E-mail: glivshyts6@math.gatech.edu

Address: Room 228, Skiles bldg, located at 686 Cherry street NW, Atlanta, GA, 30332.

- G. V. Livshyts, Maximal surface area of a convex set in Rn with respect to exponential rotation invariant measures, J. Math. Anal. Appl., 404, 2, (2013), 231-238.
- G. V. Livshyts, Maximal Surface Area of a convex set in Rn with respect to log concave rotation invariant measures, GAFA Seminar Notes, 2116, (2014), 355-384.
- G. V. Livshyts, Surface area of polytopes with respect to log-concave rotation invariant measures, Adv. Appl. Math., vol. 70, 54-69, (2015).
- G. Livshyts, A. Marsiglietti, P. Nayar, A. Zvavitch, On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric type-inequalities, to appear in the Transactions of the AMS.
- G. Livshyts, P. Paouris, P. Pivovarov, On sharp bounds for marginal densities of product measures, Israel Journal of Mathematics, (2016), 216(2), 877-889.
- A. Colesanti, G. V. Livshyts, A. Marsiglietti, The infinitesimal form of Brunn-Minkowski type inequalities, to appear in Journal of Functional Analysis.
- G. V. Livshyts, K. Tikhomirov, Randomized coverings of a convex body with its homothetic copies, and illumination, submitted.
- G. V. Livshyts, An extension of Minkowski's theorem and its applications to questions about projections for measures, to appear in Advances in Math.

Abstract. Let p be a positive number. Consider the probability measure with density c e^{− |y|^p/p}. We show that the maximal surface area of a convex body in R^n with respect to this measure is asymptotically equivalent to C(p)n^{3/4-1/p}, where the constant C(p) depends on p only. This is a generalization of results due to Ball (1993) and Nazarov (2003) in the case of the standard Gaussian measure.

Abstract. It was shown by K. Ball and F. Nazarov, that the maximal surface area of a convex set in R^n with respect to the Standard Gaussian measure is of order n^{1/4}. In the present paper we establish the analogous result for all rotation invariant log concave probability measures. Our bound is sharp, and it depends on two natural parameters: expectation and varience of the random vector distributed with respect to the measure.

Abstract. In the present paper we discuss surface area of convex polytopes with K facets. We find tight bounds on their maximal surface area with respect to log-concave rotation invariant measures in terms of K. We show that our bound is sharp up to a log factor.

Abstract. In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality \mu(t A + (1-t)B)^{1/n} \geq t \mu(A)^{1/n} + (1-t)\mu(B)^{1/n} holds true for an unconditional product measure \mu with decreasing density and a pair of unconditional convex bodies A,B in R^n. We also show that the above inequality is true for any unconditional log-concave measure \mu and unconditional convex bodies A,B in R^n. Finally, we prove that the inequality is true for a symmetric log-concave measure \mu and a pair of symmetric convex sets A,B on the plane, which, in particular, settles two-dimensional case of the conjecture for Gaussian measure proposed by R. Gardner and A. Zvavitch.

Abstract. We discuss optimal constants in a recent result of Rudelson and Vershynin on marginal densities. We show that if f is a product probability density on R^n, then the density of any marginal of f is bounded by 2^{k/2}, where k is the dimension of E. The proof relies on an adaptation of Ball's approach to cube slicing, carried out for functions. Motivated by inequalities for dual affine quermassintegrals, we also prove an isoperimetric inequality for certain averages of the marginals of such f for which the cube is the extremal case.

Abstract. Log-Brunn-Minkowski inequality was conjectured by Boroczky, Lutwak, Yang and Zhang, and it states that a certain strengthening of the classical Brunn-Minkowski inequality is admissible in the case of symmetric convex sets. In this note, we obtain stability results for Log-Brunn-Minkowski and dimensional Brunn-Minkowski inequalities for rotation invariant log-conave measures near a ball. Remarkably, the assumption of symmetry is only necessary for Log-Brunn-Minkowski stability, which emphasizes an important difference between the two conjectured inequalities. Also, we determine the infinitesimal version of the log-Brunn-Minkowski inequality. As a consequence, we obtain a strong Poincare-type inequality in the case of unconditional convex sets, as well as for symmetric convex sets on the plane. Additionally, we derive an infinitesimal equivalent version of the B-conjecture for an arbitrary measure.

