Mathematics

Interface contributions to topological entanglement in abelian Chern-Simons theory

Jackson Fliss (University of Illinois at Urbana Champaign)

Abstract: Abstract: In this talk I will discuss the entanglement entropy between two (possibly) distinct topological phases in Abelian Chern-Simons theory. At the interface between the phases, two issues must be addressed: (i) what are the boundary conditions that correspond to the interface being gapped?, and (ii) how does one define entanglement in continuum gauge theories where the Hilbert space typically does not admit a tensor product factorization? For the former question it is known that gapped interfaces are described by a class of boundary conditions known as topological boundary conditions (TBCs). I will describe how TBCs also address the latter question by isolating a unique gauge invariant state in the extended Hilbert space approach. I will show that upon computing the entanglement entropy, the universal correction to the area law retains a dependence on the choice of TBCs. This result matches previous microscopic calculations found in the condensed matter literature. Additionally, when the phases across the interface are taken to be identical, this construction provides a novel explanation of the equivalence between the left-right entanglement of (1+1)d Ishibashi states and the spatial entanglement of (2+1)d topological phases.

Visualizing Data Quality and Integrity in Clinical Trials: Clinical Data Reviewer (CDR)

Numan Karim (AbbVie Pharmaceutical Research and Development)

Abstract: AbbVie is a global, research driven, bio-pharmaceutical company tackling the world’s toughest health challenges. As a company we have over a hundred ongoing clinical trials at any given time. With such a large scale of clinical trials in conduct, there is a dire need to develop and deploy data and statistics driven solution to ensure clinical data quality. The Clinical Data Reviewer (CDR) is a new, yet integral role within Data and Statistical Sciences (DSS) dedicated to anticipate, troubleshoot, and resolve data integrity issues. Traditionally, data management relied heavily on simple edit checks and manual Excel listings. The release of ICHE6 R2 guidelines for good clinical practice set a new and higher standard of scientific quality for conducting, recording, and reporting clinical trials. This stressed the need for centralized monitoring and aggregate data review. The CDR role utilizes advanced statistical methodologies to perform data review and enable data-driven decisions during the life of a clinical trial. CDRs operate by the following principle: The integrity of the research depends on the integrity of the data. By exploring data trends and behaviors that cannot be captured in programmed edit checks, we ensure the broader integrity of the data. The team leverages TIBCO Spotfire, a data visualization tool, to create generic (demographics, adverse events, pharmacokinetics) and study specific (Endpoints, ePRO, drug accountability) visualizations, at a subject, site, or study level, to identify potential issues. These include, but are not limited to, scatter plots, box plots, heat maps, cross tables, pivot tables, or a combination of the above. CDRs also use R, a traditional statistical programming tool, to wrangle or analyze data and perform more complex checks across multiple data sources. Currently we are exploring univariate, unsupervised Machine Learning techniques to streamline automation of outlier and anomaly detection in Clinical Trials. Together, these techniques support a holistic review of clinical data and ensure the highest quality database from study start up to final lock.

On the three matrix analog of Gerstenhaber's theorem

Jenna Rajchgot (Jenna Rajchgot)

Abstract: In 1960, Gerstenhaber proved that the algebra generated by two commuting dxd matrices has vector space dimension at most d. The analog of this statement for four or more commuting matrices is easily seen to be false. The three matrix version remains open. After providing some history and context, I'll translate this three commuting matrix statement into an equivalent statement about certain maps between modules, and prove that this commutative-algebraic reformulation is true in special cases. I'll end with some combinatorial questions about three dimensional analogs of Young diagrams, which, if answered, would settle the three matrix analog of Gerstenhaber's theorem for other infinite families of examples. This is joint work with Matthew Satriano.

Reed-Muller Codes Achieve Capacity on Erasure Channels

Hsin-Po Wang (UIUC)

Abstract: We will talk about this https://arxiv.org/abs/1601.04689. Reed-Muller Codes generalize Reed–Solomon codes (used on CD/DVD/etc) and Hamming codes (used on RAM/etc). For such a family of codes one may ask whether it asymptotically achieves the capacity in the sense of Shannon's information theory. The answer is yes, on certain channels, and we will go through the ideas in the paper.

Abelian and non-abelian BF theory on cobordisms endowed with cellular decomposition

Pavel Mnev (Notre Dame)

Abstract: We will present an example of a topological field theory living on cobordisms endowed with CW decomposition (this example corresponds to the so-called BF theory in its abelian and non-abelian variants), which satisfies the Batalin-Vilkovisky master equation, satisfies (a version of) Segal's gluing axiom w.r.t. concatenation of cobordisms and is compatible with cellular aggregations. In non-abelian case, the action functional of the theory is constructed out of local unimodular L-infinity algebras on cells; the partition function carries the information about the Reidemeister torsion, together with certain information pertaining to formal geometry of the moduli space of local systems. This theory provides an example of the BV-BFV programme for quantization of field theories on manifolds with boundary in cohomological formalism. This is a joint work with Alberto S. Cattaneo and Nicolai Reshetikhin.

Toroidal actions on fermionic q-Fock space II

Joshua Wen (Illinois)

Abstract: Continuing from before, I’ll describe how our two actions on finite q-wedges can each be stabilized in the semi-infinite limit and how they can glued into a level (0,1) action of the quantum toroidal algebra by ‘rotating the Dynkin diagram’. The description of the level 0 action as quantum affine Schur-Weyl duality applied to the polynomial representation of the affine Hecke algebra acting via Macdonald operators allows one to use nonsymmetric Macdonald polynomials to construct interesting bases of q-Fock space. Switching gears, I’ll present Nagao’s work relating the toroidal action on q-Fock space with that on K-theory of cyclic quiver varieties, where such a basis is mapped to the fixed point classes. This assignment of some kind of Macdonald polynomial to a fixed point class in a Nakajima quiver variety should ring a bell for Procesi bundle enthusiasts, and if time permits, I’ll present possible directions along that thread.

Applications of functoriality

Emily Riehl (Johns Hopkins University)

Abstract: Abstract: This talk will introduce the concept of a functor from category theory assuming no prior acquaintance and highlight some applications of this notion. In the first part we will conceive of a functor as a bridge between two different mathematical theories. As a prototypical example we will consider the fundamental group construction from algebraic topology and explain how its functoriality leads to a proof of the Brouwer fixed point theorem. In the second part, we will explain how the search for a functorial clustering algorithm lead to a breakthrough in topological data analysis and speculate how a similar functors might be used to define combinatorial models of metric spaces.

Summer Illinois Math (SIM) Camp Information Session

  [email]

Abstract: The Summer Illinois Math (SIM) Camp is a week long camp hosted by the department, where campers will see the creative, discovery driven side of math. We are looking for graduate students to design and teach courses during camp. These will be paid positions. In 2018, there will be three camps: one for rising 8th and 9th grade students June 25-29, one for rising 9th and 10th grade students July 9-13, and one for rising 10th-12th grade students July 23-27. This information session if for those who are interested in learning more about the instructor position or SIM Camp in general. If you cannot make it, or have other questions, you can email us at math-simcamp@illinois.edu.