Events

The Littlewood Conjecture states that for all pairs of real numbers \((\alpha, \beta)\) the product \(\mid q\mid \mid q\alpha+p_1\mid\mid q\beta+p_2\mid\) becomes arbitrarily close to \(0\) when the vector \((q, p_1, p_2)\) ranges in \( \mathbb{Z}^3 \) and \( q \neq 0\). To date, despite much progress, it is not known whether this statement is true. In this talk, I will discuss a partial converse of the Littlewood conjecture, where the factor \(\mid q\mid\) is replaced by an increasing function \(f(\mid q\mid).\) More specifically, following up on the work of Badziahin and Velani, I will be interested in determining functions \(f\) for which the above product and its higher dimensional generalisations stay bounded away from \(0\) for at least one pair \((\alpha, \beta) \in\, \mathbb{R}^2.\) This problem happens to be intimately connected with the equidistribution rate of certain segments on the expanding torus in \( SL_3(\mathbb{R})/SL_3(\mathbb{Z})\) under the action of the full diagonal group.

Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces X and Y. We study the problem of determining the degree of approximation of a such operators on a compact subset KX X using a finite amount of information. If F : KX ! KY, a well established strategy to approximate F(F) for some F 2 KX is to encode F (respectively, F(F)) in terms of a  finite number d (respectively m) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of m functions on a compact subset of a high dimensional Euclidean space Rd, equivalently, the unit sphere Sd embedded in Rd+1. The problem is challenging because d, m, as well as the complexity of the approximation on Sd are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; i.e., with the constants involved in the estimates on the approximation on Sd being O(d1=6). We study different smoothness classes for the operators, and also propose a method for approximation of F(F) using only information in a small neighborhood of F, resulting in an effective reduction in the number of parameters involved.To further mitigate the problem of large number of parameters, we propose prefabricated networks, resulting in a substantially smaller number of effective parameters. The problem is studied in both deterministic and probabilistic settings.

We discuss exotic Lagrangian tori in dimension greater than or equal to six. First, we give another approach to Auroux's result that there are infinitely many tori in $\mathbb{R}^6$ which are distinct up to symplectomorphisms of the ambient space. The exotic tori we construct naturally appear in a two-parameter family, some of which are not monotone. Second, we show that our construction can be carried out purely locally in a Darboux chart and that (provided the ambient symplectic manifold is tame) the resulting tori are also distinct up to symplectomorphisms.

In enumerative geometry, Virasoro constraints first appeared in the context of moduli of stable curves and maps. These constraints provide a rich set of conjectural relations among Gromov-Witten descendent invariants. Recently, the analogous constraints were formulated in several sheaf theoretic contexts; stable pairs on 3-folds, Hilbert scheme of points on surfaces, higher rank sheaves on surfaces with only (p,p)-cohomology. In joint work with A. Bojko, M. Moreira, we extend and reinterpret Virasoro constraints in sheaf theory using Joyce's vertex algebra. This new interpretation yields a proof of Virasoro constraints for curves and surfaces with only (p,p) cohomology by means of wall-crossing formulas.

An introduction to Quantum Computation, the first Quantum Algorithm and how to use it to protect the integrity of Italian wine culture.

I will explain some new results showing that the Chow and cohomology rings of moduli spaces of stable curves in relatively low genus and low number of marked points are isomorphic and equal to the tautological ring. These computations involve both concrete geometric techniques in order to explicitly study various strata in the moduli spaces and more abstract techniques relating the computations in the Chow ring to those in cohomology. This part is joint work with Hannah Larson. Next, I will explain a surprising extended application of these results to the vanishing of the eleventh cohomology of moduli spaces of pointed stable curves of genus g at least 2. This part is joint work with Hannah Larson and Sam Payne.