Abstract. We present a probabilistic model of illuminating a convex body by independently distributed light sources. In addition to recovering C.A. Rogers' upper bounds for the illumination number, we improve previous estimates of J. Januszewski and M. Naszodi for a generalized version of the illumination parameter.

Abstract. Minkowski’s Theorem asserts that every centered measure on the sphere which is not concentrated on a great subsphere is the surface area measure of some convex body, and, moreover, the surface area measure determines a convex body uniquely. In this manuscript we prove an extension of Minkowski’s theorem. Consider a measure μ on Rn with positive degree of concavity and positive degree of homogeneity. We show that a surface area measure of a convex set K, weighted with respect to μ, determines a convex body uniquely up to μ-measure zero. We also establish an existence result under natural conditions including symmetry. We apply this result to extend the solution to classical Shephard’s problem, which asks the following: if one convex body in Rn has larger projections than another convex body in every direction, does it mean that the volume of the first convex body is also greater? The answer to this question is affirmative when n ≤ 2 and negative when n ≥ 3. In this paper we introduce a new notion which relates projections of convex bodies to a given measure μ, and is a direct generalization of the Lebesgue area of a projection. Using this notion we state a generalization of the Shephard problem to measures and prove that the answer is affirmative for n ≤ 2 and negative for n ≥ 3 for measures which have a positive degree of homogeneity and a positive degree of concavity. We also prove stability and separation results, and establish useful corollaries. Finally, we describe two types of uniqueness results which follow from the extension of Minkowski’s theorem.

- A. Colesanti, G. V. Livshyts, The log-Minkowski problem near a ball.
- G.V. Livshyts, On a dual isoperimetric inequality for log-concave measures.
- G. V. Livshyts, On the Gaussian concentration inequality and its relation to the Gaussian surface area, a letter.

### Introduction to Probability and Statistics (Fall 2015, undergraduate)

### Probability I (Fall 2016, graduate)

#### Home work 1 (Probability I)

#### Home work 2 (Probability I)

#### Home work 3 (Probability I)

#### Home work 4 (Probability I)

#### Home work 5 (Probability I)

#### Test 1 (Probability I)

#### Test 2 (Probability I)

#### Midterm (Probability I)

### PUTNAM preparation (Fall 2016, undergraduate)

The William Lowell Putnam Mathematical Competition is an annual mathematics competition for undergraduate college students enrolled at institutions of higher learning in the United States and Canada. It awards a scholarship and cash prizes ranging from $250 to $2,500 for the top students and $5,000 to $25,000 for the top schools. It is widely considered to be the most prestigious university-level mathematics examination in the world. The competition was founded in 1927 by Elizabeth Lowell Putnam in memory of her husband William Lowell Putnam. The exam has been offered annually since 1938 and is administered by the Mathematical Association of America. See the oficial webpage of the PUTNAM competition, as well as the page containing PUTNAM problems and solutions of the recent years. All of the Georgia Tech students interested in participating PUTNAM, and/or joining my class aimed to prepare for the competition, which runs on Tuesdays 3:05-4:55 pm at 171 skiles, are more then welcome to get in touch with me via e-mail, or stop by my office 228 Skiles, or to just show up in class! Below please see some materials for the course.

#### PUTNAM exam 2015

#### Solutions to PUTNAM exam 2015

#### Excersize set 1 (PUTNAM preparation)

#### Problem set 1 (PUTNAM preparation)

#### Problem set 2 (PUTNAM preparation)

#### Problem set 3 (PUTNAM preparation)

#### Problem set 4 (PUTNAM preparation)

#### Mini olympiad (PUTNAM preparation)

#### Problem set 6 (PUTNAM preparation)

#### Problem set 7 (PUTNAM preparation)

#### Problem set 8 (PUTNAM preparation)

#### Problem set 9 (PUTNAM preparation)

#### Problem set 10 (PUTNAM preparation)