Past

Perverse sheaves are an indispensable tool in geometric representation theory that can be used to construct representations and compute composition multiplicities. These ‘sheaves’ live in a certain $\ell$-adic derived category. In this talk we will discuss a beautiful construction of this category based on the pro-étale topology and explore some applications in representation theory.

In his pioneering and celebrated 1968 paper on the elementary theory of finite fields Ax asked if the theory of the class of all the finite rings $\mathbb{Z}/m\mathbb{Z}$, for all $m>1$, is decidable. In that paper, Ax proved that the existential theory of this class is decidable via his result that the theory of the class of all the rings $\mathbb{Z}/p^n\mathbb{Z}$ (with $p$ and $n$ varying) is decidable. This used Chebotarev’s Density Theorem and model theory of pseudo-finite fields.

I will talk about a recent solution jointly with Angus Macintyre of Ax’s Problem using model theory of the ring of adeles of the rational numbers.

Let K be a number field and let L/K be an abelian extension. The genus field of L/K is the largest extension of L which is unramified at all places of L and abelian as an extension of K. The genus group is its Galois group over L, which is a quotient of the class group of L, and the genus number is the size of the genus group. We study the quantitative behaviour of genus numbers as one varies over abelian extensions L/K with fixed Galois group. We give an asymptotic formula for the average value of the genus number and show that any given genus number appears only 0% of the time. This is joint work with Christopher Frei and Daniel Loughran.

Learning dynamical models from data plays a vital role in engineering design, optimization, and predictions. Building models describing the dynamics of complex processes (e.g., weather dynamics, reactive flows, brain/neural activity, etc.) using empirical knowledge or first principles is frequently onerous or infeasible. Therefore, system identification has evolved as a scientific discipline for this task since the 1960ies. Due to the obvious similarity of approximating unknown functions by artificial neural networks, system identification was an early adopter of machine learning methods. In the first part of the talk, we will review the development in this area until now.

For complex systems, identifying the full dynamics using system identification may still lead to high-dimensional models. For engineering tasks like optimization and control synthesis as well as in the context of digital twins, such learned models might still be computationally too challenging in the aforementioned multi-query scenarios. Therefore, it is desirable to identify compact approximate models from the available data. In the second part of this talk, we will therefore exploit that the dynamics of high-fidelity models often evolve in lowdimensional manifolds. We will discuss approaches learning representations of these lowdimensional manifolds using several ideas, including the lifting principle and autoencoders. In particular, we will focus on learning state-space representations that can be used in classical tools for computational engineering. Several numerical examples will illustrate the performance and limitations of the suggested approaches.

This talk will explore a series of topics and example applications at the interface of graph theory and dynamics, from synchronization and diffusion dynamics on networks, to graph-based data clustering, to graph convolutional neural networks. The underlying links are provided by concepts in spectral graph theory.

Group cohomology is a powerful tool which has found many applications in modern group theory. It can be calculated and interpreted through geometric, algebraic and topological means, and as such it encodes the relationships between these different aspects of infinite groups. The aim of this talk is to introduce a circle of ideas which link group cohomology with the theory of BNS invariants, and the property of being subgroup separable. No prior knowledge of any of these topics will be assumed.

In the operator algebraic approach to quantum field theory, the DHR category is a braided tensor category describing topological point defects of a theory with at least 1 (+1) dimensions. A single von Neumann algebra with no extra structure can be thought of as a 0 (+1) dimensional quantum field theory. In this case, we would not expect a braided tensor category of point defects since there are not enough dimensions to implement a braiding. We show, however, that one can think of central sequence algebras as operators localized ``at infinity", and apply the DHR recipe to obtain a braided tensor category of bimodules of a von Neumann algebra M, which is a Morita invariant. When M is a II_1 factor, the braided subcategory of automorphic objects recovers Connes' chi(M) and Jones' kappa(M). We compute this for II_1 factors arising naturally from subfactor theory and show that any Drinfeld center of a fusion category can be realized. Based on joint work with Quan Chen and Dave Penneys.

