Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, November 28, 2017.

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Tuesday, November 28, 2017

11:00 am in 345 Altgeld Hall,Tuesday, November 28, 2017

Colimits, descent and equifibrant replacement

Egbert Rijke (CMU Philosophy)

Abstract: Homotopy type theory is an emerging field of mathematics, based on Martin-Löf's constructive theory of types. We think of types as spaces, and type families as fibrations. With the addition of the univalence axiom and higher inductive types doing homotopy theory in type theory (and in a proof assistant!) then becomes feasible. (Reflexive) coequalizers can be used to define a many homotopy colimits in type theory. The case of reflexive coequalizers is interesting because classically the topos of reflexive graphs is cohesive over the topos of sets. I will present analogous results in homotopy type theory.

12:00 pm in 243 Altgeld Hall,Tuesday, November 28, 2017

On Cayley and Langlands type correspondences for Higgs bundles

Laura Schaposnik   [email] (UIC)

Abstract: The Hitchin fibration is a natural tool through which one can understand the moduli space of Higgs bundles and its interesting subspaces (branes). After reviewing the type of questions and methods considered in the area, we shall dedicate this talk to the study of certain branes which lie completely inside the singular fibres of the Hitchin fibrations. Through Cayleyand Langlands type correspondences, we shall provide a geometric description of these objects, and consider the implications of our methods in the context of representation theory, Langlands duality, and within a more generic study of symmetries on moduli spaces.

1:00 pm in 345 Altgeld Hall,Tuesday, November 28, 2017

Differential-henselizations and maximality

Nigel Pynn-Coates (Illinois Math)

Abstract: Henselian fields play an important role in valuation theory and the model theory of valued fields. Two useful results are that every valued field has a henselization, and that henselianity is equivalent to algebraic maximality (at least in equicharacteristic 0). I will state my analogous results for asymptotic valued differential fields (where "differential" is put in the right places) and hope to say something about the proof and possible applications in ongoing work.

1:00 pm in 347 Altgeld Hall,Tuesday, November 28, 2017

Decoupling for Parsell-Vinogradov Systems

Ruixiang Zhang (IAS)

Abstract: Parsell-Vinogradov (P-V) systems are "high dimensional" generalizations of the relevant system in Vinogradov's Mean Value Theorem (VMVT) and can be generalized to the so-called translation-dilation-invariant (TDI) systems. For P-V systems, there is an easy lower bound for the number of integer solutions inside a box. It is conjectured that this lower bound is also an upper bound with at most an $N^{\varepsilon}$-loss. In the VMVT case this was proved by Wooley and Bourgain-Demeter-Guth. Two parallel theories, namely efficient congruencing and decoupling, have been developed to attack counting questions for such systems. I'll present the decoupling approach in this talk, starting with the Bourgain-Demeter-Guth work on the one-dimensional system in VMVT case. Then I'll explain some new phenomena in high dimensional setting. We will see how the new difficulties are largely related to combinatorics and explain how we can address them. We hope the framework here will also work for general TDI systems. This is joint work in progress with Shaoming Guo.

2:00 pm in 347 Altgeld Hall,Tuesday, November 28, 2017

Spine decompositions and limit theorems for a class of critical superprocesses

Zhenyao Sun (Peking University, China)

Abstract: In this talk, we first discuss a spine decomposition theorem and a 2-spine decomposition theorem for some critical superprocesses. These two kinds of decompositions are unified as a decomposition theorem for size-biased Poisson random measures. Then we use these decompositions to give probabilistic proofs of the asymptotic behavior of the survival probability and Yaglom's exponential limit law for some critical superprocesses. This is a joint work with Yan-Xia Ren and Renming Song.

2:00 pm in 241 Altgeld Hall,Tuesday, November 28, 2017

Polynomial methods in number theory

Dong Dong   [email] (UIUC)

Abstract: Many recent breakthroughs in number theory, combinatorics, harmonic analysis, and PDE involve "polynomial method". What is a polynomial method? In fact, there are different types of polynomial methods. In this talk, I will introduce one simple but extremely powerful such method (other types will be introduced in subsequential talks). Knowledge of Taylor expansion (or mean value theorem) is required to attend this talk.

