Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, April 20, 2017.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, April 20, 2017

11:00 am in 241 Altgeld Hall,Thursday, April 20, 2017

#### Unexpected biases in the distribution of consecutive primes

###### Robert Lemke Oliver (Tufts University)

Abstract: While the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of consecutive primes among the permissible pairs of reduced residue classes (mod q) is surprisingly erratic. We propose a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures, which fits the observed data very well. We also study the distribution of the terms predicted by the conjecture, which proves to be surprisingly subtle. This is joint work with Kannan Soundararajan.

12:00 pm in 243 Altgeld Hall,Thursday, April 20, 2017

#### Special subgroups of Bianchi groups

###### Michelle Chu (UT Austin Math)

Abstract: The study of (virtually) special cube complexes and groups played a key role in the resolution of the virtual Haken and virtual fibering conjectures. Recently there has been interest in determining the index of special subgroups of virtually special groups. In this talk, we answer this question in the case of the Bianchi groups. We find special congruence subgroups for each Bianchi group $PSL(2,O_d)$ with uniformly bounded index independent of d.

12:30 pm in 464 Loomis Laboratory,Thursday, April 20, 2017

#### Bulk locality from modular flow

###### Aitor Lewkowycz (Stanford Physics)

1:00 pm in 243 Altgeld Hall,Thursday, April 20, 2017

#### Compressed Learning

###### Marius Junge

Abstract: This is an introduction on machine learning for SVM's with compressed sensing.

1:00 pm in 347 Altgeld Hall,Thursday, April 20, 2017

#### The eye as a window on the body: mathematical modeling of ocular biomechanics, fluid-dynamics and oxygenation

###### Giovanna Guidoboni

Abstract: The eye is the only place in the human body where blood flow and systemic vascular features can be observed and measured easily and non-invasively down to the capillary level. Numerous clinical studies have shown correlations between alterations in ocular blood flow and ocular diseases (e.g. glaucoma, age-related macular degeneration, diabetic retinopathy), neurodegenerative diseases (e.g. Alzheimer’s disease, Parkinson’s disease) and other systemic diseases (e.g. hypertension, diabetes). Thus, deciphering the mechanisms governing ocular blood flow could be the key to the use of eye examinations as a non-invasive approach to the diagnosis and continuous monitoring for many patients. However, many factors influence ocular hemodynamics, including intraocular pressure (IOP), blood pressure and blood flow autoregulation, and it is extremely challenging to single out their individual contributions during clinical and animal studies. In the recent years, we have been developing mathematical models and computational methods to aid the interpretation of clinical data. In this talk, we will present models describing (i) the blood flow in the ocular macro- and micro-vasculature, accounting for the IOP-induced deformation of the vessel walls; (ii) the regulation of blood flow in the retina, accounting for the myogenic, shear-stress, carbon dioxide (CO2) and oxygen (O2) responses, as well as the role of nitric oxide (NO) in mediating neural signals to the vessel walls; (iii) O2 transport, diffusion and consumption in the retinal vasculature and tissue. Results will show how the synergy between mathematical modeling and clinical data allowed us to estimate the relative contribution of IOP, blood pressure and blood flow autoregulation on ocular tissue perfusion and vessel mechanics, and to distinguish disease mechanisms in different subgroups of glaucoma patients. Current efforts in translating this multiscale/multiphysics modeling methods into clinical resources to be used for individualized approaches to disease diagnosis and treatment will also be discussed.

2:00 pm in 241 Altgeld Hall,Thursday, April 20, 2017

#### Ranks of elliptic curves, Selmer groups, and Tate-Shafarevich groups

###### Robert Lemke Oliver (Tuft University)

Abstract: A big problem in number theory is how to access the rank of an elliptic curve, i.e. the minimal number of points needed to generate the full set of rational points. Assuming the generalized Riemann hypothesis and the Birch and Swinnerton-Dyer conjectures, an algorithm exists that will determine the rank of any specific elliptic curve, but this says nothing about what ranks are typically like. While an analytic mindset is useful for thinking about how ranks "should" behave, almost all actual theorems, from Mordell-Weil to the recent work of Bhargava and Shankar, passes through an algebraic gadget called the Selmer group. This is given by a somewhat complicated definition in terms of Galois cohomology, which is intimidating and unilluminating for people who are more comfortable with classical analytic number theory and L-functions. This talk will aim to make Selmer groups somewhat less mystifying, and along the way we will discuss some of the speaker's forthcoming work with Bhargava, Klagsbrun, and Shnidman.

2:00 pm in 243 Altgeld Hall,Thursday, April 20, 2017

#### A Hilbert bundle description of differential K-theory

###### Alexander Gorokhovsky (University of Colorado Boulder)

Abstract: We give a description of differential K-theory in terms of infinite dimensional Hilbert bundles. As an application we propose a construction of twisted differential K-theory. This is a joint work with J. Lott.

3:00 pm in 347 Altgeld Hall,Thursday, April 20, 2017

#### A Murnaghan-Nakayama rule for quantum cohomology of the flag manifold

###### Carolina Benedetti (Fields Institute and York University)

Abstract: Given k less than n and a hook \lambda inside a k,(n-k) box, Mészáros et. al. made use of right operators to provide a rule for the expansion of the quantum Schur polynomial s_{\lambda} in term of generators in the quantum Fomin-Kirillov algebra. In this talk, we will make use of Mészáros et. al. result to provide a different combinatorial interpretation of such expansion, using left operators. As a consequence, we will derive a combinatorial rule for the expansion of quantum power-sum polynomials. This is current work with N. Bergeron, L. Colmenarejo, F. Saliola, F. Sottile.

4:00 pm in 245 Altgeld Hall,Thursday, April 20, 2017

#### Tarski numbers

###### Mark Sapir (Centennial Professor, Vanderbilt University, and George A. Miller Visiting Professor, Univ. of Illinois)

Abstract: It is known since Hausdorff, Banach and Tarski that one can decompose a 2-sphere into 4 pieces, move the pieces using rotations of the sphere, and obtain two spheres of the same radius (assuming the Axiom of Choice). Thus a sphere with the group of rotations acting on it has a paradoxical decomposition with 4 pieces. In general if we have a group G acting on a set X, then the Tarski number of X is the minimal number of pieces in a paradoxical decomposition of X. For example, if G acts on itself by left multiplication, then we can talk about the Tarski number of G. I will show how to use Golod-Shafarevich groups to prove that the set of possible Tarski numbers of groups is infinite. I will also show how to use l_2-Betti numbers of groups and cost of group actions to construct groups with Tarski numbers 5 and 6. Note that 4, 5 and 6 are the only numbers that are currently known to be Tarski numbers of groups. This is a joint work with Gili Golan and Mikhail Ershov.

5:00 pm in Alice Campbell Alumni Center ,Thursday, April 20, 2017

#### LAS Teaching Excellence Awards

Abstract: The LAS Teaching Excellence Awards ceremony will be held from 5-7 pm on Thursday, April 20, 2017, at the Alice Campbell Alumni Center.