Department of

# Mathematics

Seminar Calendar
for Graduate Analysis Seminar events the year of Monday, October 18, 2021.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    September 2021          October 2021          November 2021
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Friday, February 12, 2021

2:00 pm in Contact for zoom link,Friday, February 12, 2021

#### The Largest Sum-free Subsets of Integers and its Generalization

###### Shukun Wu (UIUC Math)

Abstract: An old conjecture in additive combinatorics asks: what is the largest sum-free subset of any set of N positive integers? Here the word "largest" should be understood in terms of cardinality. For example, the largest sum-free subset of the first N positive integers has cardinality [(N+1)/2], which is the number of odd integers smaller than N, as well as the number of integers that lie in the interval [N/2,N]. In this talk, I will discuss a special case of the sum-free conjecture and the analogous conjecture on (k,l)-sum-free sets. This is a joint work with Yifan Jing

Friday, February 19, 2021

2:00 pm in Contact for zoom link,Friday, February 19, 2021

#### Expanding Thurston Maps and Visual Spheres

###### Stathis Chrontsios (UIUC Math)

Abstract: Complex Dynamics study iterations of entire functions and/or rational maps. While doing so in the latter case, the properties of the Riemann sphere are being used (e.g. conformal structure). In this talk, the protagonists will be Thurston maps, generalizations of rational maps that are defined on topological spheres. Connections with their complex analytic counterparts will be discussed, as well as questions from a metric-analytic point of view about visual spheres, the metric spaces the dynamics of these maps induce.

Friday, February 26, 2021

2:00 pm in Contact for zoom link,Friday, February 26, 2021

#### Minimizing Eigenvalues of the Laplacian

###### Scott Harman (UIUC Math)

Abstract: pectral theory is concerned with studying the eigenvalues of partial differential operators. The most canonical example is the second order Laplacian . In this talk, we are concerned mostly with minimizing eigenvalues of the Laplacian on a bounded domain with and has zero boundary condition. To phrase it another way, what shape of domain will yield the smallest frequencies? We will give a brief overview of spectral theory, a proof of the Faber-Krahn inequality which states the ball minimizes the first eigenvalue, and conjectures for higher eigenvalues and the shape of the domain in the limiting case.

Friday, March 5, 2021

2:00 pm in Contact for zoom link,Friday, March 5, 2021

#### $L^{2}$ decoupling: An Introduction and Applications

###### Ryan McConnell (UIUC Math)

Abstract: Suppose one wants to decompose a function on into parts that have disjoint Fourier supports. Due to linearity of the Fourier transform, the sum of these parts must be the original function. Suppose, however, that you wish to understand how the norms of these parts relates to the norm of the original function. While this task isn’t so straight forward, Bourgain and Demeter (2015) were able to resolve a conjecture related to this: the $L^{2}$ Decoupling Conjecture.In this talk we will introduce the conjecture, provide applications of it to both PDEs and Number Theory, prove an earlier result of Bourgain for a smaller range of admissible exponents, and (time permitting) comment on how to adapt the ideas presented to prove the full result.

Friday, March 12, 2021

2:00 pm in Contact for zoom link,Friday, March 12, 2021

#### Circle Packings and a discrete Riemann Mapping Theorem

###### Kesav Krishnan (UIUC Math)

Abstract: I will introduce Circle Packings as defined by Thurston, and describe how they provide a discrete analogue of complex analysis and conformal maps. I will also introduce the discrete analogue of the Riemann Mapping Theorem, and if time permits I will talk about how discrete complex function theory plays a crucial role in certain 2-dimensional models in probability theory.

Friday, March 19, 2021

2:00 pm in Contact for zoom link,Friday, March 19, 2021

#### The descriptive complexity of classes of Banach lattices

###### Mary Angelica Gramcko-Tursi (UIUC Math)

Abstract: The descriptive complexity of a class of spaces gives us an initial sense of how "nice" the class is. When a set is not "nice," this fact can also be used to derive results related to the existence of universal spaces. In this talk, we present some examples of Borel and non-Borel classes of Banach lattices and some related results derived from their descriptive complexity. We also show areas where the descriptive complexity results on Banach lattices compare and contrast with previously done analogous work on Banach spaces.

Friday, April 2, 2021

2:00 pm in Contact for zoom link,Friday, April 2, 2021

#### Understanding Cyclicity in Spaces of Analytic Functions

###### Jeet Sampat (Washington University St. Louis)

Abstract: TBA

Friday, September 3, 2021

2:00 pm in 347 Altgeld Hall,Friday, September 3, 2021

#### How Jean Bourgain Revolutionized the Study of Dispersive PDEs

###### Ryan McConnell (UIUC)

Abstract: Around the time that he won the Fields medal ('94) until about 2000, Jean Bourgain made remarkable progress in the theory of well-posedenss and global well-posedness in dispersive PDEs. Paper after paper Professor Bourgain established (for anyone else) career defining results whose effects are still very much felt to this day. In this talk, I want to motivate and go over some of his results from this period with the hope of impressing just how amazing (compared to prior results) some of his work was. The talk will be introductory (in a sense), with the hope of gently kicking off the seminar. As always, there will be cookies.

