in these two lectures we plan to survey some known results about complete noncompact shrinking ricci solitons in dimension four. the main goal is to understand the geometry at infinity of these manifolds, and for that we address the following basic questions:
*estimate the number of ends of 4d shrinkers.
*understand the asymptotic geometry along each end.
*uniqueness results.
*open problems/conjectures.

holonomy vectors of translation surfaces provide a geometric generalization for higher genus surfaces of (primitive) integer lattice points. the counting and distribution properties of holonomy vectors on translation surfaces have been studied extensively. a natural question to ask is: how random are the holonomy vectors of a translation surface? we motivate the gap distribution of slopes of holonomy vectors as a measure of randomness and compute the gap distribution for the class of translation surfaces given by gluing two identical tori along a slit. no prior background on translation surfaces or gap distributions will be assumed.

this talk concerns a classical problem in computational wave propagations: how does one truncate an infinite domain to a finite size without introducing reflection waves from the artificial boundaries? the state-of-the-art approach is attaching to those boundaries a perfectly matched layer (pml). in the continuous theory, pmls are subject to an analytically continued wave equation that damps all incident waves without creating any interfacial reflection. however, it is believed that ``numerical reflections'' are unavoidable after discretization. in this talk, i will demonstrate a truly reflectionless discrete pml. the key is to uncover the geometric mechanism hidden in the differential calculus formalism; in discretizing the theory, approximations are the best one can hope for the latter, while the former often admits exact discretization.

i will talk about why some classical theorems (e.g. waldhausen, laudenbach-poenaru) can be modified to hold for non-orientable 3- and 4-manifolds. as a consequence, we can draw kirby diagrams and trisection diagrams and conclude the usual main trisection theorems for non-orientable 4-manifolds. this is joint work with patrick naylor (university of waterloo).

dear zft attendee, has your research ever come down to proving a polynomial inequality? if so, would you want to use boring old analysis to come up with a five line proof? or would you rather spend two days creating an algorithm to find the exact factorization of the polynomial? the latter is the obvious choice. come on over to zft this week and see how clarkson's inequality can be proved in a special case by using linear recursion and binomial coefficients! i guarantee that you will think of the polynomial $y^2-2y+1$ differently for the rest of your life!

roth famously used fourier analysis to upper bound the size of a set of integers without 3-term arithmetic progressions. one might hope that similar techniques can be used for 4 or more term progressions, and some simple examples demonstrate otherwise. however, gowers (2001) introduced a ``higher order'' fourier analysis which generalizes roth's proof to longer arithmetic progressions. in this talk, we will give a combinatorial sketch of the methods of higher order fourier analysis culminating in the key problem of the field, the inverse conjecture for the gowers norms.

an expander graph is a type of graph that is sparse yet highly-connected. they have found uses in many subjects including computer science, topology, algebra, number theory and algebraic geometry. we will talk about the basics of expander graphs and how they can be used to give quantitative versions of algebraic statements regarding interesting groups

ricci soliton is a self-similar solution to the ricci flow and arises naturally in the singularity analysis of the flow. steady ricci soliton is a kind of soliton whose associated ricci flow evolves by reparametrizing a fixed metric. it is closely related to the type ii limit solution to the ricci flow. steady ricci soliton with integrable scalar curvature was studied by deruelle in 2012, later by catino-mastrolia-monticelli in 2016, munteanu-sung-wang in 2019, deng-zhu in 2020. in this talk, we shall discuss a classification result on steady kaehler ricci soliton with nonnegative ricci curvature and integrable scalar curvature. we then apply the result to study the steady kaehler ricci soliton with subquadratic volume growth or fast curvature decay.

in these two lectures we plan to survey some known results about complete noncompact shrinking ricci solitons in dimension four. the main goal is to understand the geometry at infinity of these manifolds, and for that we address the following basic questions:
*estimate the number of ends of 4d shrinkers.
*understand the asymptotic geometry along each end.
*uniqueness results.
*open problems/conjectures.