In recent years, duality approaches have yielded new results about the high-dimensional cohomology of several groups and moduli spaces, such as $\operatorname{SL}_n(\mathbb{Z})$ and $\mathcal{M}_g$. I will explain the general strategy of these approaches and survey results that have been obtained so far. To give an example, I will first explain how Borel-Serre duality can be used to show that the rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ vanishes near its virtual cohomological dimension. This is based on joint work with Miller-Patzt-Sroka-Wilson and builds on results by Church-Farb-Putman. I will then put this into a more general context by giving an overview of analogous results for mapping class groups of surfaces, automorphism groups of free groups and further arithmetic groups such as $\operatorname{SL}_n(\mathcal{O}_K)$ and $\operatorname{Sp}_{2n}(\mathbb{Z})$.

We investigate the theoretical properties of subsampling and hashing as tools for approximate Euclidean norm-preserving embeddings for vectors with (unknown) additive Gaussian noises. Such embeddings are called Johnson-Lindenstrauss embeddings due to their celebrated lemma. Previous work shows that as sparse embeddings, if a comparable embedding dimension to the Gaussian matrices is required, the success of subsampling and hashing closely depends on the $l_\infty$ to $l_2$ ratios of the vectors to be mapped. This paper shows that the presence of noise removes such constrain in high-dimensions; in other words, sparse embeddings such as subsampling and hashing with comparable embedding dimensions to dense embeddings have similar norm-preserving dimensionality-reduction properties, regardless of the $l_\infty$ to $l_2$ ratios of the vectors to be mapped. The key idea in our result is that the noise should be treated as information to be exploited, not simply a nuisance to be removed. Numerical illustrations show better performances of sparse embeddings in the presence of noise.

The triangle removal lemma of Ruzsa and Szemerédi is a fundamental result in extremal graph theory; very roughly speaking, it says that if a graph is "far" from triangle-free, then it contains "many" triangles. Despite decades of research, there is still a lot that we don't understand about this simple statement; for example, our understanding of the quantitative dependencies is very poor.

In this talk, I will discuss asymmetric versions of the triangle removal lemma, where in some cases we can get almost optimal quantitative bounds. The proofs use a mix of ideas coming from graph theory, number theory, probabilistic combinatorics, and Ramsey theory.

Based on joint work with Lior Gishboliner and Asaf Shapira.

In the last few years, major advances have been made in our understanding of the representation theory of reductive algebraic groups over algebraically closed fields of positive characteristic. Four key tools which are central to this progress are highest weight theory, reduction to the principal block, wall-crossing functors, and tilting modules. When considering instead the representation theory of the Lie algebras of these algebraic groups, more subtleties arise. If we look at those modules whose p-character is in so-called standard Levi form we are able to recover the four tools mentioned above, but they have been less well-studied in this setting. In this talk, we will explore the similarities and differences which arise when employing these tools for the Lie algebras rather than the algebraic groups. This research is funded by a research fellowship from the Royal Commission for the Exhibition of 1851.

In this talk we are presenting a new approach for compatible finite element discretisations for atmospheric flows on a terrain following mesh. In classical compatible finite element discretisations, the H(div)-velocity space involves the application of Piola transforms when mapping from a reference element to the physical element in order to guarantee normal continuity. In the case of a terrain following mesh, this causes an undesired coupling of the horizontal and vertical velocity components. We are proposing a new finite element space, that drops the Piola transform. For solving the equations we introduce a hybridisable formulation with trace variables supported on horizontal cell faces in order to enforce the normal continuity of the velocity in the solution. Alongside the discrete formulation for various fluid equations we discuss solver approaches that are compatible with them and present our latest numerical results.

In this talk I will discuss Onsager's conjecture for energy conservation. Moreover, in 1949 Onsager conjectured that weak solutions to the incompressible Euler equations, that were Hölder continuous with Hölder exponent greater than 1/3, conserved kinetic energy. Onsager also conjectured that there were weak solutions that were Hölder continuous with Hölder exponent less than 1/3 that didn't conserve kinetic energy. I will discuss the results regarding the former, focusing mainly on the case where the spacial domain is bounded with C^2 boundary, as proved by Bardos and Titi.

