i will describe an ongoing project to study slices to schubert varieties in the affine grassmannian. these are poisson varieties, and we will be mainly interested in quantizing them. the resulting algebras, called truncated shifted yangians, have a beautiful representation theory. we will discuss this and also mention some connections to nakajima quiver varieties which were recently discovered by braverman-finkelberg-nakajima and webster. in the pre-talk i'll define the affine grassmannian and discuss its role in geometric representation theory.

over 75$\%$ of ovarian cancers are detected in late stage disease with poor prognosis while if detected in early stage prognosis is often excellent. despite therapeutic advances the mortality rate has not changed over the past 50 years. this makes early detection an appealing approach to investigate for its potential to reduce ovarian cancer mortality.

screening trials starting in 1985 tested serum ca125 annually, a newly discovered blood test for monitoring ovarian cancer therapy. women with ca125 exceeding a threshold were referred to transvaginal ultrasound and additional ca125 tests. this multi-modal approach attained an acceptable positive predictive value (ppv) however greater sensitivity for early stage disease remained a significant concern. statistical analysis of longitudinal ca125 values from these trials indicated that most cases had exponentially rising ca125 from a baseline while most non-cases had relatively flat ca125 profiles. the challenge for the statistician was to devise a screening approach that leveraged the information in longitudinal ca125 values to increase sensitivity while maintaining the same ppv. statistical modeling of these data led to a calculation of the risk of having a change-point given age and one or more serial ca125 values, essentially using each woman as her own control. the basis for the risk calculation was a hierarchical longitudinal change-point mixture model. this risk estimate incorporating longitudinal information is a surrogate for the risk of having undetected ovarian cancer which is the optimal information on which screening decisions should be based.

in 1996, the first of five screening trials implemented this risk calculation in general population postmenopausal women and in women at increased genetic risk. trials in the general population measured ca125 annually and the algorithm referred women with intermediate risks to an additional ca125 test in 3 months, and women with elevated risks to an immediate ultrasound. all published trials implementing the risk of ovarian cancer algorithm showed an increase in early stage detection. no other ovarian cancer screening trials in the general or high risk populations have achieved this result.

statisticians were also crucial in the design and analyses of these screening trials. the largest trial had ovarian cancer mortality as the endpoint and showed a mortality difference (p $\leq$ 0.05) with a 28$\%$ reduction in the second half. this reduction was seen in the 80\% of cases where the first ca125 test preceded the change-point (incident cases) enabling such cases to be their own control. however, further follow-up is needed for definitive conclusions. (jacobs menon et. al. the lancet 2015).

one way of modeling phenomena in typical physical settings is to study pdes in random environments. the subject of stochastic homogenization is concerned with identifying the asymptotic behavior of solutions to pdes with random coefficients. specifically, we are interested in the following: if the random effects are microscopic compared to the length scale at which we observe the phenomena, can we predict the behavior which takes place on average? for certain models of pdes and under suitable hypotheses on the environment, the answer is affirmative. in this talk, i will focus on the stochastic homogenization for reaction-diffusion equations with both kpp and ignition nonlinearities. in the large-scale-large-time limit, the behavior of typical solutions is governed by a simple deterministic hamilton-jacobi equation modeling front propagation. in particular, we prove the existence of deterministic asymptotic speeds of propagation for reaction-diffusion equations in random media with both compactly supported and front-like initial data. such models are relevant for predicting the evolution of a population or the spread of a fire in a heterogeneous environment. this talk is based on joint work with andrej zlatos.

in this talk, i will introduce a new geometric inequality: the sphere covering inequality. the inequality states that the total area of two {\it distinct} surfaces with gaussian curvature less than 1, which are also conformal to the euclidean unit disk with the same conformal factor on the boundary, must be at least $4 \pi$. in other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. we apply the sphere covering inequality to show the best constant of a moser-trudinger type inequality conjectured by a. chang and p. yang. other applications of this inequality include the classification of certain onsager vortices on the sphere, the radially symmetry of solutions to gaussian curvature equation on the plane, classification of solutions for mean field equations on flat tori and the standard sphere, etc. the resolution of several open problems in these areas will be presented. the talk is based on joint work with amir moradifam from uc riverside.

the study of partial sums of iid sequences is tightly connected to that of iterated convolutions. in this talk, i will discuss results that resemble local limit theorems for iterated convolution of complex valued functions in the case of $\mathbb z$ and $\mathbb z^d$. similarities and differences with the probability densities will be in the spotlight.

the faltings height is a useful invariant for addressing questions in arithmetic geometry. in his celebrated proof of the mordell and shafarevich conjectures, faltings shows the faltings height satisfies a
certain northcott property, which allows him to deduce his finiteness
statements. in this work we prove a new northcott property for the
faltings height. namely we show, assuming the colmez conjecture and the
artin conjecture, that there are finitely many cm abelian varieties of a
fixed dimension which have bounded faltings height. the technique
developed uses new tools from integral p-adic hodge theory to study the
variation of faltings height within an isogeny class of cm abelian
varieties. in special cases, we are able to use these techniques to
moreover develop new colmez-type formulas for the faltings height.

in recent years the sequence of integers cat(n)=(n choose 2)/(n+1) called ``catalan numbers'' has been extended to a family of integers cat(a,b)=(a+b choose a)/(a+b) that is parametrized by rational numbers a/b. these numbers originally showed up as the number of lattice paths in an axb rectangle that stay above the diagonal. in 2002, jaclyn anderson gave a bijection between these paths and so-called (a,b)-core partitions. these are integer partitions in which no cell has hook length divisible by a or b. this result unlocked many new ideas in the area between combinatorics and representation theory. on the one hand, there have been many combinatorial conjectures and slightly fewer proofs. on the other hand, it seems that the numbers cat(a,b) ultimately come from the representation theory of rational cherednik algebras. the existence of a symmetric (q,t)- graded version of the numbers cat(a,b) suggests that there should be a ``rational'' generalization of mark haiman's results on the hilbert scheme of points in $c^2$.