Friday, September 10, 2021

2:00 pm in 347 Altgeld Hall,Friday, September 10, 2021

#### To Be Announced

###### Scott Harman (UIUC)

Abstract: Scott will talk about his VERY COOL research! Cookies will be provided to those cool enough to attend.

Friday, September 17, 2021

2:00 pm in 347 Altgeld Hall,Friday, September 17, 2021

#### Heuristics and LIES

###### Ryan McConnell (UIUC)

Abstract: In this talk I will discuss several heuristics used in determining "optimal" ranges for well-posedness in dispersive PDEs, which will include several methods for disproving well-posedness. I will then rant about how, much like statistics, people use these heuristics to LIE to their unsuspecting readers. As always, there will be cookies.

Friday, September 24, 2021

2:00 pm in 347 Altgeld Hall,Friday, September 24, 2021

#### You won't BELIEVE what Connor has to say about the Atiyah-Singer Index Theorem

Abstract: In his debut Graduate Analysis Seminar talk, UIUC's very own Connor Grady will be speaking very analytically about one of the COOLEST theorems: The Atiyah-Singer Index Theorem. The Atiyah-Singer Index Theorem is a central theorem in differential geometry, connecting the topology of manifolds to the analysis of differential operators. In this talk, we will be looking at this theorem mostly from the point of view of geometry and topology, while trying to do our best to see why analysts should care about it. (There will be cookies even if you don't care about it.)

Friday, October 1, 2021

2:00 pm in Altgeld Hall,Friday, October 1, 2021

#### How to not do analysis

Abstract: Today we’ll be making simple problems as difficult as possible.

Friday, October 8, 2021

2:00 pm in 347 Altgeld Hall,Friday, October 8, 2021

#### Discrete Methods for the Non Linear Schrödinger Equation

###### Kesav Krishnan (UIUC)

Abstract: The Non Linear Schrödinger Equation (NLS) has been the subject of intense study by the Harmonic Analysis and PDE community. Primarily this is because it is one of the canonical examples of dispersive PDEs that have Soliton solutions. In this talk I will survey results for the discrete analogue of the focusing NLS, and the use of the discrete equation to construct invariant measures. I will conclude with a discussion of Sourav Chatterjee's work on resolving the Soliton resolution conjecture.

Friday, October 29, 2021

2:00 pm in 347 Altgeld Hall,Friday, October 29, 2021

#### Localization and Laplacians

###### Gayana Jayasinghe (UIUC)

Abstract: Localization can be broadly described as a phenomenon where certain integrands turn out to only be supported near special points. The classic result of this form is stationary phase. I will talk about different instances of localization and the analysis involved. I'll begin with stationary phase and move onto Atiyah Bott localization, the Duistermaat-Heckman theorem, and Witten deformation. For more on Witten deformation, check out the Graduate Geometry and Topology Seminar this week.

Friday, November 12, 2021

2:00 pm in 347 Altgeld Hall,Friday, November 12, 2021

#### Functional & Geometric Inequalities: An Introduction

###### Rachel Barnett (UIUC)

Abstract: Abstract: This talk introduces functional and geometric inequalities, opening with a discussion about what they are and key questions that frame their study. We prove the Polya-Szego inequality and then use it to establish the classical Faber-Krahn inequality, which states that balls minimize the first eigenvalue of the Dirichlet Laplacian among all sets of the same volume. We conclude by outlining what a more general Faber-Krahn inequality looks like and how one might go about deriving it. This talk is based on a paper by Brasco and De Phillips (2016).

Friday, November 19, 2021

2:00 pm in 347 Altgeld Hall,Friday, November 19, 2021

#### Prekopa-Leindler inquality and the sharp Sobolev inequality

###### Daesung Kim   [email] (UIUC)

Abstract: Abstract: Prekopa-Leindler inequality is a functional version of Brunn-Minkowski inequality, and has a close relation to many interesting inequalities. We discuss the proof of Prekopa-Leindler inequality using optimal transport technique. As an application, we derive the sharp Sobolev inequality.

Friday, December 3, 2021

2:00 pm in 347 Altgeld Hall,Friday, December 3, 2021

#### The Calderon projector in boundary value problems

###### Karthik Vasu (UIUC)

Abstract: Usually elliptic operators on manifolds with boundary are not Fredholm without any restriction. When some boundary conditions are imposed it does become Fredholm. The goal will be to "parametrize" the solutions using boundary values. The Calderon projector will be a useful tool in determining which boundary conditions are Fredholm and in finding the index.