This talk will be devoted to our recent developments in the analysis of emerging models for complex flows. I will start from presenting a general PDE system describing two-fluid flows, for which we prove existence of global in time weak solutions for arbitrary large initial data. I will explain where the famous approach of Lions developed for the compressible Navier-Stokes equations fails and how to use a more direct, weighted Kolmogorov criterion to prove compactness of approximating sequences of solutions. Through a formal limit, I will link the two-fluid model to the constrained two-phase models. Applications of such models include modelling of granular flows, crowd motion, or shallow water flow through a channel. The last part of my talk will focus on the rigorous derivation of these models from the compressible Navier-Stokes equations via the vanishing singular pressure or viscosity limit.

In 2018, Keating, Rodgers, Roditty-Gershon and Rudnick conjectured that the variance of sums of the divisor
function in short intervals is described by a certain piecewise polynomial coming from a unitary matrix integral. That is
to say, this conjecture ties a straightforward arithmetic problem to random matrix theory. They supported their
conjecture by analogous results in the setting of polynomials over a finite field rather than in the integer setting. In this
talk, we'll discuss arithmetic problems over F_q[T] and their connections to matrix integrals, focusing on variations on
the divisor function problem with symplectic and orthogonal distributions. Joint work with Matilde Lalín.

In 2018, Keating, Rodgers, Roditty-Gershon and Rudnick conjectured that the variance of sums of the divisor function in short intervals is described by a certain piecewise polynomial coming from a unitary matrix integral. That is to say, this conjecture ties a straightforward arithmetic problem to random matrix theory. They supported their conjecture by analogous results in the setting of polynomials over a finite field rather than in the integer setting. In this talk, we'll discuss arithmetic problems over F_q[T] and their connections to matrix integrals, focusing on variations on the divisor function problem with symplectic and orthogonal distributions. Joint work with Matilde Lalín.

I shall discuss my recent work showing that the Bogomolov-Tschinkel universality conjecture holds if and only if the mapping class groups of a punctured surface is large. One consequence of this result is that all genus 2 surface-by-surface (and all genus 2 surface-by-free) groups are virtually algebraically fibered. Moreover, I will explain why simple curve homology does not always generate homology of finite covers of closed surface. I will also mention my work with O. Tosic regarding the Putman-Wieland conjecture, and explain the partial solution to the Prill's problem about algebraic curves.

 

In this talk I will present an exclusion process with different types of particles: A, B and C. This last type can be understood as holes. Two scaling limits will be discussed: hydrodynamic limits in the boundary driven setting; and equilibrium fluctuations for an evolution on the torus. In the later case, we distinguish several cases, that depend on the choice of the jump rates, for which we get in the limit either the stochastic Burgers equation or the Ornstein-Uhlenbeck equation. These results match with predictions from non-linear fluctuating hydrodynamics. 
(Joint work with G. Cannizzaro, A. Occelli, R. Misturini).

Associated to any smooth projective curve C is its degree d Jacobian variety, parametrising isomorphism classes of degree d line bundles on C. Letting the curve vary as well, one is led to the universal Jacobian stack. This stack admits several compactifications over the stack of marked stable curves $\overline{\mathcal{M}}_{g,n}$, depending on the choice of a stability condition. In this talk I will introduce these compactified universal Jacobians, and explain how their moduli spaces can be constructed using Geometric Invariant Theory (GIT). This talk is based on arXiv:2210.11457.

Characterising intractable high-dimensional random variables is one of the fundamental challenges in stochastic computation. We develop a deep transport map that is suitable for sampling concentrated distributions defined by an unnormalised density function. We approximate the target distribution as the push-forward of a reference distribution under a composition of order-preserving transformations, in which each transformation is formed by a tensor train decomposition. The use of composition of maps moving along a sequence of bridging densities alleviates the difficulty of directly approximating concentrated density functions. We propose two bridging strategies suitable for wide use: tempering the target density with a sequence of increasing powers, and smoothing of an indicator function with a sequence of sigmoids of increasing scales. The latter strategy opens the door to efficient computation of rare event probabilities in Bayesian inference problems.

Numerical experiments on problems constrained by differential equations show little to no increase in the computational complexity with the event probability going to zero, and allow to compute hitherto unattainable estimates of rare event probabilities for complex, high-dimensional posterior densities.
 

