abel's theorem (1824) that the generic polynomial of degree n is
solvable in radicals if and only if $n$ $\<$ 5 is well-known. however, the
classical works of bring (1786) and klein (1884) give solutions of the
generic quintic polynomial by allowing certain other ``nice'' algebraic
functions of one variable. for the sextic, it is conjectured that any
solution requires algebraic functions of two variables. in this talk, we
will examine and relate the intrinsic geometries of the known solutions
of the sextic in two variables, extending the work of green (1978).

an important result due to sanderson and ion says that characters of level one demazure modules are specialized macdonald polynomials. in this talk, i will introduce a new class of symmetric polynomials indexed by a pair of dominant weights in $\mathfrak{sl}_{n+1}$ which is expressed as a linear combination of specialized symmetric macdonald polynomials with coefficients defined recursively. these polynomials arose in my own work while investigating the characters of higher level demazure modules. using representation theory, we will see that this new family of polynomials interpolates between characters of level one and level two demazure modules for affine $\mathfrak{sl}_{n+1}$ and give rise to new results in the representation theory of current algebras as a corollary. this is based on joint work with vyjayanthi chari, peri shereen, and jeffrey wand.

in this talk, we will first briefly review the generic chaining method and dudley's inequality. then, we will show the application of chaining methods in random dimensionality reduction. we will mainly consider dimensionality reduction with johnson-lindenstrauss type embeddings, which are random matrix constructions to reduce the dimension while approximately preserving euclidean inter-point distances in the data set. in particular, we will consider ``fast'' and ``sparse'' johnson-lindenstrauss embeddings. finally, we summarize some applications of chaining methods in data streaming algorithms and dictionary learning. the material of this talk is based on some lecture notes by jelani nelson and some research work by sjoerd dirksen, jean bourgain and jelani nelson.

a key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. we study the dynamics of n interacting species that live in a stochastic environment. our models are described by $n$ dimensional piecewise deterministic markov processes. these are processes $(x(t), r(t))$ where the vector $x$ denotes the density of the $n$ species and $r(t)$ is a finite state space process which keeps track of the environment. in any fixed environment the process follows the flow given by a system of ordinary differential equations. the randomness comes from the changes or switches in the environment, which happen at random times. we give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. as an example we look at the competitive exclusion principle from ecology and show how the random switching can `rescue' species from extinction.

let $x$ be a smooth affine variety over a finite field of characteristic
$p$. the dwork-monsky trace formula is a fundamental tool in
understanding the $l$-functions of $p$-adic representations of
$\pi_1(x)$. we extend this result to the study of representations valued
in a higher-dimensional local ring $r$. the special case
$r=\mathbb{z}_p[[t]]$ arises naturally in the study
of \'etale $\mathbb{z}_p$-towers over $x$. time permitting, we discuss
some spectral-halo type results and conjectures describing the $p$-adic
variation of slopes in certain $\mathbb{z}_p$-towers.

change point detection is a popular tool for identifying locations in a data sequence where an abrupt change occurs in the data distribution and has been widely studied for euclidean data. modern data very often is non- euclidean, for example distribution valued data or network data. change point detection is a challenging problem when the underlying data space is a metric space where one does not have basic algebraic operations like addition of the data points and scalar multiplication.
in this talk, i propose a method to infer the presence and location of change points in the distribution of a sequence of independent data taking values in a general metric space. change points are viewed as locations at which the distribution of the data sequence changes abruptly in terms of either its fr\'echet mean or fr\'echet variance or both. the proposed method is based on comparisons of fr\'echet variances before and after putative change point locations. first, i will establish that under the null hypothesis of no change point the limit distribution of the proposed scan function is the square of a standardized brownian bridge. it is well known that such con- vergence is rather slow in moderate to high dimensions. for more accurate results in finite sample applications, i will provide a theoretically justified bootstrap-based scheme for testing the presence of change points. next, i will show that when a change point exists, (1) the proposed test is con- sistent under contiguous alternatives and (2) the estimated location of the change-point is consistent. all of the above results hold for a broad class of metric spaces under mild entropy conditions. examples include the space of univariate probability distributions and the space of graph laplacians for networks. i will illustrate the efficacy of the proposed approach in empirical studies and in real data applications with sequences of maternal fertility distributions. finally, i will talk about some future extensions and other related research directions, for instance, when one has samples of dynamic metric space data. this talk is based on joint work with prof. hans-georg m"uller.

the complement system plays a major role in the immune system to recognize and clear invading pathogens. although activation of the complement system is tightly controlled, dysregulation leads to a cascade of events that is implicated in autoimmune disorders and infectious diseases. here, to gain a systems-level understanding of the complement system, we developed two comprehensive quantitative models that describe the biochemical reactions of the complement system under a renal disorder known as c3 glomerulopathy (c3g) and bacterial infection by neisseria meningitidis. our c3g model is composed of (odes) 290 ordinary differential equations with 142 kinetic parameters that describe the state of complement system under homeostasis and disease (c3g). furthermore, we introduced therapy states by modeling known inhibitors of the complement system, a compstatin variant (c3 inhibitor) and eculizumab (c5 inhibitor). we then evaluate our system by generating concentration-time profiles of biomarkers such as c3, c3a-desarg, c5, and fc5b-9. our model shows compstatin treatment to have strong restorative effects on early-stage biomarkers such as c3 and c3a-desarg, whereas eculizumab has strong restorative effects on late-stage biomarkers c5 and fc5b-9. these results also implicate the need for patient-tailored therapies that target early stage complement activation under c3g and that treatment may depend on the specific manifestations of a patient's genetic profile in complement regulatory function. after our modeling efforts in c3g, we continued complement modeling for neisseria meningitidis. this pathogen can cause meningococcal disease and studies have shown individuals with deficiencies in the complement system, notably the membrane attack complex (mac), have a 7,000- to 10,000-fold higher risk of developing meningococcal disease. we subsequently developed a quantitative biochemical model to assess dynamics of mac production. bacterial infection by composed of 670 odes with 328 kinetic parameters our model shows highest mac deposition on neisseria meningitidis is mainly dependent on a concentration barrier where immune activators are at least three orders of magnitude higher than regulators. this makes rising levels of immune regulators as early intervention markers for the sporadic meningococcal disease. altogether, our models serve as frameworks to simulate disease-specific scenarios. subsequently, this can lead to early diagnosis, patient-specific treatments, and aid in drug discovery to identify novel inhibitory sites.

consider a smooth connected closed two-dimensional riemannian manifold
$\sigma$ with positive gauss curvature. if $u$ is a $c^2$ isometric
embedding of $\sigma$, then $u (\sigma)$ is convex. in the fifties nash
and kuiper showed, astonishingly, that this is not necessarily true when
the map is $c^1$. it is expected that the threshold at which isometric
embeddings "change nature" is the $\frac{1}{2}$-hoelder continuity of
their derivatives, a conjecture which shares a striking similarity with
a (recently solved) problem in the theory of fully developed turbulence.

in my talk i will review several plausible reasons for the threshold
and a very recent work, joint with dominik inauen, which indeed shows a
suitably weakened form of the conjecture.

we will define the p-adic numbers and discuss some basic properties,
and in the process try to give the audience a sense of how analysis over
the p-adics compares to (and differs from) analysis over the real or
complex numbers. in particular, we hope to convey why a nice theory of
analytic geometry over the p-adics might be hard to come by, and then
discuss some of the more sophisticated methods created in order to develop
such a theory.