith (University of Auckland)

**Abstract**

Solving systems of linear equations $Ax=b$ is easy, but how can we solve such a system when given a "noisy" version of $b$? Over the reals one can use the least squares method, but the problem is harder when working over a finite field. Recently this subject has become very important in cryptography, due to the introduction of new cryptosystems with interesting properties.

The talk will survey work in this area. I will discuss connections with coding theory and cryptography. I will also explain how Fourier analysis in finite groups can be used to solve variants of this problem, and will briefly describe some other applications of Fourier analysis in cryptography. The talk will be accessible to a general mathematical audience.

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**15 October 2015**

16:00 (Room L5)

**Title:**Sum of Seven Cubes.

**by:**Samir Siksek (University of Warwick)

**Abstract**

In 1851, Carl Jacobi made the experimental observation that all integers are sums of seven non-negative cubes, with precisely 17 exceptions, the largest of which is 454. Building on previous work by Maillet, Landau, Dickson, Linnik, Watson, Bombieri, Ramaré, Elkies and many others, we complete the proof of Jacobi's observation.

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**11 June 2015**

16:00 (Room L5)

**Title:**Moduli stacks of potentially Barsotti-Tate Galois representations.

**by:**Toby Gee (Imperial College)

**Abstract**

I will discuss joint work with Ana Caraiani, Matthew Emerton and David Savitt, in which we construct moduli stacks of two-dimensional potentially Barsotti-Tate Galois representations, and study the relationship of their geometry to the weight part of Serre's conjecture.

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**04 June 2015**

16:00 (Room L5)

**Title:**Bounded Intervals containing many primes.

**by:**Roger Baker (Brigham Young University)

**Abstract**

I describe joint work with Alastair Irving in which we improve a result of D.H.J. Polymath on the length of intervals in $[N,2N]$ that can be shown to contain m primes. Here $m$ should be thought of as large but fixed, while $N$ tends to infinity. The Harman sieve is the key to the improvement. The preprint will be available on the Math ArXiv before the date of the talk.

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**28 May 2015**

16:00 (Room L5)

**Title:**Cubic hypersurfaces over global fields.

**by:**Pankaj Vishe (University of York)

**Abstract**

Let $X$ be a smooth cubic hypersurface of dimension $m$ defined over a global field $K$. A conjecture of Colliot-Thelene(02) states that $X$ satisfies the Hasse Principle and Weak approximation as long as $m\geq3$. We use a global version of Hardy-Littlewood circle method along with the theory of global $L$-functions to establish this for $m\geq6$, in the case $K=\mathbb{F}_{q}(T)$, where $\text{char}(\mathbb{F}_{q})>3$.

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**21 May 2015**

16:00 (Room L5)

**Title:**Anabelian Geometry with étale homotopy types.

**by:**Jakob Stix (Goethe Universität - Frankfurt)

**Abstract**

Classical anabelian geometry shows that for hyperbolic curves the etale fundamental group encodes the curve provided the base field is sufficiently arithmetic. In higher dimensions it is natural to replace the etale fundamental group by the etale homotopy type. We will report on progress obtained in this direction in a recent joint work with Alexander Schmidt.

**Joint seminar with Logic.

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**14 May 2015**

16:00 (Room L6)

**Title:**Equidistribution of Eisenstein series.

**by:**Matthew Young (Texas A&M University)

**Abstract**

I will discuss some recent results on the distribution of the real-analytic Eisenstein series on thin sets, such as a geodesic segment. These investigations are related to mean values of the Riemann zeta function, and have connections to quantum chaos.

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**07 May 2015**

16:00 (Room L5)

**Title:**Heuristics for distributions of Arakelov class groups.

**by:**Alex Bartel (University of Warwick)

**Abstract**

The Cohen-Lenstra heuristics, postulated in the early 80s, conceptually explained numerous phenomena in the behaviour of ideal class groups of number fields that had puzzled mathematicians for decades, by proposing a probabilistic model: the probability that the class group of an imaginary quadratic field is isomorphic to a given group $A$ is inverse proportional to $\#\text{Aut}(A)$. This is a very natural model for random algebraic objects. But the probability weights for more general number fields, while agreeing well with experiments, look rather mysterious. I will explain how to recover the original heuristic in a very conceptual way by phrasing it in terms of Arakelov class groups instead. The main difficulty that one needs to overcome is that Arakelov class groups typically have infinitely many automorphisms. We build up a theory of commensurability of modules, of groups, and of rings, in order to remove this obstacle. This is joint work with Hendrik Lenstra.

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**30 April 2015**

16:00 (Room L6)

**Title:**Quadratic Weyl Sums, Automorphic Functions, and Invariance Principles.

**by:**Jens Marklof (University of Bristol)

**Abstract**

Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. In the present study we construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, i.e., there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation here is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion, non-differentiability), although time increments are not independent and the value distribution at each fixed time is distinctly different from a normal distribution. Joint work with Francesco Cellarosi.

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**12 March 2015**

16:00 (Room L5)

**Title:**Arithmetic Statistics in Function Fields.

**by:**Jon Keating (University of Bristol)

**Abstract**

I will review some classical problems in number theory concerning the statistical distribution of the primes, square-free numbers and values of the divisor function; for example, fluctuations in the number of primes in short intervals and in arithmetic progressions. I will then explain how analogues of these problems in the function field setting can be resolved by expressing them in terms of matrix integrals.

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**05 March 2015**

16:00 (Room L2)

**Title:**Some density results in number theory.

**by:**John Cremona (University of Warwick)

**Abstract**

I will describe joint work with Manjul Bhargava (Princeton) and Tom Fisher (Cambridge) in which we determine the probability that random equation from certain families has a solution either locally (over the reals or the $p$-adics), everywhere locally, or globally. Three kinds of equation will be considered: quadratics in any number of variables, ternary cubics and hyperelliptic quartics.

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**26 February 2015**

16:00 (Room L5)

**Title:**Restriction of Banach representations of $GL_{2}(\mathbb{Q}_{p})$ to $GL_{2}(\mathbb{Z}_{p})$.

**by:**Gabriel Dospinescu (ENS Lyon)

**Abstract**

Thanks to the $p$-adic local Langlands correspondence for $GL_{2}(\mathbb{Q}_{p})$, one "knows" all admissible unitary topologically irreducible representations of $GL_{2}(\mathbb{Z}_{p})$. In this talk I will focus on some elementary properties of their restriction to $GL_{2}(\mathbb{Z}_{p})$: for instance, to what extent does the restriction to $GL_{2}(\mathbb{Z}_{p})$ allow one to recover the original representation, when is the restriction of finite length, etc.