2:00 pm in 241 Altgeld Hall,Tuesday, November 28, 2017

Polynomial methods in number theory

Dong Dong   [email] (UIUC)

Abstract: Many recent breakthroughs in number theory, combinatorics, harmonic analysis, and PDE involve "polynomial method". What is a polynomial method? In fact, there are different types of polynomial methods. In this talk, I will introduce one simple but extremely powerful such method (other types will be introduced in subsequential talks). Knowledge of Taylor expansion (or mean value theorem) is required to attend this talk.

3:00 pm in 241 Altgeld Hall,Tuesday, November 28, 2017

Hypergraphs not containing a given tight tree with a small core

Alexandr Kostochka (Illinois Math)

Abstract: An $r$-uniform hypergraph ($r$-graph, for short) is a tight $r$-tree if its edges can be ordered $e_1,...,e_m$ so that for every $i$ edge $e_i$ contains a vertex $v_i$ that does not belong to any preceding edge and the set $e_i-v_i$ lies in some preceding edge. An $r$-graph $G$ is $H$-free for an $r$-graph $H$ if $G$ does not contain $H$. A long-standing conjecture of Kalai generalizing the Erdős-Sós Conjecture for trees, asserts that for all $r\geq 2$ if $T$ is a tight $r$-tree with $t$ edges and $G$ is an $n$-vertex $T$-free $r$-graph then $e(G)\leq \frac{t-1}{r}{n \choose r-1}$. We establish an asymptotic (in $|T|$) form of Kalai's Conjecture for $r$-trees $T$ that have a bounded size core, i.e. a tight subtree $T'$ of $T$ such that each edge in $T-T'$ shares $r-1$ vertices with an edge in $T'$. We also prove the conjecture for a class of tight $3$-trees. This is joint work with Zoltan Füredi, Tao Jiang, Dhruv Mubayi and Jacques Verstraёte.

3:00 pm in 345 Altgeld Hall,Tuesday, November 28, 2017

Pareto Optimality and Robust Optimisation

Vali Asimit (Cass Business School, London, UK)

Abstract: The optimal reinsurance contract has been a topic of great interest for many decades in the insurance literature. The primary insurer aims to share the risk with one or many other reinsurers for a premium. Finding the optimal contract has been a topic for a long time in insurance economics literature. Depending on how the problem is put, the solutions are different, but layer reinsurance appears to be optimal contract, which confirms the existing empirical evidence. In this talk, we will focus on finding the optimal contract from the perspective of primary reinsurer or from the perspective of all insurers. Interestingly, all solutions are optimal Pareto when all risk preferences have the translation invariance property. The robustness of the reinsurance contract is also discussed. Finally, we will show that introducing of some constraints, the optimal contract is no longer a layer reinsurance and instead, a proportional reinsurance contract appears to be the optimal one.

3:00 pm in Altgeld Hall 243,Tuesday, November 28, 2017

Higher rank Clifford indices for curves on K3 surfaces

Chunyi Li (University of Warwick)

Abstract: The Clifford index $Cliff_1(C)$ of curve $C$ is the second most important invariant of $C$ after the genus, measuring the complexity of the curve in its moduli space. The celebrated work by Lazarsfeld indicates that $Cliff_1(C)$ is $g-1-[g/2]$ when $C\in |H|$ is on a polarized K3 surface $(X,H)$. Inspired by the work of Mercat, an adequate generalization $Cliff_r(C)$ for higher rank vector bundles has been defined by Lange and Newstead. Via the tool of Bridgeland stability condition, for curves on generic K3 surfaces we compute that $Cliff_r(C)=2(g-1-[g/r])/r$, when $g\geq r^2\geq 4$. In the talk, I will explain more details on this classical topic and how does stability condition help to solve them. This is a joint work with S. Feyzbakhsh.