Much of the arithmetic behaviour of an elliptic curve can be understood by examining its mod p reduction at some prime p. In this talk, we will aim to explain some of the ways we can define the mod p reduction, and the classifications of which reduction types occur.

Topics to be covered include the classical reduction types (good/multiplicative/additive), the Kodaira-Neron reduction types that refine them, and the Raynaud parametrisation of a semistable abelian variety. Time permitting, we may also discuss joint work with Vladimir Dokchitser classifying the semistable reduction types of 2-dimensional abelian varieties.

I will discuss the many appearances of the class of IP sets in classical theorems of combinatorial number theory and ergodic theory. Our point of departure will be the celebrated theorem of Hindman on partition regularity of IP sets, which is crucial for the introduction of IP-limits. We then discuss how existence of certain IP-limits translates into recurrence statements, which in turn give rise to results in number theory via the Furstenberg correspondence principle. Throughout the talk, the methods of ergodic theory will play an important role - however, no prior familiarity with them is required.

Starting from Hilbert's 10th problem, I will explain how to characterise the set of integers by non-solubility of a set of polynomial equations and discuss related challenges. The methods needed are almost entirely elementary; ingredients from algebraic number theory will be explained as we go along. No knowledge of first-order logic is necessary.

I will give a historical overview of some of the theorems proved by the
Ancient Greeks, which are now taken for granted but were, and are,
landmarks in the history of mathematics. Particular attention will be
given to the calculation of areas, including theorems of Hippocrates,
Euclid and Archimedes.

The moduli space of tropical curves (and its variants) is one of the most-studied objects in tropical geometry. So far this moduli space has only been considered as an essentially set-theoretic coarse moduli space (sometimes with additional structure). As a consequence of this restriction, the tropical forgetful map does not define a universal curve
(at least in the positive genus case). The classical work of Knudsen has resolved a similar issue for the algebraic moduli space of curves by considering the fine moduli stacks instead of the coarse moduli spaces. In this talk I am going to give an introduction to these fascinating tropical moduli spaces and report on ongoing work with R. Cavalieri, M. Chan, and J. Wise, where we propose the notion of a moduli stack of tropical curves as a geometric stack over the category of rational polyhedral cones. Using this framework one can give a natural interpretation of the forgetful morphism as a universal curve. The coarse moduli space arises as the set of $\mathbb{R}_{\geq 0}$-valued points of the moduli stack. Given time, I will also explain how the process of tropicalization for these moduli stacks can be phrased in a more fundamental way using the language of logarithmic algebraic stacks.
 

It has been observed long time ago (by many people) that singular affine symplectic varieties come in pairs; that is often to an affine singular symplectic variety $X$ one can associate a dual variety $X^!$; the geometries of $X$ and $X^!$ (and their quantizations) are related in a non-trivial way. The purpose of the talk will be 3-fold:

1) Explain a set of conjectures of Braden, Licata, Proudfoot and Webster which provide an exact formulation of the relationship between $X$ and $X^!$

2) Present a list of examples of symplectically dual pairs (some of them are very recent); in particular, we shall explain how the symplectic duals to Nakajima quiver varieties look like.

3) Give a new approach to the construction of $X^!$ and a proof of the conjectures from part 1).

The talk is based on a work in progress with Finkelberg and Nakajima.

Let F be a global field and let A denote its adele ring. The usual Tamagawa number formula computes the (suitably normalized) volume of the quotient G(A)/G(F) in terms of values of the zeta-function of F at the exponents of G; here G is simply connected semi-simple group. When F is functional field, this computation is closely related to the Atiyah-Bott computation of the cohomology of the moduli space of G-bundles on a smooth projective curve.

I am going to present a (somewhat indirect) generalization of the Tamagawa formula to the case when G is an affine Kac-Moody group and F is a functional fiend. Surprisingly, the proof heavily uses the so called Macdonald constant term identity. We are going to discuss possible (conjectural) geometric interpretations of this formula (related to moduli spaces of bundles on surfaces).

This is joint work with D.Kazhdan.

The question that the Crepant Resolution Conjecture (CRC) wants to address is: given an orbifold X that admits a repant resolution Y, can we systematically compare the Gromov-Witten theories of the two spaces? That this should happen was first observed by physicists and the question was imported into mathematics by Y.Ruan, who posited it as the search for an isomorphism in the quantum cohomologies of the two spaces. In the last fifteen years this question has evolved and found different formulations which various degree of generality and validity. Perhaps the most powerful approach to the CRC is through Givental's formalism. In this case, Coates, Corti, Iritani and Tseng propose that the CRC should consist of the natural comparison of geometric objects constructed from the GW potential fo the space. We explore this approach in the setting of open GW invariants. We formulate an open version of the CRC using this formalism, and make some verifications. Our approach is well tuned with Iritani's approach to the CRC via integral structures, and it seems to suggest that open invariants should play a prominent role in mirror symmetry. 

I will discuss a notoriously hard problem in group theory known as the flat closing conjecture. This states that a group with a finite classifying space is either hyperbolic or contains a Baumslag-Solitar Subgroup. I will give some strategies to try and create a counterexample to this conjecture. 

Due to its use in cryptographic protocols such as the Diffie--Hellman

key exchange, the discrete logarithm problem attracted a considerable

amount of attention in the past 40 years. In this talk, we summarize

the key technical ideas and their evolution for the case of discrete

logarithms in small characteristic finite fields. This road leads from

the original belief that this problem was hard enough for

cryptographic purpose to the current state of the art where the

algorithms are so efficient and practical that the problem can no

longer be considered for cryptographic use.