Cup product in second (co)homology of a 4-manifold tells us a lot about its topology, even determining the manifold up to homeomorphism under certain assumptions. A geometrical question that asks what is the minimal genus of an embedded surface that represents a chosen class in the second homology, goes one step further, and gives essential information on the smooth structure. In joint work with Marco Marengon we present a simple but flexible method to remove multiple singularities of immersed surfaces in 4-manifolds. One concrete consequence is that 'many knots bound disks in 4-manifolds' - if we take a small ball in a 4-manifold, we can show that many knots on its boundary will bound disks in the interior of the manifold. Another consequence I will try to address is a work in progress regarding the above mentioned minimal genus problem in manifolds which are connected sums of complex projective planes.

We describe a fast method for solving elliptic PDEs with uncertain coefficients using kernel-based interpolation over a rank-1 lattice point set. By representing the input random field of the system using a model proposed by Kaarnioja, Kuo, and Sloan (2020), in which a countable number of independent random variables enter the random field as periodic functions, it is shown that the kernel interpolant can be constructed for the PDE solution (or some quantity of interest thereof) as a function of the stochastic variables in a highly efficient manner using fast Fourier transform. The method works well even when the stochastic dimension of the problem is large, and we obtain rigorous error bounds which are independent of the stochastic dimension of the problem. We also outline some techniques that can be used to further improve the approximation error. This talk is based on joint work with Yoshihito Kazashi, Frances Kuo, Fabio Nobile, and Ian Sloan.

In higher gauge theory, gerbes with connection are the higher structures categorifying the role of principal bundles with connection in classical gauge theory. Gerbes have been extensively studied in the past and the notion of a 2-connection on a non-abelian gerbe has been introduced via transport functors in the work of Schreiber-Waldorf. At the heart of this construction lies the familiar concept that a connection can be seen equivalently as a parallel transport system. The functorial nature of parallel transport suggests that non-abelian gerbes with connection can be realized as field theories from the geometric cobordism category introduced by Grady-Pavlov. We will focus on describing the construction of Grady-Pavlov’s geometric cobordism category, with an outlook on a possible construction of a field theory encoding parallel transport.

The automatic continuity problem asks the following question: Given two topological groups G and H and an algebraic homomorphism φ : G -> H, can we find conditions on G, H and φ ensuring that φ is continuous. The first result in this direction was proved by Dudley in the 1960s and says that any abstract homomorphism from a locally compact Hausdorff group to a free (abelian) group is continuous. In this talk I want to motivate related questions for groups originating from geometric group theory and talk about recent developments.

Algorithmic trading has become increasingly influential in financial industry. However, not all algorithmic trading is successful. We will analyze important ingredients of a typical algorithmic trading strategy: find a trading signal, backtest, money management, validation, and suspension. Risk management methods play important roles in algorithmic trading. In this introductory talk we will focus on the main ideas and limiting the technical details so that the talk is accessible to undergraduate students.

I will discuss joint work with Jonas Bergström and Sam Payne. Arbarello and Cornalba have shown that H^1, H^3, and H^5 vanish for the moduli spaces of stable n-pointed curves of genus g. We show that H^7 and H^9 vanish as well. It is known that H^{11} doesn't vanish in general, so the result is sharp. To obtain this result, we need the vanishing for genus 4 and n at most 3. We actually determine the full cohomology in these cases. The main method is approximate counts over finite fields. I will also discuss the connection with recent results of Canning, Larson, and Payne.

In randomized experiments, we randomly assign the treatment that each experimental subject receives. Randomization can help us accurately estimate the difference in treatment effects with high probability. It also helps ensure that the groups of subjects receiving each treatment are similar. If we have already measured characteristics of our subjects that we think could influence their response to treatment, then we can increase the precision of our estimates of treatment effects by balancing those characteristics between the groups. We show how to use the recently developed Gram-​Schmidt Walk algorithm of Bansal, Dadush, Garg, and Lovett to efficiently assign treatments to subjects in a way that balances known characteristics without sacrificing the benefits of randomization. These allow us to obtain more accurate estimates of treatment effects to the extent that the measured characteristics are predictive of treatment effects, while also bounding the worst-​case behavior when they are not. This is joint work with Chris Harshaw, Fredrik Sävje, and Peng Zhang.