We study supersymmetric gauge theories with 8 supercharges in d=3,4,5,6. For these theories, one can perform Higgsings by turning on VEVs of scalar fields. However, this process can often be difficult when dealing with superconformal field theories (SCFTs) where the Lagrangian is often not known. Using techniques of magnetic quivers and a new algorithm we call "Inverted Quiver Subtraction", we show how one can easily obtain the SCFT(s) after Higgsing. This technique can be equally well applied to SCFTs in d=3,4,5,6. 

The basic question in prime number theory is to try to understand the number of primes in some interesting set of integers. Unfortunately many of the most basic and natural examples are famous open problems which are over 100 years old!

We aim to give an accessible survey of (a selection of) the main results and techniques in prime number theory. In particular we highlight progress on some of these famous problems, as well as a selection of our favourite problems for future progress.

The basic question in prime number theory is to try to understand the number of primes in some interesting set of integers. Unfortunately many of the most basic and natural examples are famous open problems which are over 100 years old!

We aim to give an accessible survey of (a selection of) the main results and techniques in prime number theory. In particular we highlight progress on some of these famous problems, as well as a selection of our favourite problems for future progress.

Reaction-diffusion waves have long been used to describe the growth and spread of populations undergoing a spatial range expansion. Such waves are generally classed as either pulled, where the dynamics are driven by the very tip of the front and stochastic fluctuations are high, or pushed, where cooperation in growth or dispersal results in a bulk-driven wave in which fluctuations are suppressed. These concepts have been well studied experimentally in populations where the cooperation leads to a density-dependent growth rate. By contrast, relatively little is known about experimental populations that exhibit a density-dependent dispersal rate.

Using bacteriophage T7 as a test organism, we present novel experimental measurements that demonstrate that the diffusion of phage T7, in a lawn of host E. coli, is hindered by steric interactions with host bacteria cells. The coupling between host density, phage dispersal and cell lysis caused by viral infection results in an effective density-dependent diffusion rate akin to cooperative behavior. Using a system of reaction-diffusion equations, we show that this effect can result in a transition from a pulled to pushed expansion. Moreover, we find that a second, independent density-dependent effect on phage dispersal spontaneously emerges as a result of the viral incubation period, during which phage is trapped inside the host unable to disperse. Our results indicate both that bacteriophage can be used as a controllable laboratory population to investigate the impact of density-dependent dispersal on evolution, and that the genetic diversity and adaptability of expanding viral populations could be much greater than is currently assumed.

Néron models are mathematical objects which play a very important role in contemporary arithmetic geometry. However, they usually behave badly, particularly in respect of exact sequences and base change, which makes most problems regarding their behaviour very delicate. Chai introduced the base change conductor, a rational number associated with a semiabelian variety $G$ which measures the failure of the Néron model of $G$ to commute with (ramified) base change. Moreover, Chai conjectured that this invariant is additive in certain exact sequences. We shall introduce a new method to study the Néron models of Jacobians of proper (possibly singular) curves, and sketch a proof of Chai's conjecture for semiabelian varieties which are also Jacobians. 

In order for one to infer reasonable predictions from a model, it must be calibrated to reproduce observations in the market. We use the semimartingale optimal transport methodology to formulate this calibration problem into a constrained optimisation problem, with our model calibrated using a finite number of European options observed in the market as constraints. Given such a PDE formulation, we are able to then derive a dual formulation involving an HJB equation which we can numerically solve. We focus on two cases: (1) The stochastic interest rate is known and perfectly matches the observed term structure in the market, however the asset local volatility and correlation are not known and must be calibrated; (2) The dynamics of both the stochastic interest rate and the underlying asset are unknown, and we must jointly calibrate both to European options on the interest rate and on the asset.

ife forms have devised impressive and subtle ways to exploit fluid flows. I’ll talk about birds as flying machines whose behaviors can give surprising insights into flow physics. One story explains how flocking interactions can help to bring flapping flyers into orderly formations. A second story involves the more subtle role of aerodynamics in the highly efficient breathing of birds, which is thought to be critical to their ability to fly.

Tensors, also known as multiway arrays, have become ubiquitous as representations for operators or as convenient schemes for storing data. Yet, when it comes to compressing these objects or analyzing the data stored in them, the tendency is to ``flatten” or ``matricize” the data and employ traditional linear algebraic tools, ignoring higher dimensional correlations/structure that could have been exploited. Impediments to the development of equivalent tensor-based approaches stem from the fact that familiar concepts, such as rank and orthogonal decomposition, have no straightforward analogues and/or lead to intractable computational problems for tensors of order three and higher.