the multiple access channel (mac) is one of the most well studied and understood network information theoretic models, describing a scenario where k users wish to deliver their information message to one receiver, sharing the same transmission channel. beyond the multitude of practical and somewhat heuristic mac protocols (e.g., tdma, fdma, cdma, csma, aloha, and variations thereof), the information theoretic capacity region is well understood under many situations of interest, and in particular in the gaussian case, modeling the uplink of a wireless system with one access point or base station (receiver) and several users, sharing the same frequency band.
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more recently, a variant of this model has been proposed for a situation where a very large (virtually unlimited) number of users wish to communicate only very sporadically, such that at any point in time only a finite and relatively small number of users are ``active''. this scenario is appropriate for machine-type communications and internet of things, where a multitude of sensors and objects have only rather sporadic data to send, but they need to send them when they are created, at random times.
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the identification of the active user set, or the active message set (the list of messages transmitted, irrespectively of who is transmitting them) has some points in common with a compressed sensing problem, where the activity vector (entry 1 if a user/message is active and 0 otherwise) is the key object to be estimated at the receiver side.
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in this talk we review the basic mac model and results, a variant for massive random access called ``unsourced random access'' where all users use the same codebook, and related recent results and algorithms. including some capacity scaling for this model, which remains quite open as far as a full information theoretic characterization is concerned.

this will be a mainly expository talk about some recent applications of number theory to topology. the crux of these applications is the construction of a cohomology theory called topological modular forms (tmf) out of the moduli of elliptic curves. i'll explain what tmf is, what we have been doing with it, and what we'd still like to know; i'll also discuss more recent attempts to extend the theory using level structures, higher-dimensional abelian varieties, and k3 surfaces. time permitting, i'll talk about my work with dominic culver on some partial number-theoretic interpretations of tmf co-operations.

this will be a mainly expository talk about some recent
applications of number theory to topology. the crux of these
applications is the construction of a cohomology theory called
topological modular forms (tmf) out of the moduli of elliptic curves.
i'll explain what tmf is, what we have been doing with it, and what we'd
still like to know; i'll also discuss more recent attempts to extend the
theory using level structures, higher-dimensional abelian varieties, and
k3 surfaces. time permitting, i'll talk about my work with dominic
culver on some partial number-theoretic interpretations of tmf
co-operations.

morphogen gradients play a vital role in developmental biology by enabling embryonic cells to infer their spatial location and determine their developmental fate accordingly. the standard mechanism for generating a morphogen gradient involves a morphogen being produced from a localized source and subsequently degrading. while this mechanism is effective over the length and time scales of tissue development, it fails over typical subcellular length scales due to the rapid dissipation of spatial asymmetries. single-particle tracking experiments have recently found that c. elegans zygotes rely on space-dependent switching diffusivities to form intracellular gradients during cell polarization. we analyze a model of switching diffusivities to determine its role in protein concentration gradient formation. in particular, we determine how the presence of switching diffusivities modifies the standard theory and show that space-dependent switching diffusivities can yield a gradient in the absence of a localized source. our mathematical analysis yields explicit formulas for the intracellular concentration gradient which closely match the results of previous experiments and numerical simulations. we further consider how this mechanism of switching diffusive states interacts with a locally varying periodic microstructure in the cell, and use homogenization theory to show that at the typical cellular scales involved, such a microstructure does not necessarily need to be resolved in fine detail in order to accurately capture the dynamics of the system.

many familiar objects in algebra, geometry and topology belong to
2-calabi-yau categories, e.g. coherent sheaves on a k3 or abelian
surface, higgs bundles on a smooth and proper curve, local systems on a
riemann surface, and representations of a fixed preprojective algebra.
in each case the borel-moore homology of the stack of objects carries a
hall algebra structure, which in the preprojective algebra case contains
all of the raising operators on the cohomology of nakajima quiver
varieties, and in other cases produces a kind of nonlinear/global
analogue of this theory. by reducing everything locally to the case of
representations of a preprojective algebra, i'll show that in each case
the borel-moore homology of the stack of objects carries a perverse
filtration. i'll explain how this filtration can be used to produce
generators for the borel-moore homology of the stack of objects coming
from intersection cohomology of the coarse moduli space, along with
tautological classes. in joint work with sjoerd beentjes, we show that
in all but one of the examples in the first sentence of this abstract
this borel-moore homology is pure. in the k3 case this proves a
conjecture of halpern-leistner.

given a class of finite structures, is there a way to assemble them into a
countably infinite limit object in a canonical way? for example, consider
the collection of finite linear orders. clearly the limit object should be
some infinite linearly ordered set, but there are lots to choose from.
should we use the order of the naturals? the integers? the rationals?
similarly, consider the class of finite graphs. there are lots of
countably infinite graphs which embed every finite graph. which one is
best? fraisse theory allows us to answer this question; for certain
collections of finite structures, we can consider the ``generic''
countably infinite structure which embeds every member of the finite
class.