Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, April 21, 2015.

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Tuesday, April 21, 2015

11:00 am in 243 Altgeld Hall,Tuesday, April 21, 2015

Rational homology of configuration spaces via factorization homology

Benjamin Knudsen   [email] (Northwestern)

Abstract: The study of configuration spaces is particularly tractable over a field of characteristic zero, and there has been great success over the years in producing complexes simple enough for explicit computations, formulas for Betti numbers, and descriptive results. I will discuss recent work identifying the rational homology of the configuration spaces of an arbitrary manifold with the homology of a Lie algebra constructed from its cohomology. The aforementioned results follow immediately from this identification, albeit with hypotheses removed; in particular, one obtains a new, elementary proof of homological stability for configuration spaces.

11:00 am in 241 Altgeld Hall,Tuesday, April 21, 2015

The overpartition function modulo 16 revived

Xinhua Xiong (China Three Gorges University)

Abstract: I will give a complete determination of the overpartition function modulo 16, this is the generalization of the result of Byungchan Kim on overpartition function modulo 8. Moreover, I will give infinitely many new Ramanujan-like congruences for overpartition function modulo 16.

1:00 pm in Altgeld Hall 243,Tuesday, April 21, 2015

Homogeneous dynamics and applications to number theory

Jayadev Athreya (UIUC Math)

Abstract: To set the stage for Margulis' Tondeur lectures, I will survey some problems in homogeneous dynamics and their relationship to number theory. I will focus on the Oppenheim and Littlewood conjectures.

1:00 pm in 347 Altgeld Hall,Tuesday, April 21, 2015

Manifold decompositions and indices of Schrödinger operators

Graham Cox (UNC Chapel Hill)

Abstract: When finding the eigenvalues of a differential operator, it is often convenient to partition the spatial domain and then compute the spectrum on each component. This is useful when the operator has localized structure, such as a compactly supported defect superimposed over a known background. In this case the partition decouples the defect and the background. Then one must determine how the spectrum on the original domain is related to the spectra on the subdomains. In this talk I will describe such spectral decompositions using the Maslov index, a generalized winding number for paths of Lagrangian subspaces. In particular, the Morse index (number of negative eigenvalues) of the original boundary value problem is given by the sum of the Morse indices on each subdomain plus a "coupling term" that depends on the Dirichlet-to-Neumann maps for the common boundary. An immediate corollary is the nodal domain theorem of Courant, with an explicit formula for the nodal deficiency. I will also discuss applications to periodic boundary conditions and Morse decompositions of manifolds.

1:00 pm in 345 Altgeld Hall,Tuesday, April 21, 2015

Hrushovski constructions and green fields

Travis Nell (UIUC)

Abstract: We will consider work of Caycedo and Zilber giving an analytic example of a green field. A green field is a structure $(F,+,\cdot,0,1,G)$, where $F$ is an algebraically closed field and $G$ is a predicate for a divisible multiplicative subgroup of $F$, where the Morley rank of the whole structure is $\omega*2$, while the Morley rank of $G$ is $\omega$. These fields were originally constructed by Poizat with a Hrushovski construction. We will consider this construction, then give some remarks about how Caycedo and Zilber's example models the theory of such an object.

2:00 pm in 347 Altgeld Hall,Tuesday, April 21, 2015

On the chemical distance in critical percolation

Philippe Sosoe (Harvard University)

Abstract: In two-dimensional critical percolation, the works of Aizenman-Burchard and Kesten-Zhang imply that macroscopic distances inside percolation clusters are bounded below by a power of the Euclidean distance greater than (1+\epsilon), for some positive $\epsilon$. No more precise lower bound has been given so far. Conditioned on the existence of an open crossing of a box of side length $n$, there is a distinguished open path which can be characterized in terms of arm exponents: the lowest open path crossing the box. This clearly gives an upper bound for the shortest path. The lowest crossing was shown by Zhang and Morrow to have volume $n^{4/3+o(1)}$ on the triangular lattice. We compare the length of shortest circuit in an annulus to that of the innermost circuit (defined analogously to the lowest crossing), and show that the ratio of the expected length of the shortest circuit to the expected length of the innermost crossing tends to zero as the size of the annulus grows. Our methods also allow us to show that, conditioned on the existence of a crossing, the length of the shortest crossing divided by that of the lowest crossing tends to zero in probability, answering a question of Kesten and Zhang. Joint work with Jack Hanson and Michael Damron.

3:00 pm in 241 Altgeld Hall,Tuesday, April 21, 2015

Largest families of sets, under conditions defined by a given poset

Gyula O.H. Katona   [email] (MTA Rényi Institute, Budapest)

Abstract: Let $[n]=\{ 1,2,\ldots , n\}$ and let $\mathcal{F} \subset 2^{[n]}$ be a family of its subsets. Sperner proved in 1928 that if $\mathcal{F}$ contains no pair of members in inclusion that is $F_1, F_2\in \mathcal{F}$ implies $F_1\not\subset F_2$ then $|\mathcal{F}|\leq \binom{n}{\lfloor n/2\rfloor}$. Inspired by an application, Erdős generalized this theorem in the following way. Suppose that $\mathcal{F}$ contains no $k+1$ distinct members $F_1\subset F_2\subset \ldots \subset F_{k+1}$ then $|\mathcal{F}|$ is at most the sum of the $k$ largest binomial coefficients of order $n$. This bound is, of course, tight. We will survey results of this type: determine the largest family of subsets of $[n]$ under a certain condition forbidding a given configuration described solely by inclusion among the members. This maximum is denoted by La$(n,P)$ where $P$ is the forbidden configuration. The final (hopelessly difficult) conjecture is that La$(n,P)$ is asymptotically equal to the sum of the $k$ largest binomial coefficients where the $k$ largest levels contain no configuration $P$, but $k+1$ levels do. Another related problem is the following one. A copy of the given poset $P$ is a family $\mathcal{F}$ of subsets of $[n]$ where the embedding $f: P \rightarrow \mathcal{F}$ maps comparable elements of $P$ into comparable subsets. Two copies $\mathcal{F}_1, \mathcal{F}_2$ of $P$ are incomparable if no member of $\mathcal{F}_1$ is a subset of a member of $\mathcal{F}_2$ and no member of $\mathcal{F}_2$ is a subset of a member of $\mathcal{F}_1$. The maximum number of incomparable copies of $P$ is denoted by LA$(n.P)$. This quantity is asymptotically determined for all $P$, unlike La$(n,P)$.

4:00 pm in 314 Altgeld Hall,Tuesday, April 21, 2015

Oppenheim conjecture and related problems

Gregory Margulis (Yale)

Abstract: Students should attend the Tondeur Lectures being presented by Gregory Margulis (Yale) April 21-23.

4:00 pm in 314 Altgeld Hall,Tuesday, April 21, 2015

Oppenheim conjecture and related problems

Gregory Margulis (Yale)

Abstract: The Oppenheim conjecture proved in mideighties states that the set of values at integral points of an irrational indefinite quadratic form in n>2 variables is dense in R. I will talk about the history and proof of the conjecture and will also talk about various problems related to the conjecture. Video of lecture at https://youtu.be/joe7vROMG-c

A reception will be held following this lecture in 239 Altgeld Hall from 5 to 6:30 pm.