Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, February 9, 2016.

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Tuesday, February 9, 2016

11:00 am in 345 Altgeld Hall,Tuesday, February 9, 2016

The EHP sequence in A1 algebraic topology

Ben Williams (UBC)

Abstract: The classical EHP sequence is a partial answer to the question of how far the unit map of the loop-suspension adjunction fails to be a weak equivalence. It can be used to move information from stable to unstable homotopy theory. I will explain why there is an EHP sequence in A1 algebraic topology, and what implications this has for the unstable A1 homotopy groups of spheres.

12:00 pm in Altgeld Hall 243,Tuesday, February 9, 2016

L^2-torsion of free-by-cyclic groups

Matthew Clay (University of Arkansas)

Abstract: I will provide an upper bound on the $L^2$-torsion of a free-by-cyclic group, $-p^{(2)}(G_\Phi)$, in terms of a relative train-track representative for $\Phi$ in Aut(F). This result shares features with a theorem of Luck-Schick computing the $L^2$-torsion of the fundamental group of a 3-manifold that fibers over the circle in that it shows that the $L^2$-torsion is determined by the exponential dynamics of the monodromy. In light of the result of Luck-Schick, a special case of this bound is analogous to the bound on the volume of a 3-manifold that fibers over the circle with pseudo-Anosov monodromy by the normalized entropy recently demonstrated by Kojima-McShane.

1:00 pm in 243 Altgeld Hall,Tuesday, February 9, 2016

Instability of steep ocean waves and whitecapping

Sergey Dyachenko (UIUC Math)

Abstract: Wave breaking in deep oceans is a challenge that still defies complete scientific understanding. Sailors know that at wind speeds of approximately 5m/sec, the random looking windblown surface begins to develop patches of white foam (’whitecaps’) near sharply angled wave crests. We idealize such a sea locally by a family of close to maximum amplitude Stokes waves and show, using highly accurate simulation algorithms based on a conformal map representation, that perturbed Stokes waves develop the universal feature of an overturning plunging jet. We analyze both the cases when surface tension is absent and present. In the latter case, we show the plunging jet is regularized by capillary waves which rapidly become nonlinear Crapper waves in whose trough pockets whitecaps may be spawned.

1:00 pm in 345 Altgeld Hall,Tuesday, February 9, 2016

Borel equivalence relations and cardinal algebras (Part II) by Alexander S. Kechris and Henry L. Macdonald

Anton Bernshteyn (UIUC)

Abstract: For a Borel equivalence relation $E$ on a Borel space $X$, let $nE$ denote the direct sum of $n$ copies of $E$, i.e., the equivalence relation $F$ on $X \times \{1, \ldots, n\}$ such that $(x, i)F(y,j)$ if and only if $i = j$ and $xEy$. Suppose that $E$ and $F$ are countable Borel equivalence relations such that $nE$ is Borel reducible to $nF$. Does this imply that $E$ is Borel reducible to $F$? It turns out that the answers to this and several other questions about the Borel reducibility preorder can be obtained purely algebraically using the theory of cardinal algebras. This theory was developed by Tarski in the late 1940s in order to study the properties of cardinal addition in the absence of the full Axiom of Choice and has been largely forgotten since then. In this series of talks we will survey Tarski's results on cardinal algebras and see how they transfer to the Borel reducibility setting.

2:00 pm in 347 Altgeld Hall,Tuesday, February 9, 2016

The stochastic area and Hopf fibrations

Jing Wang   [email] (UIUC Math)

Abstract: In this talk we study the stochastic area processes that are associated with the Brownian motions on complex symmetric spaces $\mathbb{CP}^n$ and $\mathbb{CH}^n$, and obtain the limiting laws of these processes. The connections to the Brownian motions on $\mathbb{CR}$ sphere $\mathbb{S}^{2n+1}$ and anti-de Sitter space $\mathbb{H}^{2n+1}$ through Hopf fibrations are also obtained. This is a joint work with F. Baudoin.

3:00 pm in 241 Altgeld Hall,Tuesday, February 9, 2016

The Multicolour Ramsey Number of a Long Odd Cycle

Jozef Skokan   [email] (Department of Mathematics London School of Economics)

Abstract: Abstract: For a graph $G$, the $k$-colour Ramsey number $R_k(G)$ is the least integer $N$ such that every $k$-colouring of the edges of the complete graph $K_N$ contains a monochromatic copy of $G$. Let $C_n$ denote the cycle on $n$ vertices. We show that for fixed $k\ge 3$ and $n$ odd and sufficiently large, $$R_k(C_n)=2^{k-1}(n-1)+1.$$ This generalises a result of Kohayakawa, Simonovits and Skokan and resolves a conjecture of Bondy and Erd\H{o}s for large $n$. We also establish a surprising correspondence between extremal $k$-colourings for this problem and perfect matchings in the hypercube $Q_k$. This allows us to in fact prove a stability-type generalisation of the above. The proof is analytic in nature, the first step of which is to use the Regularity Lemma to relate this problem in Ramsey theory to one in convex optimisation. This is joint work with Matthew Jenssen.

3:00 pm in 243 Altgeld Hall,Tuesday, February 9, 2016

Factorization structures on the Hilbert scheme of points of a smooth variety

Emily Cliff (University of Oxford)

Abstract: Let X be a smooth complex variety of arbitrary dimension d. We define the notion of a factorization algebra over X, as well as its non-linear analogue, a factorization space. We also introduce chiral algebras over X, which are equivalent to factorization algebras. Both of these objects are geometric versions of vertex algebras. We show how we can use the Hilbert scheme of points of X to construct a factorization space on X, and hence, by linearizing, a factorization algebra or equivalently a chiral algebra on X. We conclude with a discussion of some of the properties of this factorization algebra.

4:00 pm in 243 Altgeld Hall,Tuesday, February 9, 2016

Notorious LPC: computing Lichnerowicz-Poisson cohomology

Melinda Lanius (UIUC Math)

Abstract: Poisson cohomology is an important invariant in the study of poisson structures. Unfortunately, the computation of Poisson cohomology is quite difficult in general and explicit results are known in only very select cases. We will present a new way of computing Poisson cohomology for a particular class of minimally degenerate Poisson structures that is inspired by the isomorphism between a symplectic form and a non-degenerate Poisson bi-vector.