L6

Leandro Sánchez Betancourt
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The Cost of Latency: Improving Fill Ratios in Foreign Exchange Markets

Latency is the time delay between an exchange streaming market data to a trader, the trader processing information and deciding to trade, and the exchange receiving the order from the trader.  Liquidity takers  face  a  moving target problem as a consequence of their latency in the marketplace -- they send marketable orders that aim at a price and quantity they observed in the LOB, but by the time their order was processed by the Exchange, prices (and/or quantities) may have worsened, so the  order  cannot  be  filled. If liquidity taking orders can walk the limit order book (LOB), then orders that arrive late may still be filled at worse prices. In this paper we show how to optimally choose the discretion of liquidity taking orders to walk the LOB. The optimal strategy balances the tradeoff between the costs of walking the LOB and targeting  a desired percentage of filled orders over a period of time.  We employ a proprietary data set of foreign exchange trades to analyze the performance of the strategy. Finally, we show the relationship between latency and the percentage of filled orders, and showcase the optimal strategy as an alternative investment to reduce latency.

Jasdeep Kalsi
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An SPDE model for the Limit Order Book

I will introduce a microscopic model for the Limit Order Book in a static setting i.e. in between price movements. Here, order flow at different price levels is given by Poisson processes which depend on the relative price and the depth of the book. I will discuss how reflected SPDEs can be obtained as scaling limits of such models. This motivates an SPDE with reflection and a moving boundary as a model for the dynamic Order Book. An outline for how to prove existence and uniqueness for the equation will be presented, as well as some simple simulations of the model.

We generalise the existence of combinatorial designs to the setting of subset sums in lattices with coordinates indexed by labelled faces of simplicial complexes. This general framework includes the problem of decomposing hypergraphs with extra edge data, such as colours and orders, and so incorporates a wide range of variations on the basic design problem, notably Baranyai-type generalisations, such as resolvable hypergraph designs, large sets of hypergraph designs and decompositions of designs by designs. Our method also gives approximate counting results, which is new for many structures whose existence was previously known, such as high dimensional permutations or Sudoku squares.

Breakthroughs in machine learning have led to impressive results in numerous fields in recent years. I will review some of the best-known results on the computer science side, provide simple ways to think about the associated techniques, discuss possible applications in string theory, and present some applications in string theory where they already exist. One promising direction is using machine learning to generate conjectures that are then proven by humans as theorems. This method, sometimes referred to as intelligible AI, will be exemplified in an enormous ensemble of F-theory geometries that will be featured throughout the talk.

My talk is based on joint work with Claire Debord (Univ. Auvergne).
We will explain why Lie groupoids are very naturally linked to Atiyah-Singer index theory.
In our approach -originating from ideas of Connes, various examples of Lie groupoids
- allow to generalize index problems,
- can be used to construct the index of pseudodifferential operators without using the pseudodifferential calculus,
- give rise to proofs of index theorems, 
- can be used to construct the pseudodifferential calculus.

We give an alternative, statistical physics based proof of the Ω(d2^{-d}) lower bound for the maximum sphere packing density in dimension d by showing that a random configuration from the hard sphere model has this density in expectation. While the leading constant we achieve is not the best known, we do obtain additional geometric information: we prove a lower bound on the entropy density of sphere packings at this density, a measure of how plentiful such packings are. This is joint work with Felix Joos and Will Perkins.

The construction of permutation functions of a finite field is a task of great interest in cryptography and coding theory. In this talk we describe a method which combines Chebotarev density theorem with elementary group theory to produce permutation rational functions over a finite field F_q. Our method is entirely constructive and as a corollary we get the classification of permutation polynomials up to degree 4 over any finite field of odd characteristic.

This is a joint work with Andrea Ferraguti.
 

After reviewing old work with Teitelbaum, in which we constructed the character variety X of the additive group o_L in a finite extension L/Q_p and established the Fourier isomorphism for the distribution algebra of o_L, I will briefly report on more recent work with Berger and Xie, in which we establish the theory of (\varphi_L,\Gamma_L)-modules over X and relate it to Galois representations. Then I will discuss an ongoing project with Venjakob. Our goal is to use this theory over X for Iwasawa theory.

An interval exchange transformation is a map  of an 
interval to 
itself that rearranges a finite number of intervals by translations.  They 
appear among other places in the 
subject of rational billiards and flows of translation surfaces. An 
interesting phenomenon is that an IET may have dense orbits that are not 
uniformly distributed, a property known as non unique ergodicity.  I will 
talk about this phenomenon and present some new results about how common 
this is. Joint work with Jon Chaika.

I will talk about some recent work with Chantal David and Matilde Lalin about the mean value of L-functions associated to cubic characters over F_q[t] when q=1 (mod 3). I will explain how to obtain an asymptotic formula with a (maybe a little surprising) main term, which relies on using results from the theory of metaplectic Eisenstein series about cancellation in averages of cubic Gauss sums over functions fields.