the classical hele-shaw flow describes the motion of incompressible viscous fluid, which occupies part of the space between two parallel, nearby plates. with source and drift, the equation is used in models of tumor growth where cells evolve with contact inhibition, and congested population dynamics. we consider the flow with hölder continuous source and lipschitz continuous drift. we show that if the free boundary of the solution is locally close to a lipschitz graph, then it is indeed lipschitz, given that the lipschitz constant is small. this is joint work with inwon kim.

non-unital operator systems are norm closed subspaces of b(h) that are closed under the involution map $x \mapsto x^*$. for example, c*-algebras are examples of non-unital operator systems. much like gelfand duality, a result of kadison from the 80s shows that operator systems generated by commuting elements are categorically dual to a class of geometric structures, namely compact convex sets. unlike the c*-theory, a remarkable result due to webster-winkler and davidson-kennedy shows that kadison's duality theorem readily generalizes to the non-commutative unital setting as well. in our talk, we discuss kadison’s original duality as well as the nc convex duality for non-commutative operator systems due to myself, matt kennedy, and nick manor. finally, we will have a discussion of some results on the side of operator systems that this illuminates.

applied benchmark tests for the famous `subgraph isomorphism problem' empirically discovered interesting phase transitions in random graphs. this motivates our rigorous study of two variants of the induced subgraph isomorphism problem for two independent binomial random graphs with constant edge-probabilities p_1,p_2. in particular, (i) we prove a sharp threshold result for the appearance of g_{n,p_1} as an induced subgraph of g_{n,p_2}, (ii) we show two-point concentration of the size of the maximum common induced subgraph of g_{n, p_1} and g_{n,p_2}, and (iii) we show that the number of induced copies of g_{n,p_1} in g_{n,p_2} has a `squashed lognormal' limiting distribution. these results confirm simulation-based phase transition predictions of mccreesh-prosser-solnon-trimble, and resolve several open problems of chatterjee-diaconis.

the proofs are based on careful refinements of the first and second moment method, using several extra twists to (a) take the non-standard behavior into account, and (b) work around the large variance issues that prevent standard applications of the second moment method, using in particular pseudorandom properties and multi-round exposure arguments to tame the variance.

based on joint work with erlang surya and emily zhu (both ucsd).

variational integrators are a class of geometric structure-preserving numerical integrators for variational differential equations that are based on a discretization of hamilton’s variational principle. we will present our construction of geometric variational integrators for multisymplectic lagrangian and hamiltonian pdes, as well as our construction of geometric variational integrators for adjoint systems arising in optimization and optimal control. for multisymplectic pdes, we will examine their continuous multisymplectic structures and variational principles, and utilize these to develop a discrete variational principle, leading to variational integrators which preserve their multisymplectic structure as well as satisfy a discrete noether’s theorem. for adjoint systems, we will introduce novel type ii variational principles for adjoint systems on vector spaces and lie groups. we utilize these to develop a discrete variational principle for adjoint systems, leading to symplectic and presymplectic integrators for adjoint systems which allow one to compute exact gradients of cost functions in optimization and optimal control problems. numerical examples will be presented throughout.

take a look at the mighty tetrahedron and its mirror image. explore the subgroup lattice of s4 and get a geometric perspective on why some of its subgroups are normal while others have conjugates.

see the sylow-3 and sylow-5 subgroups of a5, and learn how counting to 20 in greek can help you remember how many of them there are. it's quite simple, actually - but we'll finish off with the following not-entirely-trivial question: can you see s5 by holding a dodecahedron up to a mirror?

 asymptotic counting of geometric objects has a long history. in the case of flat surfaces, various works by masur and veech showed the quadratic asymptotic growth of the number of saddle connections of bounded length. in this spirit, athreya, fairchild, and masur showed that, for almost any flat surface, the number of pairs of saddle connection of bounded length and bounded virtual area increases quadratically with the constraint on length. in this case the « almost all » is with respect to the so-called masur-veech measure.

to demonstrate this, they use tools of ergodic theory (hence the result is true almost everywhere). this result can be extended in several ways, giving an error term or extending it to almost any sl(2,r)-invariant measure. we will present several useful tools for tackling these questions.

transportation of measure underlies many contemporary methods in machine learning and statistics. sampling, which is a fundamental building block in computational science, can be done efficiently given an appropriate measure-transport map. we ask: what is the effect of using approximate maps in such algorithms?

we propose a new framework to analyze the approximation power of measure transport. this framework applies to existing algorithms, but also suggests new ones. at the core of our analysis is the theory of optimal transport regularity, approximation theory, and an emerging class of inequalities, previously studied in the context of uncertainty quantification (uq).

imagine a population evolving over time, with genetic information being passed down from generation to generation, while evolution shapes it. the inherently random nature of this process makes it an ideal subject to be studied using stochastic processes, particularly markov processes. now, imagine if the system had memory, meaning genetic information could be inherited from many generations in the past. in such cases, we refer to a "seed bank."

seed banks can break the markovian nature of the process. in this presentation, we will explain how to overcome this difficulty. furthermore, we will describe the connections between seed bank models, stochastic delay differential equations and the fractional brownian motion.

this talk will focus on böhm and lafuente 2017's work on immortal homogeneous ricci flow, where they prove that any sequence of blow-downs of such flow will subconverges to an expanding homogeneous ricci soliton. for a \mathfrac{g}-homogeneous space m, ricci flow of g-invariant metrics can be shown to be equivalent to a flow on ad(h) invariant "bracket." we will show the existence of stratification of the space of brackets that induces curvature estimate on each strata, motivated by geometric invariant theory. the sharp case of the inequality corresponds to the limit case of an expanding g-invariant ricci soliton, and we will show that as immortal homogeneous ricci flow is of type iii, the blowdown will subconverges to such limit case. finally if time allows, we will discuss böhm and lafuente recent proof of alekseevskii's conjecture and its implication to homogeneous ricci soliton.

 in his seminal work on modular curves and the eisenstein ideal, mazur studied the existence of congruences between certain eisenstein series and newforms, proving that eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal.
in this talk, we'll explore generalizations of mazur's result to squarefree level, focusing on recent work, joint with p. wake and c. wang-erickson, about a non-optimal level n that is the product of two distinct primes and where the galois deformation ring is not expected to be gorenstein. first, we will outline a galois-theoretic criterion for the deformation ring to be as small as possible, and when this criterion is satisfied, deduce an r=t theorem. then we'll discuss some of the techniques required to computationally verify the criterion.

it is rumoured that zariski was rather annoyed with grothendieck, since prior to the latter's introduction of sheaves and cohomology into algebraic geometry, only zariski could do the subject whereas subsequntly any idiot could. equally, much of grothendieck's enterprise is purely categorical so we could reasonably expect similar results in, for example, dynamical systems. an example in the litterature that was crying out for such an interpretation was adam epstein's various generalisations of thurston's rigidity of post critically finite rational maps, which we'll put, and improve, in their natural context of a dynamical topus. en passant this also leads to some highly clarifying results about real blow ups.