Department of

Mathematics

Seminar Calendar
for events the day of Tuesday, November 1, 2016.

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events for the
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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, November 1, 2016

11:00 am in 345 Altgeld Hall,Tuesday, November 1, 2016

Homotopy probability theory on a Riemannian manifold

Gabriel Drummond-Cole (IBS Center for Geometry and Physics, Pohang, S. Korea)

Abstract: Homotopy probability theory is a homological enrichment of algebraic probability theory, a toy model for the algebra of observables in a quantum field theory. I will introduce the basics of the theory and use it to describe a reformulation of fluid flow equations on a compact Riemannian manifold.

12:30 pm in 464 Loomis Laboratory,Tuesday, November 1, 2016

Arpan Bhattacharyya (Fudan University Physics)

Abstract: In this talk I will explore recently proposed tensor networks/AdS correspondence based on the quantum error correcting model. Coxeter group is a useful tool to describe tensor networks in a negatively curved space. Studying generic tensor network populated by perfect tensors, we will find that the physical wave function generically do not admit any connected correlation functions of local operators. As we know computation of correlation functions plays an important role in establishing AdS/CFT duality, so we will like to remedy this problem. In this talk, I will give a proposal for computing connected two and three point functions and we will see there is a possibility of the emergence of the analogues of Witten diagrams in the tensor network. Using the Coxeter construction, I will describe how to construct the tensor network counterpart of the BTZ black hole, by orbifolding the discrete lattice on which the network resides and show that the construction naturally reproduces some of the salient features of the BTZ black hole, such as the appearance of RT surfaces that could wrap the horizon, depending on the size of the entanglement region. Also I will briefly mention how to construct the bulk isometry generators in terms of Coxeter tessellations thereby taking a step towards establishing the bulk/boundary symmetry matching and solidifying the AdS/CFT dictionary.

1:00 pm in 345 Altgeld Hall,Tuesday, November 1, 2016

A local Ramsey theory for block sequences

Iian Smythe (Cornell Math)

Abstract: Gowers proved an approximate Ramsey theorem for analytic partitions of the space of block sequences in a Banach space. Exact, discretized, versions of this result were later given by Rosendal. We isolate the combinatorial properties of the space of block sequences which enable these constructions and prove that they can be carried out within certain subfamilies, analogous to selective coideals and the role they play in Mathias’ local form of Silver’s theorem for analytic partitions of $[\mathbb{N}]^\infty$. We consider applications of these results to understanding the combinatorial structure of projections in Calkin algebra. Under large cardinal assumptions, these results are extended to partitions in $\mathbf{L}(\mathbb{R})$.

1:00 pm in 7 Illini Hall,Tuesday, November 1, 2016

An introduction to Vinogradov's mean value theorem

George Shakan   [email] (UIUC Math)

Abstract: Recent progress in harmonic analysis (Bourgain, Demeter and Guth) has given rise to a proof of the 80 year old open problem: the so-called Vinogradov mean value theorem. First, I'll lay down the principles that allow one to study number theoretic problems with analysis (Fourier). Then I will attempt to give multiple perspectives for the appearance of the critical exponent. If time permits, I will talk about how the the theorem can be interpreted as a stronger form of orthogonality. What I plan to talk about is contained in my most recent blog post found at gshakan.wordpress.com.

2:00 pm in 241 Altgeld Hall,Tuesday, November 1, 2016

An adaptation of the Langlands Correspondence in the contexts of Geometry and Quantum Physics

Georgios Kydonakis (UIUC )

Abstract: The Langlands Program emerged in the late 1960s as a web of conjectures relating deep questions in number theory, algebraic geometry and the theory of automorphic forms. While in the case of number fields the Langlands Correspondence is still a conjecture (except for special cases), in the case of function fields it is a theorem. A geometric reformulation of the correspondence was suggested in the 1980s, which would make sense over algebraic curves defined over $\mathbb{C}$, i.e. Riemann Surfaces. More recently, a relation between Langlands duality in Mathematics and S-duality in Quantum Field Theory has been established. We will discuss elements of these adaptations through the prism of "analogy" in Mathematics, the way A. Weil described it in his famous 1940 letter from jail.

3:00 pm in 241 Altgeld Hall,Tuesday, November 1, 2016

Reconstruction from $k$-decks for graphs with maximum degree 2

Hannah Spinoza (Illinois Math)

Abstract: The $k$-deck of a graph is its multiset of induced subgraphs on $k$ vertices. In Problem 11898 of the American Mathematical Monthly, Richard Stanley posed a question that begins to suggest the difficulty of reconstructing $2$-regular graphs from their $k$-decks. We prove that $n$-vertex graphs with maximum degree $2$ have the same $k$-decks if each cycle has at least $k+1$ vertices, each path component has at least $k-1$ vertices, and the number of edges is the same. Using this for lower bounds, we obtain for each graph with maximum degree at most $2$ the least $k$ such that it is determined by its $k$-deck. For the $n$-vertex cycle this value is $\lfloor n/2 \rfloor$, and for the $n$-vertex path it is $\lfloor n/2 \rfloor+1$. This is joint work with Douglas West

3:00 pm in 243 Altgeld Hall,Tuesday, November 1, 2016

An action of the cactus group on crystals

Iva Halacheva (University of Lancaster)

Abstract: Any Lie algebra g which is complex, finite-dimensional, and semisimple has an associated group J(g) built out of its Dynkin diagram, and known as the cactus group. Another type of objects related to g are crystals, each encoding the information of a corresponding g-representation. We describe two realizations of an action of the cactus group on any g-crystal. The first is combinatorial, via the so called Schützenberger involutions. The second is geometric, and comes from the monodromy action for a De Concini-Procesi moduli space, induced by a family of maximal commutative subalgebras in U(g).

4:00 pm in 1 Illini Hall,Tuesday, November 1, 2016

Stable Risk Sharing and Its Monotonicity

Xin Chen (Industrial and Enterprise Systems Engineering, UIUC)

Abstract: We consider a risk sharing problem in which agents pool their random costs together and seek an allocation rule to redistribute the risk back to each agent. The problem is put into a cooperative game framework and we focus on two salient properties of an allocation rule: stability and monotonicity employing concepts of core and population monotonicity from cooperative game theory. When the risks of the agents are measured by coherent risk measures, we construct a risk allocation rule based on duality theory and establish its stability. When restricting the risk measures to the class of distortion risk measures, the duality-based risk allocation rule is population monotonic if the random costs are independent and log-concave. For the case with dependent normally distributed random costs, a simple condition on the dependence structure is identified to ensure the monotonicity property.