In this talk, we will review some of the common tensor decompositions and discuss their theoretical and practical limitations. We then discuss a family of tensor algebras based on a new definition of tensor-tensor products. Unlike other tensor approaches, the framework we derive based around this tensor-tensor product allows us to generalize in a very elegant way all classical algorithms from linear algebra. Furthermore, under our framework, tensors can be decomposed in a natural (e.g. ‘matrix-mimetic’) way with provable approximation properties and with provable benefits over traditional matrix approximation. In addition to several examples from recent literature illustrating the advantages of our tensor-tensor product framework in practice, we highlight interesting open questions and directions for future research.

In this seminar, we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporated the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self-propulsion. The main result of this work is the incompressible limit of this model, which builds a bridge between the density-based model and a geometry free-boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.

Condensed Mathematics is a tool recently developed by Clausen and Scholze and it is proving fruitful in many areas of algebra and geometry. In this talk, we will cover the definition of condensed sets, the analogues of topological spaces in the condensed setting. We will also talk about condensed modules over a ring and some of their nice properties like forming an abelian category. Finally, we'll discuss some recent results that have been obtained through the application of Condensed Mathematics.

We will be joined by Professor Kobi Kremnizer, who is a trained mental health first-aider, to discuss ways to protect your mental health this season.

In view of Takesaki-Takai duality, we can go back and forth between C*-dynamical systems of an abelian group and ones of its Pontryagin dual by taking crossed products. In this talk, I present a similar duality between actions on C*-algebras of two constructions of locally compact quantum groups: one is the bicrossed product due to Vaes-Vainerman, and the other is the double crossed product due to Baaj-Vaes. I will explain the situation by illustrating the example coming from groups. If time permits, I will also discuss its consequences in the case of quantum doubles.

A metric is said to be (globally) median,  if any three points have a unique “median” which  lies  between  any  two  points  from  the  triple.  
Such  spaces  arise  naturally  in  many different contexts.  The property of being locally median can be viewed as a kind of
non-positive curvature condition.  We show that a complete uniformly locally median space is
globally median if and only if it is simply connected.  This is an analogue of the well known Cartan-Hadamard Theorem for non-positively curved manifolds, or more generally CAT(0) spaces.  However it leaves open a number of interesting questions.

We consider a bond percolation on the hypercube in the supercritical regime. We derive vertex-expansion properties of the giant component. As a consequence we obtain upper bounds on the diameter of the giant component and the mixing time of the lazy random walk on the giant component. This talk is based on joint work with Joshua Erde and Michael Krivelevich.

We discuss the Local Langlands correspondence in connection with the Bernstein center and the Stable Bernstein center. We also give an example of stable Bernstein center as a stable essentially compact invariant distribution.

I will present a model for an optimal portfolio allocation and consumption problem for a portfolio composed of a risk-free bond and two illiquid assets. Two forms of illiquidity are presented, both illiquidities based on Lévy processes. The goal of the investor is to maximise a certain utility function, and the optimal utility is found as a solution of a nonlinear PIDE of the Hamilton-Jacobi-Bellman kind.

The Dedekind zeta function generalises the Riemann zeta
function to other number fields than the rationals. The analytic class number
formula says that the leading term of the Dedekind zeta function is a
product of invariants of the number field. I will say some things
about the class number formula, about L-functions, and about Stark's
conjecture which generalises the class number formula.

The chromatic polynomial \chi(Q) can be defined for any graph, such that for Q integer it counts the number of colourings. It has many remarkable properties, and I describe several that are derived easily by using fusion categories, familiar from topological quantum field theory. In particular, I define the chromatic algebra, a planar algebra whose evaluation gives the chromatic polynomial. Linear identities of the chromatic polynomial at certain values of Q then follow from the Jones-Wenzl projector of the associated category. An unusual non-linear one called Tutte's golden identity relates \chi(\phi+2) for planar triangulations to the square of \chi(\phi+1), where \phi is the golden mean. Tutte's original proof is purely combinatorial. I will give here an elementary proof by manipulations of a topological invariant related to the Jones polynomial. Time permitting, I will also mention analogous identities for graphs on more general surfaces. Based on work with Slava Krushkal.

