In this talk, we shall introduce various identities among partitions of integers, and how these can be expressed via formal power series. In particular, we shall look at the Rogers Ramanujan identities of power series, and discuss possible combinatorial proofs using partitions and Durfree squares.

# Forthcoming Seminars

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Graph products are a class of groups that 'interpolate' between direct and free products, and generalise the notion of right-angled Artin groups. Given a property that free products (and maybe direct products) are known to satisfy, a natural question arises: do graph products satisfy this property? For instance, it is known that graph products act on tree-like spaces (quasi-trees) in a nice way (acylindrically), just like free products. In the talk we will discuss a construction of such an action and, if time permits, its relation to solving systems of equations over graph products.

In this talk I will present the recent developments in the topic of existence of solutions to the two-fluid systems. I will discuss the application of approach developed by P.-L. Lions and E. Feireisl and explain the limitations of this technique in the context of multi-component flow models. A particular example of such a model is two-fluids Stokes system with single velocity field and two densities, and with an algebraic pressure law closure. The existence result that uses the compactness criterion introduced for the Navier-Stokes system by D. Bresch and P.-E. Jabin will be presented. I will also mention an innovative construction of solutions relying on the G. Crippa and C. DeLellis stability estimates for the transport equation.

We will consider an extension of the Eisenberg-Noe model of financial contagion to allow for time dynamics in both discrete and continuous time. Mathematical results on existence and uniqueness of firm wealths under discrete and continuous-time will be provided. The financial implications of time dynamics will be considered, with focus on how the dynamic clearing solutions differ from those of the static Eisenberg-Noe model.

Since the legendary 1972 encounter of H. Montgomery and F. Dyson at tea time in Princeton, a statistical correspondence of the non-trivial zeros of the Riemann Zeta function with eigenvalues of high-dimensional random matrices has emerged. Surrounded by many deep conjectures, there is a striking analogyto the energy levels of a quantum billiard system with chaotic dynamics. Thanks

to extensive calculation of Riemann zeros by A. Odlyzko, overwhelming numerical evidence has been found for the quantum analogy. The statistical accuracy provided by an enormous dataset of more than one billion zeros reveals distinctive finite size effects. Using the physical analogy, a precise prediction of these effects was recently accomplished through the numerical evaluation of operator determinants and their perturbation series (joint work with P. Forrester and A. Mays, Melbourne).

We will see some results and conjectures on the zeta and multizeta values in the function field context, and see how they relate to homological-homotopical objects, such as t-motives, iterated extensions, and to Hopf algebras, big Galois representations.

Conical symplectic resolutions are one of the main objects in the contemporary mix of algebraic geometry and representation theory,

known as geometric representation theory. They cover many interesting families of objects such as quiver varieties and hypertoric

varieties, and some simpler such as Springer resolutions. The last findings [Braverman, Finkelberg, Nakajima] say that they arise

as Higgs/Coulomb moduli spaces, coming from physics. Most of the gadgets attached to conical symplectic resolutions are rather

algebraic, such as their quatizations and $\mathcal{O}$-categories. We are rather interested in the symplectic topology of them, in particular

finding smooth exact Lagrangians that appear in the central fiber of the (defining) resolution, as they are objects of the Fukaya category.

Many types of patterns emerging spontaneously can be observed in systems involving thin elastic plates and subjected to external or internal stresses (compression, differential growth, shearing, tearing, etc.). These mechanical systems can sometime be seen as model systems for more complex natural systems and allow to study in detail elementary emerging patterns. One of the simplest among such systems is a bilayer composed of a thin plate resting on a thick deformable substrate. Upon slight compression, periodic undulations (wrinkles) with a well-defined wavelength emerge at the level of the thin layer. We will show that, as the compression increases, this periodic state is unstable and that a second order transition to a localized state (fold) occurs when the substrate is a dense fluid.

In this talk, I will present an incomplete equilibrium model to determine the price of an annuity. A finite number of agents receive stochastic income streams and choose between consumption and investment in the traded annuity. The novelty of this model is its ability to handle running consumption and general income streams. In particular, the model incorporates mean reverting income, which is empirically relevant but historically too intractable in equilibrium. The model is set in a Brownian framework, and equilibrium is characterized and proven to exist using a system of fully coupled quadratic BSDEs. This work is joint with Gordan Zitkovic.

At first glance the Interdistrict shipping problem resembles a transportation problem. N sources with M destinations with k Stock keeping units (SKU’s); however, we want to solve for the optimal shipping frequency between each node while determining the flow of each SKU across the network. As the replenishment quantity goes up, the shipping frequency goes down and the inventory holding cost goes up (AWI = Replenishment Qty/2 + SS). Safety stock also increases as frequency decreases. The relationship between replenishment quantity and shipping frequency is non-linear (frequency = annual demand/replenishment qty). The trucks which are used to transfer the product have finite capacity and the cost to drive the truck between 2 locations is constant regardless of how many containers are actually on the truck up to the max capacity. Each product can have a different footprint of truck capacity. Cross docking is allowed. (i.e. a truck may travel from Loc A to loc B carrying products X and Y. At loc B, the truck unloads product X, picks up product Z, and continues to location C. The key here is that product Y does not incur any handling costs at Loc B while products X and Z do.)

The objective function seeks to minimize the total costs ( distribution + handling + inventory holding costs) for all locations, for all SKU’s, while determining how much of each product should flow across each arc such that all demand is satisfied.