The signature is a non-commutative exponential that appeared in the foundational work of K-T Chen in the 1950s. It is also a fundamental object in the theory of rough paths (Lyons, 1998). More recently, it has been proposed, and used, as part of a practical methodology to give a way of summarising multimodal, possibly irregularly sampled, time-ordered data in a way that is insensitive to its parameterisation. A key property underpinning this approach is the ability of linear functionals of the signature to approximate arbitrarily any compactly supported and continuous function on (unparameterised) path space. We present some new results on the properties of a selection of topologies on the space of unparameterised paths. We discuss various applications in this context.
This is based on joint work with William Turner.
 

The moduli space M of stable bundles on a Riemann surface possesses a natural family of holomorphic trivector fields. The talk will introduce these objects with examples and then use them to gain information about the Hochschild cohomology of M.

K-homology is the dual theory to K-theory for C*-algebras.  I will show how under appropriate quasi-diagonality and countability assumptions K-homology (more generally, KK-theory) can be realized by completely positive and contractive, and approximately multiplicative, maps to matrix algebras modulo an appropriate equivalence relation.  I’ll briefly explain some connections to manifold topology and existence / uniqueness theorems in C*-algebra classification theory (due to Dadarlat and Eilers).

Some of this is based on joint work with Guoliang Yu, and some is work in progress

While scattering problems are posed on unbounded domains, volumetric discretizations typically require truncating the domain at a finite distance, closing the system with some sort of boundary condition.  These conditions typically suffer from some deficiency, such as perturbing the boundary value problem to be solved or changing the character of the operator so that the discrete system is difficult to solve with iterative methods.

We introduce a new technique for the Helmholtz problem, based on using the Green formula representation of the solution at the artificial boundary.  Finite element discretization of the resulting system gives optimal convergence estimates.  The resulting algebraic system can be solved effectively with a matrix-free GMRES implementation, preconditioned with the local part of the operator.  Extensions to the Morse-Ingard problem, a coupled system of pressure/temperature equations arising in modeling trace gas sensors, will also be given.

Minimal submanifolds are the critical points of the volume functional. If the second derivative of the volume is nonnegative, we say that such a minimal submanifold is stable.

After reviewing some basics of minimal submanifolds in a generic Riemannian manifold, I will give some motivations behind the Lawson--Simons conjecture, which claims that there are no stable minimal submanifolds in 1/4-pinched spheres. Finally, I will discuss my recent work with Giada Franz on the nonexistence of stable minimal submanifolds in conformal pinched spheres.

2:00 Julian Sieber

On the (Non-)stationary density of fractional SDEs

I will present a novel approach for studying the density of SDEs driven by additive fractional Brownian motion. It allows us to establish smoothness and Gaussian-type upper and lower bounds for both the non-stationary as well as the stationary density. While the stationary density has not been studied in any previous works, the former was the subject of multiple articles by Baudoin, Hairer, Nualart, Ouyang, Pillai, Tindel, among others. The common theme of all of these works is to obtain the results through bounds on the Malliavin derivative. The main disadvantage of this approach lies in the non-optimal regularity conditions on the SDE's coefficients. In case of additive noise, the equation is known to be well-posed if the drift is merely sublinear and measurable (resp. Holder continuous). Relying entirely on classical methods of stochastic analysis (avoiding any Malliavin calculus), we prove the aforementioned Gaussian-type bounds under optimal regularity conditions.

The talk is based on a joint work with Xue-Mei Li and Fabien Panloup.

2:45 Thomas Tendron

A central limit theorem for a spatial logistic branching process in the slow coalescence regime

We study the scaling limits of a spatial population dynamics model which describes the sizes of colonies located on the integer lattice, and allows for branching, coalescence in the form of local pairwise competition, and migration. When started near the local equilibrium, the rates of branching and coalescence in the particle system are both linear in the local population size - we say that the coalescence is slow. We identify a rescaling of the equilibrium fluctuations process under which it converges to an infinite dimensional Ornstein-Uhlenbeck process with alpha-stable driving noise if the offspring distribution lies in the domain of attraction of an alpha-stable law with alpha between one and two.

3:30 Break

4:00-5:30 Careers Discussion